Add Batch 775dfc5c-40ee-4168-b1c9-6ba86a0b5230 data

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@@ -2810,3 +2810,67 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
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2023/Fantasia3D_ Disentangling Geometry and Appearance for High-quality Text-to-3D Content Creation/5c5d4c5d-3820-44df-916a-bfeef9c352be_content_list.json

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+    {
+        "type": "text",
+        "text": "Fantasia3D: Disentangling Geometry and Appearance for High-quality Text-to-3D Content Creation",
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+    {
+        "type": "text",
+        "text": "Rui Chen,\\* Yongwei Chen,\\* Ningxin Jiao, Kui Jia\\* South China University of Technology",
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+    {
+        "type": "text",
+        "text": "Abstract",
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+        "text": "Automatic 3D content creation has achieved rapid progress recently due to the availability of pre-trained, large language models and image diffusion models, forming the emerging topic of text-to-3D content creation. Existing text-to-3D methods commonly use implicit scene representations, which couple the geometry and appearance via volume rendering and are suboptimal in terms of recovering finer geometries and achieving photorealistic rendering; consequently, they are less effective for generating high-quality 3D assets. In this work, we propose a new method of Fantasia3D for high-quality text-to-3D content creation. Key to Fantasia3D is the disentangled modeling and learning of geometry and appearance. For geometry learning, we rely on a hybrid scene representation, and propose to encode surface normal extracted from the representation as the input of the image diffusion model. For appearance modeling, we introduce the spatially varying bidirectional reflectance distribution function (BRDF) into the text-to-3D task, and learn the surface material for photorealistic rendering of the generated surface. Our disentangled framework is more compatible with popular graphics engines, supporting relighting, editing, and physical simulation of the generated 3D assets. We conduct thorough experiments that show the advantages of our method over existing ones under different text-to-3D task settings. Project page and source codes: https://fantasia3d.github.io/.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
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+    {
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+        "text": "Automatic 3D content creation [43, 18, 33, 44] powered by large language models has drawn significant attention recently, due to its convenience to entertaining and gaming industries, virtual/augmented reality, and robotic applications. The traditional process of creating 3D assets typically involves multiple, labor-intensive stages, including geome",
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+        "image_caption": [
+            "soft body simulation"
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+        "image_caption": [
+            "cloth simulation"
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+        "image_caption": [
+            "material editing",
+            "Figure 1. Provided with a textual description of \"a highly detailed stone bust of Theodoros Kolokotronis\", our method produces high-quality geometry as well as disentangled materials, and enables photorealistic rendering."
+        ],
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+    {
+        "type": "text",
+        "text": "try modeling, shape baking, UV mapping, material creation, and texturing, as described in [15], where different software tools and the expertise of skilled artists are often required. Imperfections would also accumulate across these stages, resulting in low-quality 3D assets. It is thus desirable to automate such a process, and ideally to generate high-quality 3D assets that have geometrically fair surfaces, rich materials and textures, and support photorealistic rendering under arbitrary views.",
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+    {
+        "type": "text",
+        "text": "In this work, we focus on automatic 3D content creation given text prompts encoded by large language models, i.e., the text-to-3D tasks [18, 33]. Text-to-3D is inspired by the tremendous success of text-to-image research [34, 36, 30, 35]. To enable 3D generation, most existing methods [33, 23] rely on the implicit scene modeling of Neural Radiance Field (NeRF) [25, 4, 27], and learn the NeRFs by back-propagating the supervision signals from image diffusion models. However, NeRF modeling is less effective for surface recovery [43, 46], since it couples the learning of surface geometry with that of pixel colors via volume rendering. Consequently, 3D creation based on",
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+    {
+        "type": "header",
+        "text": "CVF",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "* Equal contribution.",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "† Corresponding author.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "22246",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "NeRF modeling is less effective for recovery of both the fine surface and its material and texture. In the meanwhile, explicit and hybrid scene representations [43, 46, 38] are proposed to improve over NeRF by modeling the surface explicitly and performing view synthesis via surface rendering.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we are motivated to use 3D scene representations that are more amenable to the generation of high-quality 3D assets given text prompts. We present an automatic text-to-3D method called Fantasia3D. Key to Fantasia3D is a disentangled learning of geometry and appearance models, such that both a fine surface and a rich material/texture can be generated. To enable such a disentangled learning, we use the hybrid scene representation of DMTET [38], which maintains a deformable tetrahedral grid and a differentiable mesh extraction layer; deformation can thus be learned through the layer to explicitly control the shape generation. For geometry learning, we technically propose to encode a rendered normal map, and use shape encoding of the normal as the input of a pre-trained, image diffusion model; this is in contrast different from existing methods that commonly encode rendered color images. For appearance modeling, we introduce, for the first time, the spatially varying Bidirectional Reflectance Distribution Function (BRDF) into the text-to-3D task, thus enabling material learning that supports photorealistic rendering of the learned surface. We implement the geometry model and the BRDF appearance model as simple MLPs. Both models are learned through the pre-trained image diffusion model, using a loss of Score Distillation Sampling (SDS) [33]. We use the pre-trained stable diffusion [35, 40] as the image generation model in this work.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We note that except for text prompts, our method can also be triggered with additional inputs of users' preferences, such as a customized 3D shape or a generic 3D shape of a certain object category; this is flexible for users to better control what content is to be generated. In addition, given the disentangled generation of geometry and appearance, it is convenient for our method to support relighting, editing, and physical simulation of the generated 3D assets. We conduct thorough experiments to verify the efficacy of our proposed methods. Results show that our proposed Fantasia3D outperforms existing methods for high-quality and diverse 3D content creation. We summarize our technical contributions as follows.",
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+    {
+        "type": "text",
+        "text": "- We propose a novel method, termed Fantasia3D, for high-quality text-to-3D content creation. Our method disentangles the modeling and learning of geometry and appearance, and thus enables both a fine recovery of geometry and photorealistic rendering of per-view images.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "- For geometry learning, we use a hybrid representation",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "of DMTET, which supports learning surface deformation via a differentiable mesh extraction; we propose to render and encode the surface normal extracted from DMTET as the input of the pre-trained image diffusion model, which enables more subtle control of shape generation.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "- For appearance modeling, to the best of our knowledge, we are the first to introduce the full BRDF learning into text-to-3D content creation, facilitated by our proposed geometry-appearance disentangled framework. BRDF modeling promises high-quality 3D generation via photorealistic rendering.",
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+    {
+        "type": "text",
+        "text": "2. Related work",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Text-to-3D content creation. Motivated by the desire to generate high-quality 3D content from simple semantics such as text prompts, text-to-3D has drawn considerable attention in recent years [33, 13, 18]. Existing methods either use pre-trained 2D text-to-image models [35, 3, 36], together with score distillation sampling [33], to generate 3D geometries [18] or synthesize novel views [33], or train a text-conditioned 3D generative model from scratch [39, 49, 17, 29]. These methods generate 3D geometries with little exploration of generating high-quality lighting and surface materials. On the other hand, TANGO [8] is able to generate high-quality surface materials given text prompts; unfortunately, the method requires as input a 3D surface mesh. Our proposed method addresses the shortcomings of the above methods, and is able to generate both high-quality surface geometries and their corresponding materials, both of which are crucial for photorealistic rendering of the generated 3D content. Our method thus, for the first time, closes the loop of object-level text-to-3D content creation.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Surface material estimation. The estimation of surface materials is a long-standing challenge in computer vision and graphics research. Earlier methods [2, 45] focus on recovering physically based materials under known lighting conditions, whose usefulness is, however, limited in real-world scenarios. Subsequent methods [12, 11, 5, 1] try to estimate materials under natural lighting conditions, assuming the availability of complete geometry information. More recently, the joint reconstruction of geometry and materials is proposed given calibrated multi-view images [6, 7, 19, 48]. Alternative to these methods, we explore the novel creation of surface materials and geometries from trained language models.",
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+        "text": "3. Preliminary",
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+        "text": "In this section, we present a few preliminaries that are necessary for presenting our proposed method in Section 4.",
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+        "image_caption": [
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+        "image_caption": [
+            "\"A German Sheperd\""
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+        "image_caption": [
+            "\"A tarantula, highly detailed\""
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+        "image_caption": [
+            "Figure 2. Results of our method. The upper portion of this figure showcases the generation results obtained from solely text prompts. The lower portion showcases user-guided generation results given guiding meshes with the corresponding textual descriptions."
+        ],
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+        "text": "3.1. Score distillation sampling",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "DreamFusion [33] presents a method that optimizes 3D scene parameters and synthesizes novel views from tex-",
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+    {
+        "type": "text",
+        "text": "tual descriptions, by employing a pre-trained 2D diffusion model. The scene is represented as a differentiable image parameterization [26], where a differentiable generator  $g$",
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+    {
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+        "image_caption": [
+            "geometry",
+            "(a) Disentangled representation"
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+        "image_caption": [
+            "\" a highly detailed stone bust of Theodoros Kolokotronis \"",
+            "appearance"
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "$k_{n}$"
+        ],
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+        ],
+        "page_idx": 3
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+        "img_path": "images/b6ace02311bd3129b3d5d532a4b11cc19e4eb0dd146283cd0e4242099d495721.jpg",
+        "image_caption": [
+            "(b) Geometry modeling"
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+    {
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+        "image_caption": [
+            "(c) Appearance modeling",
+            "Figure 3. Overview of our method. Our method can generate disentangled geometry and appearance given a text prompt (cf. figure (a)), which are produced by (b) geometry modeling and (c) appearance modeling, respectively. (b) We employ DMTET as our 3D geometry representation, which is initialized as a 3D ellipsoid here. To optimize the parameters of DMTET, we render the normal map (and the object mask in the early training phase) of the extracted mesh from DMTet as the shape encoding of stable diffusion [35, 40]. (c) For appearance modeling, we introduce the spatially-varying Bidirectional Reflectance Distribution Function (BRDF) modeling into text-to-3D generation, and learn to predict three components (namely,  $k_d$ ,  $k_{rm}$ , and  $k_n$ ) of the appearance. Both geometry and appearance modeling are supervised by Score Distillation Sampling (SDS) loss [33]."
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+    {
+        "type": "text",
+        "text": "renders 2D images  $x = g(\\theta)$  from a modified Mip-NeRF [4] parameterized as  $\\theta$ . DreamFusion leverages a diffusion model  $\\phi$  (Imagen [36] in this instance) to provide a score function  $\\hat{\\epsilon}_{\\phi}(x_t; y, t)$ , which predicts the sampled noise  $\\epsilon$  given the noisy image  $x_t$ , text-embedding  $y$ , and noise level  $t$ . This score function guides the direction of the gradient for updating the scene parameters  $\\theta$ , and the gradient is calculated by Score Distillation Sampling (SDS):",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bigtriangledown_ {\\theta} \\mathcal {L} _ {\\mathrm {S D S}} (\\phi , x) = \\mathbb {E} _ {t, \\epsilon} \\left[ w (t) \\left(\\hat {\\epsilon} _ {\\phi} \\left(x _ {t}; y, t\\right) - \\epsilon\\right) \\frac {\\partial x}{\\partial \\theta} \\right], \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "while  $w(t)$  is a weighting function. Since Imagen is not publicly accessible, in this work, we use the released latent space diffusion model of Stable Diffusion [35, 40] as our guidance model, and revise the SDS loss (1) accordingly. Details are given in Section 4.2.",
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+    },
+    {
+        "type": "text",
+        "text": "3.2.DMTET",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implicit surface representations [25, 31, 22] are popularly used in novel view synthesis and 3D reconstruction, due to their capabilities to represent complex scenes. However, surfaces of lower quality may be obtained [28] by extracting explicit meshes from these implicit representations using marching cubes [20]. Instead, Shen et al. [38] propose a hybrid representation, termed DMTET, that has two key features, i.e., a deformable tetrahedral grid and a differentiable Marching Tetrahedral (MT) layer. The deformable tetrahedral grid  $(V_{T}, T)$  has vertices  $V_{T}$  in the tetrahedral grid  $T$ . For each vertex  $v_{i} \\in V_{T}$ , the proposed method predicts the Signed Distance Function (SDF) value  $s(v_{i})$  and a position offset  $\\triangle v_{i}$  by:",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n(s (v _ {i}), \\triangle v _ {i}) = \\Psi (v _ {i}; \\psi), \\tag {2}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $\\psi$  is the parameters of a network  $\\Psi$ , enabling the extraction of explicit meshes through MT layer during each",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "iteration of training. In this work, We use DMTET as our geometry representation and render the mesh extracted from MT layer iteratively by a differentiable renderer[16, 28].",
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+    {
+        "type": "text",
+        "text": "4. The Proposed Method",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we present our proposed method of Fantasia3D for high-quality text-to-3D object generation, by disentangling the modeling and learning of geometry and appearance. For geometry modeling, we rely on the hybrid surface representation of DMTET, and parameterize the 3D geometry as an MLP  $\\Psi$  that learns to predict the SDF value and position offset for each vertex in the deformable tetrahedral grid of DMTET; in contrast to previous methods, we propose to use the rendered normal map (and the object mask in the early training phase) of the extracted mesh from DMTET as the input of shape encoding. For appearance modeling, we introduce, for the first time, the full Bidirectional Reflectance Distribution Function (BRDF) modeling into text-to-3D generation, and learn an MLP  $\\Gamma$  that predicts parameters of surface material and supports high-quality 3D generation via photorealistic rendering. Given the disentangled modeling of  $\\Psi$  and  $\\Gamma$ , the whole pipeline is learned with SDS supervision, and the gradients are backpropagated through the pre-trained stable diffusion model. Our pipeline is initialized either as a 3D ellipsoid or as a customized 3D model provided by users. Fig. 3 gives an illustration of our proposed method. In contrast, previous methods couple the geometry and appearance learning, and are suboptimal in terms of leveraging the powerful pretrained 2D image diffusion models via SDS loss. Details of the proposed Fantasia3D are presented as follows.",
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+    {
+        "type": "page_number",
+        "text": "22249",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1. DMTET initialization",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We adopt DMTET as our 3D scene representation, which is parameterized as the MLP network  $\\Psi$ . For each vertex  $v_{i} \\in V_{T}$  of the tetrahedral grid  $(V_{T}, T)$ ,  $\\Psi$  is trained to predict the SDF value  $s(v_{i})$  and the deformation offset  $\\triangle v_{i}$ . A triangular mesh can be extracted from  $(V_{T}, T)$  using the MT layer, whose procedure is also differentiable w.r.t. the parameters of  $\\Psi$ . We initialize DMTET either as a 3D ellipsoid or as a customized 3D model provided by users; the latter choice is useful when the text-to-3D task is to be conditioned on users' preferences. In either case, we initialize  $\\Psi$  by the following fitting procedure. We sample a point set  $\\{p_{i} \\in \\mathbb{R}^{3}\\}$  whose points are in close proximity to the initialized 3D ellipsoid or customized model, and compute their SDF values, resulting in  $\\{SDF(p_{i})\\}$ ; we use the following loss to optimize the parameters  $\\psi$  of  $\\Psi$ :",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {S D F}} = \\sum_ {p _ {i} \\in P} \\| s (p _ {i}; \\psi) - S D F (p _ {i}) \\| _ {2} ^ {2}. \\tag {3}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2. Geometry modeling",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Previous text-to-3D methods [33, 13] commonly use NeRF [25, 4] as the implicit scene representation, which couples the geometry with color/appearance and use volume rendering for view synthesis. Since NeRF modeling is less effective for surface reconstruction [43, 46], these methods are consequently less effective to generate high-quality 3D surfaces by back-propagating the supervision of SDS loss through pre-trained text-to-image models. As a remedy, the method [18] uses a second stage of the refined generation that is based on scene modeling of explicit surfaces.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose to decouple the generation of geometry from that of appearance, based on the hybrid scene representation of DMTET, which enables photorealistic surface rendering to make better use of the powerful pre-trained text-to-image models. More specifically, given the current DMTET with MLP parameters  $\\psi$ , we generate a normal map  $n$ , together with an object mask  $o$ , as:",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n(n, o) = g _ {n} (\\psi , c), \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $g_{n}$  is a differentiable render (e.g., nvidiffrast [16]), and  $c$  is a sampled camera pose. We randomly sample the camera poses in the spherical coordinate system to ensure that the camera poses are distributed uniformly on the sphere. We propose to use the generated  $n$  (and  $o$ ) as the input of shape encoding to connect with stable diffusion. To update  $\\psi$ , we again employ SDS loss that computes the gradient w.r.t.  $\\psi$  as:",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bigtriangledown_ {\\psi} \\mathcal {L} _ {\\mathrm {S D S}} (\\phi , \\tilde {n}) = \\mathbb {E} \\left[ w (t) \\left(\\hat {\\epsilon} _ {\\phi} \\left(z _ {t} ^ {\\tilde {n}}; y, t\\right) - \\epsilon\\right) \\frac {\\partial \\tilde {n}}{\\partial \\psi} \\frac {\\partial z ^ {\\tilde {n}}}{\\partial \\tilde {n}} \\right], \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/a9292b7196f2979ecd8e71863033957bbc30c2bcbd57e8c1788c7ee7c010a3c1.jpg",
+        "image_caption": [
+            "Figure 4. Three components of the material model, namely the diffuse term  $k_{d}$ , the roughness and metallic term  $k_{rm}$ , and the normal variation term  $k_{n}$ ."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/6f644163abc5fb23b018f1347ed4afa289c4ffc69a52407db83b1811defae0e5.jpg",
+        "image_caption": [
+            "Figure 5. A comparison of UV edge padding (cf. the left column) and original texturing (cf. the right column). UV edge padding removes the white seams that appear in the right rendering."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\phi$  parameterizes the pre-trained stable diffusion model,  $\\tilde{n}$  denotes concatenation of the normal  $n$  with the mask  $o$ ,  $z^{\\tilde{n}}$  is the latent code of  $\\tilde{n}$  via shape encoding,  $\\hat{\\epsilon}_{\\phi}(z_t^{\\tilde{n}};y,t)$  is the predicted noise given text embedding  $y$  and noise level  $t$ , and  $\\epsilon$  is the noise added in  $z_t^{\\tilde{n}}$ . In practice, we utilize a coarse-to-fine strategy to model the geometry. During the early phase of training, we use the downsampled  $\\tilde{n}$  as the latent code, which is inspired by [23], to rapidly update  $\\Psi$  and attain a coarse shape. However, a domain gap exists between  $\\tilde{n}$  and the latent space data distribution learned by the VAE encoder in the stable diffusion, which may lead to a mismatch of the generated geometry from the textual description. To mitigate this discrepancy, we implement a data augmentation technique by introducing random rotations to  $\\tilde{n}$ . Our experimental observations reveal that this technique enhances the alignment between the generated geometry and the provided textual description. In the later phase of training, aiming to capture finer geometric details with greater precision, we encode the high-resolution normal  $n$  (without the mask  $o$ ) to derive  $z^n$ , using the pretrained image encoder in stable diffusion.",
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+    },
+    {
+        "type": "page_number",
+        "text": "22250",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3. Appearance modeling",
+        "text_level": 1,
+        "bbox": [
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+            90,
+            281,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Given a learned DMTET with geometry parameters  $\\psi$ , we aim for photorealistic surface rendering to better leverage the powerful image diffusion model. This is achieved by introducing into text-to-3D generation the Physically-Based Rendering (PBR) material model. As illustrated in Fig. 4, the material model we use [21] comprises three components, namely the diffuse term  $k_{d} \\in \\mathbb{R}^{3}$ , the roughness and metallic term  $k_{rm} \\in \\mathbb{R}^{2}$ , and the normal variation term  $k_{n} \\in \\mathbb{R}^{3}$ . The  $k_{d}$  term denotes the diffuse value, while the  $k_{rm}$  term encompasses the roughness  $r$  and metalness  $m$ ; the roughness  $r$  serves as a metric for measuring the extent of specular reflection and a parameter of the GGX [42] normal distribution function in the rendering equation. The metalness parameter  $m$  and diffuse value  $k_{d}$  can be used to compute the specular term  $k_{s}$  using  $k_{s} = (1 - m) \\cdot 0.04 + m \\cdot k_{d}$ . Furthermore, we use a normal variation  $k_{n}$  in tangent space to enhance the surface lighting effect and introduce additional geometric variations.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use an MLP  $\\Gamma$  as our parameterized material model.  $\\Gamma$  is learned to predict spatially varying material parameters, which are subsequently used to render the surface extracted from DMTET. More specifically, for any point  $p\\in \\mathbb{R}^3$  on the surface, we use hash-grid positional encoding [27], and generate the diffuse term  $k_{d}$ , the specular term  $k_{rm}$ , and the tangent space normal  $k_{n}$  as:",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(k _ {d}, k _ {r m}, k _ {n}\\right) = \\Gamma (\\beta (p); \\gamma), \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $\\beta$  is the positional encoding of  $p$ , and  $\\gamma$  parameterizes the MLP  $\\Gamma$ . The basic rendering equation suggests that each image pixel at a specific viewing direction can be rendered as",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL (p, \\omega) = \\int_ {\\Omega} L _ {i} (p, \\omega_ {i}) f (p, \\omega_ {i}, \\omega) (\\omega_ {i} \\cdot n _ {p}) \\mathrm {d} \\omega_ {i}, \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            608,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $L$  is the rendered pixel color along the direction  $\\omega$  from the surface point  $p$ ,  $\\Omega = \\{\\omega_i : \\omega_i \\cdot n_p \\geq 0\\}$  denotes a hemisphere with the incident direction  $\\omega_i$  and surface normal  $n_p$  at  $p$ ,  $L_i$  is the incident light that is represented by an off-the-shelf environment map [32], and  $f$  is the BRDF determined by the material parameters  $(k_d, k_{rm}, k_n)$  predicted by (6). We note that  $L$  is the summation of diffuse intensity  $L_d$  and specular intensity  $L_s$ , and the two terms can be computed as follows:",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} L (p, \\omega) = L _ {d} (p) + L _ {s} (p, \\omega), \\\\ L _ {d} (p) = k _ {d} (1 - m) \\int_ {\\Omega} L _ {i} (p, \\omega_ {i}) (\\omega_ {i} \\cdot n _ {p}) \\mathrm {d} \\omega_ {i}, \\\\ L _ {s} (p, \\omega) = \\int_ {\\Omega} \\frac {D F G}{4 (\\omega \\cdot n _ {p}) \\left(\\omega_ {i} \\cdot n _ {p}\\right)} L _ {i} \\left(p, \\omega_ {i}\\right) \\left(\\omega_ {i} \\cdot n _ {p}\\right) \\mathrm {d} \\omega_ {i}, \\tag {8} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            88,
+            803,
+            468,
+            901
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $F$ ,  $G$ , and  $D$  represent the Fresnel term, the shadowing-mask term, and the GGX distribution of normal, respectively. Following [28], the hemisphere integration can be calculated using the split-sum method.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "By aggregating the rendered pixel colors along the direction  $\\omega$  (i.e., camera pose), we have the rendered image  $x = \\{L(p,\\omega)\\}$  that connects with the image encoder of the pre-trained stable diffusion model. We update the parameters  $\\gamma$  by computing the gradient of the SDS loss w.r.t.  $\\gamma$ :",
+        "bbox": [
+            496,
+            151,
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+            227
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\nabla_ {\\gamma} \\mathcal {L} _ {\\mathrm {S D S}} (\\phi , x) = \\mathbb {E} \\left[ w (t) \\left(\\hat {\\epsilon} _ {\\phi} \\left(z _ {t} ^ {x}; y, t\\right) - \\epsilon\\right) \\frac {\\partial x}{\\partial \\gamma} \\frac {\\partial z ^ {x}}{\\partial x} \\right]. \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            508,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Notations in (9) are similarly defined as those in (5).",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Texturing. Given the trained  $\\Gamma$ , we proceed by sampling the generated appearance as 2D texture maps, in accordance with the UV map generated by the xatlas [47]. Note that texture seams would emerge by direct incorporation of the sampled 2D textures into graphics engines (e.g., Blender [10]). We instead employ the UV edge padding technique [9], which involves expanding the boundaries of UV islands and filling empty regions. As illustrated in Fig. 5, this padding technique removes background pixels in the texture map and also removes the seams in the resulting renderings.",
+        "bbox": [
+            496,
+            304,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details. We implement the network  $\\Psi$  as a three-layer MLP with 32 hidden units, and implement  $\\Gamma$  as a two-layer MLP with 32 hidden units. Our method is optimized on 8 Nvidia RTX 3090 GPUs for about 15 minutes for learning  $\\Psi$  and about 16 minutes for learning  $\\Gamma$ , respectively, where we use AdamW optimizer with the respective learning rates of  $1 \\times 10^{-3}$  and  $1 \\times 10^{-2}$ . For each iteration, we uniformly sample 24 camera poses for the rendering of normal maps and colored images. More details of our implementation are available in the supplemental material. In geometry modeling, we set  $\\omega(t) = \\sigma^2$  during the early phase and then transition to  $w(t) = \\sigma^2 \\sqrt{1 - \\sigma^2}$  as we progress to the later phase. In appearance modeling, we apply  $w(t) = \\sigma^2 \\sqrt{1 - \\sigma^2}$  during the early phase, followed by a shift to  $1 / \\sigma^2$  as we enter the later phase. This approach mitigates the issue related to over-saturated color within the appearance modeling.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5. Experiments",
+        "text_level": 1,
+        "bbox": [
+            500,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present comprehensive experiments to evaluate the efficacy of our proposed method for text-to-3D content creation. We first conduct ablation studies in Section 5.1 that verify the importance of our key design of disentangling geometry and appearance for text-to-3D generation. In Section 5.2, we show the efficacy of our method for the generation of 3D models with PBR materials from arbitrary text prompts, where we also compare with two recent state-of-the-art methods (namely, Magic3D [18] and DreamFusion [33]). In Section 5.3, we present our results",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "22251",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "under the setting of user-guided generation and compare them with Latent-NeRF [23]. We finally demonstrate in Section 5.4 that the 3D assets generated by our method are readily compatible with popular graphics engines such as Blender [10], thus facilitating relighting, editing, and physical simulation of the resulting 3D models.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1. Ablation studies",
+        "text_level": 1,
+        "bbox": [
+            80,
+            195,
+            235,
+            209
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We use an example text prompt of \"a highly detailed stone bust of Theodoros Kolokotronis\" for the ablation studies. Results of alternative settings are given in Fig. 6. As the reference, the first two columns show the geometry and appearance results of our Fantasia3D rendered from the front and back views. To verify the effectiveness of the disentangled design in Fantasia3D, we also conduct experiments as follows: in each iteration, we render shaded images of the mesh extracted from DMTET, and learn to update the parameters of a network responsible for both the geometry and material, using the gradient computed by SDS loss. Results in the second two columns show that such an entangled learning fails to generate plausible results. Previous methods (e.g., DreamFusion [33] and Magic3D [18]) couple the geometry and appearance generation together, following NeRF [25]. Our adoption of the disentangled representation is mainly motivated by the difference of problem nature for generating surface geometry and appearance. In fact, when dealing with finer recovery of surface geometry from multi-view images, methods (e.g., VolSDF [46], ndiffrec [28], etc) that explicitly take the surface modeling into account triumph; our disentangled representation enjoys the benefit similar to these methods. The disentangled representation also enables us to include the BRDF material representation in the appearance modeling, achieving better photo-realistic rendering by the BRDF physical prior.",
+        "bbox": [
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+        "type": "text",
+        "text": "To investigate how shape encoding of the normal map plays role in Fantasia3D, we replace the normal map with an image that is shaded on the mesh extracted from DMTET using fixed material parameters; results in the third two columns become weird with twisted geometries. This is indeed one of the key factors that makes the success of Fantasia3D. Our initial hypothesis is that shape information contained in normal and mask images could be beneficial to geometry learning, and as such, we further observe that the value range of normal maps is normalized in  $(-1, 1)$ , which aligns with the data range required for latent space diffusion; our empirical studies verify our hypothesis. Our hypothesis is further corroborated by observing that the LAION-5B [37] dataset used for training Stable Diffusion contains normals (referring to website for retrieval of normal data in LAION-5B [37]), which allows Stable Diffusion to handle the optimization of normal maps effectively. To deal with rough and coarse geometry in the early stage of learning, we use the concatenated  $64 \\times 64 \\times 4$  (normal,",
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+        "type": "text",
+        "text": "mask) images for better convergence. As the learning progresses, it becomes essential to render the  $512 \\times 512 \\times 3$  high-resolution normal images for capturing finer geometry details, and we choose to use normal images only in the later stage. This strategy strikes an accuracy-efficiency balance throughout the geometry optimization process.",
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+        "text": "Finally, we replace the full BRDF in Fantasia3D with a simple diffuse color rendering; the results in the last two columns become less realistic and are short of reflection effects when rendered from different views.",
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+        "text": "5.2. Zero-shot generation",
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+        "text": "In this section, we evaluate our method for generating 3D assets from solely natural language descriptions (i.e., the setting of zero-shot generation), by comparing with two state-of-the-art methods, namely DreamFusion [33] and Magic3D [18]. Fig. 7 gives the comparative results given the same text prompts. Since DreamFusion and Magic3D do not have released codes, their results are obtained by downloading from their project pages. Comparing our method with Magic3D, we observe that our results are more photorealistic with competitive geometries. We consistently outperform DreamFusion in both appearance and geometry generation. Notably, our method also offers the convenience of easy geometry extraction and editing, as demonstrated in 5.4, which are less obvious from DreamFusion or Magic3D. More of our results are given in the top half of Fig. 2 and Fig. 8. Furthermore, we compare our appearance modeling stage with several mesh stylization methods, namely Text2mesh [24], CLIP-Mesh [14] and Latent-NeRF [23]. Fig. 9 shows that our method excels in generating more realistic appearances, outperforming the other competitors. We present additional results and comparisons in the supplemental materials.",
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+        "text": "5.3. User-guided generation",
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+        "text": "In addition to zero-shot generation, our method is flexible to accept a customized 3D model as the initialization, alternative to a 3D ellipsoid, thereby facilitating user-guided asset generation. As shown in the lower half of Fig. 2, our method is able to generate rich details in both the geometry and appearance when provided with low-quality 3D models for initialization. The three meshes used in the experiments are from Text2Mesh [24], Latent-NeRF [23], and the creation of Stable DreamFusion [41], respectively. We also compare with the state-of-the-art approach of Latent-NeRF [23] under this user-guided generation setting; results are given in Fig. 10. Our method outperforms Latent-NeRF [23] in both the geometry and texture generation when given the same input meshes and text prompts.",
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+        "img_path": "images/90fc4edd80a9d14b2ecc57ad26065df184394285f382d351e77384ce0fffec43.jpg",
+        "image_caption": [
+            "Figure 6. Ablation studies of our method. The text prompt is \"a highly detailed stone bust of Theodoros Kolokotronis\". Please refer to Section 5.1 for specific settings of individual columns. Please refer to the video results in the supplemental materials for better comparisons."
+        ],
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+        "image_caption": [
+            "A 3D model of an adorable cottage with a thatched roof"
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+        "image_caption": [
+            "Figure 8. Geometries beyond genus-zero ones. DMTET can deform to any topologies, which enables Fantasia3D to generate complex geometries, including those beyond genus-zero ones."
+        ],
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+        "text": "5.4. Scene editing and simulation",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given the disentangled design, our method produces high-quality generation of both the surface and appearance, and is compatible with popular graphics engines for scene editing and physical simulation. In Figure 1, we import into",
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+        "image_caption": [
+            "Figure 7. Comparison of zero-shot generation. Since DreamFusion and Magic3D do not have released codes, their results are obtained by downloading from their project pages. More results and comparisons are available in the supplemental materials.",
+            "Figure 9. Comparison of Texturing. We compare the appearance stage of Fantasia3D with three text-driven texturing methods, using the geometry generated by the geometry stage of Fantasia3D."
+        ],
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+    {
+        "type": "text",
+        "text": "Blender a stone statue generated by our method from the text prompt of \"a highly detailed stone bust of Theodoros Kolokotronis\", where soft body and cloth simulations are performed along with material editing. Given the high qual",
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+        "image_caption": [
+            "Latent -NeRF",
+            "Ours",
+            "A robot hand, realistic",
+            "A teddy bear in a tuxedo",
+            "A German Shepherd"
+        ],
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+        "img_path": "images/513fdc5ce70fe11e9496aa15d0c0f0d58257daca1feb5ce76727407754fcf6b2.jpg",
+        "image_caption": [
+            "Figure 10. Comparison of user-guided generation. The top row shows the input meshes provided by users, the bottom row gives the input text prompts, and the middle two rows show the results from the comparative methods.",
+            "Figure 11. Scene Editing. Three generated objects, namely crown, tiger, and ice cream, are imported into the Cornell Box scene. The scene is then rendered using the Cycles path tracer in Blender, producing natural shadows and reflectance effects."
+        ],
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+        "image_caption": [
+            "Original",
+            "Figure 12. Relighting. Our generated Iron Man model is relit using different lighting setups in Blender, producing diverse reflectance effects on the armor."
+        ],
+        "image_footnote": [],
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+        "img_path": "images/71c6448fe863f12eecc4d8b63406fa896669848e8a5d58691f969869ad572f7f.jpg",
+        "image_caption": [
+            "Factory"
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+        "image_caption": [
+            "Night"
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+    {
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+        "img_path": "images/7d58f17661be479ef40debeb6fdf66026eb1da9fb8e12a33fff9e2cee3bcf8be.jpg",
+        "image_caption": [
+            "Studio"
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+    {
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+        "img_path": "images/610b8842a83c9c64698a995d209d2817682a8c779cc1878256c466a66e074071.jpg",
+        "image_caption": [
+            "Sunset"
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "ity of the generated surface geometry (e.g., no holes on the surface), the simulations are of high-level physical accuracy, as shown in the accompanying supplemental video.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Moreover, we showcase the ability to modify the material of the statue by replacing it with another wood material downloaded from the Internet [32], which poses a challenge for comparative methods such as DreamFusion [33]. Additionally, in Fig. 11, by importing generated ice cream, tiger, and crown into the Cornell Box, we demonstrate the plausible physical interaction between our generated results and the scene, with natural shadows being cast. Finally, Fig. 12 illustrates the replacement of the HDR environment map to produce diverse lighting and corresponding reflectance effects on the generated iron man.",
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+    {
+        "type": "text",
+        "text": "6. Limitations",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While Fantasia3D demonstrates promising advancements in the realm of generating photorealistic 3D assets from text prompts, several limitations remain. For instance, while our method successfully produces loose visual effects, it remains a significant challenge to generate corresponding loose geometries, such as hair, fur, and grass, solely based on text prompts. Additionally, our method primarily emphasizes object generation, thereby lacking the capacity to generate complete scenes with background from text prompts. Consequently, our future research endeavors will be dedicated to addressing these limitations by focusing on the generation of complete scenes and intricate loose geometries.",
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+    {
+        "type": "text",
+        "text": "7. Conclusion",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present Fantasia3D, a new method for automatic text-to-3D generation. Fantasia3D uses disentangled modeling and learning of geometry and appearance, and is able to generate both the fine surface and rich material/texture. Fantasia3D is based on the hybrid scene representation of DMTET. For geometry learning, we propose to encode a rendered normal map, and use shape encoding of the normal as the input of the pre-trained, stable diffusion model. For appearance modeling, we introduce the spatially varying BRDF into the text-to-3D task, thus enabling material learning that supports photorealistic rendering of the learned surface. Expect for text prompts, our method can be triggered with a customized 3D shape as well; this is flexible for users to better control what content is to be generated. Our method is also convenient to support relighting, editing, and physical simulation of the generated 3D assets. Our method is based on pre-trained image diffusion models (i.e., the stable diffusion). In future research, we are interested in learning 3D diffusion directly from the large language models.",
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+    {
+        "type": "text",
+        "text": "Acknowledgements. This work is supported in part by Program for Guangdong Introducing Innovative and Entrepreneurial Teams (No.: 2017ZT07X183) and Guangdong R&D key project of China (No.: 2019B010155001).",
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+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            78,
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+        "page_idx": 9
+    },
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2023/Fantasia3D_ Disentangling Geometry and Appearance for High-quality Text-to-3D Content Creation/5c5d4c5d-3820-44df-916a-bfeef9c352be_origin.pdf

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2023/Fantasia3D_ Disentangling Geometry and Appearance for High-quality Text-to-3D Content Creation/full.md

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+# Fantasia3D: Disentangling Geometry and Appearance for High-quality Text-to-3D Content Creation
+
+Rui Chen,\* Yongwei Chen,\* Ningxin Jiao, Kui Jia\* South China University of Technology
+
+# Abstract
+
+Automatic 3D content creation has achieved rapid progress recently due to the availability of pre-trained, large language models and image diffusion models, forming the emerging topic of text-to-3D content creation. Existing text-to-3D methods commonly use implicit scene representations, which couple the geometry and appearance via volume rendering and are suboptimal in terms of recovering finer geometries and achieving photorealistic rendering; consequently, they are less effective for generating high-quality 3D assets. In this work, we propose a new method of Fantasia3D for high-quality text-to-3D content creation. Key to Fantasia3D is the disentangled modeling and learning of geometry and appearance. For geometry learning, we rely on a hybrid scene representation, and propose to encode surface normal extracted from the representation as the input of the image diffusion model. For appearance modeling, we introduce the spatially varying bidirectional reflectance distribution function (BRDF) into the text-to-3D task, and learn the surface material for photorealistic rendering of the generated surface. Our disentangled framework is more compatible with popular graphics engines, supporting relighting, editing, and physical simulation of the generated 3D assets. We conduct thorough experiments that show the advantages of our method over existing ones under different text-to-3D task settings. Project page and source codes: https://fantasia3d.github.io/.
+
+# 1. Introduction
+
+Automatic 3D content creation [43, 18, 33, 44] powered by large language models has drawn significant attention recently, due to its convenience to entertaining and gaming industries, virtual/augmented reality, and robotic applications. The traditional process of creating 3D assets typically involves multiple, labor-intensive stages, including geome
+
+![](images/96bd34b20c430d182c285af1cf737506089d17e9a8f1432f9d0beaf516feea90.jpg)  
+generation
+
+![](images/5c460cdc38d10888d583404ce2a6d3c9b8d9fdbeab1eadde14427d7b3bf3d0f5.jpg)  
+soft body simulation
+
+![](images/81e844df219938fd9a1c280d47899aa8d89aa4fd386b6e849704c8af9889bb14.jpg)  
+cloth simulation
+
+![](images/54615609df4386e2bd48d11e5254e4d2b89e52bda915aa9211934a0fee8d4a9a.jpg)  
+material editing  
+Figure 1. Provided with a textual description of "a highly detailed stone bust of Theodoros Kolokotronis", our method produces high-quality geometry as well as disentangled materials, and enables photorealistic rendering.
+
+![](images/8b094c93b38ea99503b7da547a637fe2052d8d49bf6179f21364e32256e7b4da.jpg)  
+material
+
+![](images/be0ba8595891df06b02efd6cc89aa298288081bfb2422bcaa638683f2c9b5141.jpg)  
+geometry
+
+try modeling, shape baking, UV mapping, material creation, and texturing, as described in [15], where different software tools and the expertise of skilled artists are often required. Imperfections would also accumulate across these stages, resulting in low-quality 3D assets. It is thus desirable to automate such a process, and ideally to generate high-quality 3D assets that have geometrically fair surfaces, rich materials and textures, and support photorealistic rendering under arbitrary views.
+
+In this work, we focus on automatic 3D content creation given text prompts encoded by large language models, i.e., the text-to-3D tasks [18, 33]. Text-to-3D is inspired by the tremendous success of text-to-image research [34, 36, 30, 35]. To enable 3D generation, most existing methods [33, 23] rely on the implicit scene modeling of Neural Radiance Field (NeRF) [25, 4, 27], and learn the NeRFs by back-propagating the supervision signals from image diffusion models. However, NeRF modeling is less effective for surface recovery [43, 46], since it couples the learning of surface geometry with that of pixel colors via volume rendering. Consequently, 3D creation based on
+
+NeRF modeling is less effective for recovery of both the fine surface and its material and texture. In the meanwhile, explicit and hybrid scene representations [43, 46, 38] are proposed to improve over NeRF by modeling the surface explicitly and performing view synthesis via surface rendering.
+
+In this work, we are motivated to use 3D scene representations that are more amenable to the generation of high-quality 3D assets given text prompts. We present an automatic text-to-3D method called Fantasia3D. Key to Fantasia3D is a disentangled learning of geometry and appearance models, such that both a fine surface and a rich material/texture can be generated. To enable such a disentangled learning, we use the hybrid scene representation of DMTET [38], which maintains a deformable tetrahedral grid and a differentiable mesh extraction layer; deformation can thus be learned through the layer to explicitly control the shape generation. For geometry learning, we technically propose to encode a rendered normal map, and use shape encoding of the normal as the input of a pre-trained, image diffusion model; this is in contrast different from existing methods that commonly encode rendered color images. For appearance modeling, we introduce, for the first time, the spatially varying Bidirectional Reflectance Distribution Function (BRDF) into the text-to-3D task, thus enabling material learning that supports photorealistic rendering of the learned surface. We implement the geometry model and the BRDF appearance model as simple MLPs. Both models are learned through the pre-trained image diffusion model, using a loss of Score Distillation Sampling (SDS) [33]. We use the pre-trained stable diffusion [35, 40] as the image generation model in this work.
+
+We note that except for text prompts, our method can also be triggered with additional inputs of users' preferences, such as a customized 3D shape or a generic 3D shape of a certain object category; this is flexible for users to better control what content is to be generated. In addition, given the disentangled generation of geometry and appearance, it is convenient for our method to support relighting, editing, and physical simulation of the generated 3D assets. We conduct thorough experiments to verify the efficacy of our proposed methods. Results show that our proposed Fantasia3D outperforms existing methods for high-quality and diverse 3D content creation. We summarize our technical contributions as follows.
+
+- We propose a novel method, termed Fantasia3D, for high-quality text-to-3D content creation. Our method disentangles the modeling and learning of geometry and appearance, and thus enables both a fine recovery of geometry and photorealistic rendering of per-view images.
+
+- For geometry learning, we use a hybrid representation
+
+of DMTET, which supports learning surface deformation via a differentiable mesh extraction; we propose to render and encode the surface normal extracted from DMTET as the input of the pre-trained image diffusion model, which enables more subtle control of shape generation.
+
+- For appearance modeling, to the best of our knowledge, we are the first to introduce the full BRDF learning into text-to-3D content creation, facilitated by our proposed geometry-appearance disentangled framework. BRDF modeling promises high-quality 3D generation via photorealistic rendering.
+
+# 2. Related work
+
+Text-to-3D content creation. Motivated by the desire to generate high-quality 3D content from simple semantics such as text prompts, text-to-3D has drawn considerable attention in recent years [33, 13, 18]. Existing methods either use pre-trained 2D text-to-image models [35, 3, 36], together with score distillation sampling [33], to generate 3D geometries [18] or synthesize novel views [33], or train a text-conditioned 3D generative model from scratch [39, 49, 17, 29]. These methods generate 3D geometries with little exploration of generating high-quality lighting and surface materials. On the other hand, TANGO [8] is able to generate high-quality surface materials given text prompts; unfortunately, the method requires as input a 3D surface mesh. Our proposed method addresses the shortcomings of the above methods, and is able to generate both high-quality surface geometries and their corresponding materials, both of which are crucial for photorealistic rendering of the generated 3D content. Our method thus, for the first time, closes the loop of object-level text-to-3D content creation.
+
+Surface material estimation. The estimation of surface materials is a long-standing challenge in computer vision and graphics research. Earlier methods [2, 45] focus on recovering physically based materials under known lighting conditions, whose usefulness is, however, limited in real-world scenarios. Subsequent methods [12, 11, 5, 1] try to estimate materials under natural lighting conditions, assuming the availability of complete geometry information. More recently, the joint reconstruction of geometry and materials is proposed given calibrated multi-view images [6, 7, 19, 48]. Alternative to these methods, we explore the novel creation of surface materials and geometries from trained language models.
+
+# 3. Preliminary
+
+In this section, we present a few preliminaries that are necessary for presenting our proposed method in Section 4.
+
+![](images/efaab1140d7d7f4a549406d3df2f1028a6f1033fa7525bba6a62538effcb4176.jpg)  
+A vintage record player
+
+![](images/3f889ac1671c93b008d312a19598a61610b7addb47c9ce49f911c5aac8555964.jpg)  
+An ice cream sundae
+
+![](images/cfca4ad8b7ef41fadb3d6ad5549e9176b8bc843d0750cfc0c983f7244c7f16fb.jpg)  
+A red rotary telephone
+
+![](images/977880aae93c629c67dcc75f318615a9b246639d3f95e30f9fe08575e5d687f0.jpg)  
+A fresh cinnamon roll covered in glaze, high resolution
+
+![](images/7af8a7efb801773ee139c1360c344b8d25d45c6bb4096c181d435d1c16154f78.jpg)  
+A delicious croissant
+
+![](images/310577bb002178b94ee712d1d777efb063e59e21fdbfbd1199192d36d1efc293.jpg)  
+A golden goblet
+
+![](images/cd68ae1ee09e6f898f35af71ea0dfdd3b0ca618bfd1037c9cd1186d95e935c43.jpg)  
+The leaning tower of Pisa
+
+![](images/1eb0689c79bf1bca25041fbb85d0300a09952c6d63d3e35e85b8ebbfd2466aca.jpg)  
+A highly detailed sandcastle
+
+![](images/5abca5ddc0ed465e757f91032dd28e56402b1e523568177593cd245c62de9c8a.jpg)  
+A car made out of cheese
+
+![](images/d09aaba1c901dcde7f947e7903bbe0cbc20734adf99d9b8d10db3179f1ab3478.jpg)  
+"A photo of the ironman"
+
+![](images/5e1735da28df1709758d6e240912baa9d45a642fe7ef93f7ce74681eb6e89a6f.jpg)  
+"A German Sheperd"
+
+![](images/4fb5d3401db09062591137976c2f8cd89f14a718c1cbaa21227dd4c08b98e1f1.jpg)  
+"A tarantula, highly detailed"
+
+![](images/088851f3e6e8f887a07e1b6e33ecbb56c95e1f0cf0cd08d8f33aa68a57ec7b0c.jpg)  
+Figure 2. Results of our method. The upper portion of this figure showcases the generation results obtained from solely text prompts. The lower portion showcases user-guided generation results given guiding meshes with the corresponding textual descriptions.
+
+![](images/2338b92c54b6d8bd4ccff44292cfd3c790d41827bb693b50d4f9d361cec8a077.jpg)
+
+![](images/167cc07b1c69f662772595ee0b3ac9718c35d761290ae6ff3f616694bcaf65de.jpg)
+
+![](images/c68b5c6e1eb8913bbc3c6a44d382eec2d3a267066c912865c70dbac964f42d87.jpg)
+
+![](images/2affc69c296889b59bb6986e48f8bfaa71b3c05d57faffc770ad0b83b8b65db6.jpg)
+
+# 3.1. Score distillation sampling
+
+DreamFusion [33] presents a method that optimizes 3D scene parameters and synthesizes novel views from tex-
+
+tual descriptions, by employing a pre-trained 2D diffusion model. The scene is represented as a differentiable image parameterization [26], where a differentiable generator  $g$
+
+![](images/5625c428fe0b0a1b3863b553f7e9b35fc62664aa5c343b0e0bfc93e0575054d4.jpg)  
+geometry  
+(a) Disentangled representation
+
+![](images/e20e35b90ad9c479de972d8428518ae7051ac5c2c4e6e4b07a96c85b26c51dfc.jpg)  
+" a highly detailed stone bust of Theodoros Kolokotronis "  
+appearance
+
+![](images/84227980e03bb45164660a3bdc801221765cda8fcba743f6f4ad29c0ed530a19.jpg)  
+$k_{n}$
+
+![](images/b6ace02311bd3129b3d5d532a4b11cc19e4eb0dd146283cd0e4242099d495721.jpg)  
+(b) Geometry modeling
+
+![](images/22deeb2b6fd38fc676c0bc2dac6f5e39cafab9efb4d8db75093505ae48e12505.jpg)  
+(c) Appearance modeling  
+Figure 3. Overview of our method. Our method can generate disentangled geometry and appearance given a text prompt (cf. figure (a)), which are produced by (b) geometry modeling and (c) appearance modeling, respectively. (b) We employ DMTET as our 3D geometry representation, which is initialized as a 3D ellipsoid here. To optimize the parameters of DMTET, we render the normal map (and the object mask in the early training phase) of the extracted mesh from DMTet as the shape encoding of stable diffusion [35, 40]. (c) For appearance modeling, we introduce the spatially-varying Bidirectional Reflectance Distribution Function (BRDF) modeling into text-to-3D generation, and learn to predict three components (namely,  $k_d$ ,  $k_{rm}$ , and  $k_n$ ) of the appearance. Both geometry and appearance modeling are supervised by Score Distillation Sampling (SDS) loss [33].
+
+renders 2D images  $x = g(\theta)$  from a modified Mip-NeRF [4] parameterized as  $\theta$ . DreamFusion leverages a diffusion model  $\phi$  (Imagen [36] in this instance) to provide a score function  $\hat{\epsilon}_{\phi}(x_t; y, t)$ , which predicts the sampled noise  $\epsilon$  given the noisy image  $x_t$ , text-embedding  $y$ , and noise level  $t$ . This score function guides the direction of the gradient for updating the scene parameters  $\theta$ , and the gradient is calculated by Score Distillation Sampling (SDS):
+
+$$
+\bigtriangledown_ {\theta} \mathcal {L} _ {\mathrm {S D S}} (\phi , x) = \mathbb {E} _ {t, \epsilon} \left[ w (t) \left(\hat {\epsilon} _ {\phi} \left(x _ {t}; y, t\right) - \epsilon\right) \frac {\partial x}{\partial \theta} \right], \tag {1}
+$$
+
+while  $w(t)$  is a weighting function. Since Imagen is not publicly accessible, in this work, we use the released latent space diffusion model of Stable Diffusion [35, 40] as our guidance model, and revise the SDS loss (1) accordingly. Details are given in Section 4.2.
+
+# 3.2.DMTET
+
+Implicit surface representations [25, 31, 22] are popularly used in novel view synthesis and 3D reconstruction, due to their capabilities to represent complex scenes. However, surfaces of lower quality may be obtained [28] by extracting explicit meshes from these implicit representations using marching cubes [20]. Instead, Shen et al. [38] propose a hybrid representation, termed DMTET, that has two key features, i.e., a deformable tetrahedral grid and a differentiable Marching Tetrahedral (MT) layer. The deformable tetrahedral grid  $(V_{T}, T)$  has vertices  $V_{T}$  in the tetrahedral grid  $T$ . For each vertex  $v_{i} \in V_{T}$ , the proposed method predicts the Signed Distance Function (SDF) value  $s(v_{i})$  and a position offset  $\triangle v_{i}$  by:
+
+$$
+(s (v _ {i}), \triangle v _ {i}) = \Psi (v _ {i}; \psi), \tag {2}
+$$
+
+where  $\psi$  is the parameters of a network  $\Psi$ , enabling the extraction of explicit meshes through MT layer during each
+
+iteration of training. In this work, We use DMTET as our geometry representation and render the mesh extracted from MT layer iteratively by a differentiable renderer[16, 28].
+
+# 4. The Proposed Method
+
+In this section, we present our proposed method of Fantasia3D for high-quality text-to-3D object generation, by disentangling the modeling and learning of geometry and appearance. For geometry modeling, we rely on the hybrid surface representation of DMTET, and parameterize the 3D geometry as an MLP  $\Psi$  that learns to predict the SDF value and position offset for each vertex in the deformable tetrahedral grid of DMTET; in contrast to previous methods, we propose to use the rendered normal map (and the object mask in the early training phase) of the extracted mesh from DMTET as the input of shape encoding. For appearance modeling, we introduce, for the first time, the full Bidirectional Reflectance Distribution Function (BRDF) modeling into text-to-3D generation, and learn an MLP  $\Gamma$  that predicts parameters of surface material and supports high-quality 3D generation via photorealistic rendering. Given the disentangled modeling of  $\Psi$  and  $\Gamma$ , the whole pipeline is learned with SDS supervision, and the gradients are backpropagated through the pre-trained stable diffusion model. Our pipeline is initialized either as a 3D ellipsoid or as a customized 3D model provided by users. Fig. 3 gives an illustration of our proposed method. In contrast, previous methods couple the geometry and appearance learning, and are suboptimal in terms of leveraging the powerful pretrained 2D image diffusion models via SDS loss. Details of the proposed Fantasia3D are presented as follows.
+
+# 4.1. DMTET initialization
+
+We adopt DMTET as our 3D scene representation, which is parameterized as the MLP network  $\Psi$ . For each vertex  $v_{i} \in V_{T}$  of the tetrahedral grid  $(V_{T}, T)$ ,  $\Psi$  is trained to predict the SDF value  $s(v_{i})$  and the deformation offset  $\triangle v_{i}$ . A triangular mesh can be extracted from  $(V_{T}, T)$  using the MT layer, whose procedure is also differentiable w.r.t. the parameters of  $\Psi$ . We initialize DMTET either as a 3D ellipsoid or as a customized 3D model provided by users; the latter choice is useful when the text-to-3D task is to be conditioned on users' preferences. In either case, we initialize  $\Psi$  by the following fitting procedure. We sample a point set  $\{p_{i} \in \mathbb{R}^{3}\}$  whose points are in close proximity to the initialized 3D ellipsoid or customized model, and compute their SDF values, resulting in  $\{SDF(p_{i})\}$ ; we use the following loss to optimize the parameters  $\psi$  of  $\Psi$ :
+
+$$
+\mathcal {L} _ {\mathrm {S D F}} = \sum_ {p _ {i} \in P} \| s (p _ {i}; \psi) - S D F (p _ {i}) \| _ {2} ^ {2}. \tag {3}
+$$
+
+# 4.2. Geometry modeling
+
+Previous text-to-3D methods [33, 13] commonly use NeRF [25, 4] as the implicit scene representation, which couples the geometry with color/appearance and use volume rendering for view synthesis. Since NeRF modeling is less effective for surface reconstruction [43, 46], these methods are consequently less effective to generate high-quality 3D surfaces by back-propagating the supervision of SDS loss through pre-trained text-to-image models. As a remedy, the method [18] uses a second stage of the refined generation that is based on scene modeling of explicit surfaces.
+
+In this work, we propose to decouple the generation of geometry from that of appearance, based on the hybrid scene representation of DMTET, which enables photorealistic surface rendering to make better use of the powerful pre-trained text-to-image models. More specifically, given the current DMTET with MLP parameters  $\psi$ , we generate a normal map  $n$ , together with an object mask  $o$ , as:
+
+$$
+(n, o) = g _ {n} (\psi , c), \tag {4}
+$$
+
+where  $g_{n}$  is a differentiable render (e.g., nvidiffrast [16]), and  $c$  is a sampled camera pose. We randomly sample the camera poses in the spherical coordinate system to ensure that the camera poses are distributed uniformly on the sphere. We propose to use the generated  $n$  (and  $o$ ) as the input of shape encoding to connect with stable diffusion. To update  $\psi$ , we again employ SDS loss that computes the gradient w.r.t.  $\psi$  as:
+
+$$
+\bigtriangledown_ {\psi} \mathcal {L} _ {\mathrm {S D S}} (\phi , \tilde {n}) = \mathbb {E} \left[ w (t) \left(\hat {\epsilon} _ {\phi} \left(z _ {t} ^ {\tilde {n}}; y, t\right) - \epsilon\right) \frac {\partial \tilde {n}}{\partial \psi} \frac {\partial z ^ {\tilde {n}}}{\partial \tilde {n}} \right], \tag {5}
+$$
+
+![](images/a9292b7196f2979ecd8e71863033957bbc30c2bcbd57e8c1788c7ee7c010a3c1.jpg)  
+Figure 4. Three components of the material model, namely the diffuse term  $k_{d}$ , the roughness and metallic term  $k_{rm}$ , and the normal variation term  $k_{n}$ .
+
+![](images/6f644163abc5fb23b018f1347ed4afa289c4ffc69a52407db83b1811defae0e5.jpg)  
+Figure 5. A comparison of UV edge padding (cf. the left column) and original texturing (cf. the right column). UV edge padding removes the white seams that appear in the right rendering.
+
+where  $\phi$  parameterizes the pre-trained stable diffusion model,  $\tilde{n}$  denotes concatenation of the normal  $n$  with the mask  $o$ ,  $z^{\tilde{n}}$  is the latent code of  $\tilde{n}$  via shape encoding,  $\hat{\epsilon}_{\phi}(z_t^{\tilde{n}};y,t)$  is the predicted noise given text embedding  $y$  and noise level  $t$ , and  $\epsilon$  is the noise added in  $z_t^{\tilde{n}}$ . In practice, we utilize a coarse-to-fine strategy to model the geometry. During the early phase of training, we use the downsampled  $\tilde{n}$  as the latent code, which is inspired by [23], to rapidly update  $\Psi$  and attain a coarse shape. However, a domain gap exists between  $\tilde{n}$  and the latent space data distribution learned by the VAE encoder in the stable diffusion, which may lead to a mismatch of the generated geometry from the textual description. To mitigate this discrepancy, we implement a data augmentation technique by introducing random rotations to  $\tilde{n}$ . Our experimental observations reveal that this technique enhances the alignment between the generated geometry and the provided textual description. In the later phase of training, aiming to capture finer geometric details with greater precision, we encode the high-resolution normal  $n$  (without the mask  $o$ ) to derive  $z^n$ , using the pretrained image encoder in stable diffusion.
+
+# 4.3. Appearance modeling
+
+Given a learned DMTET with geometry parameters  $\psi$ , we aim for photorealistic surface rendering to better leverage the powerful image diffusion model. This is achieved by introducing into text-to-3D generation the Physically-Based Rendering (PBR) material model. As illustrated in Fig. 4, the material model we use [21] comprises three components, namely the diffuse term  $k_{d} \in \mathbb{R}^{3}$ , the roughness and metallic term  $k_{rm} \in \mathbb{R}^{2}$ , and the normal variation term  $k_{n} \in \mathbb{R}^{3}$ . The  $k_{d}$  term denotes the diffuse value, while the  $k_{rm}$  term encompasses the roughness  $r$  and metalness  $m$ ; the roughness  $r$  serves as a metric for measuring the extent of specular reflection and a parameter of the GGX [42] normal distribution function in the rendering equation. The metalness parameter  $m$  and diffuse value  $k_{d}$  can be used to compute the specular term  $k_{s}$  using  $k_{s} = (1 - m) \cdot 0.04 + m \cdot k_{d}$ . Furthermore, we use a normal variation  $k_{n}$  in tangent space to enhance the surface lighting effect and introduce additional geometric variations.
+
+We use an MLP  $\Gamma$  as our parameterized material model.  $\Gamma$  is learned to predict spatially varying material parameters, which are subsequently used to render the surface extracted from DMTET. More specifically, for any point  $p\in \mathbb{R}^3$  on the surface, we use hash-grid positional encoding [27], and generate the diffuse term  $k_{d}$ , the specular term  $k_{rm}$ , and the tangent space normal  $k_{n}$  as:
+
+$$
+\left(k _ {d}, k _ {r m}, k _ {n}\right) = \Gamma (\beta (p); \gamma), \tag {6}
+$$
+
+where  $\beta$  is the positional encoding of  $p$ , and  $\gamma$  parameterizes the MLP  $\Gamma$ . The basic rendering equation suggests that each image pixel at a specific viewing direction can be rendered as
+
+$$
+L (p, \omega) = \int_ {\Omega} L _ {i} (p, \omega_ {i}) f (p, \omega_ {i}, \omega) (\omega_ {i} \cdot n _ {p}) \mathrm {d} \omega_ {i}, \tag {7}
+$$
+
+where  $L$  is the rendered pixel color along the direction  $\omega$  from the surface point  $p$ ,  $\Omega = \{\omega_i : \omega_i \cdot n_p \geq 0\}$  denotes a hemisphere with the incident direction  $\omega_i$  and surface normal  $n_p$  at  $p$ ,  $L_i$  is the incident light that is represented by an off-the-shelf environment map [32], and  $f$  is the BRDF determined by the material parameters  $(k_d, k_{rm}, k_n)$  predicted by (6). We note that  $L$  is the summation of diffuse intensity  $L_d$  and specular intensity  $L_s$ , and the two terms can be computed as follows:
+
+$$
+\begin{array}{l} L (p, \omega) = L _ {d} (p) + L _ {s} (p, \omega), \\ L _ {d} (p) = k _ {d} (1 - m) \int_ {\Omega} L _ {i} (p, \omega_ {i}) (\omega_ {i} \cdot n _ {p}) \mathrm {d} \omega_ {i}, \\ L _ {s} (p, \omega) = \int_ {\Omega} \frac {D F G}{4 (\omega \cdot n _ {p}) \left(\omega_ {i} \cdot n _ {p}\right)} L _ {i} \left(p, \omega_ {i}\right) \left(\omega_ {i} \cdot n _ {p}\right) \mathrm {d} \omega_ {i}, \tag {8} \\ \end{array}
+$$
+
+where  $F$ ,  $G$ , and  $D$  represent the Fresnel term, the shadowing-mask term, and the GGX distribution of normal, respectively. Following [28], the hemisphere integration can be calculated using the split-sum method.
+
+By aggregating the rendered pixel colors along the direction  $\omega$  (i.e., camera pose), we have the rendered image  $x = \{L(p,\omega)\}$  that connects with the image encoder of the pre-trained stable diffusion model. We update the parameters  $\gamma$  by computing the gradient of the SDS loss w.r.t.  $\gamma$ :
+
+$$
+\nabla_ {\gamma} \mathcal {L} _ {\mathrm {S D S}} (\phi , x) = \mathbb {E} \left[ w (t) \left(\hat {\epsilon} _ {\phi} \left(z _ {t} ^ {x}; y, t\right) - \epsilon\right) \frac {\partial x}{\partial \gamma} \frac {\partial z ^ {x}}{\partial x} \right]. \tag {9}
+$$
+
+Notations in (9) are similarly defined as those in (5).
+
+Texturing. Given the trained  $\Gamma$ , we proceed by sampling the generated appearance as 2D texture maps, in accordance with the UV map generated by the xatlas [47]. Note that texture seams would emerge by direct incorporation of the sampled 2D textures into graphics engines (e.g., Blender [10]). We instead employ the UV edge padding technique [9], which involves expanding the boundaries of UV islands and filling empty regions. As illustrated in Fig. 5, this padding technique removes background pixels in the texture map and also removes the seams in the resulting renderings.
+
+Implementation Details. We implement the network  $\Psi$  as a three-layer MLP with 32 hidden units, and implement  $\Gamma$  as a two-layer MLP with 32 hidden units. Our method is optimized on 8 Nvidia RTX 3090 GPUs for about 15 minutes for learning  $\Psi$  and about 16 minutes for learning  $\Gamma$ , respectively, where we use AdamW optimizer with the respective learning rates of  $1 \times 10^{-3}$  and  $1 \times 10^{-2}$ . For each iteration, we uniformly sample 24 camera poses for the rendering of normal maps and colored images. More details of our implementation are available in the supplemental material. In geometry modeling, we set  $\omega(t) = \sigma^2$  during the early phase and then transition to  $w(t) = \sigma^2 \sqrt{1 - \sigma^2}$  as we progress to the later phase. In appearance modeling, we apply  $w(t) = \sigma^2 \sqrt{1 - \sigma^2}$  during the early phase, followed by a shift to  $1 / \sigma^2$  as we enter the later phase. This approach mitigates the issue related to over-saturated color within the appearance modeling.
+
+# 5. Experiments
+
+In this section, we present comprehensive experiments to evaluate the efficacy of our proposed method for text-to-3D content creation. We first conduct ablation studies in Section 5.1 that verify the importance of our key design of disentangling geometry and appearance for text-to-3D generation. In Section 5.2, we show the efficacy of our method for the generation of 3D models with PBR materials from arbitrary text prompts, where we also compare with two recent state-of-the-art methods (namely, Magic3D [18] and DreamFusion [33]). In Section 5.3, we present our results
+
+under the setting of user-guided generation and compare them with Latent-NeRF [23]. We finally demonstrate in Section 5.4 that the 3D assets generated by our method are readily compatible with popular graphics engines such as Blender [10], thus facilitating relighting, editing, and physical simulation of the resulting 3D models.
+
+# 5.1. Ablation studies
+
+We use an example text prompt of "a highly detailed stone bust of Theodoros Kolokotronis" for the ablation studies. Results of alternative settings are given in Fig. 6. As the reference, the first two columns show the geometry and appearance results of our Fantasia3D rendered from the front and back views. To verify the effectiveness of the disentangled design in Fantasia3D, we also conduct experiments as follows: in each iteration, we render shaded images of the mesh extracted from DMTET, and learn to update the parameters of a network responsible for both the geometry and material, using the gradient computed by SDS loss. Results in the second two columns show that such an entangled learning fails to generate plausible results. Previous methods (e.g., DreamFusion [33] and Magic3D [18]) couple the geometry and appearance generation together, following NeRF [25]. Our adoption of the disentangled representation is mainly motivated by the difference of problem nature for generating surface geometry and appearance. In fact, when dealing with finer recovery of surface geometry from multi-view images, methods (e.g., VolSDF [46], ndiffrec [28], etc) that explicitly take the surface modeling into account triumph; our disentangled representation enjoys the benefit similar to these methods. The disentangled representation also enables us to include the BRDF material representation in the appearance modeling, achieving better photo-realistic rendering by the BRDF physical prior.
+
+To investigate how shape encoding of the normal map plays role in Fantasia3D, we replace the normal map with an image that is shaded on the mesh extracted from DMTET using fixed material parameters; results in the third two columns become weird with twisted geometries. This is indeed one of the key factors that makes the success of Fantasia3D. Our initial hypothesis is that shape information contained in normal and mask images could be beneficial to geometry learning, and as such, we further observe that the value range of normal maps is normalized in  $(-1, 1)$ , which aligns with the data range required for latent space diffusion; our empirical studies verify our hypothesis. Our hypothesis is further corroborated by observing that the LAION-5B [37] dataset used for training Stable Diffusion contains normals (referring to website for retrieval of normal data in LAION-5B [37]), which allows Stable Diffusion to handle the optimization of normal maps effectively. To deal with rough and coarse geometry in the early stage of learning, we use the concatenated  $64 \times 64 \times 4$  (normal,
+
+mask) images for better convergence. As the learning progresses, it becomes essential to render the  $512 \times 512 \times 3$  high-resolution normal images for capturing finer geometry details, and we choose to use normal images only in the later stage. This strategy strikes an accuracy-efficiency balance throughout the geometry optimization process.
+
+Finally, we replace the full BRDF in Fantasia3D with a simple diffuse color rendering; the results in the last two columns become less realistic and are short of reflection effects when rendered from different views.
+
+# 5.2. Zero-shot generation
+
+In this section, we evaluate our method for generating 3D assets from solely natural language descriptions (i.e., the setting of zero-shot generation), by comparing with two state-of-the-art methods, namely DreamFusion [33] and Magic3D [18]. Fig. 7 gives the comparative results given the same text prompts. Since DreamFusion and Magic3D do not have released codes, their results are obtained by downloading from their project pages. Comparing our method with Magic3D, we observe that our results are more photorealistic with competitive geometries. We consistently outperform DreamFusion in both appearance and geometry generation. Notably, our method also offers the convenience of easy geometry extraction and editing, as demonstrated in 5.4, which are less obvious from DreamFusion or Magic3D. More of our results are given in the top half of Fig. 2 and Fig. 8. Furthermore, we compare our appearance modeling stage with several mesh stylization methods, namely Text2mesh [24], CLIP-Mesh [14] and Latent-NeRF [23]. Fig. 9 shows that our method excels in generating more realistic appearances, outperforming the other competitors. We present additional results and comparisons in the supplemental materials.
+
+# 5.3. User-guided generation
+
+In addition to zero-shot generation, our method is flexible to accept a customized 3D model as the initialization, alternative to a 3D ellipsoid, thereby facilitating user-guided asset generation. As shown in the lower half of Fig. 2, our method is able to generate rich details in both the geometry and appearance when provided with low-quality 3D models for initialization. The three meshes used in the experiments are from Text2Mesh [24], Latent-NeRF [23], and the creation of Stable DreamFusion [41], respectively. We also compare with the state-of-the-art approach of Latent-NeRF [23] under this user-guided generation setting; results are given in Fig. 10. Our method outperforms Latent-NeRF [23] in both the geometry and texture generation when given the same input meshes and text prompts.
+
+![](images/90fc4edd80a9d14b2ecc57ad26065df184394285f382d351e77384ce0fffec43.jpg)  
+Figure 6. Ablation studies of our method. The text prompt is "a highly detailed stone bust of Theodoros Kolokotronis". Please refer to Section 5.1 for specific settings of individual columns. Please refer to the video results in the supplemental materials for better comparisons.
+
+![](images/26f31b77563130a7e73b932d38d49e3bed73d3e99e6a89ee6de0cdec610cc77f.jpg)  
+A 3D model of an adorable cottage with a thatched roof
+
+![](images/9c2b524a175e69f80b8cb6910c8cf53e1df5785e7315e4f90669cf3e229aeb87.jpg)  
+Figure 8. Geometries beyond genus-zero ones. DMTET can deform to any topologies, which enables Fantasia3D to generate complex geometries, including those beyond genus-zero ones.
+
+# 5.4. Scene editing and simulation
+
+Given the disentangled design, our method produces high-quality generation of both the surface and appearance, and is compatible with popular graphics engines for scene editing and physical simulation. In Figure 1, we import into
+
+![](images/1bcca3ebbd3a06cd4d488e7d69f1bbaf2d9c320b2f3acf54c4b72a97fb4de7c3.jpg)  
+Figure 7. Comparison of zero-shot generation. Since DreamFusion and Magic3D do not have released codes, their results are obtained by downloading from their project pages. More results and comparisons are available in the supplemental materials.  
+Figure 9. Comparison of Texturing. We compare the appearance stage of Fantasia3D with three text-driven texturing methods, using the geometry generated by the geometry stage of Fantasia3D.
+
+Blender a stone statue generated by our method from the text prompt of "a highly detailed stone bust of Theodoros Kolokotronis", where soft body and cloth simulations are performed along with material editing. Given the high qual
+
+![](images/2fefd7402f365968e37abfa2fdd69ac05811d5df7ff2c82ee6639f71c540c7a0.jpg)  
+Latent -NeRF  
+Ours  
+A robot hand, realistic  
+A teddy bear in a tuxedo  
+A German Shepherd
+
+![](images/513fdc5ce70fe11e9496aa15d0c0f0d58257daca1feb5ce76727407754fcf6b2.jpg)  
+Figure 10. Comparison of user-guided generation. The top row shows the input meshes provided by users, the bottom row gives the input text prompts, and the middle two rows show the results from the comparative methods.  
+Figure 11. Scene Editing. Three generated objects, namely crown, tiger, and ice cream, are imported into the Cornell Box scene. The scene is then rendered using the Cycles path tracer in Blender, producing natural shadows and reflectance effects.
+
+![](images/865ce3ebe886c160b4d5062fc459c395e17bd01f9eb6eda8e9519790a5ccf4df.jpg)  
+Original  
+Figure 12. Relighting. Our generated Iron Man model is relit using different lighting setups in Blender, producing diverse reflectance effects on the armor.
+
+![](images/71c6448fe863f12eecc4d8b63406fa896669848e8a5d58691f969869ad572f7f.jpg)  
+Factory
+
+![](images/c96d51c45b96de43aef0245606b668a1ff4e25e1bf42f95eecee602db3a4a5fe.jpg)  
+Night
+
+![](images/7d58f17661be479ef40debeb6fdf66026eb1da9fb8e12a33fff9e2cee3bcf8be.jpg)  
+Studio
+
+![](images/610b8842a83c9c64698a995d209d2817682a8c779cc1878256c466a66e074071.jpg)  
+Sunset
+
+ity of the generated surface geometry (e.g., no holes on the surface), the simulations are of high-level physical accuracy, as shown in the accompanying supplemental video.
+
+Moreover, we showcase the ability to modify the material of the statue by replacing it with another wood material downloaded from the Internet [32], which poses a challenge for comparative methods such as DreamFusion [33]. Additionally, in Fig. 11, by importing generated ice cream, tiger, and crown into the Cornell Box, we demonstrate the plausible physical interaction between our generated results and the scene, with natural shadows being cast. Finally, Fig. 12 illustrates the replacement of the HDR environment map to produce diverse lighting and corresponding reflectance effects on the generated iron man.
+
+# 6. Limitations
+
+While Fantasia3D demonstrates promising advancements in the realm of generating photorealistic 3D assets from text prompts, several limitations remain. For instance, while our method successfully produces loose visual effects, it remains a significant challenge to generate corresponding loose geometries, such as hair, fur, and grass, solely based on text prompts. Additionally, our method primarily emphasizes object generation, thereby lacking the capacity to generate complete scenes with background from text prompts. Consequently, our future research endeavors will be dedicated to addressing these limitations by focusing on the generation of complete scenes and intricate loose geometries.
+
+# 7. Conclusion
+
+In this paper, we present Fantasia3D, a new method for automatic text-to-3D generation. Fantasia3D uses disentangled modeling and learning of geometry and appearance, and is able to generate both the fine surface and rich material/texture. Fantasia3D is based on the hybrid scene representation of DMTET. For geometry learning, we propose to encode a rendered normal map, and use shape encoding of the normal as the input of the pre-trained, stable diffusion model. For appearance modeling, we introduce the spatially varying BRDF into the text-to-3D task, thus enabling material learning that supports photorealistic rendering of the learned surface. Expect for text prompts, our method can be triggered with a customized 3D shape as well; this is flexible for users to better control what content is to be generated. Our method is also convenient to support relighting, editing, and physical simulation of the generated 3D assets. Our method is based on pre-trained image diffusion models (i.e., the stable diffusion). In future research, we are interested in learning 3D diffusion directly from the large language models.
+
+Acknowledgements. This work is supported in part by Program for Guangdong Introducing Innovative and Entrepreneurial Teams (No.: 2017ZT07X183) and Guangdong R&D key project of China (No.: 2019B010155001).
+
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+        "text": "{a2pal, hichristensen}@ucsd.edu,{sahilwa, ayujaisw, xzhnamz, wuyue, rakhada, natarap}@amazon.com",
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+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
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+        "text": "Multi-turn textual feedback-based fashion image retrieval focuses on a real-world setting, where users can iteratively provide information to refine retrieval results until they find an item that fits all their requirements. In this work, we present a novel memory-based method, called FashionNTM, for such a multi-turn system. Our framework incorporates a new Cascaded Memory Neural Turing Machine (CM-NTM) approach for implicit state management, thereby learning to integrate information across all past turns to retrieve new images, for a given turn. Unlike vanilla Neural Turing Machine (NTM), our CM-NTM operates on multiple inputs, which interact with their respective memories via individual read and write heads, to learn complex relationships. Extensive evaluation results show that our proposed method outperforms the previous state-of-the-art algorithm by  $50.5\\%$ , on Multi-turn FashionIQ [60] – the only existing multi-turn fashion dataset currently, in addition to having a relative improvement of  $12.6\\%$  on Multi-turn Shoes – an extension of the single-turn Shoes dataset [5] that we created in this work. Further analysis of the model in a real-world interactive setting demonstrates two important capabilities of our model – memory retention across turns, and agnosticity to turn order for non-contradictory feedback. Finally, user study results show that images retrieved by FashionNTM were favored by  $83.1\\%$  over other multi-turn models.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Image retrieval has been extensively studied in the computer vision community, both using classical approaches [10, 52, 25] and recently, using learning-based techniques [2, 15, 41, 43, 23]. Existing works can be grouped based on input queries considered – from image-only queries, commonly known as Content-Based Image Retrieval (CBIR)",
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+        "img_path": "images/8ecb308d0ed8fef39963a7d3739c45ca2d5a58f521f5981f8419e3ca53342207.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of multi-turn fashion image retrieval. Initially (Turn 1), the system receives an initial query image, and a textual feedback mentioning the user's desired changes. The model then retrieves a ranked list of closest matching images. Subsequently, the user keeps refining their choice by providing more feedback, while the model retrieves newer images by considering both current and past feedback. This continues until the multi-turn system has successfully obtained the final retrieved image (Turn N) with all the desired properties mentioned across every past turn."
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+        "text": "[35, 39, 48], to attributes [18], sketches [44], and natural language [32, 61]. However, most of these methods do not incorporate interactive user feedback, which is necessary for a personalized task such as fashion retrieval.",
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+        "type": "text",
+        "text": "Textual feedback-based fashion image retrieval allows users to refine online shopping search results by providing information about how the results differ from their desired product (e.g., \"a dress like this but darker in color\"). Several approaches for implementing such a system have been proposed recently [14, 7, 31, 56, 17, 60, 63], which involve learning a joint representation via multi-modal information fusion across the query (reference) image and the associated feedback, and using it to retrieve the closest matching image in the database (product catalog) as the target.",
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+    {
+        "type": "text",
+        "text": "Popular methods for fashion retrieval task [14, 31, 7,",
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+    {
+        "type": "header",
+        "text": "CVF",
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+    },
+    {
+        "type": "header",
+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
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+        "type": "page_footnote",
+        "text": "*Work primarily done during internship at Amazon. Additional details are available at https://sites.google.com/eng.ucsd.edu/fashionntm.",
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+    {
+        "type": "page_number",
+        "text": "11323",
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+    {
+        "type": "text",
+        "text": "4, 23] involve single-turn exchange of information, where users provide feedback exactly once to update the search results. However, this is not characteristic of the real-world setting as online shopping customers typically start with a general idea of what they want and iteratively update the requirements until they find something that matches their desired features. This usually involves providing additional attributes, or modifying previously specified features in each turn to refine the search results. The ideal feedback-based fashion image retrieval system is, hence, inherently multi-turn, as illustrated in Figure 1.",
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+    {
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+        "text": "There are two major challenges associated with multi-turn image retrieval. First, there is a lack of sufficient training and evaluation datasets - despite the abundance of single-turn fashion retrieval datasets, to the best of our knowledge, there is only one publicly available multi-turn fashion image retrieval dataset [60] currently. This is because labeling a sequence of images while ensuring continuity, consistency, and uni-directional information flow is a difficult problem. Thus, to facilitate research in this domain, we created a new fashion image retrieval dataset to allow for further benchmarking. Second, generalizing performance to real-world dynamic user interactive cases is non-trivial - as this is still a relatively new research domain, most existing algorithms do not generalize beyond the training dataset to consider multiple turns of interactive feedback. In this work, we propose a novel memory-based framework to explicitly consider sequential feedback from users across multiple turns to retrieve desired items, both for the static image datasets, as well as real-world dynamic users.",
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+    {
+        "type": "text",
+        "text": "Sequential modeling is a relatively mature field of research. However, a majority of the existing approaches [46, 27, 49, 20, 9] do not maintain an explicit memory, and therefore cannot learn long and complex information. Vanilla memory network-based methods, which explicitly maintain an external memory, could be used for retaining past information, but they do not provide a robust mechanism to iteratively update their memory [55, 51]. In contrast, Neural Turing Machines (NTMs) [16] provide a fully differentiable model with sophisticated read and write operations to extract and update historical information in its explicit memory via an attention mechanism. Therefore, in this work, we build on NTMs to develop a novel framework for the multi-turn retrieval task. We further propose a novel Cascaded Memory Neural Turing Machine (CM-NTM) that allows us to encode multiple relationships from the features of a particular turn and store them over time across multiple memories in a multi-turn setting. This is similar to how multi-head attention (MHA) operates for transformers [53]. To ensure that the individual memories effectively utilize each other's information, we link them together in a cascaded fashion. Evaluation results demonstrate that our proposed approach improves the retrieval performance as com",
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+        "text": "pared to the previous state-of-the-art by  $50.5\\%$  on Multi-turn FashionIQ, and by  $12.6\\%$  on Multi-turn Shoes.",
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+        "text": "In summary, we make the following contributions. First, we propose a state-of-the-art memory-based framework, called FashionNTM, for multi-turn feedback-based fashion image retrieval, that uses an external memory to learn complex long-term relationships. Second, we develop a novel Cascaded Memory Neural Turing Machine (CM-NTM), that extends NTM to learn relationships across multiple inputs via additional controllers and read/write heads in a cascaded fashion. Third, we conduct experiments to show that the proposed approach outperforms existing state-of-the-art retrieval models by  $50.5\\%$  on Multi-turn FashionIQ [60], and around  $12.6\\%$  on the multi-turn version of Shoes dataset [5] respectively. Additionally, by performing an interactive analysis, we demonstrated two important capabilities of our multi-turn system - memory retention across turns, and agnosticity to turn order for non-contradictory feedback. Finally, a user study result shows that on an average, the images retrieved by our model are preferred  $83.1\\%$  more than those from other multi-turn methods.",
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+    {
+        "type": "text",
+        "text": "2. Related Work",
+        "text_level": 1,
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+        "text": "Single turn feedback-based fashion image retrieval - Previous works in feedback-based fashion image retrieval have primarily focused on the single-turn scenario [14, 40, 31, 7, 54, 4, 3, 59, 6, 36, 22], where a model is provided with a reference image along with an associated feedback text highlighting the desired attribute changes. The typical approach is to encode the multi-modal image and text input using pre-trained visual feature extractors [19, 24, 34] and sequential natural language processors [20, 12], respectively. More recently, pre-trained extractors such as Contrastive Language-Image Pre-training (CLIP) [45] have also been used [4, 3]. This is then followed by a transformer-based decoder network [53] for generating information-rich features for image retrieval from the database. Although these methods perform well in single-turn settings, they cannot be directly used for real-world applications that deal with multiple turns of information exchange.",
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+        "text": "Multi-turn visuo-linguistic methods - A few methods have been proposed recently for fusing visual and textual input across multiple turns of information exchange. A popular application has been the video dialog task [42, 21, 33, 30, 17], where a trained system is asked to answer questions based on an ongoing video dialog. However, these kinds of dialogs are primarily text-based for both the questioner and the answering agent, without any interaction across image and text inputs. In the fashion domain, there have been some early works for the multi-turn retrieval task. Guo et al. [17] proposed a model that uses convolutional neural networks (CNNs) for encoding images and text, followed by a recurrent neural network for aggregating sequences. Then,",
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+        "image_caption": [
+            "Single-turn model; fine-tuned on multi-turn data"
+        ],
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+        "image_caption": [
+            "Figure 2: The complete FashionNTM framework. During training, the model receives an input query image,  $I_{n}$ , and associated feedback,  $U_{n}$ . From these, query features  $f_{qry}(n)$  are computed using an off-the-shelf single-turn fashion image retrieval model. These features are then fed into the memory network which learns to retain useful information from the current turn, and combine it with information from previous turns by interacting with an external memory bank, via read/write operations. The memory-modified features  $f_{qry}'(n)$  are compared with the target index features  $f_{tar}(n)$ , and other negative samples  $f_{neg}(n)$ , to compute the training loss. At inference, the model receives a series of multi-modal inputs turn-wise and computes the final modified feature  $f_{qry}'(N)$  using the trained memory network. This is compared with features  $f_{cand}$  derived from different candidate images in the database to retrieve the closest matching images."
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+        "image_caption": [
+            " Cascaded Memory Neural Turing Machine"
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+        "image_caption": [
+            "Feature vectors"
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+        "image_caption": [
+            "Similarity metric"
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+        "type": "text",
+        "text": "a  $k$ -Nearest Neighbor search is performed across sampled images to get the closest match. The entire model is trained end-to-end using reinforcement learning (RL). Inspired by this, Zhang et al. [63, 62] proposed two approaches for enhancing the text-image feature fusion by adding constraints, and using offline interactive recommendation. Recently, Yuan et al. [60] released the first multi-turn fashion image retrieval dataset, based on the original single turn FashionIQ [56]. They also proposed a state-of-the-art model, which we directly compare with our approach.",
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+    {
+        "type": "text",
+        "text": "Memory networks for vision and language - Memory networks have been widely used for a number of natural language processing and computer vision applications. Some works [55, 51, 13, 11] utilize it for Sentence Video Questions and Answering (QA) task. Another popular application is video object segmentation [8, 58, 50, 37]. Recently, there have been some works on including memory in transformer architectures [26, 47, 57, 38]. However, these approaches design their memory to be used only for specific tasks, and hence cannot be directly compared with ours. In this work, we propose a memory network based method for the multi-turn fashion image retrieval task.",
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+    {
+        "type": "text",
+        "text": "3. FashionNTM",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The multi-turn feedback-based image retrieval task can be viewed as a series of information exchange transactions. We define a transaction as one session of query context comprising a query image and the associated feedback",
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+    {
+        "type": "text",
+        "text": "text. Notationally, an  $N$ -turn transaction is represented as  $T = [(I_1,U_1),(I_2,U_2),\\dots ,(I_N,U_N)]$ , where  $I_{n}$  and  $U_{n}$  correspond to the query image and feedback utterance respectively, at turn  $n\\in [1,N]$ . Given such a transaction, the aim of a multi-turn model is to iteratively retrieve the final desired target image  $I_{tar}(N)$  by ranking candidates in the fashion image database based on a matching score.",
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+        "text": "The overall pipeline of our approach, called Fashion-NTM, is illustrated in Figure 2. We start with a single-turn feature extraction module to encode the multi-modal image and text inputs of each turn  $n$  in a multi-turn transaction. It comprises two parallel blocks - (i) a query feature extractor (QE), for processing  $I_{n}$  and  $U_{n}$  to generate the joint query representation  $f_{qry}(n)$ , and (ii) a target feature extractor (TE), for encoding all the images in the database into their corresponding index features. For the ground-truth target image  $I_{tar}(n)$ , we call these features  $f_{tar}(n)$ , while for every other sample  $I_{neg}(n)$  in the batch, we name it  $f_{neg}(n)$ . The query feature  $f_{qry}(n)$  is fed turn-wise to the Cascaded Memory Neural Turing Machine (CM-NTM) block. CM-NTM first computes several derived features from the original query feature. Subsequently, the original query feature and each of the derived features interact with their own controllers, read/write heads, and sequentially update the memories in a cascaded manner, i.e. output of one memory goes as input to the next. Ultimately, we get the enhanced feature  $f_{qry}'(n)$  as the final output, which is then compared with the target feature  $f_{tar}(n)$  using a similarity score-based batch",
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+    {
+        "type": "page_number",
+        "text": "11325",
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+    {
+        "type": "image",
+        "img_path": "images/5d5325cefed9cbf5fae460ef795f671d31d32e65cacf004d4db81605f92a3c41.jpg",
+        "image_caption": [
+            "Figure 3: Cascaded Memory Neural Turing Machine (CM-NTM). It consists of  $C$  controllers; one for the query feature and  $C - 1$  for the derived features. Each controller interacts with its own memory using its own read and write heads. The controllers are linked with each other, forming a cascaded chain. The modified features from the last memory are output as the final embedding."
+        ],
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+    {
+        "type": "text",
+        "text": "loss function. This loss treats  $f_{tar}(n)$  as a positive sample and every other feature in the batch,  $f_{neg}(n)$ , as a negative sample. We use cosine-similarity [54, 31, 14] between feature vectors as the matching score. During inference, the model receives a sequence of multi-modal turn-wise inputs and computes the final modified feature  $f_{qry}'(N)$ . This is compared with feature  $f_{cand}$  derived from different candidate images in the database, and the closest match is retrieved as output. In the following sections, we describe the key components of our framework.",
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+    {
+        "type": "text",
+        "text": "Single-turn feature extractor - We utilize the state-of-the-art single-turn image retrieval model, FashionVLP [14], for extracting multi-modal text and image features. FashionVLP extracts image embeddings at multiple levels of granularity and incorporates a vision-language pre-trained transformer for fusing these encodings with text feedback to obtain multi-modal query features. It adopts a convolutional neural network (CNN) architecture with contextual attention for fusing fashion-contextual image features of target images. Specifically, for each turn of a multi-turn transaction, we obtain query features  $f_{qry}$  and target features  $f_{tar}$ . An  $N$ -turn retrieval transaction is of the form  $T_{retr} = [f_{qry}(1), f_{qry}(2), \\dots, f_{qry}(N)]$ , with the target feature representation given by  $f_{tar}(N)$ .",
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+        "type": "text",
+        "text": "3.1. Cascaded Memory Neural Turing Machine",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To learn relationships across transactions, we propose a novel Cascaded Memory Neural Turing Machine (CM-NTM) module. CM-NTM allows multiple inputs to interact with their own memories using individual read and write heads to learn multiple complex relationships in the input data. We achieve this by deriving several features from the original  $n$ -th turn query feature  $f_{qry}(n)$  using a projection function  $\\mathcal{F}$ , which comprises a fully-connected (FC) layer with batch normalization. Specifically, for a  $C$ -stage cascaded CM-NTM, we obtain the derived feature",
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+    {
+        "type": "text",
+        "text": "$f_{drv}^{c} = \\mathrm{BatchNorm}(\\mathrm{FC}_{\\mathrm{c}}(f_{qry}(n)))$  where  $c \\in [1, C - 1]$ , and  $\\mathrm{FC}_{\\mathrm{c}}(\\cdot)$  is the FC layer for the  $c$ -th stage. Having obtained  $C$  inputs comprising the original query and  $C - 1$  derived features, we pass them sequentially to our memory network. Figure 3 presents our memory network architecture. It consists of three main components: Controller, Read/Write heads, and Memory blocks.",
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+        "type": "text",
+        "text": "Controller - The vanilla NTM has a single controller which takes the query feature at turn  $n$ , along with the previous turn's read vector, and emits an intermediate controller output. This is used by the read and write heads to compute the current turn's attention weights, which are then used to update the memory. In our work, we introduce  $C$  different controllers - one for each of our inputs. Each controller,  $c \\in [1, C]$  can therefore interactively update its memory via individual read and write heads, allowing it to learn multiple complex relationships in each turn.",
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+        "type": "text",
+        "text": "Read/Write heads - The controller output is fed to these heads. The write head learns to generate erase and add parameters, which are used to update the current memory. Similarly, the read head generates an attention weight vector, which is used to obtain a weighted sum over the memory locations to get the read vector  $\\mathbf{r}_{out}(n)$ . In our work, we have separate read and write heads for each memory.",
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+        "type": "text",
+        "text": "Memory block - This is represented as a 2-D matrix of the form  $N \\times M$ , where  $N$  corresponds to memory locations and  $M$  to the vector size at each location. The output of each memory block is a fused representation of the controller output and the read vector. In our cascaded multi-memory setup, the controllers are linked in a chain, such that the memory-modified features are sequentially propagated. Specifically, for controller  $c \\in [1, C]$  at turn  $n$ , the input is given by  $\\mathrm{input}^c(n) = [\\mathbf{r}_{out}^{c-1}(n); f_{drv}^c; \\mathbf{r}_{out}^c(n-1)]$ , where  $\\mathbf{r}_{out}^0(n) = \\mathbf{r}_{out}^C(n-1)$ ,  $f_{drv}^C = f_{qry}(n)$ , and ; represents concatenation. The final output,  $f_{qry}'(n)$ , is the fused representation of the last controller output  $\\mathrm{output}_{ctrl}^C$  and final read vector  $\\mathbf{r}_{out}^C$ . We experiment with a different number of memories  $C$  and memory sizes in Section 4.7.",
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+    {
+        "type": "text",
+        "text": "3.2. Loss function",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We adopt a batch cross-entropy loss [14], where each entry in a batch acts as a negative sample for all other entries. In the multi-turn setting, we compute the loss function turnwise. Given a batch size  $B$ , the loss between predicted feature  $\\mathbf{x}_n = [{}^1 f_{qry}'(n), {}^2 f_{qry}'(n), \\ldots, {}^B f_{qry}'(n)]$ , and ground-truth  $\\mathbf{y}_n = [{}^1 f_{tar}(n), {}^2 f_{tar}(n), \\ldots, {}^B f_{tar}(n)]$ , at turn  $n$  is of the form",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} (\\mathbf {x} _ {n}, \\mathbf {y} _ {n}) = \\frac {1}{B} \\sum_ {i = 1} ^ {B} - \\log \\frac {e ^ {\\kappa (^ {i} f _ {q r y} ^ {\\prime} (n) , ^ {i} f _ {t a r} (n))}}{\\sum_ {j = 1} ^ {B} e ^ {\\kappa (^ {i} f _ {q r y} ^ {\\prime} (n) , ^ {j} f _ {t a r} (n))}}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>We mainly discuss our novel changes to the NTM design. For details regarding the vanilla architecture, please refer to the original paper [16].",
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+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "11326",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/1bf8c356c197e52a9b09ae5a2f50c37978a6d2e84276af064db1bd1f744ad94d.jpg",
+        "table_caption": [
+            "Table 1: Statistics of the datasets used in this work. The top 2 rows are for the recently proposed Multi-turn FashionIQ dataset [60]. The bottom two rows are for the Multi-turn version of the Shoes [5] dataset, which we created as part of our work."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td rowspan=\"2\">Split</td><td colspan=\"3\">Number of transactions with</td><td rowspan=\"2\">Total images</td></tr><tr><td>2-turns</td><td>3-turns</td><td>4-turns</td></tr><tr><td rowspan=\"2\">Multi-turn FashionIQ [60]</td><td>Train</td><td>6,897</td><td>1,733</td><td>475</td><td>10,438</td></tr><tr><td>Test</td><td>1,752</td><td>483</td><td>165</td><td>6,274</td></tr><tr><td rowspan=\"2\">Multi-turn Shoes (ours)</td><td>Train</td><td>1,659</td><td>1,036</td><td>982</td><td>11,030</td></tr><tr><td>Test</td><td>296</td><td>96</td><td>28</td><td>4,631</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\kappa$  is the cosine similarity metric [14]. In this way, each  $f_{tar}(n)$  in the batch,  $\\forall j\\in [1,B]$  and  $j\\neq i$  , serves as a negative sample  $f_{neg}(n)$  for a given  $f_{qry}^{\\prime}(n)$  . The turn-wise retrieval loss function is represented as  $\\mathcal{L}_{retr}^{n} =$ $\\mathcal{L}(\\mathbf{x}_n,\\mathbf{y}_n)$  , with the overall loss function for our proposed multi-turn model given by  $\\mathcal{L} = \\frac{1}{N}\\sum_{n = 1}^{N}\\mathcal{L}_{retr}^{n}$",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4. Experimental Evaluation",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1. Datasets",
+        "text_level": 1,
+        "bbox": [
+            76,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Multi-turn FashionIQ [60]: To the best of our knowledge, this is the only existing fashion dataset with multi-turn sessions, where each turn is derived from the original single-turn FashionIQ [56] dataset. It comprises of 11,505 sessions across three clothing types - dress, top-tee, and shirt. The dataset is split into transactions of 2-turns, 3-turns, and 4-turns. In each turn, the data is represented as a pair  $(I_n, U_n)$ , where  $I_n$  and  $U_n$  correspond to the query image and the feedback text for turn  $n$ .",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Multi-turn Shoes: The original Shoes dataset [5] contains images of 10 categories of women's shoes obtained from the web along with automatic labeling of attributes. Guo et al. [17] provide additional natural language descriptions of the images to make them suitable for single-turn feedback-based image retrieval. This resulted in about 10k training pairs and 4.6k test queries. In this work, we create a multi-turn extension of this dataset to further research in this domain. Like the approach described in [60], we concatenated several single-turn transactions by matching the target image of one session to the query of another. To maintain",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "consistency with Multi-turn FashionIQ, we also developed transactions of 2-turns, 3-turns, and 4-turns. Table 1 provides the statistics of both the datasets.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2. Implementation Details",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Single-turn feature extractor pre-training - We use the recently proposed single turn (ST) fashion image retrieval model, FashionVLP [14], to extract the query and target features for both datasets. To ensure a fair comparison with other algorithms, we re-train FashionVLP only on those single-turn queries that are part of the multi-turn dataset. The implementation details and hyperparameters are similar to those mentioned in Goenka et al. [14].",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "CM-NTM training - We build our CM-NTM model using the open-source implementation [28] of NTM [16]. We implement a separate controller for each memory in our cascaded design using Long Short Term Memory (LSTM) networks [20]. The read and write heads are composed of multi-layer perceptrons (MLPs) and attention networks. We use  $C = 4$  for Shoes, and  $C = 8$  memory stages for FashionIQ, as it is a larger dataset. To ensure equal turn lengths for batch training, we pad the short turn transactions by repeating the last transaction similar to Yuan et al. [60]. We train our models for 100 epochs with a batch size of 80 and a learning rate of 1e-4. We used the PyTorch framework with Adam [29] optimizer for training.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3. Baselines and Previous State-of-the-art",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We compare the performance of our model with six other single-turn and multi-turn approaches.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Single-turn methods - In these methods, we aggregate the multi-turn query data into a single feature without iterating over them turn-wise. The first baseline is  $\\mathrm{ST} + \\mathrm{avg}(\\mathrm{all~turns})$ , where we take the mean of all the  $N$  query features in a transaction to get a single mean query  $\\bar{f}_{qry}(N)$ . This is compared with the candidate target features  $f_{tar}(N)$ . Next, we have  $\\mathrm{ST} + \\mathrm{cat}(\\mathrm{all~captions})$ , where all the captions of a multi-turn transaction are concatenated into one long caption, along with initial reference image, to get query features  $\\hat{f}_{qry}(N)$ . This is compared with target features  $f_{tar}(N)$  to retrieve the final images.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/66f68bd13d416788b92343d9503a1e7837205d74db189d4d101da74d88354806.jpg",
+        "table_caption": [
+            "Table 2: Quantitative results on Multi-turn FashionIQ [60]. We compare with multiple single-turn and multi-turn baselines, and state-of-the-art works [17, 60]. Results show the superior performance of our proposed approach on the popularly used recall rate evaluation metric."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">Dress</td><td colspan=\"2\">Toptee</td><td colspan=\"2\">Shirt</td><td colspan=\"3\">Overall</td></tr><tr><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>Mean</td></tr><tr><td colspan=\"10\">Single-turn</td></tr><tr><td>ST + avg(all turns)</td><td>25.6</td><td>32.5</td><td>32.1</td><td>38.1</td><td>27.0</td><td>32.3</td><td>28.2</td><td>34.3</td><td>31.3</td></tr><tr><td>ST + cat(all captions)</td><td>30.2</td><td>36.3</td><td>36.0</td><td>44.1</td><td>35.3</td><td>42.5</td><td>33.8</td><td>41.0</td><td>37.4</td></tr><tr><td colspan=\"10\">Multi-turn</td></tr><tr><td>Dialog Manager [17]</td><td>12.7</td><td>16.7</td><td>11.6</td><td>15.8</td><td>13.9</td><td>17.7</td><td>13.1</td><td>15.2</td><td>14.2</td></tr><tr><td>CFIR [60]</td><td>29.8</td><td>33.5</td><td>29.4</td><td>33.6</td><td>30.5</td><td>34.1</td><td>30.3</td><td>33.4</td><td>31.9</td></tr><tr><td>ST + EWMA (ours)</td><td>42.0</td><td>48.4</td><td>43.8</td><td>50.9</td><td>36.9</td><td>44.2</td><td>40.9</td><td>47.8</td><td>44.4</td></tr><tr><td>ST + LSTM (ours)</td><td>47.8</td><td>52.5</td><td>44.4</td><td>50.5</td><td>41.6</td><td>47.6</td><td>44.6</td><td>50.2</td><td>47.4</td></tr><tr><td>FashionNTM (ours)</td><td>48.3</td><td>52.8</td><td>45.1</td><td>49.8</td><td>43.8</td><td>48.8</td><td>45.7</td><td>50.4</td><td>48.1</td></tr></table>",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "11327",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Multi-turn methods - In these methods, the multi-turn data is fed turn-wise to the model. The first method is Dialog Manager [17], which employs reinforcement learning (RL) framework to learn relationships between turns. Next, we have the previous state-of-the-art Conversational Fashion Image Retrieval CFIR method [60], which encodes only the text using a transformer network [53], and uses a simple Gated Recurrent Unit (GRU) layer for multi-turn image retrieval. The third baseline is  $\\mathrm{ST} + \\mathrm{EWMA}$ , where we use exponential weighted moving average [1] to heuristically aggregate past turns, with more weights given to recent history. Finally, we have the  $\\mathrm{ST} + \\mathrm{LSTM}$  model that uses a single-layer LSTM [20] with a hidden size of 100 for aggregating past information.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.4. Results",
+        "text_level": 1,
+        "bbox": [
+            76,
+            311,
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+            325
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics - Following [14, 60, 17], we evaluate models using the standard top-K recall (i.e. R@K) for image retrieval. Overall performance are compared specifically on the average of R@5 and R@8.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Quantitative results on multi-turn datasets - In Table 2, we compare the results of different methods on the multi-turn FashionIQ dataset. We observe that the multi-turn baselines generally perform better than single-turn methods. This is expected as aggregating data by naively averaging/concatenating loses feedback content and turn-order information and hence is likely to miss out on important cues. Our memory-based approach outperforms all the other multi-turn baselines by a large margin. This shows that the memory network can store and retrieve useful information to and from the memory between intermediate turns, which allows it to keep track of past information better than other networks that do not use explicit memory. The  $50.5\\%$  performance gain over the previous state-of-the-art highlights our model's capability to learn meaningful representations over multiple turns of conversational feedback.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Table 3, we provide a similar comparison for the Multi-turn Shoes dataset. Consistent with results for Multi-turn FashionIQ, the multi-turn baselines perform better than",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Comparison with existing single-turn and multi-turn models on the multi-turn version of the Shoes [5] dataset. We compare with multiple single-turn, and multi-turn baselines. Comparative analysis shows the superior performance of our proposed approach using the popular recall rate evaluation metric.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/bc2f8144efc625fca9621db7e46ddeaa6a24bbb1167c3fcbdca6480848fa57ec.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>R@5</td><td>R@8</td><td>Mean</td></tr><tr><td colspan=\"4\">Single-turn</td></tr><tr><td>ST + avg(all turns)</td><td>12.4</td><td>17.6</td><td>15.0</td></tr><tr><td>ST + cat(all captions)</td><td>10.7</td><td>13.6</td><td>12.2</td></tr><tr><td colspan=\"4\">Multi-turn</td></tr><tr><td>ST + EWMA (ours)</td><td>18.3</td><td>23.8</td><td>21.1</td></tr><tr><td>ST + LSTM (ours)</td><td>23.3</td><td>32.1</td><td>27.7</td></tr><tr><td>FashionNTM (ours)</td><td>26.7</td><td>35.7</td><td>31.2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "all the single-turn ones, and our memory network-based approach performs the best with a relative improvement of  $12.6\\%$ . The difference in performance across different models is more pronounced in these results than in Table 2. This is possibly because the annotations are cleaner and more consistent in this dataset as compared to Multi-turn FashionIQ [60]. For instance, multiple images in the FashionIQ dataset can match a particular query, but only one of them is labeled as the ground-truth.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "An important property of a good multi-turn system is that performance should be robust to the length of the historical information (number of past turns). For example, even for a large number of previous turns considered, the model should efficiently retain desirable details, while filtering out unnecessary information. In Table 4, we analyze this property for models with and without memory. A single-turn model that assumes ground-truth information for all past turns, and evaluated only on the final-turn, is taken as reference. This is expected to be the upper-bound on the performance in a single-turn setting, as perfect information about the history is guaranteed. We vary the number of past turns included in the input transaction history (versus treated as ground-truth) for the multi-turn models (with or without memory) in order to evaluate their effectiveness at capturing and utilizing past information. As seen from the table, for a model without memory, the performance depreciates significantly with each additional turn from the",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Importance of aggregating historical data using memory-based vs non-memory approach. In the first row, we show the result of a model using only the final turn information of a multi-turn transaction. This assumes the groundtruth retrieval for all previous turns, and therefore provides the upper-bound on single-turn performance for final retrieval. Subsequently, we include additional information from the history, and compare performance across models with and without memory. As seen from the non-memory case, the performance depreciates a lot ( $\\approx$  64.9% difference compared to the final turn's performance), as we try to aggregate longer historical information. In contrast, for the memory network model, it can be seen that the performance is fairly consistent ( $\\approx$  0.5% difference) across the turn length.",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0c73d11e836bd8ed07209ff79930e73a576d31b9d80ceddb47f2c5e628137385.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Memory usage</td><td rowspan=\"2\">Turn configuration</td><td colspan=\"2\">Dress</td><td colspan=\"2\">Toptee</td><td colspan=\"2\">Shirt</td><td colspan=\"3\">Overall</td><td rowspan=\"2\">Difference from final turn</td></tr><tr><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>Mean</td></tr><tr><td>-</td><td>only final turn</td><td>77.9</td><td>77.9</td><td>84.0</td><td>84.0</td><td>74.1</td><td>77.8</td><td>78.7</td><td>79.9</td><td>79.3</td><td>-</td></tr><tr><td colspan=\"12\">Experiments with data aggregated across multiple-turns</td></tr><tr><td rowspan=\"3\">Only single turn (w/o memory)</td><td>last two turns</td><td>51.3</td><td>58.4</td><td>56.0</td><td>76.0</td><td>44.4</td><td>48.1</td><td>50.6</td><td>60.9</td><td>55.8</td><td>-29.6%</td></tr><tr><td>last three turns</td><td>33.6</td><td>43.4</td><td>32.0</td><td>44.0</td><td>29.6</td><td>40.7</td><td>31.8</td><td>42.7</td><td>37.3</td><td>-53.0%</td></tr><tr><td>all turns</td><td>18.6</td><td>28.3</td><td>32.0</td><td>32.0</td><td>25.9</td><td>29.6</td><td>25.5</td><td>30.0</td><td>27.8</td><td>-64.9%</td></tr><tr><td rowspan=\"3\">FashionNTM (with memory)</td><td>last two turns</td><td>76.1</td><td>77.9</td><td>84.0</td><td>84.0</td><td>77.8</td><td>77.8</td><td>79.3</td><td>79.9</td><td>79.6</td><td>+0.4%</td></tr><tr><td>last three turns</td><td>77.0</td><td>77.9</td><td>84.0</td><td>84.0</td><td>77.8</td><td>77.8</td><td>79.6</td><td>79.9</td><td>79.8</td><td>+0.6%</td></tr><tr><td>all turns</td><td>76.1</td><td>77.9</td><td>84.0</td><td>84.0</td><td>77.8</td><td>77.8</td><td>79.3</td><td>79.9</td><td>79.6</td><td>+0.4%</td></tr></table>",
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+    },
+    {
+        "type": "page_number",
+        "text": "11328",
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+    {
+        "type": "image",
+        "img_path": "images/4f2d844c6cd723d0ff98953db67cb4307fcd4ceaf7fa767f0e827d0e310f3ba6.jpg",
+        "image_caption": [
+            "(a) Multi-turn FashionIQ. From left to right, the images belong to the Dress, Shirt, and Toptee categories respectively."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f7685d81cb666dc03f51972cdba64f2870b305f1d65baa6d982d579c99787d1.jpg",
+        "image_caption": [
+            "(b) Multi-turn Shoes. From left to right, we have three different samples of shoes from the dataset.",
+            "Figure 4: Top-5 final image retrievals on the evaluated multi-turn datasets. The top row illustrates 3 sets of multi-turn query session. We consider three different multi-turn models - EWMA, LSTM, and our proposed FashionNTM. As seen from the retrievals, our proposed model correctly predicts the target image in all 3 cases for the FashionIQ dataset, and in 2 out of 3 cases for the Shoes dataset."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "past treated as input rather than ground-truth. As we go further back in history, the performance consistently reduces. However, for our multi-turn model with memory, the performance does not change appreciably as the number of turns change. This shows that our proposed approach can successfully retain/filter out past data based on their relevance, across various lengths of history.",
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+        "type": "text",
+        "text": "An interesting observation is that having only the final turn with memory does not yield a good result. This is possibly due to the initialization method of the memory network, which is random. Hence, in absence of a history (only single turn case), the only past features to be aggregated are the random initialization features.",
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+    {
+        "type": "text",
+        "text": "Qualitative results - In addition to the quantitative experiments described above, we also present some qualitative final image retrieval results of our evaluated models on both the multi-turn datasets. The first set of results are shown in Figures 4a and 4b for the Multi-turn FashionIQ [60] and the",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Multi-turn Shoes datasets, respectively<sup>2</sup>. We compare the top-5 predicted results from our proposed model with two other multi-turn baselines, ST + EWMA and ST + LSTM. As seen in the figures, our approach can correctly predict the desired target image for both datasets with higher ranks as compared to other baselines. More specifically, we see that in Figure 4a, even though all the three multi-turn models can infer the general sense of desired attributes, such as \"is longer\" in the left block and \"has short sleeves\" in the right, only our model can capture complex and detailed properties, e.g., \"gray and not red\" in the left block, and \"red color with center logo\" in the middle block. Furthermore, our model can retrieve multiple desirable products, as seen by the first four images in the middle block, and four out of five images in the right block. Similar results are observed for Shoes in Figure 4b, where our model correctly predicts",
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+    {
+        "type": "page_footnote",
+        "text": "2Please refer to the supplementary material for additional results, along with the differences in annotation quality between the two datasets.",
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+        "text": "11329",
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+    {
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+        "img_path": "images/f7eb55bc6d7b6c2e082298f5770b585b7874676b8514e0f827e55a0542b77e65.jpg",
+        "image_caption": [
+            "Figure 5: Memory retention capability of our proposed approach. Given an initial query image, we take two interactive user feedbacks in turns. On the left side, we have the single-turn model which only learns to retrieve an image using a single dialog exchange. As a result, none of the retrieved images in turn 2 are \"green in color\", which was desired in turn 1. In contrast, our proposed approach on the right can learn to retain data from both the turns, and therefore retrieves desirable product in 2 out of 3 cases."
+        ],
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+    {
+        "type": "text",
+        "text": "the desired target as rank-1 in two out of three examples, whereas the other models fail to retrieve meaningful results.",
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+    {
+        "type": "text",
+        "text": "4.5. Model Analysis in Interactive Settings",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In addition to the results above for the static dataset, we also performed some interactive experiments to evaluate whether our model can adapt to real-world dynamic use cases beyond the trained datasets.",
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+    {
+        "type": "text",
+        "text": "Memory retention - In this experiment, we demonstrate the memory retention capability of our proposed model by comparing it with a single-turn network, which does not retain historical information. We start with an initial query image retrieved via a single-turn model. Subsequently, we take user input for the next two turns to retrieve newer sets of images. As seen in Figure 5, for the single-turn model (left side), none of the retrieved images in turn 2 are green in color. This is because the \"green in color\" attribute was a desired property in turn 1, which the model without memory could not recover. In contrast, for our memory network approach (right side) both the top-2 retrieved images for turn 2 are \"green in color\" in addition to having \"a solid color and small image\". Thus, our model can learn to retain information from previous turns.",
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+    {
+        "type": "text",
+        "text": "Agnosticity to turn order - Ideally, a deployed multi-turn image retrieval system should be independent of the order of feedbacks provided, as long as they are non-contradictory. This is demonstrated in the experiment conducted in Figure 6. In the first case, we take two text inputs as feedbacks from a user and present them to the model. In the second case, we reverse the order of the feedbacks. As shown in the figure, for the flipped case, our proposed FashionNTM model retrieves similar looking final products, even though the intermediate retrievals are very different.",
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+    {
+        "type": "text",
+        "text": "It is to be noted that for both the experiments depicted in Figures 5 and 6, the feedback is taken from a dynamic user, thereby establishing the interactive capability of our",
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+    {
+        "type": "image",
+        "img_path": "images/9d9d51acf4766157a663d3f12189104f011612f36551dadc14201d924d0e795e.jpg",
+        "image_caption": [
+            "Figure 6: Turn order independence feature of a memory network. In this experiment, we start with an initial query image, and take two non-contradictory user feedbacks. In one case (left), we let the model retrieve images based on the original order of the feedbacks, whilst in the other (right), we flip the order of the feedbacks. Our proposed approach adapts to presented input, and retrieves similar looking final results for both the cases, even though the intermediate outputs are quite different."
+        ],
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+    {
+        "type": "text",
+        "text": "proposed model beyond the training dataset.",
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+    {
+        "type": "text",
+        "text": "4.6. User Study",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Fashion image retrieval is inherently a subjective task, where the task success heavily relies on the satisfaction of a customer. Thus, we conducted a small human-preference survey among 5 participants (not associated with the paper in any way). To each user, we showed the final top-1 retrieved image by the 3-best multi-turn models on the FashionIQ [60] dataset from Table 2 - EWMA, LSTM, and our proposed FashionNTM. To ensure consistency, we generated 45 queries for this study, whose results are shown in Figure 7. The  $y$ -axis shows the number of preferred retrieval results for each user. Results show that each of the 5 users preferred images retrieved by the proposed FashionNTM model, on an average  $83.1\\%$  more as compared to other multi-turn methods. In the supplementary material, we include some examples of the user interface shown to",
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+    {
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+        "image_caption": [
+            "Figure 7: Human preference study of top multi-turn systems."
+        ],
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+        "text": "11330",
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+    {
+        "type": "table",
+        "img_path": "images/fbb40748c5145270ff5fb3cf0a47bb229d7afc5f072bf68ce478e4d871530e18.jpg",
+        "table_caption": [
+            "Table 5: Different number of memories for the proposed approach by fixing the memory size to  $4 \\times 768$ . We select the mean value for comparison and pick the best one."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Number of memories (C)</td><td>R@5</td><td>R@8</td><td>Mean</td><td>% increase</td></tr><tr><td>ST+v-NTM</td><td>1</td><td>24.5</td><td>30.5</td><td>27.5</td><td>-</td></tr><tr><td rowspan=\"4\">FashionNTM</td><td>2</td><td>26.4</td><td>33.8</td><td>30.1</td><td>9.5</td></tr><tr><td>4</td><td>27.6</td><td>35.5</td><td>31.5</td><td>14.5</td></tr><tr><td>8</td><td>26.9</td><td>35.2</td><td>31.1</td><td>13.1</td></tr><tr><td>16</td><td>26.4</td><td>33.8</td><td>30.1</td><td>9.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "the participants during this study.",
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+    },
+    {
+        "type": "text",
+        "text": "4.7. Ablation Studies",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We perform multiple ablation studies to gain insights on how changing different configurations of the memory network affect the overall performance. We perform these ablations on the multi-turn Shoes dataset as it contains more realistic and consistent feedback texts.<sup>3</sup>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Number of memories in CM-NTM - This experiment involves varying the number of memories  $C$  in our cascaded memory architecture. In Table 5, we observe that the cascaded memory CM-NTM models perform significantly better than vanilla NTM, which has only one memory. We hypothesize that having inputs from multiple turns interacting with the same memory could eventually lead to saturation as we get additional data which could be alleviated if there are multiple memories in the network to recover the past which might help in capturing multiple complex relationships in multi-turn interactions better. Additionally, multiple memory networks can help in learning diverse representations of the input using derived features, which is not possible with a single memory network. As seen in Table 5 for Shoes dataset, the performance increases with the number of memories, peaking at  $C = 4$ , and then gradually decreases as the model starts to overfit.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Inference time with multiple memories - In this experiment, we study the performance of FashionNTM in terms of the mean of R@5 and R@8 recall rates along with the time taken to process one multi-turn transaction. For each  $C$ , we evaluate the performance across four different memory sizes. As seen from Figure 8, the inference time increases with additional memories. Configurations in the green and blue clusters are desirable, as they provide a good trade-off between recall performance and computation time, while the pink and red clusters are undesirable due to poor performance and longer inference time, respectively.",
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+    {
+        "type": "text",
+        "text": "5. Conclusion and Future Work",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we presented a novel cascaded Neural Turing Machine-based approach, called FashionNTM, for multi-turn feedback-based fashion image retrieval. Multiple",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/f18ce485451afd9b388c4009f09403a9674bffd8188731395ffe15d2ab912694.jpg",
+        "image_caption": [
+            "Figure 8: Scatter plot showing recall versus inference time for multiple memories in our CM-NTM for the Multi-turn Shoes dataset. The configurations belonging to the green cluster give the best recall overall, while having a high inference time. The configurations in the blue cluster provide a suitable alternative with quicker inference at the cost of lower recall. The magenta and red clusters are undesirable configurations due to poor performance and long inference time, respectively."
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "memories in our model allow it to effectively retain and recall a number of complex relationships across transactions in the multi-turn setting, and multiple controllers help in assigning relative importance to each feature stored in the memory. This aids in attending to different parts of the input at different levels, thus leading to better performance across datasets. We also performed extensive experiments to compare our performance with baselines and previous state-of-the-art and observed that our multi-memory model significantly outperforms previous works [60], with up to  $50.5\\%$  relative improvement on Multi-turn FashionIQ, and by  $12.6\\%$  on the multi-turn Shoes dataset, which we created in this work. We further demonstrated that our model can generalize beyond the trained setting to dynamically interact with real-world users to retrieve meaningful final product images. Finally, a user preference study reveals that our model is preferred by human participants on an average  $83.1\\%$  more as compared to other multi-turn methods.",
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+    {
+        "type": "text",
+        "text": "Despite promising results, there are a few limitations that make multi-turn image retrieval a hard problem to solve. Firstly, there is dearth of high quality and diverse multi-turn image retrieval datasets in the fashion domain which hinders comprehensive studies in this field. Additionally, deploying current approaches to real-world scenarios (e.g., virtual private assistants) becomes a challenge due to computational requirements. Lastly, selecting the right configuration for different components such as memory size, number of memories, etc. in a memory-based network is not a trivial task and depends a lot on the use case. Nevertheless, for future work, our approach could be further extended to non-fashion domains where multi-turn feedback-based information retrieval solutions are required.",
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+    {
+        "type": "page_footnote",
+        "text": "3For a similar study on the Multi-turn Fashion-IQ dataset, please refer to the supplementary material.",
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+    },
+    {
+        "type": "text",
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2023/FashionNTM_ Multi-turn Fashion Image Retrieval via Cascaded Memory/96c19156-eda3-4f76-b877-c0799e40849d_origin.pdf

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2023/FashionNTM_ Multi-turn Fashion Image Retrieval via Cascaded Memory/full.md

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+# FashionNTM: Multi-turn Fashion Image Retrieval via Cascaded Memory
+
+Anwesan Pal\*, Sahil Wadhwa?, Ayush Jaiswal?, Xu Zhang?, Yue Wu?, Rakesh Chada?, Pradeep Natarajan?, and Henrik I. Christensen
+
+$^{1}$ UC San Diego,  $^{2}$ Amazon Alexa Natural Understanding,
+
+{a2pal, hichristensen}@ucsd.edu,{sahilwa, ayujaisw, xzhnamz, wuyue, rakhada, natarap}@amazon.com
+
+# Abstract
+
+Multi-turn textual feedback-based fashion image retrieval focuses on a real-world setting, where users can iteratively provide information to refine retrieval results until they find an item that fits all their requirements. In this work, we present a novel memory-based method, called FashionNTM, for such a multi-turn system. Our framework incorporates a new Cascaded Memory Neural Turing Machine (CM-NTM) approach for implicit state management, thereby learning to integrate information across all past turns to retrieve new images, for a given turn. Unlike vanilla Neural Turing Machine (NTM), our CM-NTM operates on multiple inputs, which interact with their respective memories via individual read and write heads, to learn complex relationships. Extensive evaluation results show that our proposed method outperforms the previous state-of-the-art algorithm by  $50.5\%$ , on Multi-turn FashionIQ [60] – the only existing multi-turn fashion dataset currently, in addition to having a relative improvement of  $12.6\%$  on Multi-turn Shoes – an extension of the single-turn Shoes dataset [5] that we created in this work. Further analysis of the model in a real-world interactive setting demonstrates two important capabilities of our model – memory retention across turns, and agnosticity to turn order for non-contradictory feedback. Finally, user study results show that images retrieved by FashionNTM were favored by  $83.1\%$  over other multi-turn models.
+
+# 1. Introduction
+
+Image retrieval has been extensively studied in the computer vision community, both using classical approaches [10, 52, 25] and recently, using learning-based techniques [2, 15, 41, 43, 23]. Existing works can be grouped based on input queries considered – from image-only queries, commonly known as Content-Based Image Retrieval (CBIR)
+
+![](images/8ecb308d0ed8fef39963a7d3739c45ca2d5a58f521f5981f8419e3ca53342207.jpg)  
+Figure 1: Illustration of multi-turn fashion image retrieval. Initially (Turn 1), the system receives an initial query image, and a textual feedback mentioning the user's desired changes. The model then retrieves a ranked list of closest matching images. Subsequently, the user keeps refining their choice by providing more feedback, while the model retrieves newer images by considering both current and past feedback. This continues until the multi-turn system has successfully obtained the final retrieved image (Turn N) with all the desired properties mentioned across every past turn.
+
+[35, 39, 48], to attributes [18], sketches [44], and natural language [32, 61]. However, most of these methods do not incorporate interactive user feedback, which is necessary for a personalized task such as fashion retrieval.
+
+Textual feedback-based fashion image retrieval allows users to refine online shopping search results by providing information about how the results differ from their desired product (e.g., "a dress like this but darker in color"). Several approaches for implementing such a system have been proposed recently [14, 7, 31, 56, 17, 60, 63], which involve learning a joint representation via multi-modal information fusion across the query (reference) image and the associated feedback, and using it to retrieve the closest matching image in the database (product catalog) as the target.
+
+Popular methods for fashion retrieval task [14, 31, 7,
+
+4, 23] involve single-turn exchange of information, where users provide feedback exactly once to update the search results. However, this is not characteristic of the real-world setting as online shopping customers typically start with a general idea of what they want and iteratively update the requirements until they find something that matches their desired features. This usually involves providing additional attributes, or modifying previously specified features in each turn to refine the search results. The ideal feedback-based fashion image retrieval system is, hence, inherently multi-turn, as illustrated in Figure 1.
+
+There are two major challenges associated with multi-turn image retrieval. First, there is a lack of sufficient training and evaluation datasets - despite the abundance of single-turn fashion retrieval datasets, to the best of our knowledge, there is only one publicly available multi-turn fashion image retrieval dataset [60] currently. This is because labeling a sequence of images while ensuring continuity, consistency, and uni-directional information flow is a difficult problem. Thus, to facilitate research in this domain, we created a new fashion image retrieval dataset to allow for further benchmarking. Second, generalizing performance to real-world dynamic user interactive cases is non-trivial - as this is still a relatively new research domain, most existing algorithms do not generalize beyond the training dataset to consider multiple turns of interactive feedback. In this work, we propose a novel memory-based framework to explicitly consider sequential feedback from users across multiple turns to retrieve desired items, both for the static image datasets, as well as real-world dynamic users.
+
+Sequential modeling is a relatively mature field of research. However, a majority of the existing approaches [46, 27, 49, 20, 9] do not maintain an explicit memory, and therefore cannot learn long and complex information. Vanilla memory network-based methods, which explicitly maintain an external memory, could be used for retaining past information, but they do not provide a robust mechanism to iteratively update their memory [55, 51]. In contrast, Neural Turing Machines (NTMs) [16] provide a fully differentiable model with sophisticated read and write operations to extract and update historical information in its explicit memory via an attention mechanism. Therefore, in this work, we build on NTMs to develop a novel framework for the multi-turn retrieval task. We further propose a novel Cascaded Memory Neural Turing Machine (CM-NTM) that allows us to encode multiple relationships from the features of a particular turn and store them over time across multiple memories in a multi-turn setting. This is similar to how multi-head attention (MHA) operates for transformers [53]. To ensure that the individual memories effectively utilize each other's information, we link them together in a cascaded fashion. Evaluation results demonstrate that our proposed approach improves the retrieval performance as com
+
+pared to the previous state-of-the-art by  $50.5\%$  on Multi-turn FashionIQ, and by  $12.6\%$  on Multi-turn Shoes.
+
+In summary, we make the following contributions. First, we propose a state-of-the-art memory-based framework, called FashionNTM, for multi-turn feedback-based fashion image retrieval, that uses an external memory to learn complex long-term relationships. Second, we develop a novel Cascaded Memory Neural Turing Machine (CM-NTM), that extends NTM to learn relationships across multiple inputs via additional controllers and read/write heads in a cascaded fashion. Third, we conduct experiments to show that the proposed approach outperforms existing state-of-the-art retrieval models by  $50.5\%$  on Multi-turn FashionIQ [60], and around  $12.6\%$  on the multi-turn version of Shoes dataset [5] respectively. Additionally, by performing an interactive analysis, we demonstrated two important capabilities of our multi-turn system - memory retention across turns, and agnosticity to turn order for non-contradictory feedback. Finally, a user study result shows that on an average, the images retrieved by our model are preferred  $83.1\%$  more than those from other multi-turn methods.
+
+# 2. Related Work
+
+Single turn feedback-based fashion image retrieval - Previous works in feedback-based fashion image retrieval have primarily focused on the single-turn scenario [14, 40, 31, 7, 54, 4, 3, 59, 6, 36, 22], where a model is provided with a reference image along with an associated feedback text highlighting the desired attribute changes. The typical approach is to encode the multi-modal image and text input using pre-trained visual feature extractors [19, 24, 34] and sequential natural language processors [20, 12], respectively. More recently, pre-trained extractors such as Contrastive Language-Image Pre-training (CLIP) [45] have also been used [4, 3]. This is then followed by a transformer-based decoder network [53] for generating information-rich features for image retrieval from the database. Although these methods perform well in single-turn settings, they cannot be directly used for real-world applications that deal with multiple turns of information exchange.
+
+Multi-turn visuo-linguistic methods - A few methods have been proposed recently for fusing visual and textual input across multiple turns of information exchange. A popular application has been the video dialog task [42, 21, 33, 30, 17], where a trained system is asked to answer questions based on an ongoing video dialog. However, these kinds of dialogs are primarily text-based for both the questioner and the answering agent, without any interaction across image and text inputs. In the fashion domain, there have been some early works for the multi-turn retrieval task. Guo et al. [17] proposed a model that uses convolutional neural networks (CNNs) for encoding images and text, followed by a recurrent neural network for aggregating sequences. Then,
+
+![](images/c762c27b3c1f5f4a7dbe63b4bc7c1448a76711883d925d4bb33cd25b057f45a7.jpg)  
+Single-turn model; fine-tuned on multi-turn data
+
+![](images/b7b27d1364b087b746b6837d63caf9ae1703c7b378da4e6f7aa7e2cdde645c88.jpg)  
+Figure 2: The complete FashionNTM framework. During training, the model receives an input query image,  $I_{n}$ , and associated feedback,  $U_{n}$ . From these, query features  $f_{qry}(n)$  are computed using an off-the-shelf single-turn fashion image retrieval model. These features are then fed into the memory network which learns to retain useful information from the current turn, and combine it with information from previous turns by interacting with an external memory bank, via read/write operations. The memory-modified features  $f_{qry}'(n)$  are compared with the target index features  $f_{tar}(n)$ , and other negative samples  $f_{neg}(n)$ , to compute the training loss. At inference, the model receives a series of multi-modal inputs turn-wise and computes the final modified feature  $f_{qry}'(N)$  using the trained memory network. This is compared with features  $f_{cand}$  derived from different candidate images in the database to retrieve the closest matching images.
+
+![](images/0a93bcd3c48137cf8df02f4721975b3ffd4112d58b27586905fe1bab10ae0a7a.jpg)  
+Multi-turn dataset
+
+![](images/03ed5232c9152ec6d51e339eca8d452d04d34d2a622ff422ec516e29ad11407e.jpg)
+
+![](images/daa67cb922be2029d925d4dfe9531c9a80373c998bf45ca6a8fe118af4323b91.jpg)  
+ Cascaded Memory Neural Turing Machine
+
+![](images/ce2c536c100748bb3484cf9df099bea6109ecb2f6c1d61338fd3af15f74b4023.jpg)  
+Feature vectors
+
+![](images/dd26d56686c72ab14bb528c677466893b7539f0e8b650a492c4ca0d86a1372b9.jpg)  
+Similarity metric
+
+a  $k$ -Nearest Neighbor search is performed across sampled images to get the closest match. The entire model is trained end-to-end using reinforcement learning (RL). Inspired by this, Zhang et al. [63, 62] proposed two approaches for enhancing the text-image feature fusion by adding constraints, and using offline interactive recommendation. Recently, Yuan et al. [60] released the first multi-turn fashion image retrieval dataset, based on the original single turn FashionIQ [56]. They also proposed a state-of-the-art model, which we directly compare with our approach.
+
+Memory networks for vision and language - Memory networks have been widely used for a number of natural language processing and computer vision applications. Some works [55, 51, 13, 11] utilize it for Sentence Video Questions and Answering (QA) task. Another popular application is video object segmentation [8, 58, 50, 37]. Recently, there have been some works on including memory in transformer architectures [26, 47, 57, 38]. However, these approaches design their memory to be used only for specific tasks, and hence cannot be directly compared with ours. In this work, we propose a memory network based method for the multi-turn fashion image retrieval task.
+
+# 3. FashionNTM
+
+The multi-turn feedback-based image retrieval task can be viewed as a series of information exchange transactions. We define a transaction as one session of query context comprising a query image and the associated feedback
+
+text. Notationally, an  $N$ -turn transaction is represented as  $T = [(I_1,U_1),(I_2,U_2),\dots ,(I_N,U_N)]$ , where  $I_{n}$  and  $U_{n}$  correspond to the query image and feedback utterance respectively, at turn  $n\in [1,N]$ . Given such a transaction, the aim of a multi-turn model is to iteratively retrieve the final desired target image  $I_{tar}(N)$  by ranking candidates in the fashion image database based on a matching score.
+
+The overall pipeline of our approach, called Fashion-NTM, is illustrated in Figure 2. We start with a single-turn feature extraction module to encode the multi-modal image and text inputs of each turn  $n$  in a multi-turn transaction. It comprises two parallel blocks - (i) a query feature extractor (QE), for processing  $I_{n}$  and  $U_{n}$  to generate the joint query representation  $f_{qry}(n)$ , and (ii) a target feature extractor (TE), for encoding all the images in the database into their corresponding index features. For the ground-truth target image  $I_{tar}(n)$ , we call these features  $f_{tar}(n)$ , while for every other sample  $I_{neg}(n)$  in the batch, we name it  $f_{neg}(n)$ . The query feature  $f_{qry}(n)$  is fed turn-wise to the Cascaded Memory Neural Turing Machine (CM-NTM) block. CM-NTM first computes several derived features from the original query feature. Subsequently, the original query feature and each of the derived features interact with their own controllers, read/write heads, and sequentially update the memories in a cascaded manner, i.e. output of one memory goes as input to the next. Ultimately, we get the enhanced feature  $f_{qry}'(n)$  as the final output, which is then compared with the target feature  $f_{tar}(n)$  using a similarity score-based batch
+
+![](images/5d5325cefed9cbf5fae460ef795f671d31d32e65cacf004d4db81605f92a3c41.jpg)  
+Figure 3: Cascaded Memory Neural Turing Machine (CM-NTM). It consists of  $C$  controllers; one for the query feature and  $C - 1$  for the derived features. Each controller interacts with its own memory using its own read and write heads. The controllers are linked with each other, forming a cascaded chain. The modified features from the last memory are output as the final embedding.
+
+loss function. This loss treats  $f_{tar}(n)$  as a positive sample and every other feature in the batch,  $f_{neg}(n)$ , as a negative sample. We use cosine-similarity [54, 31, 14] between feature vectors as the matching score. During inference, the model receives a sequence of multi-modal turn-wise inputs and computes the final modified feature  $f_{qry}'(N)$ . This is compared with feature  $f_{cand}$  derived from different candidate images in the database, and the closest match is retrieved as output. In the following sections, we describe the key components of our framework.
+
+Single-turn feature extractor - We utilize the state-of-the-art single-turn image retrieval model, FashionVLP [14], for extracting multi-modal text and image features. FashionVLP extracts image embeddings at multiple levels of granularity and incorporates a vision-language pre-trained transformer for fusing these encodings with text feedback to obtain multi-modal query features. It adopts a convolutional neural network (CNN) architecture with contextual attention for fusing fashion-contextual image features of target images. Specifically, for each turn of a multi-turn transaction, we obtain query features  $f_{qry}$  and target features  $f_{tar}$ . An  $N$ -turn retrieval transaction is of the form  $T_{retr} = [f_{qry}(1), f_{qry}(2), \dots, f_{qry}(N)]$ , with the target feature representation given by  $f_{tar}(N)$ .
+
+# 3.1. Cascaded Memory Neural Turing Machine
+
+To learn relationships across transactions, we propose a novel Cascaded Memory Neural Turing Machine (CM-NTM) module. CM-NTM allows multiple inputs to interact with their own memories using individual read and write heads to learn multiple complex relationships in the input data. We achieve this by deriving several features from the original  $n$ -th turn query feature  $f_{qry}(n)$  using a projection function  $\mathcal{F}$ , which comprises a fully-connected (FC) layer with batch normalization. Specifically, for a  $C$ -stage cascaded CM-NTM, we obtain the derived feature
+
+$f_{drv}^{c} = \mathrm{BatchNorm}(\mathrm{FC}_{\mathrm{c}}(f_{qry}(n)))$  where  $c \in [1, C - 1]$ , and  $\mathrm{FC}_{\mathrm{c}}(\cdot)$  is the FC layer for the  $c$ -th stage. Having obtained  $C$  inputs comprising the original query and  $C - 1$  derived features, we pass them sequentially to our memory network. Figure 3 presents our memory network architecture. It consists of three main components: Controller, Read/Write heads, and Memory blocks.
+
+Controller - The vanilla NTM has a single controller which takes the query feature at turn  $n$ , along with the previous turn's read vector, and emits an intermediate controller output. This is used by the read and write heads to compute the current turn's attention weights, which are then used to update the memory. In our work, we introduce  $C$  different controllers - one for each of our inputs. Each controller,  $c \in [1, C]$  can therefore interactively update its memory via individual read and write heads, allowing it to learn multiple complex relationships in each turn.
+
+Read/Write heads - The controller output is fed to these heads. The write head learns to generate erase and add parameters, which are used to update the current memory. Similarly, the read head generates an attention weight vector, which is used to obtain a weighted sum over the memory locations to get the read vector  $\mathbf{r}_{out}(n)$ . In our work, we have separate read and write heads for each memory.
+
+Memory block - This is represented as a 2-D matrix of the form  $N \times M$ , where  $N$  corresponds to memory locations and  $M$  to the vector size at each location. The output of each memory block is a fused representation of the controller output and the read vector. In our cascaded multi-memory setup, the controllers are linked in a chain, such that the memory-modified features are sequentially propagated. Specifically, for controller  $c \in [1, C]$  at turn  $n$ , the input is given by  $\mathrm{input}^c(n) = [\mathbf{r}_{out}^{c-1}(n); f_{drv}^c; \mathbf{r}_{out}^c(n-1)]$ , where  $\mathbf{r}_{out}^0(n) = \mathbf{r}_{out}^C(n-1)$ ,  $f_{drv}^C = f_{qry}(n)$ , and ; represents concatenation. The final output,  $f_{qry}'(n)$ , is the fused representation of the last controller output  $\mathrm{output}_{ctrl}^C$  and final read vector  $\mathbf{r}_{out}^C$ . We experiment with a different number of memories  $C$  and memory sizes in Section 4.7.
+
+# 3.2. Loss function
+
+We adopt a batch cross-entropy loss [14], where each entry in a batch acts as a negative sample for all other entries. In the multi-turn setting, we compute the loss function turnwise. Given a batch size  $B$ , the loss between predicted feature  $\mathbf{x}_n = [{}^1 f_{qry}'(n), {}^2 f_{qry}'(n), \ldots, {}^B f_{qry}'(n)]$ , and ground-truth  $\mathbf{y}_n = [{}^1 f_{tar}(n), {}^2 f_{tar}(n), \ldots, {}^B f_{tar}(n)]$ , at turn  $n$  is of the form
+
+$$
+\mathcal {L} (\mathbf {x} _ {n}, \mathbf {y} _ {n}) = \frac {1}{B} \sum_ {i = 1} ^ {B} - \log \frac {e ^ {\kappa (^ {i} f _ {q r y} ^ {\prime} (n) , ^ {i} f _ {t a r} (n))}}{\sum_ {j = 1} ^ {B} e ^ {\kappa (^ {i} f _ {q r y} ^ {\prime} (n) , ^ {j} f _ {t a r} (n))}}
+$$
+
+Table 1: Statistics of the datasets used in this work. The top 2 rows are for the recently proposed Multi-turn FashionIQ dataset [60]. The bottom two rows are for the Multi-turn version of the Shoes [5] dataset, which we created as part of our work.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Split</td><td colspan="3">Number of transactions with</td><td rowspan="2">Total images</td></tr><tr><td>2-turns</td><td>3-turns</td><td>4-turns</td></tr><tr><td rowspan="2">Multi-turn FashionIQ [60]</td><td>Train</td><td>6,897</td><td>1,733</td><td>475</td><td>10,438</td></tr><tr><td>Test</td><td>1,752</td><td>483</td><td>165</td><td>6,274</td></tr><tr><td rowspan="2">Multi-turn Shoes (ours)</td><td>Train</td><td>1,659</td><td>1,036</td><td>982</td><td>11,030</td></tr><tr><td>Test</td><td>296</td><td>96</td><td>28</td><td>4,631</td></tr></table>
+
+where  $\kappa$  is the cosine similarity metric [14]. In this way, each  $f_{tar}(n)$  in the batch,  $\forall j\in [1,B]$  and  $j\neq i$  , serves as a negative sample  $f_{neg}(n)$  for a given  $f_{qry}^{\prime}(n)$  . The turn-wise retrieval loss function is represented as  $\mathcal{L}_{retr}^{n} =$ $\mathcal{L}(\mathbf{x}_n,\mathbf{y}_n)$  , with the overall loss function for our proposed multi-turn model given by  $\mathcal{L} = \frac{1}{N}\sum_{n = 1}^{N}\mathcal{L}_{retr}^{n}$
+
+# 4. Experimental Evaluation
+
+# 4.1. Datasets
+
+Multi-turn FashionIQ [60]: To the best of our knowledge, this is the only existing fashion dataset with multi-turn sessions, where each turn is derived from the original single-turn FashionIQ [56] dataset. It comprises of 11,505 sessions across three clothing types - dress, top-tee, and shirt. The dataset is split into transactions of 2-turns, 3-turns, and 4-turns. In each turn, the data is represented as a pair  $(I_n, U_n)$ , where  $I_n$  and  $U_n$  correspond to the query image and the feedback text for turn  $n$ .
+
+Multi-turn Shoes: The original Shoes dataset [5] contains images of 10 categories of women's shoes obtained from the web along with automatic labeling of attributes. Guo et al. [17] provide additional natural language descriptions of the images to make them suitable for single-turn feedback-based image retrieval. This resulted in about 10k training pairs and 4.6k test queries. In this work, we create a multi-turn extension of this dataset to further research in this domain. Like the approach described in [60], we concatenated several single-turn transactions by matching the target image of one session to the query of another. To maintain
+
+consistency with Multi-turn FashionIQ, we also developed transactions of 2-turns, 3-turns, and 4-turns. Table 1 provides the statistics of both the datasets.
+
+# 4.2. Implementation Details
+
+Single-turn feature extractor pre-training - We use the recently proposed single turn (ST) fashion image retrieval model, FashionVLP [14], to extract the query and target features for both datasets. To ensure a fair comparison with other algorithms, we re-train FashionVLP only on those single-turn queries that are part of the multi-turn dataset. The implementation details and hyperparameters are similar to those mentioned in Goenka et al. [14].
+
+CM-NTM training - We build our CM-NTM model using the open-source implementation [28] of NTM [16]. We implement a separate controller for each memory in our cascaded design using Long Short Term Memory (LSTM) networks [20]. The read and write heads are composed of multi-layer perceptrons (MLPs) and attention networks. We use  $C = 4$  for Shoes, and  $C = 8$  memory stages for FashionIQ, as it is a larger dataset. To ensure equal turn lengths for batch training, we pad the short turn transactions by repeating the last transaction similar to Yuan et al. [60]. We train our models for 100 epochs with a batch size of 80 and a learning rate of 1e-4. We used the PyTorch framework with Adam [29] optimizer for training.
+
+# 4.3. Baselines and Previous State-of-the-art
+
+We compare the performance of our model with six other single-turn and multi-turn approaches.
+
+Single-turn methods - In these methods, we aggregate the multi-turn query data into a single feature without iterating over them turn-wise. The first baseline is  $\mathrm{ST} + \mathrm{avg}(\mathrm{all~turns})$ , where we take the mean of all the  $N$  query features in a transaction to get a single mean query  $\bar{f}_{qry}(N)$ . This is compared with the candidate target features  $f_{tar}(N)$ . Next, we have  $\mathrm{ST} + \mathrm{cat}(\mathrm{all~captions})$ , where all the captions of a multi-turn transaction are concatenated into one long caption, along with initial reference image, to get query features  $\hat{f}_{qry}(N)$ . This is compared with target features  $f_{tar}(N)$  to retrieve the final images.
+
+Table 2: Quantitative results on Multi-turn FashionIQ [60]. We compare with multiple single-turn and multi-turn baselines, and state-of-the-art works [17, 60]. Results show the superior performance of our proposed approach on the popularly used recall rate evaluation metric.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Dress</td><td colspan="2">Toptee</td><td colspan="2">Shirt</td><td colspan="3">Overall</td></tr><tr><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>Mean</td></tr><tr><td colspan="10">Single-turn</td></tr><tr><td>ST + avg(all turns)</td><td>25.6</td><td>32.5</td><td>32.1</td><td>38.1</td><td>27.0</td><td>32.3</td><td>28.2</td><td>34.3</td><td>31.3</td></tr><tr><td>ST + cat(all captions)</td><td>30.2</td><td>36.3</td><td>36.0</td><td>44.1</td><td>35.3</td><td>42.5</td><td>33.8</td><td>41.0</td><td>37.4</td></tr><tr><td colspan="10">Multi-turn</td></tr><tr><td>Dialog Manager [17]</td><td>12.7</td><td>16.7</td><td>11.6</td><td>15.8</td><td>13.9</td><td>17.7</td><td>13.1</td><td>15.2</td><td>14.2</td></tr><tr><td>CFIR [60]</td><td>29.8</td><td>33.5</td><td>29.4</td><td>33.6</td><td>30.5</td><td>34.1</td><td>30.3</td><td>33.4</td><td>31.9</td></tr><tr><td>ST + EWMA (ours)</td><td>42.0</td><td>48.4</td><td>43.8</td><td>50.9</td><td>36.9</td><td>44.2</td><td>40.9</td><td>47.8</td><td>44.4</td></tr><tr><td>ST + LSTM (ours)</td><td>47.8</td><td>52.5</td><td>44.4</td><td>50.5</td><td>41.6</td><td>47.6</td><td>44.6</td><td>50.2</td><td>47.4</td></tr><tr><td>FashionNTM (ours)</td><td>48.3</td><td>52.8</td><td>45.1</td><td>49.8</td><td>43.8</td><td>48.8</td><td>45.7</td><td>50.4</td><td>48.1</td></tr></table>
+
+Multi-turn methods - In these methods, the multi-turn data is fed turn-wise to the model. The first method is Dialog Manager [17], which employs reinforcement learning (RL) framework to learn relationships between turns. Next, we have the previous state-of-the-art Conversational Fashion Image Retrieval CFIR method [60], which encodes only the text using a transformer network [53], and uses a simple Gated Recurrent Unit (GRU) layer for multi-turn image retrieval. The third baseline is  $\mathrm{ST} + \mathrm{EWMA}$ , where we use exponential weighted moving average [1] to heuristically aggregate past turns, with more weights given to recent history. Finally, we have the  $\mathrm{ST} + \mathrm{LSTM}$  model that uses a single-layer LSTM [20] with a hidden size of 100 for aggregating past information.
+
+# 4.4. Results
+
+Evaluation Metrics - Following [14, 60, 17], we evaluate models using the standard top-K recall (i.e. R@K) for image retrieval. Overall performance are compared specifically on the average of R@5 and R@8.
+
+Quantitative results on multi-turn datasets - In Table 2, we compare the results of different methods on the multi-turn FashionIQ dataset. We observe that the multi-turn baselines generally perform better than single-turn methods. This is expected as aggregating data by naively averaging/concatenating loses feedback content and turn-order information and hence is likely to miss out on important cues. Our memory-based approach outperforms all the other multi-turn baselines by a large margin. This shows that the memory network can store and retrieve useful information to and from the memory between intermediate turns, which allows it to keep track of past information better than other networks that do not use explicit memory. The  $50.5\%$  performance gain over the previous state-of-the-art highlights our model's capability to learn meaningful representations over multiple turns of conversational feedback.
+
+In Table 3, we provide a similar comparison for the Multi-turn Shoes dataset. Consistent with results for Multi-turn FashionIQ, the multi-turn baselines perform better than
+
+Table 3: Comparison with existing single-turn and multi-turn models on the multi-turn version of the Shoes [5] dataset. We compare with multiple single-turn, and multi-turn baselines. Comparative analysis shows the superior performance of our proposed approach using the popular recall rate evaluation metric.
+
+<table><tr><td>Model</td><td>R@5</td><td>R@8</td><td>Mean</td></tr><tr><td colspan="4">Single-turn</td></tr><tr><td>ST + avg(all turns)</td><td>12.4</td><td>17.6</td><td>15.0</td></tr><tr><td>ST + cat(all captions)</td><td>10.7</td><td>13.6</td><td>12.2</td></tr><tr><td colspan="4">Multi-turn</td></tr><tr><td>ST + EWMA (ours)</td><td>18.3</td><td>23.8</td><td>21.1</td></tr><tr><td>ST + LSTM (ours)</td><td>23.3</td><td>32.1</td><td>27.7</td></tr><tr><td>FashionNTM (ours)</td><td>26.7</td><td>35.7</td><td>31.2</td></tr></table>
+
+all the single-turn ones, and our memory network-based approach performs the best with a relative improvement of  $12.6\%$ . The difference in performance across different models is more pronounced in these results than in Table 2. This is possibly because the annotations are cleaner and more consistent in this dataset as compared to Multi-turn FashionIQ [60]. For instance, multiple images in the FashionIQ dataset can match a particular query, but only one of them is labeled as the ground-truth.
+
+An important property of a good multi-turn system is that performance should be robust to the length of the historical information (number of past turns). For example, even for a large number of previous turns considered, the model should efficiently retain desirable details, while filtering out unnecessary information. In Table 4, we analyze this property for models with and without memory. A single-turn model that assumes ground-truth information for all past turns, and evaluated only on the final-turn, is taken as reference. This is expected to be the upper-bound on the performance in a single-turn setting, as perfect information about the history is guaranteed. We vary the number of past turns included in the input transaction history (versus treated as ground-truth) for the multi-turn models (with or without memory) in order to evaluate their effectiveness at capturing and utilizing past information. As seen from the table, for a model without memory, the performance depreciates significantly with each additional turn from the
+
+Table 4: Importance of aggregating historical data using memory-based vs non-memory approach. In the first row, we show the result of a model using only the final turn information of a multi-turn transaction. This assumes the groundtruth retrieval for all previous turns, and therefore provides the upper-bound on single-turn performance for final retrieval. Subsequently, we include additional information from the history, and compare performance across models with and without memory. As seen from the non-memory case, the performance depreciates a lot ( $\approx$  64.9% difference compared to the final turn's performance), as we try to aggregate longer historical information. In contrast, for the memory network model, it can be seen that the performance is fairly consistent ( $\approx$  0.5% difference) across the turn length.
+
+<table><tr><td rowspan="2">Memory usage</td><td rowspan="2">Turn configuration</td><td colspan="2">Dress</td><td colspan="2">Toptee</td><td colspan="2">Shirt</td><td colspan="3">Overall</td><td rowspan="2">Difference from final turn</td></tr><tr><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>R@5</td><td>R@8</td><td>Mean</td></tr><tr><td>-</td><td>only final turn</td><td>77.9</td><td>77.9</td><td>84.0</td><td>84.0</td><td>74.1</td><td>77.8</td><td>78.7</td><td>79.9</td><td>79.3</td><td>-</td></tr><tr><td colspan="12">Experiments with data aggregated across multiple-turns</td></tr><tr><td rowspan="3">Only single turn (w/o memory)</td><td>last two turns</td><td>51.3</td><td>58.4</td><td>56.0</td><td>76.0</td><td>44.4</td><td>48.1</td><td>50.6</td><td>60.9</td><td>55.8</td><td>-29.6%</td></tr><tr><td>last three turns</td><td>33.6</td><td>43.4</td><td>32.0</td><td>44.0</td><td>29.6</td><td>40.7</td><td>31.8</td><td>42.7</td><td>37.3</td><td>-53.0%</td></tr><tr><td>all turns</td><td>18.6</td><td>28.3</td><td>32.0</td><td>32.0</td><td>25.9</td><td>29.6</td><td>25.5</td><td>30.0</td><td>27.8</td><td>-64.9%</td></tr><tr><td rowspan="3">FashionNTM (with memory)</td><td>last two turns</td><td>76.1</td><td>77.9</td><td>84.0</td><td>84.0</td><td>77.8</td><td>77.8</td><td>79.3</td><td>79.9</td><td>79.6</td><td>+0.4%</td></tr><tr><td>last three turns</td><td>77.0</td><td>77.9</td><td>84.0</td><td>84.0</td><td>77.8</td><td>77.8</td><td>79.6</td><td>79.9</td><td>79.8</td><td>+0.6%</td></tr><tr><td>all turns</td><td>76.1</td><td>77.9</td><td>84.0</td><td>84.0</td><td>77.8</td><td>77.8</td><td>79.3</td><td>79.9</td><td>79.6</td><td>+0.4%</td></tr></table>
+
+![](images/4f2d844c6cd723d0ff98953db67cb4307fcd4ceaf7fa767f0e827d0e310f3ba6.jpg)  
+(a) Multi-turn FashionIQ. From left to right, the images belong to the Dress, Shirt, and Toptee categories respectively.
+
+![](images/2f7685d81cb666dc03f51972cdba64f2870b305f1d65baa6d982d579c99787d1.jpg)  
+(b) Multi-turn Shoes. From left to right, we have three different samples of shoes from the dataset.  
+Figure 4: Top-5 final image retrievals on the evaluated multi-turn datasets. The top row illustrates 3 sets of multi-turn query session. We consider three different multi-turn models - EWMA, LSTM, and our proposed FashionNTM. As seen from the retrievals, our proposed model correctly predicts the target image in all 3 cases for the FashionIQ dataset, and in 2 out of 3 cases for the Shoes dataset.
+
+past treated as input rather than ground-truth. As we go further back in history, the performance consistently reduces. However, for our multi-turn model with memory, the performance does not change appreciably as the number of turns change. This shows that our proposed approach can successfully retain/filter out past data based on their relevance, across various lengths of history.
+
+An interesting observation is that having only the final turn with memory does not yield a good result. This is possibly due to the initialization method of the memory network, which is random. Hence, in absence of a history (only single turn case), the only past features to be aggregated are the random initialization features.
+
+Qualitative results - In addition to the quantitative experiments described above, we also present some qualitative final image retrieval results of our evaluated models on both the multi-turn datasets. The first set of results are shown in Figures 4a and 4b for the Multi-turn FashionIQ [60] and the
+
+Multi-turn Shoes datasets, respectively<sup>2</sup>. We compare the top-5 predicted results from our proposed model with two other multi-turn baselines, ST + EWMA and ST + LSTM. As seen in the figures, our approach can correctly predict the desired target image for both datasets with higher ranks as compared to other baselines. More specifically, we see that in Figure 4a, even though all the three multi-turn models can infer the general sense of desired attributes, such as "is longer" in the left block and "has short sleeves" in the right, only our model can capture complex and detailed properties, e.g., "gray and not red" in the left block, and "red color with center logo" in the middle block. Furthermore, our model can retrieve multiple desirable products, as seen by the first four images in the middle block, and four out of five images in the right block. Similar results are observed for Shoes in Figure 4b, where our model correctly predicts
+
+![](images/f7eb55bc6d7b6c2e082298f5770b585b7874676b8514e0f827e55a0542b77e65.jpg)  
+Figure 5: Memory retention capability of our proposed approach. Given an initial query image, we take two interactive user feedbacks in turns. On the left side, we have the single-turn model which only learns to retrieve an image using a single dialog exchange. As a result, none of the retrieved images in turn 2 are "green in color", which was desired in turn 1. In contrast, our proposed approach on the right can learn to retain data from both the turns, and therefore retrieves desirable product in 2 out of 3 cases.
+
+the desired target as rank-1 in two out of three examples, whereas the other models fail to retrieve meaningful results.
+
+# 4.5. Model Analysis in Interactive Settings
+
+In addition to the results above for the static dataset, we also performed some interactive experiments to evaluate whether our model can adapt to real-world dynamic use cases beyond the trained datasets.
+
+Memory retention - In this experiment, we demonstrate the memory retention capability of our proposed model by comparing it with a single-turn network, which does not retain historical information. We start with an initial query image retrieved via a single-turn model. Subsequently, we take user input for the next two turns to retrieve newer sets of images. As seen in Figure 5, for the single-turn model (left side), none of the retrieved images in turn 2 are green in color. This is because the "green in color" attribute was a desired property in turn 1, which the model without memory could not recover. In contrast, for our memory network approach (right side) both the top-2 retrieved images for turn 2 are "green in color" in addition to having "a solid color and small image". Thus, our model can learn to retain information from previous turns.
+
+Agnosticity to turn order - Ideally, a deployed multi-turn image retrieval system should be independent of the order of feedbacks provided, as long as they are non-contradictory. This is demonstrated in the experiment conducted in Figure 6. In the first case, we take two text inputs as feedbacks from a user and present them to the model. In the second case, we reverse the order of the feedbacks. As shown in the figure, for the flipped case, our proposed FashionNTM model retrieves similar looking final products, even though the intermediate retrievals are very different.
+
+It is to be noted that for both the experiments depicted in Figures 5 and 6, the feedback is taken from a dynamic user, thereby establishing the interactive capability of our
+
+![](images/9d9d51acf4766157a663d3f12189104f011612f36551dadc14201d924d0e795e.jpg)  
+Figure 6: Turn order independence feature of a memory network. In this experiment, we start with an initial query image, and take two non-contradictory user feedbacks. In one case (left), we let the model retrieve images based on the original order of the feedbacks, whilst in the other (right), we flip the order of the feedbacks. Our proposed approach adapts to presented input, and retrieves similar looking final results for both the cases, even though the intermediate outputs are quite different.
+
+proposed model beyond the training dataset.
+
+# 4.6. User Study
+
+Fashion image retrieval is inherently a subjective task, where the task success heavily relies on the satisfaction of a customer. Thus, we conducted a small human-preference survey among 5 participants (not associated with the paper in any way). To each user, we showed the final top-1 retrieved image by the 3-best multi-turn models on the FashionIQ [60] dataset from Table 2 - EWMA, LSTM, and our proposed FashionNTM. To ensure consistency, we generated 45 queries for this study, whose results are shown in Figure 7. The  $y$ -axis shows the number of preferred retrieval results for each user. Results show that each of the 5 users preferred images retrieved by the proposed FashionNTM model, on an average  $83.1\%$  more as compared to other multi-turn methods. In the supplementary material, we include some examples of the user interface shown to
+
+![](images/e66da7637eaa55744a96b737a57f5cae4c4cbed8b312a63d04a3288beda6ebd8.jpg)  
+Figure 7: Human preference study of top multi-turn systems.
+
+Table 5: Different number of memories for the proposed approach by fixing the memory size to  $4 \times 768$ . We select the mean value for comparison and pick the best one.  
+
+<table><tr><td>Model</td><td>Number of memories (C)</td><td>R@5</td><td>R@8</td><td>Mean</td><td>% increase</td></tr><tr><td>ST+v-NTM</td><td>1</td><td>24.5</td><td>30.5</td><td>27.5</td><td>-</td></tr><tr><td rowspan="4">FashionNTM</td><td>2</td><td>26.4</td><td>33.8</td><td>30.1</td><td>9.5</td></tr><tr><td>4</td><td>27.6</td><td>35.5</td><td>31.5</td><td>14.5</td></tr><tr><td>8</td><td>26.9</td><td>35.2</td><td>31.1</td><td>13.1</td></tr><tr><td>16</td><td>26.4</td><td>33.8</td><td>30.1</td><td>9.5</td></tr></table>
+
+the participants during this study.
+
+# 4.7. Ablation Studies
+
+We perform multiple ablation studies to gain insights on how changing different configurations of the memory network affect the overall performance. We perform these ablations on the multi-turn Shoes dataset as it contains more realistic and consistent feedback texts.<sup>3</sup>
+
+Number of memories in CM-NTM - This experiment involves varying the number of memories  $C$  in our cascaded memory architecture. In Table 5, we observe that the cascaded memory CM-NTM models perform significantly better than vanilla NTM, which has only one memory. We hypothesize that having inputs from multiple turns interacting with the same memory could eventually lead to saturation as we get additional data which could be alleviated if there are multiple memories in the network to recover the past which might help in capturing multiple complex relationships in multi-turn interactions better. Additionally, multiple memory networks can help in learning diverse representations of the input using derived features, which is not possible with a single memory network. As seen in Table 5 for Shoes dataset, the performance increases with the number of memories, peaking at  $C = 4$ , and then gradually decreases as the model starts to overfit.
+
+Inference time with multiple memories - In this experiment, we study the performance of FashionNTM in terms of the mean of R@5 and R@8 recall rates along with the time taken to process one multi-turn transaction. For each  $C$ , we evaluate the performance across four different memory sizes. As seen from Figure 8, the inference time increases with additional memories. Configurations in the green and blue clusters are desirable, as they provide a good trade-off between recall performance and computation time, while the pink and red clusters are undesirable due to poor performance and longer inference time, respectively.
+
+# 5. Conclusion and Future Work
+
+In this paper, we presented a novel cascaded Neural Turing Machine-based approach, called FashionNTM, for multi-turn feedback-based fashion image retrieval. Multiple
+
+![](images/f18ce485451afd9b388c4009f09403a9674bffd8188731395ffe15d2ab912694.jpg)  
+Figure 8: Scatter plot showing recall versus inference time for multiple memories in our CM-NTM for the Multi-turn Shoes dataset. The configurations belonging to the green cluster give the best recall overall, while having a high inference time. The configurations in the blue cluster provide a suitable alternative with quicker inference at the cost of lower recall. The magenta and red clusters are undesirable configurations due to poor performance and long inference time, respectively.
+
+memories in our model allow it to effectively retain and recall a number of complex relationships across transactions in the multi-turn setting, and multiple controllers help in assigning relative importance to each feature stored in the memory. This aids in attending to different parts of the input at different levels, thus leading to better performance across datasets. We also performed extensive experiments to compare our performance with baselines and previous state-of-the-art and observed that our multi-memory model significantly outperforms previous works [60], with up to  $50.5\%$  relative improvement on Multi-turn FashionIQ, and by  $12.6\%$  on the multi-turn Shoes dataset, which we created in this work. We further demonstrated that our model can generalize beyond the trained setting to dynamically interact with real-world users to retrieve meaningful final product images. Finally, a user preference study reveals that our model is preferred by human participants on an average  $83.1\%$  more as compared to other multi-turn methods.
+
+Despite promising results, there are a few limitations that make multi-turn image retrieval a hard problem to solve. Firstly, there is dearth of high quality and diverse multi-turn image retrieval datasets in the fashion domain which hinders comprehensive studies in this field. Additionally, deploying current approaches to real-world scenarios (e.g., virtual private assistants) becomes a challenge due to computational requirements. Lastly, selecting the right configuration for different components such as memory size, number of memories, etc. in a memory-based network is not a trivial task and depends a lot on the use case. Nevertheless, for future work, our approach could be further extended to non-fashion domains where multi-turn feedback-based information retrieval solutions are required.
+
+# References
+
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+        "text": "Fast Adversarial Training with Smooth Convergence",
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+        "text": "Mengnan Zhao, Lihe Zhang*, Yuqiu Kong and Baocai Yin",
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+        "text": "Dalian University of Technology, China",
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+        "text": "zmnwelcome@mail.dlut.edu.cn，{zhanglihe，yqkong，ybc}@dlut.edu.cn",
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+        "text": "Abstract",
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+        "text": "Fast adversarial training (FAT) is beneficial for improving the adversarial robustness of neural networks. However, previous FAT work has encountered a significant issue known as catastrophic overfitting when dealing with large perturbation budgets, i.e. the adversarial robustness of models declines to near zero during training. To address this, we analyze the training process of prior FAT work and observe that catastrophic overfitting is accompanied by the appearance of loss convergence outliers. Therefore, we argue a moderately smooth loss convergence process will be a stable FAT process that solves catastrophic overfitting. To obtain a smooth loss convergence process, we propose a novel oscillatory constraint (dubbed ConvergeSmooth) to limit the loss difference between adjacent epochs. The convergence stride of ConvergeSmooth is introduced to balance convergence and smoothing. Likewise, we design weight centralization without introducing additional hyperparameters other than the loss balance coefficient. Our proposed methods are attack-agnostic and thus can improve the training stability of various FAT techniques. Extensive experiments on popular datasets show that the proposed methods efficiently avoid catastrophic overfitting and outperform all previous FAT methods. Code is available at https://github.com/FAT-CS/ConvergeSmooth.",
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+        "text": "1. Introduction",
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+        "text": "Recent breakthroughs in deep learning [21, 32] have aroused researchers' interest in the security of neural networks [45, 48, 39]. In particular, the advanced research proves the vulnerability of deep models to adversarial attacks [8, 11, 27]. For instance, tiny crafted perturbations can fool models in various fields into making wrong decisions [15, 50, 35, 24]. Considering the security risks brought by adversarial attacks [18, 46, 33, 44], there is a quickly grow",
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+        "type": "text",
+        "text": "ing body of work [43, 46, 18] on improving the adversarial robustness of neural networks. Among them, adversarial training is widely applied by practitioners [31, 12].",
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+        "text": "In recent years, projected gradient descent based adversarial training (PGD-AT) [25, 34] has been widely employed for its stability and effectiveness. However, this mechanism is computationally expensive. It requires multiple gradient descent steps to generate the adversarial training data [38]. An alternative of PGD-AT is the fast adversarial training (FAT) [28], which only adopts a single-step fast gradient sign method (FGSM) [11] to generate training data. Compared to PGD-AT, FAT can efficiently train models, but easily falls into catastrophic overfitting [40, 29].",
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+        "text": "A number of FAT methods have been proposed to mitigate catastrophic overfitting. For example, Wong et al. [40] use randomly initialized perturbations to enhance the diversity of adversarial perturbations. Based on it, Andriushchenko et al. [2] raise a complementary regularizer named GradAlign to maximize the gradient alignment between benign and adversarial examples explicitly. Similarly, NuAT [36] and FGSM-MEP [42] adopt nuclear norm or p-norm to regularize the adversarial training, thereby increasing the prediction alignment between benign and adversarial examples. However, the above methods can only resolve catastrophic overfitting within the limited perturbation budget ( $\\xi \\leq 8/255$ ).  $\\xi$  specifies the perturbation degree of adversarial training data generated by various attacks. Besides, models trained by small perturbations are vulnerable to adversarial attacks with a large  $\\xi$ , e.g. the models trained by NuAT and FGSM-MEP at  $\\xi = 8/255$  perform  $53\\%$  and  $54\\%$  robustness against the PGD-50 attack with  $\\xi = 8/255$ , but only  $22\\%$  and  $20\\%$  robustness against the same attack with  $\\xi = 16/255$  respectively. Therefore, we aspire to prevent catastrophic overfitting to improve the adversarial robustness of neural models at larger perturbation budgets.",
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+        "text": "By analyzing adversarial training processes of representative work, we observe that catastrophic overfitting is usually accompanied by a slight fluctuation in the classification loss for benign samples and a sharp drop in the classification",
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+        "type": "header",
+        "text": "CVF",
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+    {
+        "type": "header",
+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
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+        "type": "page_footnote",
+        "text": "*Corresponding author",
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+    {
+        "type": "page_number",
+        "text": "4720",
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+    {
+        "type": "text",
+        "text": "loss for adversarial examples. This motivates us to question whether a smooth loss convergence process is also a stable FAT process. Moreover, we find that an oscillating adversarial training phase may restart the FAT process after catastrophic overfitting. Fig. 1 shows the details.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "According to these observations, we introduce an oscillatory constraint that limits the difference in loss between adjacent training epochs, called ConvergeSmooth. A dynamic convergence stride of ConvergeSmooth is designed considering the nonlinear decay rate of loss functions. Inspired by the smoothness of loss convergence, we further verify the effect of the proposed weight centralization on model stability. Weight centralization refers to taking the weights average of the previously trained models as the convergence center of the current model weight. Our proposed methods are attack-agnostic and thus can be combined with existing adversarial strategies in FAT, such as FGSM-RS and FGSM-MEP, to evaluate their performance.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The contributions are summarized in four aspects: (1) We verify that previous FAT works still suffer from catastrophic overfitting at a large  $\\xi$  and then study catastrophic overfitting from the perspective of convergence instability of loss functions; (2) We propose a smooth convergence constraint, ConvergeSmooth, and design a dynamic convergence stride for it, to help various FAT methods avoid catastrophic overfitting on different perturbation budgets; (3) The weight centralization is proposed without introducing extra hyperparameters other than the loss balance coefficient to stabilize FAT; (4) Extensive experiments show that the proposed methods outperform the state-of-the-art FAT techniques in terms of efficiency, robustness, and stability.",
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+        "text": "2. Related Work",
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+    {
+        "type": "text",
+        "text": "Adversarial attacks. Adversarial attacks are usually used to deceive deep-learning models. Goodfellow et al. [11] first discuss the adversarial attack (FGSM) within the classification task. They prove that adversarial examples  $x'$  generated by a single-step gradient backward can misclassify the model  $f(\\cdot; \\theta)$  with high confidence.  $\\theta$  denotes the fixed model weights.  $x'$  is generated by",
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+    {
+        "type": "equation",
+        "text": "\n$$\nx ^ {\\prime} = x + \\xi \\cdot \\operatorname {s g n} \\left(\\nabla_ {x} \\mathcal {L} (f (x; \\theta), y)\\right), \\tag {1}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $x$  is an input image,  $\\xi$  represents the perturbation budget,  $\\operatorname{sgn}(\\cdot)$  means the sign function,  $\\mathcal{L}(\\cdot)$  is usually the cross-entropy loss,  $\\nabla_{x}\\mathcal{L}(\\cdot)$  calculates the gradient of loss at  $x$ , and  $y$  denotes the ground truth labels of  $x$ . Following the FGSM [11], researchers propose a series of attack methods based on the iterative gradient backward, e.g. I-FGSM [22], MIM [8], and PGD [25]. Taking the PGD attack as an example, the adversarial example  $x_{t + 1}^{\\prime}$  produced in the iteration  $t + 1$  can be formulated as",
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+    {
+        "type": "equation",
+        "text": "\n$$\nx _ {t + 1} ^ {\\prime} = \\operatorname {c l i p} _ {\\xi} \\left(x _ {t} ^ {\\prime} + \\epsilon \\cdot \\operatorname {s g n} \\left(\\nabla_ {x _ {t} ^ {\\prime}} \\mathcal {L} \\left(f \\left(x _ {t} ^ {\\prime}; \\theta\\right), y\\right)\\right)\\right), \\tag {2}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "where  $\\epsilon$  denotes the single-step stride and  $\\mathrm{clip}_{\\xi}$  refers to projecting adversarial perturbations to a  $\\xi$ -ball.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Adversarial training. Madry et al. [25] formalize the adversarial training as a min-max optimization problem,",
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {\\theta} \\mathbb {E} _ {(x, y) \\sim D} [ \\max  _ {\\delta \\in [ - \\xi , \\xi ]} \\mathcal {L} (f \\left(x ^ {\\prime}; \\theta\\right), y) ], \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where  $x' = x + \\delta$ ,  $\\delta$  represents the adversarial perturbations generated by various attacks such as PGD and FGSM.  $D$  is the data generator. The internal maximization maximizes the classification loss to generate adversarial perturbations with fixed model weights. The external minimization minimizes the classification loss on the generated adversarial examples when optimizing the model weights.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Actually, there is a trade-off between computational efficiency and adversarial robustness in recent adversarial training methods. Compared to PGD-AT [25], FGSM-based fast adversarial training (FAT) accelerates the training process but damages the robustness of models for the problem of catastrophic overfitting [19, 17]. To mitigate this issue, Wong et.al [40] demonstrate that the FGSM with a random start strategy (FGSM-RS) can achieve comparable performance against the PGD-AT. ZeroGrad [10] zeroes the elements of gradient that are too small to craft the perturbations. enforces the model loss to increase with the increase in perturbation size. Besides, GradAlign [2] prevents catastrophic overfitting by maximizing the alignment between gradients of benign samples and adversarial examples.",
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+    {
+        "type": "text",
+        "text": "Similarly, NuAT [36] introduces a nuclear norm regularization between logits for benign and adversarial examples and uses the Bernoulli noise as the initial perturbation. ATAS [16] learns an instantiated adaptive step size that is inversely proportional to the gradient norm. It applies the adversarial perturbations from the previous epoch as the initialization of FGSM in the current training phase. In addition, Jia et al. [42] propose several prior-guided initialization methods to replace the random start strategy of FGSM-RS. Specifically, FGSM-BP adopts the adversarial perturbations from the previous batch as the attack initialization in the current batch. FGSM-MEP employs a momentum mechanism to combine all adversarial perturbations from the previous epochs to yield adversarial initialization.",
+        "bbox": [
+            496,
+            515,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although these FAT methods have resolved catastrophic overfitting at small perturbation budgets, they still suffer from catastrophic overfitting under larger perturbation budgets (e.g.  $\\xi = 16 / 255$ ). Unlike these methods, we revisit the catastrophic overfitting problem from the perspective of loss convergence instability and prevent this exception by limiting the magnitude of loss fluctuations during training.",
+        "bbox": [
+            496,
+            726,
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+            832
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. Proposed Method",
+        "text_level": 1,
+        "bbox": [
+            500,
+            844,
+            671,
+            862
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first study the performance of previous FAT methods when subjected to a large perturbation budget",
+        "bbox": [
+            500,
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+            890,
+            900
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "4721",
+        "bbox": [
+            480,
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+            955
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/cd9ecbc383adc2a12e7ab4b5a3256b22e64a083407a68b7b5fbdf8cacf2a38fa.jpg",
+        "image_caption": [
+            "(a) ResNet18 trained with FGSM-RS"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            84,
+            103,
+            356,
+            193
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/f2568362360d16210983537fbbc98350642e7fe840781af21c1099d1b6ebfb67.jpg",
+        "image_caption": [
+            "(b) ResNet18 $_2$  trained with FGSM-RS"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            357,
+            103,
+            620,
+            193
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/fdd50d80d63a5e6b09bf7fb39d5738ac55fa6b9d105d1b63edcf06374cb43fd0.jpg",
+        "image_caption": [
+            "(c) ResNet18 trained with FGSM-RS"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            622,
+            103,
+            885,
+            193
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/888d940547966a2de7eeb0aac3b876c80c50ae5fb0cbdda61901363669dbe86b.jpg",
+        "image_caption": [
+            "(d) ResNet18 trained with GradAlign"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            84,
+            205,
+            356,
+            296
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b673903b39c5fb94588140a51c32a46441f9a527c1e7d9af4534d7a47470ae1.jpg",
+        "image_caption": [
+            "(e) ResNet18 $_2$  trained with GradAlign"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            357,
+            205,
+            620,
+            296
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/7570403a5c25707a754349395f19cf4b9d241a35cc9aebda17ce19c1ec54d36a.jpg",
+        "image_caption": [
+            "(f) ResNet18 trained with GradAlign"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            622,
+            205,
+            885,
+            296
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/d26313a50019ee9fc986785530ffce4cb8414056df6f5225922cb675f1ba34e5.jpg",
+        "image_caption": [
+            "(g) ResNet18 trained with FGSM-MEP",
+            "Figure 1. The training process of previous FAT methods with ResNet18 on CIFAR-10. Each FAT method is trained 3 times.  $\\xi = 16 / 255$ . ADV-Loss and BEN-Loss denote the classification loss of models to adversarial and benign examples during training, respectively. ADV-Acc and BEN-Acc represent the classification accuracy of models to adversarial and benign examples during testing, respectively."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            84,
+            306,
+            356,
+            406
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/36780e4b0d277b67f249a6292c1612b47a45ac4e2d523f7bafd4a2dd23f657ef.jpg",
+        "image_caption": [
+            "(h) ResNet182 trained with FGSM-MEP"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            357,
+            306,
+            620,
+            406
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a205b7233de4b4082b6e17fbf138ff8447f36e6b7768b82dbd155c33980769a0.jpg",
+        "image_caption": [
+            "(i) ResNet18 trained with FGSM-MEP"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            622,
+            306,
+            885,
+            406
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\xi$ . Then, the training processes of these methods have been analyzed for a better understanding of catastrophic overfitting. Finally, we detail the proposed methods.",
+        "bbox": [
+            75,
+            476,
+            468,
+            522
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1. Performance of FAT methods on a large  $\\xi$",
+        "text_level": 1,
+        "bbox": [
+            76,
+            529,
+            429,
+            545
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Previous FAT techniques avoid catastrophic overfitting at small  $\\xi$  ( $\\xi \\leq 8/255$ ). Here, we investigate their performance on a large  $\\xi$  ( $\\xi = 16/255$  by default).",
+        "bbox": [
+            75,
+            551,
+            468,
+            597
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "FGSM-RS [40]: Based on Eq. (3), this method adopts the samples with uniformly random perturbations  $\\delta_0\\sim$ $\\mathcal{U}(-\\xi ,\\xi)$  as the attack initialization of FGSM,",
+        "bbox": [
+            75,
+            598,
+            468,
+            642
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {\\theta} \\mathbb {E} _ {(x, y) \\sim D} [ \\max  _ {\\delta_ {0} + \\delta \\in [ - \\xi , \\xi ]} \\mathcal {L} (f (x ^ {\\prime}; \\theta), y) ],\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            176,
+            648,
+            450,
+            674
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nx ^ {\\prime} = x + \\delta_ {0} + \\delta \\quad s. t. \\quad x ^ {\\prime} \\in [ 0, 1 ], \\quad \\delta_ {0} \\sim \\mathcal {U} (- \\xi , \\xi). \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            94,
+            676,
+            468,
+            705
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "GradAlign [2]: This approach increases the gradient alignment between benign samples  $x$  and perturbed samples  $x + \\delta_0$ , which is denoted as",
+        "bbox": [
+            75,
+            708,
+            468,
+            752
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} _ {(x, y) \\sim D} [ 1 - \\cos (\\nabla_ {x} \\mathcal {L} (x, \\theta), \\nabla_ {x + \\delta_ {0}} \\mathcal {L} (x + \\delta_ {0}, \\theta) ], \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            91,
+            758,
+            467,
+            782
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} (x, \\theta) = \\mathcal {L} (f (x; \\theta), y).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            779,
+            467,
+            795
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\cos (\\cdot)$  computes the cosine similarity between two matrices.",
+        "bbox": [
+            75,
+            801,
+            467,
+            816
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "FGSM-MEP [42]: Different from FGSM-RS, this method generates the initialization perturbations  $\\delta_0$  based on all historical adversarial perturbations from the previous epochs and introduces a regularization expressed as",
+        "bbox": [
+            75,
+            816,
+            468,
+            877
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {E} _ {(x, y) \\sim D} [ | | f (x ^ {\\prime}; \\theta) - f (x + \\delta_ {0}; \\theta) | | _ {2} ^ {2} ], \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            142,
+            883,
+            468,
+            902
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $||\\cdot ||_2^2$  denotes the squared  $\\mathcal{L}_2$  distance.",
+        "bbox": [
+            500,
+            476,
+            803,
+            492
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Fig. 1 shows their detailed adversarial training processes of ResNet18 [13] on CIFAR-10 [20]. We find that they fall into catastrophic overfitting during the  $5 \\sim 20_{\\mathrm{th}}$ ,  $40 \\sim 65_{\\mathrm{th}}$  and  $10 \\sim 15_{\\mathrm{th}}$  epochs, respectively. Graphical analysis of various models and datasets is given in the supplement.",
+        "bbox": [
+            498,
+            492,
+            890,
+            568
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2. Analysis of training process",
+        "text_level": 1,
+        "bbox": [
+            500,
+            575,
+            746,
+            590
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "From Fig. 1, we discover several typical phenomena of catastrophic overfitting: 1) Change from a smooth convergence state to an irregular fluctuation state; 2) A slight change (increase or decrease) in the standard classification loss  $\\mathcal{L}(x,\\theta)$  and a rapid descent of the adversarial classification loss  $\\mathcal{L}(x',\\theta)$ ; 3) Rapid decline in the classification accuracy of adversarial examples.",
+        "bbox": [
+            496,
+            598,
+            890,
+            704
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "After catastrophic overfitting, the methods depicted in Fig. 1 are capable of restarting a stable FAT process, even though catastrophic overfitting may occur again. The observed phenomena during the FAT restart are: 1) Change from an irregular fluctuation state to a smooth convergence state; 2) Rapid increase in both  $\\mathcal{L}(x,\\theta)$  and  $\\mathcal{L}(x',\\theta)$  after a period of decline; 3) Rapid decrease in the classification accuracy of benign samples, while the classification accuracy of adversarial examples experiences a rapid increase.",
+        "bbox": [
+            496,
+            704,
+            890,
+            839
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "On this basis, we can make the following conclusions. 1)  $\\mathcal{L}(x,\\theta)$  is more stable than  $\\mathcal{L}(x^{\\prime},\\theta)$  and models are prone to overfitting to adversarial perturbations; 2) Catastrophic overfitting is closely correlated to the convergence insta",
+        "bbox": [
+            498,
+            839,
+            890,
+            900
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "4722",
+        "bbox": [
+            480,
+            944,
+            514,
+            955
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "bility of adversarial training; 3) Exceptions in  $\\mathcal{L}(x,\\theta)$  and  $\\mathcal{L}(x^{\\prime},\\theta)$  occur simultaneously; 4) An oscillating adversarial training phase may trigger the FAT process to restart.",
+        "bbox": [
+            76,
+            90,
+            468,
+            137
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3. Smooth convergence for the stable FAT",
+        "text_level": 1,
+        "bbox": [
+            76,
+            145,
+            413,
+            161
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next, we describe the proposed method in detail.",
+        "bbox": [
+            94,
+            167,
+            419,
+            183
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Why did previous methods fail to prevent catastrophic overfitting? Despite the improvement in diversity achieved through random initialization in FGSM-RS, models are still susceptible to overfitting adversarial perturbations. GradAlign and FGSM-MEP enhance the stability of adversarial training through the constraints in Eqs. (5) and (6), respectively. However, Eq. (5) may reduce the stability of  $\\mathcal{L}(x,\\theta)$ . Meanwhile, the prediction probability  $f(x + \\delta_0;\\theta)$  in Eq. (6) is not the sweet spot to keep the FAT stable. Unlike these methods, we ensure the convergence stability of both  $\\mathcal{L}(x,\\theta)$  and  $\\mathcal{L}(x',\\theta)$ .",
+        "bbox": [
+            75,
+            184,
+            468,
+            349
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "How can catastrophic overfitting be solved? Since the catastrophic overfitting is usually accompanied by a slight change in  $\\mathcal{L}(x,\\theta)$  and a sharp decline in  $\\mathcal{L}(x',\\theta)$ , we consider a smooth loss convergence process to be a stable FAT process that resolves this issue. To this end, a complementary constraint  $\\mathcal{L}_{CS}$  for Eq. (3) is proposed,",
+        "bbox": [
+            75,
+            349,
+            468,
+            441
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {\\theta_ {t}} \\mathbb {E} _ {(x, y) \\sim D} \\left[ \\mathcal {L} \\left(x _ {t} ^ {\\prime}, \\theta_ {t}\\right) + \\mathcal {L} _ {C S} (t) \\right], \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            151,
+            450,
+            468,
+            474
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which can limit the difference in losses between adjacent epochs, expressed as",
+        "bbox": [
+            76,
+            483,
+            468,
+            513
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {C S} (t) = w _ {1} \\cdot \\left| \\mathcal {L} \\left(x _ {t} ^ {\\prime}, \\theta_ {t}\\right) - \\mathcal {L} \\left(x _ {t - 1} ^ {\\prime}, \\theta_ {t - 1}\\right) \\right| + \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            122,
+            523,
+            468,
+            547
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nw _ {2} \\cdot | \\mathcal {L} (x, \\theta_ {t}) - \\mathcal {L} (x, \\theta_ {t - 1}) |, \\quad s. t. \\quad \\mathcal {C} (x) = 1,\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            107,
+            542,
+            421,
+            558
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\theta_{t}$  represents the model weights of the  $t_{\\mathrm{th}}$  training epoch.  $w_{1}$  and  $w_{2}$  are hyper-parameters,  $w_{1}, w_{2} \\in [0,1]$ .  $|\\cdot|$  calculates the absolute value.  $x_{t}^{\\prime} = x + \\delta_{0,t} + \\delta_{t}$ .  $\\delta_{0,t}$  can be replaced by various attack initialization perturbations and  $\\delta_{t}$  is generated by",
+        "bbox": [
+            76,
+            569,
+            468,
+            645
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\max  _ {\\delta_ {t} \\in [ - \\xi , \\xi ]} \\mathcal {L} \\left(x _ {t} ^ {\\prime}, \\theta_ {t - 1}\\right), \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            655,
+            468,
+            680
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\theta_{t - 1}$  is fixed in generating  $\\delta_t$ .",
+        "bbox": [
+            76,
+            689,
+            316,
+            704
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the practical implementation of Eq. (8), storing and computing  $\\mathcal{L}(x_{t - 1}^{\\prime},\\theta_{t - 1})$  and  $\\mathcal{L}(x,\\theta_{t - 1})$  can consume a significant amount of memory. To overcome this challenge, we introduce  $u_{t - 1}^{\\prime}$  and  $u_{t - 1}$  to replace these terms, respectively.  $u_{t - 1}^{\\prime} = \\mathbb{E}_{(x,y)\\sim D}\\mathcal{L}(x_{t - 1}^{\\prime},\\theta_{t - 1})$  and  $u_{t - 1} = \\mathbb{E}_{(x,y)\\sim D}\\mathcal{L}(x,\\theta_{t - 1})$  denote the mathematical expectation of loss in the  $t - 1_{\\mathrm{th}}$  training epoch.",
+        "bbox": [
+            76,
+            704,
+            468,
+            809
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We observe that training instability is primarily caused by overfitting or underfitting of small amounts of data. Thus, in Eq. (8), the additional loss term is only applied to partial data selected by the crafted condition  $\\mathcal{C}(\\cdot)$ . Notably, exceptions in  $\\mathcal{L}(x,\\theta)$  and  $\\mathcal{L}(x',\\theta)$  occur simultaneously during training. Hence,  $\\mathcal{L}(x_t',\\theta_t)$  and  $\\mathcal{L}(x_t,\\theta_t)$  share",
+        "bbox": [
+            75,
+            810,
+            468,
+            901
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "the same condition.  $\\mathcal{C}$  is constructed by the distance between the pointwise loss  $\\mathcal{L}(x,\\theta_t)$  and the mean value  $u_{t - 1}$ . This is because over-fitted or under-fitted data often yield excessively high or low classification losses.",
+        "bbox": [
+            496,
+            90,
+            890,
+            151
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {C} (x) = \\left(\\left| \\mathcal {L} \\left(x, \\theta_ {t}\\right) - u _ {t - 1} \\right| \\geq \\gamma_ {t}\\right), \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            581,
+            165,
+            890,
+            181
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\gamma_{t} (\\gamma_{t} \\geq 0)$  represents the convergence stride to select abnormal data. It is crucial to choose an appropriate value for  $\\gamma_{t}$  to ensure effective training. When  $\\gamma_{t} = 0$ , the adversarial training process may fail to converge as it forces the predictions of all samples to remain unchanged. On the other hand, catastrophic overfitting occurs when  $\\gamma_{t} = \\infty$ , leading to undesirable outcomes. Considering that the loss difference  $d_{t-1}(d_{t-1} = |u_{t-1} - u_{t-2}|)$  between two adjacent epochs tends to decrease non-linearly,  $\\gamma_{t}$  should be a variable that varies during the training process.",
+        "bbox": [
+            496,
+            186,
+            890,
+            338
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\gamma_ {t} = \\min  \\left(\\max  \\left(d _ {t - 1}, \\gamma_ {\\text {m i n}}\\right), \\gamma_ {\\text {m a x}}\\right), \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            573,
+            353,
+            890,
+            369
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\gamma_{min}$  and  $\\gamma_{max}$  are hyper-parameters.  $\\gamma_{max}$  controls the maximum convergence speed.  $\\gamma_{min}$  ensures that the training process is not too smooth when  $d_{t - 1} \\to 0$ .",
+        "bbox": [
+            496,
+            375,
+            890,
+            419
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Example-based ConvergeSmooth. This approach adds the constraint individually to each sample, described as",
+        "bbox": [
+            496,
+            420,
+            890,
+            450
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathcal {L} _ {C S} ^ {E} (t) = w _ {1} \\cdot \\left| \\mathcal {L} \\left(x _ {t} ^ {\\prime}, \\theta_ {t}\\right) - u _ {t - 1} ^ {\\prime} \\right| + w _ {2} \\cdot \\left| \\mathcal {L} \\left(x, \\theta_ {t}\\right) - u _ {t - 1} \\right|, \\\\ s. t. \\quad | \\mathcal {L} (x, \\theta_ {t}) - u _ {t - 1} | > \\gamma_ {t}. \\tag {12} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            500,
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+            507
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Batch-based ConvergeSmooth. Likewise, we can apply the complementary constraint to a data batch,",
+        "bbox": [
+            496,
+            508,
+            890,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathcal {L} _ {C S} ^ {B} (t) = w _ {1} \\cdot \\left| u _ {B} ^ {\\prime} - u _ {t - 1} ^ {\\prime} \\right| + w _ {2} \\cdot \\left| u _ {B} - u _ {t - 1} \\right|, \\tag {13} \\\\ s. t. \\quad | u _ {B} - u _ {t - 1} | > \\gamma_ {t}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            519,
+            545,
+            890,
+            582
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $u_B^\\prime = \\mathbb{E}_{(x,y)\\sim B}\\mathcal{L}(x_t^\\prime ,\\theta_t),u_B = \\mathbb{E}_{(x,y)\\sim B}\\mathcal{L}(x,\\theta_t).$",
+        "bbox": [
+            500,
+            590,
+            879,
+            607
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Weight centralization. To mitigate the problem of manual parameter tuning, we introduce the weight centralization without requiring extra hyperparameters other than the coefficient  $w_{3}$ ,  $|\\theta_t - \\theta_{t-1}| = 0 \\rightarrow |\\mathcal{L}(x, \\theta_t) - \\mathcal{L}(x, \\theta_{t-1})| = 0$ ,",
+        "bbox": [
+            496,
+            607,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {C S} ^ {W} (t) = w _ {3} \\cdot | | \\theta_ {t} - \\frac {1}{l e n (\\phi)} \\cdot \\sum_ {j \\in \\phi} \\theta_ {j} | | _ {p}, \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            558,
+            676,
+            890,
+            714
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\| \\cdot \\|_p$  means the p-norm function  $(p = 2)$ , and  $\\phi$  denotes a set of previous model weights. The model weights  $\\theta_t$  are restricted to the center  $\\frac{1}{len(\\phi)} \\cdot \\sum_{j \\in \\phi} \\theta_j$ . The reasons behind Eq. (14) are: 1) The initial training process is stable, as indicated in Fig. 1; 2) Models at different training epochs tend to have similar weights after convergence [37].",
+        "bbox": [
+            496,
+            719,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4. Experimental Results",
+        "text_level": 1,
+        "bbox": [
+            496,
+            821,
+            705,
+            838
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1. Experimental settings",
+        "text_level": 1,
+        "bbox": [
+            500,
+            847,
+            702,
+            863
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Details. To demonstrate the effectiveness of the proposed method, we conduct comprehensive experiments on",
+        "bbox": [
+            496,
+            869,
+            890,
+            901
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4723",
+        "bbox": [
+            480,
+            944,
+            514,
+            955
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/c3c8521d1b76e9ae3048ce466e562f5ecf5e2d23f0d08eee2dc7d9aa12e229d8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">CIFAR10ξ</td><td>FGSM-RS</td><td>FGSM-MEP</td><td>B-RS</td><td>B-MEP</td></tr><tr><td>SQ↑</td><td rowspan=\"2\">16/255</td><td>19.74</td><td>22.15</td><td>28.83</td><td>29.97</td></tr><tr><td>SQ+RayS↑</td><td>18.72</td><td>20.96</td><td>27.08</td><td>28.45</td></tr></table>",
+        "bbox": [
+            78,
+            89,
+            470,
+            138
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/9cc78e2fd2dbaf9cd156da9d7c6544562d0cee308739e9288892cb3964e7ca6d.jpg",
+        "table_caption": [
+            "Table 1. Quantitative results of FAT methods against the Square and Ray-S attacks on CIFAR-10 with the backbone ResNet18."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CIFAR10</td><td>Clean</td><td>8 255</td><td>10 255</td><td>12 255</td><td>16 255</td><td>SQ+RayS</td><td>Time</td></tr><tr><td>OAAT[1]</td><td>71.30</td><td>44.29</td><td>38.56</td><td>33.68</td><td>28.48</td><td>21.06</td><td>183</td></tr><tr><td>AWP[41]</td><td>78.86</td><td>35.91</td><td>32.70</td><td>29.89</td><td>26.95</td><td>19.36</td><td>134</td></tr><tr><td>ATES[4]</td><td>74.52</td><td>43.61</td><td>38.94</td><td>35.31</td><td>30.89</td><td>22.16</td><td>-</td></tr><tr><td>ExAT[30]</td><td>80.78</td><td>49.21</td><td>42.15</td><td>36.98</td><td>31.26</td><td>23.08</td><td>70</td></tr><tr><td>E-MEP</td><td>69.84</td><td>53.54</td><td>49.26</td><td>45.89</td><td>40.78</td><td>23.78</td><td>104</td></tr><tr><td>B-MEP</td><td>63.84</td><td>50.31</td><td>46.77</td><td>43.97</td><td>40.13</td><td>28.45</td><td>101</td></tr></table>",
+        "bbox": [
+            78,
+            191,
+            475,
+            303
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 2. Quantitative results of methods against various levels of perturbation with ResNet18 as the backbone and CIFAR-10 as the dataset.  $\\sharp$  denotes the PGD-10 attack.",
+        "bbox": [
+            75,
+            311,
+            470,
+            354
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "several benchmark datasets, i.e. CIFAR-10 [20], CIFAR-100 [20], and Tiny ImageNet [7].",
+        "bbox": [
+            76,
+            367,
+            468,
+            398
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following previous works [40, 23, 49, 42], we adopt ResNet18 [13] as the backbone on the CIFAR-10 and CIFAR-100, and choose PreActResNet18 [14] on the Tiny ImageNet. In all experiments, models are optimized using the SGD optimizer with a batch size of 128, weight decay of 5e-4, and momentum of 0.9. The initial learning rates on CIFAR-10, CIFAR-100, and Tiny ImageNet are set as 0.1, 0.1, and 0.01, respectively. Then, we optimize models with a total training epoch of 110 and decay the learning rate at the  $100_{\\mathrm{th}}$  and  $105_{\\mathrm{th}}$  epoch with a factor of 0.1.",
+        "bbox": [
+            75,
+            398,
+            468,
+            549
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We apply the proposed ConvergeSmooth in combination with two attack initialization methods on CIFAR-10 and CIFAR-100, FGSM-RS [40] and FGSM-MEP [42]. Additionally, for the Tiny ImageNet, we use FGSM-BP [42] as the initialization method. As mentioned in [42], FGSM-MEP requires consuming memory to store the previous adversarial perturbations, which limits its application on large datasets. Regarding the hyperparameters, we set  $\\gamma_{max}$  in Eq. (11) as 0.03, 0.06, and 0.03 for CIFAR-10, CIFAR-100, and Tiny ImageNet, respectively.  $\\gamma_{max} = 1.5 \\cdot \\gamma_{min}$  and  $\\xi = 16/255$ . Specific details of hyperparameter settings for  $w_{1 \\sim 3}$  are given in the supplement. All experiments are conducted on a single GeForce RTX 3090 GPU.",
+        "bbox": [
+            75,
+            550,
+            468,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Baselines. We include advanced FAT methods as baselines, namely FGSM-RS [40], GradAlign [2], ZeroGrad [10], NuAT [36], ATAS [16], and FGSM-MEP [42].",
+        "bbox": [
+            76,
+            748,
+            468,
+            794
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation metrics. We follow [40, 16] and adopt FGSM [11], PGD-10 [25], PGD-20 [25], PGD-50 [25], C&W [3], APGD-CE [6] and Autoattack (AA) [6] to evaluate the adversarial robustness of models. PGD-n represents the PGD attack with n iterations. Autoattack combines APGD-CE and APGD-T (targeted APGD) as well as two complementary attacks (FAB [9] and Square attack [26]).",
+        "bbox": [
+            75,
+            795,
+            468,
+            902
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/3e97564525e8fbc6ed20f5e85e150c8587a20b3828877176d39ffb9478255b35.jpg",
+        "image_caption": [
+            "Figure 2. The training process of various FAT methods (PGD-10,  $\\xi = 16 / 255$ ) on CIFAR-10."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            514,
+            90,
+            870,
+            300
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The training easily exhibits fluctuations, and thus individual results can not objectively reflect the performance of methods. For each method, we repeat the training process three times and report the average evaluation results of the best model across the three runs (mbest). Additionally, the supplement includes the best results (best) and the average evaluation of the final model from the three runs (mfinal).",
+        "bbox": [
+            496,
+            345,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the following section, 'W-\\*', 'E-\\*', and 'B-\\*' mean 'weight centralization', 'example-based ConvergeSmooth', and 'batch-based ConvergeSmooth', respectively.",
+        "bbox": [
+            498,
+            452,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2. Significance of the Results",
+        "text_level": 1,
+        "bbox": [
+            500,
+            511,
+            736,
+            527
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Before evaluating the adversarial robustness of trained models, we demonstrate the significance of the results [1]. Specifically, we show that adversarial attacks with  $\\xi = 16/255$  rarely change the true label of input. 1) We generate adversarial examples for the FGSM-RS, FGSM-MEP, and our proposed methods on CIFAR10 and CIFAR100 datasets with  $\\xi = 16/255$  using the non-targeted attacks such as FGSM, PGD, C&W, APGD-CE, and the targeted attack APGD-T. The majority of the generated examples retain their true labels; 2) Tab. 1 examines the robustness of models against an ensemble of the Square (SQ) [26] and Ray-S [5] attacks, as these attacks generate strong oracle-invariant examples [1]; 3) We generate adversarial training data with  $\\xi = 16/255$  and test the robustness on various levels of perturbation. Tab. 2 provide comparative experiments between our proposed method and other AT techniques, including OAAT [1], ExAT [30], ATES [4] and AWP [41]. It is important to note that we re-implement these AT methods and apply them in the context of FAT. Overall, our methods achieve optimal performance on both oracle-invariant attacks and classical evaluation attacks. Namely, the classical evaluation attacks used in  $\\xi = 8/255$  are reliable and significant even when the value of  $\\xi$  is increased to 16/255.",
+        "bbox": [
+            496,
+            537,
+            890,
+            883
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Notably, our methods are plugged into FGSM-RS and",
+        "bbox": [
+            517,
+            885,
+            890,
+            901
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "4724",
+        "bbox": [
+            480,
+            944,
+            514,
+            955
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/65f4e652e24f31e3fc58298079377a3bb2b83eacc7971aa6ffe90043c2e1b25d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Methods</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10���</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Time (min)↓</td></tr><tr><td colspan=\"2\">PGD-AT [29]</td><td>65.30</td><td>46.31</td><td>40.73</td><td>35.08</td><td>33.92</td><td>30.84</td><td>33.08</td><td>26.29</td><td>370</td></tr><tr><td colspan=\"2\">FGSM-RS [40]</td><td>50.12</td><td>38.28</td><td>26.13</td><td>21.55</td><td>20.43</td><td>18.96</td><td>19.40</td><td>14.84</td><td>67</td></tr><tr><td colspan=\"2\">GradAlign [2]</td><td>58.17</td><td>39.87</td><td>33.12</td><td>26.81</td><td>24.99</td><td>22.63</td><td>23.98</td><td>17.02</td><td>135</td></tr><tr><td colspan=\"2\">ZeroGrad [10]</td><td>74.16</td><td>43.96</td><td>32.67</td><td>21.98</td><td>18.37</td><td>20.76</td><td>16.44</td><td>12.07</td><td>67</td></tr><tr><td rowspan=\"3\">Ours</td><td>W-RS</td><td>70.66</td><td>45.51</td><td>36.50</td><td>27.51</td><td>24.75</td><td>23.97</td><td>23.38</td><td>17.14</td><td>71</td></tr><tr><td>E-RS</td><td>62.38</td><td>42.07</td><td>36.78</td><td>30.80</td><td>28.90</td><td>23.64</td><td>27.71</td><td>17.55</td><td>77</td></tr><tr><td>B-RS</td><td>65.42</td><td>45.94</td><td>37.54</td><td>30.01</td><td>27.85</td><td>26.28</td><td>26.52</td><td>19.43</td><td>75</td></tr><tr><td colspan=\"2\">NuAT [36]</td><td>74.62</td><td>44.92</td><td>35.22</td><td>25.93</td><td>23.67</td><td>24.07</td><td>22.37</td><td>18.43</td><td>101</td></tr><tr><td colspan=\"2\">ATAS* [16]</td><td>64.11</td><td>-</td><td>31.39</td><td>-</td><td>28.15</td><td>-</td><td>-</td><td>21.09</td><td>-</td></tr><tr><td colspan=\"2\">FGSM-MEP [42]</td><td>53.32</td><td>36.24</td><td>31.85</td><td>27.28</td><td>26.56</td><td>22.10</td><td>26.08</td><td>18.98</td><td>92</td></tr><tr><td rowspan=\"2\">Ours</td><td>E-MEP</td><td>69.84</td><td>47.18</td><td>40.90</td><td>34.17</td><td>32.72</td><td>22.69</td><td>31.12</td><td>17.74</td><td>104</td></tr><tr><td>B-MEP</td><td>63.84</td><td>45.48</td><td>40.13</td><td>34.21</td><td>32.95</td><td>28.19</td><td>32.04</td><td>23.68</td><td>101</td></tr></table>",
+        "bbox": [
+            112,
+            87,
+            857,
+            294
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c214d09220b1bedfc314bf352f31095c408c34053f5df0faea46d6167c346efa.jpg",
+        "table_caption": [
+            "Table 3. Quantitative results of various methods ( $\\xi = 16/255$ ) on the CIFAR-10 with ResNet18 as the backbone. 'ATAS*' is the result of ATAS in [16], which is superior to our reproduction. We train each method three times. The results represent the evaluation average between the best models of three training processes. Weight centralization and regularization in MEP do not work together."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Budgets (ξ)</td><td>Methods</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Time (min)↓</td></tr><tr><td rowspan=\"4\">12/255</td><td>GradAlign [2]</td><td>66.52</td><td>43.06</td><td>31.66</td><td>27.95</td><td>26.04</td><td>27.05</td><td>25.99</td><td>21.65</td><td>135</td></tr><tr><td>NuAT [36]</td><td>72.79</td><td>51.80</td><td>41.75</td><td>38.60</td><td>37.54</td><td>35.99</td><td>36.71</td><td>32.01</td><td>101</td></tr><tr><td>FGSM-MEP [42]</td><td>74.71</td><td>52.05</td><td>38.45</td><td>36.35</td><td>35.52</td><td>33.05</td><td>33.4</td><td>27.23</td><td>92</td></tr><tr><td>B-MEP (Ours)</td><td>72.63</td><td>54.40</td><td>45.23</td><td>42.85</td><td>42.14</td><td>36.81</td><td>41.62</td><td>33.26</td><td>101</td></tr><tr><td rowspan=\"4\">10/255</td><td>GradAlign [2]</td><td>83.10</td><td>55.23</td><td>36.58</td><td>30.47</td><td>28.64</td><td>31.01</td><td>26.51</td><td>23.84</td><td>135</td></tr><tr><td>NuAT [36]</td><td>75.82</td><td>55.94</td><td>45.54</td><td>43.92</td><td>43.42</td><td>41.39</td><td>42.91</td><td>38.85</td><td>101</td></tr><tr><td>FGSM-MEP [42]</td><td>83.43</td><td>59.51</td><td>42.76</td><td>39.28</td><td>37.33</td><td>37.26</td><td>35.91</td><td>32.52</td><td>92</td></tr><tr><td>B-MEP (Ours)</td><td>75.96</td><td>57.26</td><td>47.25</td><td>45.98</td><td>45.66</td><td>41.00</td><td>45.26</td><td>39.20</td><td>101</td></tr></table>",
+        "bbox": [
+            78,
+            358,
+            895,
+            500
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/9602126751f6ae067f4fde2f01bdf75ae02067184961dfc9aa10603e85ed93dc.jpg",
+        "table_caption": [
+            "Table 4. Quantitative results of FAT methods on various  $\\xi$  with ResNet18 as the backbone and CIFAR-10 as the dataset. Models are trained and evaluated under the same  $\\xi$  . The number in bold indicates the best."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ξ = 16/255</td><td colspan=\"2\">CIFAR10</td><td colspan=\"2\">CIFAR100</td><td>Time</td></tr><tr><td>Methods</td><td>Clean</td><td>AA</td><td>Clean</td><td>AA</td><td>hours</td></tr><tr><td>GradAlign [2]</td><td>55.58</td><td>13.45</td><td>35.93</td><td>7.13</td><td>8.79</td></tr><tr><td>NuAT [36]</td><td>74.25</td><td>13.19</td><td>20.48</td><td>7.30</td><td>7.66</td></tr><tr><td>FGSM-MEP [42]</td><td>65.56</td><td>15.43</td><td>20.69</td><td>6.73</td><td>6.15</td></tr><tr><td>B-MEP (Ours)</td><td>69.94</td><td>24.27</td><td>48.64</td><td>12.05</td><td>6.72</td></tr></table>",
+        "bbox": [
+            94,
+            559,
+            454,
+            655
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 5. The adversarial accuracy of various FAT methods with WideResNet34-10 as the backbone.",
+        "bbox": [
+            76,
+            664,
+            468,
+            691
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "FGSM-MEP, which are fitted to the distribution of adversarial examples. Instead, methods such as OAAT are fitted to the distribution of benign samples and perform better on clean accuracy. Therefore, we use the settings as in OAAT and then apply OAAT and our B-OAAT to the FAT task. B-OAAT realizes  $74.12\\%$ $(+2.82\\%)$  than OAAT) clean accuracy and  $25.39\\%$  (OAAT) on SQ+RayS.",
+        "bbox": [
+            75,
+            708,
+            468,
+            815
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3. Results on CIFAR-10",
+        "text_level": 1,
+        "bbox": [
+            76,
+            829,
+            277,
+            844
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We conduct our initial experiments on CIFAR-10 using the ResNet18 backbone. The default  $\\xi$  is set to 16/255. Tab. 3 presents the main comparisons. The observations",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "are as follows: (1) Compared with previous FAT methods, the proposed approaches achieve optimal adversarial robustness against different attacks, e.g. B-RS outperforms all RS-based methods and B-MEP is superior to all other methods. Meanwhile, our methods exhibit similar performance to prior work in terms of clean classification accuracy. (2) B-MEP realizes adversarial robustness approaching PGD-AT, e.g. B-MEP performs  $32.95\\%$  robustness against the PGD-50 attack, only  $0.97\\%$  lower than PGD-AT; (3) As for time consumption, B-RS takes less time (75 minutes) than GradAlign (135 minutes) and NuAT (101 minutes). This is because ConvergeSmooth only requires the additional regularization when  $|u_{B} - u_{t - 1}| > \\gamma_{t}$  instead of adding constraints on all iterations. In addition, B-MEP (102 minutes) takes a bit more calculation cost than FGSM-MEP (92 minutes) but much less than PGD-AT (370 minutes). PGD-AT takes significantly longer than FAT works; (4) B-RS and B-MEP successfully prevent catastrophic overfitting, e.g. mfinal (in the supplement) only shows a slight reduction than mbest in terms of adversarial robustness.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Fig. 2 visualizes the training process of various FAT approaches. Compared with other methods, the proposed",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "4725",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/b4439e353d9945ce0713420cfe24fc84534981a821a69a6bbc5fd46f1fc8d393.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Methods</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Time (min)↓</td></tr><tr><td colspan=\"2\">PGD-AT</td><td>40.57</td><td>24.72</td><td>21.46</td><td>17.94</td><td>17.38</td><td>14.98</td><td>17.02</td><td>12.63</td><td>370</td></tr><tr><td colspan=\"2\">FGSM-RS [40]</td><td>30.12</td><td>15.24</td><td>12.69</td><td>10.25</td><td>9.79</td><td>8.45</td><td>9.57</td><td>6.90</td><td>67</td></tr><tr><td colspan=\"2\">GradAlign [2]</td><td>31.91</td><td>15.71</td><td>12.56</td><td>10.28</td><td>9.71</td><td>8.22</td><td>9.47</td><td>6.61</td><td>135</td></tr><tr><td colspan=\"2\">ZeroGrad [10]</td><td>47.31</td><td>23.58</td><td>17.60</td><td>12.85</td><td>11.87</td><td>11.62</td><td>11.01</td><td>7.94</td><td>67</td></tr><tr><td rowspan=\"3\">Ours</td><td>W-RS</td><td>41.97</td><td>26.30</td><td>19.09</td><td>14.77</td><td>13.72</td><td>12.82</td><td>13.36</td><td>9.67</td><td>72</td></tr><tr><td>E-RS</td><td>41.09</td><td>22.33</td><td>18.78</td><td>15.19</td><td>14.43</td><td>12.00</td><td>13.91</td><td>9.50</td><td>78</td></tr><tr><td>B-RS</td><td>41.47</td><td>25.98</td><td>19.44</td><td>15.36</td><td>14.34</td><td>12.91</td><td>13.71</td><td>9.90</td><td>76</td></tr><tr><td colspan=\"2\">NuAT [36]</td><td>31.42</td><td>20.39</td><td>16.15</td><td>13.87</td><td>13.29</td><td>11.12</td><td>12.25</td><td>8.32</td><td>101</td></tr><tr><td colspan=\"2\">ATAS [16]</td><td>55.63</td><td>30.35</td><td>15.31</td><td>10.28</td><td>8.49</td><td>10.97</td><td>6.30</td><td>5.05</td><td>70</td></tr><tr><td colspan=\"2\">FGSM-MEP [42]</td><td>21.39</td><td>13.00</td><td>11.37</td><td>9.93</td><td>9.76</td><td>7.28</td><td>9.58</td><td>6.36</td><td>92</td></tr><tr><td rowspan=\"2\">Ours</td><td>E-MEP</td><td>44.00</td><td>24.46</td><td>20.59</td><td>17.11</td><td>16.59</td><td>13.37</td><td>16.05</td><td>10.97</td><td>106</td></tr><tr><td>B-MEP</td><td>41.86</td><td>24.86</td><td>20.84</td><td>17.30</td><td>16.59</td><td>13.96</td><td>16.25</td><td>11.38</td><td>102</td></tr></table>",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/539f6af5cf612e1f4e4d111120cfdf750ae97ab0bd7105ceaf30721e19141bd8.jpg",
+        "table_caption": [
+            "Table 6. Quantitative results of various methods  $\\left( {\\xi  = {16}/{255}}\\right)$  with ResNet18 as the backbone on CIFAR-100."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Time (hour)↓</td></tr><tr><td>PGD-AT</td><td>32.52</td><td>16.47</td><td>13.40</td><td>10.63</td><td>10.24</td><td>7.95</td><td>10.00</td><td>6.41</td><td>67.2</td></tr><tr><td>FGSM-RS [40]</td><td>27.48</td><td>12.46</td><td>9.59</td><td>6.97</td><td>6.47</td><td>4.99</td><td>6.19</td><td>3.63</td><td>10.5</td></tr><tr><td>GradAlign [2]</td><td>28.65</td><td>13.80</td><td>10.40</td><td>7.78</td><td>7.08</td><td>5.75</td><td>6.59</td><td>3.90</td><td>20.9</td></tr><tr><td>ZeroGrad [10]</td><td>34.66</td><td>12.26</td><td>8.22</td><td>5.29</td><td>4.82</td><td>3.71</td><td>4.23</td><td>2.21</td><td>10.5</td></tr><tr><td>NuAT [36]</td><td>35.24</td><td>15.52</td><td>12.07</td><td>8.81</td><td>8.13</td><td>6.29</td><td>7.68</td><td>4.27</td><td>24.6</td></tr><tr><td>FGSM-BP [42]</td><td>20.02</td><td>10.18</td><td>8.38</td><td>6.98</td><td>6.77</td><td>4.56</td><td>6.57</td><td>3.62</td><td>14.3</td></tr><tr><td>B-BP (Ours)</td><td>33.51</td><td>16.32</td><td>12.92</td><td>9.72</td><td>9.23</td><td>7.26</td><td>9.32</td><td>5.95</td><td>15.4</td></tr></table>",
+        "bbox": [
+            120,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 7. Quantitative results of various methods  $\\left( {\\xi  = {16}/{255}}\\right)$  with PreActResNet18 as the backbone on Tiny ImageNet.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "B-RS achieves optimal training stability and adversarial robustness, e.g. the classification accuracy of our method fluctuates slightly during three adversarial training sessions.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Various budgets. Similar experiments are performed for budgets 10/255 and 12/255. Details are given in Tab. 4. It is evident that the proposed method achieves optimal adversarial robustness across various perturbation budgets.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Various networks. We then adopt WideResNet34 with a width factor of 10 [47] as the backbone. The results are given in Tab. 5. Our proposed B-MEP also prevents the wider architectures from catastrophic overfitting.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4. Results on CIFAR-100",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results on CIFAR-100 with the backbone ResNet-18 are presented in Tab. 6. (1) FAT methods perform similar training consumption on CIFAR-10 and CIFAR-100 as two datasets contain the same number and size of images; (2) All proposed methods can realize stable adversarial training, as evidenced by the comparable results of mbest and mfinal (in the supplement); (3) Among the RS-based FAT works, B-RS and E-RS achieve the best and second-best adversarial robustness against different attacks; (4) B-MEP outperforms all other FAT methods; (5) Although B-MEP performs slightly worse than PGD-AT, its training process is approximately 3.5 times faster than this competitor.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5. Results on Tiny ImageNet",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on the Tiny ImageNet using the PreActResNet18 backbone to demonstrate the scalability of the proposed method to large datasets. The results are given in Tab. 7. B-BP achieves higher robustness among the FAT methods and comparable robustness to PGD-AT. As for training efficiency, B-BP (15.4 hours) requires slightly more computational cost than FGSM-BP (14.3 hours), but significantly less time than PGD-AT (67.2 hours).",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.6. Ablation studies",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the following, B-RS is selected as the backbone. Figs. 3 and 4 conduct ablation studies on  $\\gamma_{max}$  and  $\\gamma_{max} / \\gamma_{min}$  respectively. In Fig. 3, the classification accuracy of both benign and adversarial examples increases with  $\\gamma_{max}$ . However, the model trained with a large  $\\gamma_{max}$  ( $>0.09$ ) suffers from catastrophic overfitting. Fig. 4 shows that the model with  $\\gamma_{max} / \\gamma_{min} = 1.5$  achieves optimal robustness.  $\\gamma_{max} / \\gamma_{min} = 1$  denotes the static convergence stride. All models can be trained stably when  $\\gamma_{max} = 0.06$ .",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Tabs. 8 and 9 provide the results of comparative experiments on  $w_{1}$  and  $w_{2}$  respectively. We empirically set  $w_{2}$  to 1 since too small  $w_{2}$  cannot prevent overfitting and too large  $w_{2}$  will affect the performance. In Tab. 8,  $w_{1} = 0$  achieves optimal adversarial robustness on CIFAR100 and",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "4726",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0dab9e0d6a952d662975b0a66f009a181a37b96231ffb04234a75806a79ef74a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>w1</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Stability</td></tr><tr><td rowspan=\"5\">CIFAR100</td><td>0.0</td><td>48.13</td><td>32.13</td><td>24.25</td><td>22.67</td><td>22.21</td><td>19.04</td><td>21.29</td><td>15.28</td><td>★★★</td></tr><tr><td>0.3</td><td>46.08</td><td>30.58</td><td>23.06</td><td>21.62</td><td>21.22</td><td>17.79</td><td>20.44</td><td>15.22</td><td>★★★</td></tr><tr><td>0.5</td><td>44.04</td><td>29.96</td><td>22.64</td><td>21.34</td><td>20.99</td><td>17.55</td><td>19.16</td><td>13.84</td><td>★★★</td></tr><tr><td>0.7</td><td>43.07</td><td>28.61</td><td>22.01</td><td>20.84</td><td>20.32</td><td>16.87</td><td>18.98</td><td>13.47</td><td>★★★</td></tr><tr><td>0.9</td><td>40.45</td><td>21.05</td><td>21.62</td><td>20.54</td><td>20.15</td><td>16.33</td><td>18.91</td><td>13.53</td><td>★★★</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/c99ac63b3639ee7e6daaff8d9f047762c021432397fc0f6553a0a2b1593e9ad7.jpg",
+        "table_caption": [
+            "Table 8. Quantitative results of the proposed method on various  $w_{1}$  with ResNet18 as the backbone and the perturbation budget 12/255. 'Stability' represents the number of times the model is stable in three training repetitions.  $\\gamma_{max}$  and  $w_{2}$  are set to 0.03 and 1, respectively."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>w2</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Stability</td></tr><tr><td rowspan=\"4\">CIFAR100</td><td>0.5</td><td>36.79</td><td>23.97</td><td>18.60</td><td>17.51</td><td>17.23</td><td>14.65</td><td>16.28</td><td>12.23</td><td>-</td></tr><tr><td>0.7</td><td>37.31</td><td>24.36</td><td>18.56</td><td>17.37</td><td>17.26</td><td>14.46</td><td>15.96</td><td>11.54</td><td>-</td></tr><tr><td>1.0</td><td>48.13</td><td>32.13</td><td>24.25</td><td>22.67</td><td>22.21</td><td>19.04</td><td>21.29</td><td>15.28</td><td>★★★</td></tr><tr><td>1.3</td><td>44.84</td><td>30.42</td><td>23.31</td><td>21.72</td><td>21.31</td><td>18.36</td><td>19.64</td><td>14.72</td><td>★★★</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 9. Quantitative results on various  ${w}_{2}\\left( {{\\gamma }_{max} = {0.03}}\\right.$  and  ${w}_{1} = 0)$  with ResNet18 as the backbone and  $\\xi  = {12}/{255}$  .",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b66648bc0d3b8cb0ffe73994f5c9c86e4af5bbb2d70966398399a8e131e75277.jpg",
+        "image_caption": [
+            "ResNet18 trained with ConvergeSmooth-RS on CIFAR-100",
+            "Figure 3. Ablation study of  $\\gamma_{max}(\\gamma_{max} / \\gamma_{min} = 1.5)$ . We provide the classification accuracy of models (ResNet-18) to benign and adversarial examples (PGD-10,  $\\xi = 16/255$ ) on CIFAR-100."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            83,
+            375,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "all settings accomplish stable FAT. According to Eqs. (7) and (8), if  $w_{1} \\neq 0$ , for the data  $x$  which satisfies  $\\mathcal{C}(x) = 1$ ,  $\\mathcal{L}(x_t', \\theta_t)$  is assigned with higher weights  $(1 + w_{1})$  or lower weights  $(1 - w_{1})$ , causing the model to excessively prioritize or neglect this portion of the data. This may weaken the performance of models. However, for Tiny ImageNet, models suffer catastrophic overfitting at  $w_{1} = 0$ . Then, we gradually increase  $w_{1}$  from 0.3 with a stride of 0.2 until the FAT process becomes stable. In addition, our weight centralization method only introduces a balance coefficient  $w_{3}$ .  $w_{3}$  is set to 0.1 since catastrophic overfitting happens when  $w_{3} = 0$  and classification performance is degraded when  $w_{3} = 0.2$ .",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5. Conclusion",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we tackle the issue of catastrophic overfitting by focusing on the convergence stability of loss func",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b22b73832bd1331752bbefd95f835fb0c27d3622c1c6426d3618be637481b38b.jpg",
+        "image_caption": [
+            "ResNet18 trained with ConvergeSmooth-RS on CIFAR-100",
+            "Figure 4. Ablation study of  $\\gamma_{max} / \\gamma_{min}$  ( $\\gamma_{max} = 0.06$ ). We provide the classification accuracy of models (ResNet-18) to benign and adversarial examples (PGD-10,  $\\xi = 16/255$ ) on CIFAR-100."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            506,
+            373,
+            880,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "tions. Through experimental analysis, we find that this issue is accompanied by abnormal convergence behavior of losses. Motivated by this phenomenon, we propose complementary constraints, namely E-ConvergeSmooth and B-ConvergeSmooth, based on the adversarial training constraint. To alleviate the burden of parameter tuning, weight centralization is designed utilizing priors from previous epochs. Extensive experiments on different network architectures and datasets show that the proposed methods effectively solve catastrophic overfitting and exhibit superior robustness against various adversarial attacks.",
+        "bbox": [
+            496,
+            666,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments. This work was supported by the National Key R&D Program of China #2018AAA0102000 and the National Natural Science Foundation of China #62276046 and #U19B2039.",
+        "bbox": [
+            498,
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+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "4727",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            78,
+            89,
+            173,
+            104
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
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2023/Fast Adversarial Training with Smooth Convergence/23b31082-56d8-432b-80e3-85fc235c4ff5_origin.pdf

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+# Fast Adversarial Training with Smooth Convergence
+
+Mengnan Zhao, Lihe Zhang*, Yuqiu Kong and Baocai Yin
+
+Dalian University of Technology, China
+
+zmnwelcome@mail.dlut.edu.cn，{zhanglihe，yqkong，ybc}@dlut.edu.cn
+
+# Abstract
+
+Fast adversarial training (FAT) is beneficial for improving the adversarial robustness of neural networks. However, previous FAT work has encountered a significant issue known as catastrophic overfitting when dealing with large perturbation budgets, i.e. the adversarial robustness of models declines to near zero during training. To address this, we analyze the training process of prior FAT work and observe that catastrophic overfitting is accompanied by the appearance of loss convergence outliers. Therefore, we argue a moderately smooth loss convergence process will be a stable FAT process that solves catastrophic overfitting. To obtain a smooth loss convergence process, we propose a novel oscillatory constraint (dubbed ConvergeSmooth) to limit the loss difference between adjacent epochs. The convergence stride of ConvergeSmooth is introduced to balance convergence and smoothing. Likewise, we design weight centralization without introducing additional hyperparameters other than the loss balance coefficient. Our proposed methods are attack-agnostic and thus can improve the training stability of various FAT techniques. Extensive experiments on popular datasets show that the proposed methods efficiently avoid catastrophic overfitting and outperform all previous FAT methods. Code is available at https://github.com/FAT-CS/ConvergeSmooth.
+
+# 1. Introduction
+
+Recent breakthroughs in deep learning [21, 32] have aroused researchers' interest in the security of neural networks [45, 48, 39]. In particular, the advanced research proves the vulnerability of deep models to adversarial attacks [8, 11, 27]. For instance, tiny crafted perturbations can fool models in various fields into making wrong decisions [15, 50, 35, 24]. Considering the security risks brought by adversarial attacks [18, 46, 33, 44], there is a quickly grow
+
+ing body of work [43, 46, 18] on improving the adversarial robustness of neural networks. Among them, adversarial training is widely applied by practitioners [31, 12].
+
+In recent years, projected gradient descent based adversarial training (PGD-AT) [25, 34] has been widely employed for its stability and effectiveness. However, this mechanism is computationally expensive. It requires multiple gradient descent steps to generate the adversarial training data [38]. An alternative of PGD-AT is the fast adversarial training (FAT) [28], which only adopts a single-step fast gradient sign method (FGSM) [11] to generate training data. Compared to PGD-AT, FAT can efficiently train models, but easily falls into catastrophic overfitting [40, 29].
+
+A number of FAT methods have been proposed to mitigate catastrophic overfitting. For example, Wong et al. [40] use randomly initialized perturbations to enhance the diversity of adversarial perturbations. Based on it, Andriushchenko et al. [2] raise a complementary regularizer named GradAlign to maximize the gradient alignment between benign and adversarial examples explicitly. Similarly, NuAT [36] and FGSM-MEP [42] adopt nuclear norm or p-norm to regularize the adversarial training, thereby increasing the prediction alignment between benign and adversarial examples. However, the above methods can only resolve catastrophic overfitting within the limited perturbation budget ( $\xi \leq 8/255$ ).  $\xi$  specifies the perturbation degree of adversarial training data generated by various attacks. Besides, models trained by small perturbations are vulnerable to adversarial attacks with a large  $\xi$ , e.g. the models trained by NuAT and FGSM-MEP at  $\xi = 8/255$  perform  $53\%$  and  $54\%$  robustness against the PGD-50 attack with  $\xi = 8/255$ , but only  $22\%$  and  $20\%$  robustness against the same attack with  $\xi = 16/255$  respectively. Therefore, we aspire to prevent catastrophic overfitting to improve the adversarial robustness of neural models at larger perturbation budgets.
+
+By analyzing adversarial training processes of representative work, we observe that catastrophic overfitting is usually accompanied by a slight fluctuation in the classification loss for benign samples and a sharp drop in the classification
+
+loss for adversarial examples. This motivates us to question whether a smooth loss convergence process is also a stable FAT process. Moreover, we find that an oscillating adversarial training phase may restart the FAT process after catastrophic overfitting. Fig. 1 shows the details.
+
+According to these observations, we introduce an oscillatory constraint that limits the difference in loss between adjacent training epochs, called ConvergeSmooth. A dynamic convergence stride of ConvergeSmooth is designed considering the nonlinear decay rate of loss functions. Inspired by the smoothness of loss convergence, we further verify the effect of the proposed weight centralization on model stability. Weight centralization refers to taking the weights average of the previously trained models as the convergence center of the current model weight. Our proposed methods are attack-agnostic and thus can be combined with existing adversarial strategies in FAT, such as FGSM-RS and FGSM-MEP, to evaluate their performance.
+
+The contributions are summarized in four aspects: (1) We verify that previous FAT works still suffer from catastrophic overfitting at a large  $\xi$  and then study catastrophic overfitting from the perspective of convergence instability of loss functions; (2) We propose a smooth convergence constraint, ConvergeSmooth, and design a dynamic convergence stride for it, to help various FAT methods avoid catastrophic overfitting on different perturbation budgets; (3) The weight centralization is proposed without introducing extra hyperparameters other than the loss balance coefficient to stabilize FAT; (4) Extensive experiments show that the proposed methods outperform the state-of-the-art FAT techniques in terms of efficiency, robustness, and stability.
+
+# 2. Related Work
+
+Adversarial attacks. Adversarial attacks are usually used to deceive deep-learning models. Goodfellow et al. [11] first discuss the adversarial attack (FGSM) within the classification task. They prove that adversarial examples  $x'$  generated by a single-step gradient backward can misclassify the model  $f(\cdot; \theta)$  with high confidence.  $\theta$  denotes the fixed model weights.  $x'$  is generated by
+
+$$
+x ^ {\prime} = x + \xi \cdot \operatorname {s g n} \left(\nabla_ {x} \mathcal {L} (f (x; \theta), y)\right), \tag {1}
+$$
+
+where  $x$  is an input image,  $\xi$  represents the perturbation budget,  $\operatorname{sgn}(\cdot)$  means the sign function,  $\mathcal{L}(\cdot)$  is usually the cross-entropy loss,  $\nabla_{x}\mathcal{L}(\cdot)$  calculates the gradient of loss at  $x$ , and  $y$  denotes the ground truth labels of  $x$ . Following the FGSM [11], researchers propose a series of attack methods based on the iterative gradient backward, e.g. I-FGSM [22], MIM [8], and PGD [25]. Taking the PGD attack as an example, the adversarial example  $x_{t + 1}^{\prime}$  produced in the iteration  $t + 1$  can be formulated as
+
+$$
+x _ {t + 1} ^ {\prime} = \operatorname {c l i p} _ {\xi} \left(x _ {t} ^ {\prime} + \epsilon \cdot \operatorname {s g n} \left(\nabla_ {x _ {t} ^ {\prime}} \mathcal {L} \left(f \left(x _ {t} ^ {\prime}; \theta\right), y\right)\right)\right), \tag {2}
+$$
+
+where  $\epsilon$  denotes the single-step stride and  $\mathrm{clip}_{\xi}$  refers to projecting adversarial perturbations to a  $\xi$ -ball.
+
+Adversarial training. Madry et al. [25] formalize the adversarial training as a min-max optimization problem,
+
+$$
+\min  _ {\theta} \mathbb {E} _ {(x, y) \sim D} [ \max  _ {\delta \in [ - \xi , \xi ]} \mathcal {L} (f \left(x ^ {\prime}; \theta\right), y) ], \tag {3}
+$$
+
+where  $x' = x + \delta$ ,  $\delta$  represents the adversarial perturbations generated by various attacks such as PGD and FGSM.  $D$  is the data generator. The internal maximization maximizes the classification loss to generate adversarial perturbations with fixed model weights. The external minimization minimizes the classification loss on the generated adversarial examples when optimizing the model weights.
+
+Actually, there is a trade-off between computational efficiency and adversarial robustness in recent adversarial training methods. Compared to PGD-AT [25], FGSM-based fast adversarial training (FAT) accelerates the training process but damages the robustness of models for the problem of catastrophic overfitting [19, 17]. To mitigate this issue, Wong et.al [40] demonstrate that the FGSM with a random start strategy (FGSM-RS) can achieve comparable performance against the PGD-AT. ZeroGrad [10] zeroes the elements of gradient that are too small to craft the perturbations. enforces the model loss to increase with the increase in perturbation size. Besides, GradAlign [2] prevents catastrophic overfitting by maximizing the alignment between gradients of benign samples and adversarial examples.
+
+Similarly, NuAT [36] introduces a nuclear norm regularization between logits for benign and adversarial examples and uses the Bernoulli noise as the initial perturbation. ATAS [16] learns an instantiated adaptive step size that is inversely proportional to the gradient norm. It applies the adversarial perturbations from the previous epoch as the initialization of FGSM in the current training phase. In addition, Jia et al. [42] propose several prior-guided initialization methods to replace the random start strategy of FGSM-RS. Specifically, FGSM-BP adopts the adversarial perturbations from the previous batch as the attack initialization in the current batch. FGSM-MEP employs a momentum mechanism to combine all adversarial perturbations from the previous epochs to yield adversarial initialization.
+
+Although these FAT methods have resolved catastrophic overfitting at small perturbation budgets, they still suffer from catastrophic overfitting under larger perturbation budgets (e.g.  $\xi = 16 / 255$ ). Unlike these methods, we revisit the catastrophic overfitting problem from the perspective of loss convergence instability and prevent this exception by limiting the magnitude of loss fluctuations during training.
+
+# 3. Proposed Method
+
+In this section, we first study the performance of previous FAT methods when subjected to a large perturbation budget
+
+![](images/cd9ecbc383adc2a12e7ab4b5a3256b22e64a083407a68b7b5fbdf8cacf2a38fa.jpg)  
+(a) ResNet18 trained with FGSM-RS
+
+![](images/f2568362360d16210983537fbbc98350642e7fe840781af21c1099d1b6ebfb67.jpg)  
+(b) ResNet18 $_2$  trained with FGSM-RS
+
+![](images/fdd50d80d63a5e6b09bf7fb39d5738ac55fa6b9d105d1b63edcf06374cb43fd0.jpg)  
+(c) ResNet18 trained with FGSM-RS
+
+![](images/888d940547966a2de7eeb0aac3b876c80c50ae5fb0cbdda61901363669dbe86b.jpg)  
+(d) ResNet18 trained with GradAlign
+
+![](images/6b673903b39c5fb94588140a51c32a46441f9a527c1e7d9af4534d7a47470ae1.jpg)  
+(e) ResNet18 $_2$  trained with GradAlign
+
+![](images/7570403a5c25707a754349395f19cf4b9d241a35cc9aebda17ce19c1ec54d36a.jpg)  
+(f) ResNet18 trained with GradAlign
+
+![](images/d26313a50019ee9fc986785530ffce4cb8414056df6f5225922cb675f1ba34e5.jpg)  
+(g) ResNet18 trained with FGSM-MEP  
+Figure 1. The training process of previous FAT methods with ResNet18 on CIFAR-10. Each FAT method is trained 3 times.  $\xi = 16 / 255$ . ADV-Loss and BEN-Loss denote the classification loss of models to adversarial and benign examples during training, respectively. ADV-Acc and BEN-Acc represent the classification accuracy of models to adversarial and benign examples during testing, respectively.
+
+![](images/36780e4b0d277b67f249a6292c1612b47a45ac4e2d523f7bafd4a2dd23f657ef.jpg)  
+(h) ResNet182 trained with FGSM-MEP
+
+![](images/a205b7233de4b4082b6e17fbf138ff8447f36e6b7768b82dbd155c33980769a0.jpg)  
+(i) ResNet18 trained with FGSM-MEP
+
+$\xi$ . Then, the training processes of these methods have been analyzed for a better understanding of catastrophic overfitting. Finally, we detail the proposed methods.
+
+# 3.1. Performance of FAT methods on a large  $\xi$
+
+Previous FAT techniques avoid catastrophic overfitting at small  $\xi$  ( $\xi \leq 8/255$ ). Here, we investigate their performance on a large  $\xi$  ( $\xi = 16/255$  by default).
+
+FGSM-RS [40]: Based on Eq. (3), this method adopts the samples with uniformly random perturbations  $\delta_0\sim$ $\mathcal{U}(-\xi ,\xi)$  as the attack initialization of FGSM,
+
+$$
+\min  _ {\theta} \mathbb {E} _ {(x, y) \sim D} [ \max  _ {\delta_ {0} + \delta \in [ - \xi , \xi ]} \mathcal {L} (f (x ^ {\prime}; \theta), y) ],
+$$
+
+$$
+x ^ {\prime} = x + \delta_ {0} + \delta \quad s. t. \quad x ^ {\prime} \in [ 0, 1 ], \quad \delta_ {0} \sim \mathcal {U} (- \xi , \xi). \tag {4}
+$$
+
+GradAlign [2]: This approach increases the gradient alignment between benign samples  $x$  and perturbed samples  $x + \delta_0$ , which is denoted as
+
+$$
+\mathbb {E} _ {(x, y) \sim D} [ 1 - \cos (\nabla_ {x} \mathcal {L} (x, \theta), \nabla_ {x + \delta_ {0}} \mathcal {L} (x + \delta_ {0}, \theta) ], \tag {5}
+$$
+
+$$
+\mathcal {L} (x, \theta) = \mathcal {L} (f (x; \theta), y).
+$$
+
+$\cos (\cdot)$  computes the cosine similarity between two matrices.
+
+FGSM-MEP [42]: Different from FGSM-RS, this method generates the initialization perturbations  $\delta_0$  based on all historical adversarial perturbations from the previous epochs and introduces a regularization expressed as
+
+$$
+\mathbb {E} _ {(x, y) \sim D} [ | | f (x ^ {\prime}; \theta) - f (x + \delta_ {0}; \theta) | | _ {2} ^ {2} ], \tag {6}
+$$
+
+where  $||\cdot ||_2^2$  denotes the squared  $\mathcal{L}_2$  distance.
+
+Fig. 1 shows their detailed adversarial training processes of ResNet18 [13] on CIFAR-10 [20]. We find that they fall into catastrophic overfitting during the  $5 \sim 20_{\mathrm{th}}$ ,  $40 \sim 65_{\mathrm{th}}$  and  $10 \sim 15_{\mathrm{th}}$  epochs, respectively. Graphical analysis of various models and datasets is given in the supplement.
+
+# 3.2. Analysis of training process
+
+From Fig. 1, we discover several typical phenomena of catastrophic overfitting: 1) Change from a smooth convergence state to an irregular fluctuation state; 2) A slight change (increase or decrease) in the standard classification loss  $\mathcal{L}(x,\theta)$  and a rapid descent of the adversarial classification loss  $\mathcal{L}(x',\theta)$ ; 3) Rapid decline in the classification accuracy of adversarial examples.
+
+After catastrophic overfitting, the methods depicted in Fig. 1 are capable of restarting a stable FAT process, even though catastrophic overfitting may occur again. The observed phenomena during the FAT restart are: 1) Change from an irregular fluctuation state to a smooth convergence state; 2) Rapid increase in both  $\mathcal{L}(x,\theta)$  and  $\mathcal{L}(x',\theta)$  after a period of decline; 3) Rapid decrease in the classification accuracy of benign samples, while the classification accuracy of adversarial examples experiences a rapid increase.
+
+On this basis, we can make the following conclusions. 1)  $\mathcal{L}(x,\theta)$  is more stable than  $\mathcal{L}(x^{\prime},\theta)$  and models are prone to overfitting to adversarial perturbations; 2) Catastrophic overfitting is closely correlated to the convergence insta
+
+bility of adversarial training; 3) Exceptions in  $\mathcal{L}(x,\theta)$  and  $\mathcal{L}(x^{\prime},\theta)$  occur simultaneously; 4) An oscillating adversarial training phase may trigger the FAT process to restart.
+
+# 3.3. Smooth convergence for the stable FAT
+
+Next, we describe the proposed method in detail.
+
+Why did previous methods fail to prevent catastrophic overfitting? Despite the improvement in diversity achieved through random initialization in FGSM-RS, models are still susceptible to overfitting adversarial perturbations. GradAlign and FGSM-MEP enhance the stability of adversarial training through the constraints in Eqs. (5) and (6), respectively. However, Eq. (5) may reduce the stability of  $\mathcal{L}(x,\theta)$ . Meanwhile, the prediction probability  $f(x + \delta_0;\theta)$  in Eq. (6) is not the sweet spot to keep the FAT stable. Unlike these methods, we ensure the convergence stability of both  $\mathcal{L}(x,\theta)$  and  $\mathcal{L}(x',\theta)$ .
+
+How can catastrophic overfitting be solved? Since the catastrophic overfitting is usually accompanied by a slight change in  $\mathcal{L}(x,\theta)$  and a sharp decline in  $\mathcal{L}(x',\theta)$ , we consider a smooth loss convergence process to be a stable FAT process that resolves this issue. To this end, a complementary constraint  $\mathcal{L}_{CS}$  for Eq. (3) is proposed,
+
+$$
+\min  _ {\theta_ {t}} \mathbb {E} _ {(x, y) \sim D} \left[ \mathcal {L} \left(x _ {t} ^ {\prime}, \theta_ {t}\right) + \mathcal {L} _ {C S} (t) \right], \tag {7}
+$$
+
+which can limit the difference in losses between adjacent epochs, expressed as
+
+$$
+\mathcal {L} _ {C S} (t) = w _ {1} \cdot \left| \mathcal {L} \left(x _ {t} ^ {\prime}, \theta_ {t}\right) - \mathcal {L} \left(x _ {t - 1} ^ {\prime}, \theta_ {t - 1}\right) \right| + \tag {8}
+$$
+
+$$
+w _ {2} \cdot | \mathcal {L} (x, \theta_ {t}) - \mathcal {L} (x, \theta_ {t - 1}) |, \quad s. t. \quad \mathcal {C} (x) = 1,
+$$
+
+where  $\theta_{t}$  represents the model weights of the  $t_{\mathrm{th}}$  training epoch.  $w_{1}$  and  $w_{2}$  are hyper-parameters,  $w_{1}, w_{2} \in [0,1]$ .  $|\cdot|$  calculates the absolute value.  $x_{t}^{\prime} = x + \delta_{0,t} + \delta_{t}$ .  $\delta_{0,t}$  can be replaced by various attack initialization perturbations and  $\delta_{t}$  is generated by
+
+$$
+\max  _ {\delta_ {t} \in [ - \xi , \xi ]} \mathcal {L} \left(x _ {t} ^ {\prime}, \theta_ {t - 1}\right), \tag {9}
+$$
+
+where  $\theta_{t - 1}$  is fixed in generating  $\delta_t$ .
+
+In the practical implementation of Eq. (8), storing and computing  $\mathcal{L}(x_{t - 1}^{\prime},\theta_{t - 1})$  and  $\mathcal{L}(x,\theta_{t - 1})$  can consume a significant amount of memory. To overcome this challenge, we introduce  $u_{t - 1}^{\prime}$  and  $u_{t - 1}$  to replace these terms, respectively.  $u_{t - 1}^{\prime} = \mathbb{E}_{(x,y)\sim D}\mathcal{L}(x_{t - 1}^{\prime},\theta_{t - 1})$  and  $u_{t - 1} = \mathbb{E}_{(x,y)\sim D}\mathcal{L}(x,\theta_{t - 1})$  denote the mathematical expectation of loss in the  $t - 1_{\mathrm{th}}$  training epoch.
+
+We observe that training instability is primarily caused by overfitting or underfitting of small amounts of data. Thus, in Eq. (8), the additional loss term is only applied to partial data selected by the crafted condition  $\mathcal{C}(\cdot)$ . Notably, exceptions in  $\mathcal{L}(x,\theta)$  and  $\mathcal{L}(x',\theta)$  occur simultaneously during training. Hence,  $\mathcal{L}(x_t',\theta_t)$  and  $\mathcal{L}(x_t,\theta_t)$  share
+
+the same condition.  $\mathcal{C}$  is constructed by the distance between the pointwise loss  $\mathcal{L}(x,\theta_t)$  and the mean value  $u_{t - 1}$ . This is because over-fitted or under-fitted data often yield excessively high or low classification losses.
+
+$$
+\mathcal {C} (x) = \left(\left| \mathcal {L} \left(x, \theta_ {t}\right) - u _ {t - 1} \right| \geq \gamma_ {t}\right), \tag {10}
+$$
+
+where  $\gamma_{t} (\gamma_{t} \geq 0)$  represents the convergence stride to select abnormal data. It is crucial to choose an appropriate value for  $\gamma_{t}$  to ensure effective training. When  $\gamma_{t} = 0$ , the adversarial training process may fail to converge as it forces the predictions of all samples to remain unchanged. On the other hand, catastrophic overfitting occurs when  $\gamma_{t} = \infty$ , leading to undesirable outcomes. Considering that the loss difference  $d_{t-1}(d_{t-1} = |u_{t-1} - u_{t-2}|)$  between two adjacent epochs tends to decrease non-linearly,  $\gamma_{t}$  should be a variable that varies during the training process.
+
+$$
+\gamma_ {t} = \min  \left(\max  \left(d _ {t - 1}, \gamma_ {\text {m i n}}\right), \gamma_ {\text {m a x}}\right), \tag {11}
+$$
+
+where  $\gamma_{min}$  and  $\gamma_{max}$  are hyper-parameters.  $\gamma_{max}$  controls the maximum convergence speed.  $\gamma_{min}$  ensures that the training process is not too smooth when  $d_{t - 1} \to 0$ .
+
+Example-based ConvergeSmooth. This approach adds the constraint individually to each sample, described as
+
+$$
+\begin{array}{l} \mathcal {L} _ {C S} ^ {E} (t) = w _ {1} \cdot \left| \mathcal {L} \left(x _ {t} ^ {\prime}, \theta_ {t}\right) - u _ {t - 1} ^ {\prime} \right| + w _ {2} \cdot \left| \mathcal {L} \left(x, \theta_ {t}\right) - u _ {t - 1} \right|, \\ s. t. \quad | \mathcal {L} (x, \theta_ {t}) - u _ {t - 1} | > \gamma_ {t}. \tag {12} \\ \end{array}
+$$
+
+Batch-based ConvergeSmooth. Likewise, we can apply the complementary constraint to a data batch,
+
+$$
+\begin{array}{l} \mathcal {L} _ {C S} ^ {B} (t) = w _ {1} \cdot \left| u _ {B} ^ {\prime} - u _ {t - 1} ^ {\prime} \right| + w _ {2} \cdot \left| u _ {B} - u _ {t - 1} \right|, \tag {13} \\ s. t. \quad | u _ {B} - u _ {t - 1} | > \gamma_ {t}, \\ \end{array}
+$$
+
+where  $u_B^\prime = \mathbb{E}_{(x,y)\sim B}\mathcal{L}(x_t^\prime ,\theta_t),u_B = \mathbb{E}_{(x,y)\sim B}\mathcal{L}(x,\theta_t).$
+
+Weight centralization. To mitigate the problem of manual parameter tuning, we introduce the weight centralization without requiring extra hyperparameters other than the coefficient  $w_{3}$ ,  $|\theta_t - \theta_{t-1}| = 0 \rightarrow |\mathcal{L}(x, \theta_t) - \mathcal{L}(x, \theta_{t-1})| = 0$ ,
+
+$$
+\mathcal {L} _ {C S} ^ {W} (t) = w _ {3} \cdot | | \theta_ {t} - \frac {1}{l e n (\phi)} \cdot \sum_ {j \in \phi} \theta_ {j} | | _ {p}, \tag {14}
+$$
+
+where  $\| \cdot \|_p$  means the p-norm function  $(p = 2)$ , and  $\phi$  denotes a set of previous model weights. The model weights  $\theta_t$  are restricted to the center  $\frac{1}{len(\phi)} \cdot \sum_{j \in \phi} \theta_j$ . The reasons behind Eq. (14) are: 1) The initial training process is stable, as indicated in Fig. 1; 2) Models at different training epochs tend to have similar weights after convergence [37].
+
+# 4. Experimental Results
+
+# 4.1. Experimental settings
+
+Details. To demonstrate the effectiveness of the proposed method, we conduct comprehensive experiments on
+
+<table><tr><td colspan="2">CIFAR10ξ</td><td>FGSM-RS</td><td>FGSM-MEP</td><td>B-RS</td><td>B-MEP</td></tr><tr><td>SQ↑</td><td rowspan="2">16/255</td><td>19.74</td><td>22.15</td><td>28.83</td><td>29.97</td></tr><tr><td>SQ+RayS↑</td><td>18.72</td><td>20.96</td><td>27.08</td><td>28.45</td></tr></table>
+
+Table 1. Quantitative results of FAT methods against the Square and Ray-S attacks on CIFAR-10 with the backbone ResNet18.  
+
+<table><tr><td>CIFAR10</td><td>Clean</td><td>8 255</td><td>10 255</td><td>12 255</td><td>16 255</td><td>SQ+RayS</td><td>Time</td></tr><tr><td>OAAT[1]</td><td>71.30</td><td>44.29</td><td>38.56</td><td>33.68</td><td>28.48</td><td>21.06</td><td>183</td></tr><tr><td>AWP[41]</td><td>78.86</td><td>35.91</td><td>32.70</td><td>29.89</td><td>26.95</td><td>19.36</td><td>134</td></tr><tr><td>ATES[4]</td><td>74.52</td><td>43.61</td><td>38.94</td><td>35.31</td><td>30.89</td><td>22.16</td><td>-</td></tr><tr><td>ExAT[30]</td><td>80.78</td><td>49.21</td><td>42.15</td><td>36.98</td><td>31.26</td><td>23.08</td><td>70</td></tr><tr><td>E-MEP</td><td>69.84</td><td>53.54</td><td>49.26</td><td>45.89</td><td>40.78</td><td>23.78</td><td>104</td></tr><tr><td>B-MEP</td><td>63.84</td><td>50.31</td><td>46.77</td><td>43.97</td><td>40.13</td><td>28.45</td><td>101</td></tr></table>
+
+Table 2. Quantitative results of methods against various levels of perturbation with ResNet18 as the backbone and CIFAR-10 as the dataset.  $\sharp$  denotes the PGD-10 attack.
+
+several benchmark datasets, i.e. CIFAR-10 [20], CIFAR-100 [20], and Tiny ImageNet [7].
+
+Following previous works [40, 23, 49, 42], we adopt ResNet18 [13] as the backbone on the CIFAR-10 and CIFAR-100, and choose PreActResNet18 [14] on the Tiny ImageNet. In all experiments, models are optimized using the SGD optimizer with a batch size of 128, weight decay of 5e-4, and momentum of 0.9. The initial learning rates on CIFAR-10, CIFAR-100, and Tiny ImageNet are set as 0.1, 0.1, and 0.01, respectively. Then, we optimize models with a total training epoch of 110 and decay the learning rate at the  $100_{\mathrm{th}}$  and  $105_{\mathrm{th}}$  epoch with a factor of 0.1.
+
+We apply the proposed ConvergeSmooth in combination with two attack initialization methods on CIFAR-10 and CIFAR-100, FGSM-RS [40] and FGSM-MEP [42]. Additionally, for the Tiny ImageNet, we use FGSM-BP [42] as the initialization method. As mentioned in [42], FGSM-MEP requires consuming memory to store the previous adversarial perturbations, which limits its application on large datasets. Regarding the hyperparameters, we set  $\gamma_{max}$  in Eq. (11) as 0.03, 0.06, and 0.03 for CIFAR-10, CIFAR-100, and Tiny ImageNet, respectively.  $\gamma_{max} = 1.5 \cdot \gamma_{min}$  and  $\xi = 16/255$ . Specific details of hyperparameter settings for  $w_{1 \sim 3}$  are given in the supplement. All experiments are conducted on a single GeForce RTX 3090 GPU.
+
+Baselines. We include advanced FAT methods as baselines, namely FGSM-RS [40], GradAlign [2], ZeroGrad [10], NuAT [36], ATAS [16], and FGSM-MEP [42].
+
+Evaluation metrics. We follow [40, 16] and adopt FGSM [11], PGD-10 [25], PGD-20 [25], PGD-50 [25], C&W [3], APGD-CE [6] and Autoattack (AA) [6] to evaluate the adversarial robustness of models. PGD-n represents the PGD attack with n iterations. Autoattack combines APGD-CE and APGD-T (targeted APGD) as well as two complementary attacks (FAB [9] and Square attack [26]).
+
+![](images/3e97564525e8fbc6ed20f5e85e150c8587a20b3828877176d39ffb9478255b35.jpg)  
+Figure 2. The training process of various FAT methods (PGD-10,  $\xi = 16 / 255$ ) on CIFAR-10.
+
+The training easily exhibits fluctuations, and thus individual results can not objectively reflect the performance of methods. For each method, we repeat the training process three times and report the average evaluation results of the best model across the three runs (mbest). Additionally, the supplement includes the best results (best) and the average evaluation of the final model from the three runs (mfinal).
+
+In the following section, 'W-\*', 'E-\*', and 'B-\*' mean 'weight centralization', 'example-based ConvergeSmooth', and 'batch-based ConvergeSmooth', respectively.
+
+# 4.2. Significance of the Results
+
+Before evaluating the adversarial robustness of trained models, we demonstrate the significance of the results [1]. Specifically, we show that adversarial attacks with  $\xi = 16/255$  rarely change the true label of input. 1) We generate adversarial examples for the FGSM-RS, FGSM-MEP, and our proposed methods on CIFAR10 and CIFAR100 datasets with  $\xi = 16/255$  using the non-targeted attacks such as FGSM, PGD, C&W, APGD-CE, and the targeted attack APGD-T. The majority of the generated examples retain their true labels; 2) Tab. 1 examines the robustness of models against an ensemble of the Square (SQ) [26] and Ray-S [5] attacks, as these attacks generate strong oracle-invariant examples [1]; 3) We generate adversarial training data with  $\xi = 16/255$  and test the robustness on various levels of perturbation. Tab. 2 provide comparative experiments between our proposed method and other AT techniques, including OAAT [1], ExAT [30], ATES [4] and AWP [41]. It is important to note that we re-implement these AT methods and apply them in the context of FAT. Overall, our methods achieve optimal performance on both oracle-invariant attacks and classical evaluation attacks. Namely, the classical evaluation attacks used in  $\xi = 8/255$  are reliable and significant even when the value of  $\xi$  is increased to 16/255.
+
+Notably, our methods are plugged into FGSM-RS and
+
+<table><tr><td colspan="2">Methods</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Time (min)↓</td></tr><tr><td colspan="2">PGD-AT [29]</td><td>65.30</td><td>46.31</td><td>40.73</td><td>35.08</td><td>33.92</td><td>30.84</td><td>33.08</td><td>26.29</td><td>370</td></tr><tr><td colspan="2">FGSM-RS [40]</td><td>50.12</td><td>38.28</td><td>26.13</td><td>21.55</td><td>20.43</td><td>18.96</td><td>19.40</td><td>14.84</td><td>67</td></tr><tr><td colspan="2">GradAlign [2]</td><td>58.17</td><td>39.87</td><td>33.12</td><td>26.81</td><td>24.99</td><td>22.63</td><td>23.98</td><td>17.02</td><td>135</td></tr><tr><td colspan="2">ZeroGrad [10]</td><td>74.16</td><td>43.96</td><td>32.67</td><td>21.98</td><td>18.37</td><td>20.76</td><td>16.44</td><td>12.07</td><td>67</td></tr><tr><td rowspan="3">Ours</td><td>W-RS</td><td>70.66</td><td>45.51</td><td>36.50</td><td>27.51</td><td>24.75</td><td>23.97</td><td>23.38</td><td>17.14</td><td>71</td></tr><tr><td>E-RS</td><td>62.38</td><td>42.07</td><td>36.78</td><td>30.80</td><td>28.90</td><td>23.64</td><td>27.71</td><td>17.55</td><td>77</td></tr><tr><td>B-RS</td><td>65.42</td><td>45.94</td><td>37.54</td><td>30.01</td><td>27.85</td><td>26.28</td><td>26.52</td><td>19.43</td><td>75</td></tr><tr><td colspan="2">NuAT [36]</td><td>74.62</td><td>44.92</td><td>35.22</td><td>25.93</td><td>23.67</td><td>24.07</td><td>22.37</td><td>18.43</td><td>101</td></tr><tr><td colspan="2">ATAS* [16]</td><td>64.11</td><td>-</td><td>31.39</td><td>-</td><td>28.15</td><td>-</td><td>-</td><td>21.09</td><td>-</td></tr><tr><td colspan="2">FGSM-MEP [42]</td><td>53.32</td><td>36.24</td><td>31.85</td><td>27.28</td><td>26.56</td><td>22.10</td><td>26.08</td><td>18.98</td><td>92</td></tr><tr><td rowspan="2">Ours</td><td>E-MEP</td><td>69.84</td><td>47.18</td><td>40.90</td><td>34.17</td><td>32.72</td><td>22.69</td><td>31.12</td><td>17.74</td><td>104</td></tr><tr><td>B-MEP</td><td>63.84</td><td>45.48</td><td>40.13</td><td>34.21</td><td>32.95</td><td>28.19</td><td>32.04</td><td>23.68</td><td>101</td></tr></table>
+
+Table 3. Quantitative results of various methods ( $\xi = 16/255$ ) on the CIFAR-10 with ResNet18 as the backbone. 'ATAS*' is the result of ATAS in [16], which is superior to our reproduction. We train each method three times. The results represent the evaluation average between the best models of three training processes. Weight centralization and regularization in MEP do not work together.  
+
+<table><tr><td>Budgets (ξ)</td><td>Methods</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Time (min)↓</td></tr><tr><td rowspan="4">12/255</td><td>GradAlign [2]</td><td>66.52</td><td>43.06</td><td>31.66</td><td>27.95</td><td>26.04</td><td>27.05</td><td>25.99</td><td>21.65</td><td>135</td></tr><tr><td>NuAT [36]</td><td>72.79</td><td>51.80</td><td>41.75</td><td>38.60</td><td>37.54</td><td>35.99</td><td>36.71</td><td>32.01</td><td>101</td></tr><tr><td>FGSM-MEP [42]</td><td>74.71</td><td>52.05</td><td>38.45</td><td>36.35</td><td>35.52</td><td>33.05</td><td>33.4</td><td>27.23</td><td>92</td></tr><tr><td>B-MEP (Ours)</td><td>72.63</td><td>54.40</td><td>45.23</td><td>42.85</td><td>42.14</td><td>36.81</td><td>41.62</td><td>33.26</td><td>101</td></tr><tr><td rowspan="4">10/255</td><td>GradAlign [2]</td><td>83.10</td><td>55.23</td><td>36.58</td><td>30.47</td><td>28.64</td><td>31.01</td><td>26.51</td><td>23.84</td><td>135</td></tr><tr><td>NuAT [36]</td><td>75.82</td><td>55.94</td><td>45.54</td><td>43.92</td><td>43.42</td><td>41.39</td><td>42.91</td><td>38.85</td><td>101</td></tr><tr><td>FGSM-MEP [42]</td><td>83.43</td><td>59.51</td><td>42.76</td><td>39.28</td><td>37.33</td><td>37.26</td><td>35.91</td><td>32.52</td><td>92</td></tr><tr><td>B-MEP (Ours)</td><td>75.96</td><td>57.26</td><td>47.25</td><td>45.98</td><td>45.66</td><td>41.00</td><td>45.26</td><td>39.20</td><td>101</td></tr></table>
+
+Table 4. Quantitative results of FAT methods on various  $\xi$  with ResNet18 as the backbone and CIFAR-10 as the dataset. Models are trained and evaluated under the same  $\xi$  . The number in bold indicates the best.  
+
+<table><tr><td>ξ = 16/255</td><td colspan="2">CIFAR10</td><td colspan="2">CIFAR100</td><td>Time</td></tr><tr><td>Methods</td><td>Clean</td><td>AA</td><td>Clean</td><td>AA</td><td>hours</td></tr><tr><td>GradAlign [2]</td><td>55.58</td><td>13.45</td><td>35.93</td><td>7.13</td><td>8.79</td></tr><tr><td>NuAT [36]</td><td>74.25</td><td>13.19</td><td>20.48</td><td>7.30</td><td>7.66</td></tr><tr><td>FGSM-MEP [42]</td><td>65.56</td><td>15.43</td><td>20.69</td><td>6.73</td><td>6.15</td></tr><tr><td>B-MEP (Ours)</td><td>69.94</td><td>24.27</td><td>48.64</td><td>12.05</td><td>6.72</td></tr></table>
+
+Table 5. The adversarial accuracy of various FAT methods with WideResNet34-10 as the backbone.
+
+FGSM-MEP, which are fitted to the distribution of adversarial examples. Instead, methods such as OAAT are fitted to the distribution of benign samples and perform better on clean accuracy. Therefore, we use the settings as in OAAT and then apply OAAT and our B-OAAT to the FAT task. B-OAAT realizes  $74.12\%$ $(+2.82\%)$  than OAAT) clean accuracy and  $25.39\%$  (OAAT) on SQ+RayS.
+
+# 4.3. Results on CIFAR-10
+
+We conduct our initial experiments on CIFAR-10 using the ResNet18 backbone. The default  $\xi$  is set to 16/255. Tab. 3 presents the main comparisons. The observations
+
+are as follows: (1) Compared with previous FAT methods, the proposed approaches achieve optimal adversarial robustness against different attacks, e.g. B-RS outperforms all RS-based methods and B-MEP is superior to all other methods. Meanwhile, our methods exhibit similar performance to prior work in terms of clean classification accuracy. (2) B-MEP realizes adversarial robustness approaching PGD-AT, e.g. B-MEP performs  $32.95\%$  robustness against the PGD-50 attack, only  $0.97\%$  lower than PGD-AT; (3) As for time consumption, B-RS takes less time (75 minutes) than GradAlign (135 minutes) and NuAT (101 minutes). This is because ConvergeSmooth only requires the additional regularization when  $|u_{B} - u_{t - 1}| > \gamma_{t}$  instead of adding constraints on all iterations. In addition, B-MEP (102 minutes) takes a bit more calculation cost than FGSM-MEP (92 minutes) but much less than PGD-AT (370 minutes). PGD-AT takes significantly longer than FAT works; (4) B-RS and B-MEP successfully prevent catastrophic overfitting, e.g. mfinal (in the supplement) only shows a slight reduction than mbest in terms of adversarial robustness.
+
+Fig. 2 visualizes the training process of various FAT approaches. Compared with other methods, the proposed
+
+<table><tr><td colspan="2">Methods</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Time (min)↓</td></tr><tr><td colspan="2">PGD-AT</td><td>40.57</td><td>24.72</td><td>21.46</td><td>17.94</td><td>17.38</td><td>14.98</td><td>17.02</td><td>12.63</td><td>370</td></tr><tr><td colspan="2">FGSM-RS [40]</td><td>30.12</td><td>15.24</td><td>12.69</td><td>10.25</td><td>9.79</td><td>8.45</td><td>9.57</td><td>6.90</td><td>67</td></tr><tr><td colspan="2">GradAlign [2]</td><td>31.91</td><td>15.71</td><td>12.56</td><td>10.28</td><td>9.71</td><td>8.22</td><td>9.47</td><td>6.61</td><td>135</td></tr><tr><td colspan="2">ZeroGrad [10]</td><td>47.31</td><td>23.58</td><td>17.60</td><td>12.85</td><td>11.87</td><td>11.62</td><td>11.01</td><td>7.94</td><td>67</td></tr><tr><td rowspan="3">Ours</td><td>W-RS</td><td>41.97</td><td>26.30</td><td>19.09</td><td>14.77</td><td>13.72</td><td>12.82</td><td>13.36</td><td>9.67</td><td>72</td></tr><tr><td>E-RS</td><td>41.09</td><td>22.33</td><td>18.78</td><td>15.19</td><td>14.43</td><td>12.00</td><td>13.91</td><td>9.50</td><td>78</td></tr><tr><td>B-RS</td><td>41.47</td><td>25.98</td><td>19.44</td><td>15.36</td><td>14.34</td><td>12.91</td><td>13.71</td><td>9.90</td><td>76</td></tr><tr><td colspan="2">NuAT [36]</td><td>31.42</td><td>20.39</td><td>16.15</td><td>13.87</td><td>13.29</td><td>11.12</td><td>12.25</td><td>8.32</td><td>101</td></tr><tr><td colspan="2">ATAS [16]</td><td>55.63</td><td>30.35</td><td>15.31</td><td>10.28</td><td>8.49</td><td>10.97</td><td>6.30</td><td>5.05</td><td>70</td></tr><tr><td colspan="2">FGSM-MEP [42]</td><td>21.39</td><td>13.00</td><td>11.37</td><td>9.93</td><td>9.76</td><td>7.28</td><td>9.58</td><td>6.36</td><td>92</td></tr><tr><td rowspan="2">Ours</td><td>E-MEP</td><td>44.00</td><td>24.46</td><td>20.59</td><td>17.11</td><td>16.59</td><td>13.37</td><td>16.05</td><td>10.97</td><td>106</td></tr><tr><td>B-MEP</td><td>41.86</td><td>24.86</td><td>20.84</td><td>17.30</td><td>16.59</td><td>13.96</td><td>16.25</td><td>11.38</td><td>102</td></tr></table>
+
+Table 6. Quantitative results of various methods  $\left( {\xi  = {16}/{255}}\right)$  with ResNet18 as the backbone on CIFAR-100.  
+
+<table><tr><td>Methods</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Time (hour)↓</td></tr><tr><td>PGD-AT</td><td>32.52</td><td>16.47</td><td>13.40</td><td>10.63</td><td>10.24</td><td>7.95</td><td>10.00</td><td>6.41</td><td>67.2</td></tr><tr><td>FGSM-RS [40]</td><td>27.48</td><td>12.46</td><td>9.59</td><td>6.97</td><td>6.47</td><td>4.99</td><td>6.19</td><td>3.63</td><td>10.5</td></tr><tr><td>GradAlign [2]</td><td>28.65</td><td>13.80</td><td>10.40</td><td>7.78</td><td>7.08</td><td>5.75</td><td>6.59</td><td>3.90</td><td>20.9</td></tr><tr><td>ZeroGrad [10]</td><td>34.66</td><td>12.26</td><td>8.22</td><td>5.29</td><td>4.82</td><td>3.71</td><td>4.23</td><td>2.21</td><td>10.5</td></tr><tr><td>NuAT [36]</td><td>35.24</td><td>15.52</td><td>12.07</td><td>8.81</td><td>8.13</td><td>6.29</td><td>7.68</td><td>4.27</td><td>24.6</td></tr><tr><td>FGSM-BP [42]</td><td>20.02</td><td>10.18</td><td>8.38</td><td>6.98</td><td>6.77</td><td>4.56</td><td>6.57</td><td>3.62</td><td>14.3</td></tr><tr><td>B-BP (Ours)</td><td>33.51</td><td>16.32</td><td>12.92</td><td>9.72</td><td>9.23</td><td>7.26</td><td>9.32</td><td>5.95</td><td>15.4</td></tr></table>
+
+Table 7. Quantitative results of various methods  $\left( {\xi  = {16}/{255}}\right)$  with PreActResNet18 as the backbone on Tiny ImageNet.
+
+B-RS achieves optimal training stability and adversarial robustness, e.g. the classification accuracy of our method fluctuates slightly during three adversarial training sessions.
+
+Various budgets. Similar experiments are performed for budgets 10/255 and 12/255. Details are given in Tab. 4. It is evident that the proposed method achieves optimal adversarial robustness across various perturbation budgets.
+
+Various networks. We then adopt WideResNet34 with a width factor of 10 [47] as the backbone. The results are given in Tab. 5. Our proposed B-MEP also prevents the wider architectures from catastrophic overfitting.
+
+# 4.4. Results on CIFAR-100
+
+The results on CIFAR-100 with the backbone ResNet-18 are presented in Tab. 6. (1) FAT methods perform similar training consumption on CIFAR-10 and CIFAR-100 as two datasets contain the same number and size of images; (2) All proposed methods can realize stable adversarial training, as evidenced by the comparable results of mbest and mfinal (in the supplement); (3) Among the RS-based FAT works, B-RS and E-RS achieve the best and second-best adversarial robustness against different attacks; (4) B-MEP outperforms all other FAT methods; (5) Although B-MEP performs slightly worse than PGD-AT, its training process is approximately 3.5 times faster than this competitor.
+
+# 4.5. Results on Tiny ImageNet
+
+We conduct experiments on the Tiny ImageNet using the PreActResNet18 backbone to demonstrate the scalability of the proposed method to large datasets. The results are given in Tab. 7. B-BP achieves higher robustness among the FAT methods and comparable robustness to PGD-AT. As for training efficiency, B-BP (15.4 hours) requires slightly more computational cost than FGSM-BP (14.3 hours), but significantly less time than PGD-AT (67.2 hours).
+
+# 4.6. Ablation studies
+
+In the following, B-RS is selected as the backbone. Figs. 3 and 4 conduct ablation studies on  $\gamma_{max}$  and  $\gamma_{max} / \gamma_{min}$  respectively. In Fig. 3, the classification accuracy of both benign and adversarial examples increases with  $\gamma_{max}$ . However, the model trained with a large  $\gamma_{max}$  ( $>0.09$ ) suffers from catastrophic overfitting. Fig. 4 shows that the model with  $\gamma_{max} / \gamma_{min} = 1.5$  achieves optimal robustness.  $\gamma_{max} / \gamma_{min} = 1$  denotes the static convergence stride. All models can be trained stably when  $\gamma_{max} = 0.06$ .
+
+Tabs. 8 and 9 provide the results of comparative experiments on  $w_{1}$  and  $w_{2}$  respectively. We empirically set  $w_{2}$  to 1 since too small  $w_{2}$  cannot prevent overfitting and too large  $w_{2}$  will affect the performance. In Tab. 8,  $w_{1} = 0$  achieves optimal adversarial robustness on CIFAR100 and
+
+<table><tr><td>Dataset</td><td>w1</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Stability</td></tr><tr><td rowspan="5">CIFAR100</td><td>0.0</td><td>48.13</td><td>32.13</td><td>24.25</td><td>22.67</td><td>22.21</td><td>19.04</td><td>21.29</td><td>15.28</td><td>★★★</td></tr><tr><td>0.3</td><td>46.08</td><td>30.58</td><td>23.06</td><td>21.62</td><td>21.22</td><td>17.79</td><td>20.44</td><td>15.22</td><td>★★★</td></tr><tr><td>0.5</td><td>44.04</td><td>29.96</td><td>22.64</td><td>21.34</td><td>20.99</td><td>17.55</td><td>19.16</td><td>13.84</td><td>★★★</td></tr><tr><td>0.7</td><td>43.07</td><td>28.61</td><td>22.01</td><td>20.84</td><td>20.32</td><td>16.87</td><td>18.98</td><td>13.47</td><td>★★★</td></tr><tr><td>0.9</td><td>40.45</td><td>21.05</td><td>21.62</td><td>20.54</td><td>20.15</td><td>16.33</td><td>18.91</td><td>13.53</td><td>★★★</td></tr></table>
+
+Table 8. Quantitative results of the proposed method on various  $w_{1}$  with ResNet18 as the backbone and the perturbation budget 12/255. 'Stability' represents the number of times the model is stable in three training repetitions.  $\gamma_{max}$  and  $w_{2}$  are set to 0.03 and 1, respectively.  
+
+<table><tr><td>Dataset</td><td>w2</td><td>Clean↑</td><td>FGSM↑</td><td>PGD-10↑</td><td>PGD-20↑</td><td>PGD-50↑</td><td>C&amp;W↑</td><td>APGD-CE↑</td><td>AA↑</td><td>Stability</td></tr><tr><td rowspan="4">CIFAR100</td><td>0.5</td><td>36.79</td><td>23.97</td><td>18.60</td><td>17.51</td><td>17.23</td><td>14.65</td><td>16.28</td><td>12.23</td><td>-</td></tr><tr><td>0.7</td><td>37.31</td><td>24.36</td><td>18.56</td><td>17.37</td><td>17.26</td><td>14.46</td><td>15.96</td><td>11.54</td><td>-</td></tr><tr><td>1.0</td><td>48.13</td><td>32.13</td><td>24.25</td><td>22.67</td><td>22.21</td><td>19.04</td><td>21.29</td><td>15.28</td><td>★★★</td></tr><tr><td>1.3</td><td>44.84</td><td>30.42</td><td>23.31</td><td>21.72</td><td>21.31</td><td>18.36</td><td>19.64</td><td>14.72</td><td>★★★</td></tr></table>
+
+Table 9. Quantitative results on various  ${w}_{2}\left( {{\gamma }_{max} = {0.03}}\right.$  and  ${w}_{1} = 0)$  with ResNet18 as the backbone and  $\xi  = {12}/{255}$  .
+
+![](images/b66648bc0d3b8cb0ffe73994f5c9c86e4af5bbb2d70966398399a8e131e75277.jpg)  
+ResNet18 trained with ConvergeSmooth-RS on CIFAR-100  
+Figure 3. Ablation study of  $\gamma_{max}(\gamma_{max} / \gamma_{min} = 1.5)$ . We provide the classification accuracy of models (ResNet-18) to benign and adversarial examples (PGD-10,  $\xi = 16/255$ ) on CIFAR-100.
+
+all settings accomplish stable FAT. According to Eqs. (7) and (8), if  $w_{1} \neq 0$ , for the data  $x$  which satisfies  $\mathcal{C}(x) = 1$ ,  $\mathcal{L}(x_t', \theta_t)$  is assigned with higher weights  $(1 + w_{1})$  or lower weights  $(1 - w_{1})$ , causing the model to excessively prioritize or neglect this portion of the data. This may weaken the performance of models. However, for Tiny ImageNet, models suffer catastrophic overfitting at  $w_{1} = 0$ . Then, we gradually increase  $w_{1}$  from 0.3 with a stride of 0.2 until the FAT process becomes stable. In addition, our weight centralization method only introduces a balance coefficient  $w_{3}$ .  $w_{3}$  is set to 0.1 since catastrophic overfitting happens when  $w_{3} = 0$  and classification performance is degraded when  $w_{3} = 0.2$ .
+
+# 5. Conclusion
+
+In this paper, we tackle the issue of catastrophic overfitting by focusing on the convergence stability of loss func
+
+![](images/b22b73832bd1331752bbefd95f835fb0c27d3622c1c6426d3618be637481b38b.jpg)  
+ResNet18 trained with ConvergeSmooth-RS on CIFAR-100  
+Figure 4. Ablation study of  $\gamma_{max} / \gamma_{min}$  ( $\gamma_{max} = 0.06$ ). We provide the classification accuracy of models (ResNet-18) to benign and adversarial examples (PGD-10,  $\xi = 16/255$ ) on CIFAR-100.
+
+tions. Through experimental analysis, we find that this issue is accompanied by abnormal convergence behavior of losses. Motivated by this phenomenon, we propose complementary constraints, namely E-ConvergeSmooth and B-ConvergeSmooth, based on the adversarial training constraint. To alleviate the burden of parameter tuning, weight centralization is designed utilizing priors from previous epochs. Extensive experiments on different network architectures and datasets show that the proposed methods effectively solve catastrophic overfitting and exhibit superior robustness against various adversarial attacks.
+
+Acknowledgments. This work was supported by the National Key R&D Program of China #2018AAA0102000 and the National Natural Science Foundation of China #62276046 and #U19B2039.
+
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+        "text": "Fast Full-frame Video Stabilization with Iterative Optimization",
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+        "text": "Weiyue Zhao $^{1}$  Xin Li $^{2}$  Zhan Peng $^{1}$  Xianrui Luo $^{1}$  Xinyi Ye $^{1}$  Hao Lu $^{1}$  Zhiguo Cao $^{1*}$ $^{1}$ Key Laboratory of Image Processing and Intelligent Control, Ministry of Education; School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China  \n $^{2}$ Department of Computer Science, University of Albany, Albany NY 12222 {zhaoweiyue, peng-zhan, xianruiluo, xinyiye, hlu, zgcao}@hust.edu.cn xli48@albany.edu",
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+        "text": "Abstract",
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+        "text": "Video stabilization refers to the problem of transforming a shaky video into a visually pleasing one. The question of how to strike a good trade-off between visual quality and computational speed has remained one of the open challenges in video stabilization. Inspired by the analogy between wobbly frames and jigsaw puzzles, we propose an iterative optimization-based learning approach using synthetic datasets for video stabilization, which consists of two interacting submodules: motion trajectory smoothing and full-frame outpainting. First, we develop a two-level (coarse-to-fine) stabilizing algorithm based on the probabilistic flow field. The confidence map associated with the estimated optical flow is exploited to guide the search for shared regions through backpropagation. Second, we take a divide-and-conquer approach and propose a novel multiframe fusion strategy to render full-frame stabilized views. An important new insight brought about by our iterative optimization approach is that the target video can be interpreted as the fixed point of nonlinear mapping for video stabilization. We formulate video stabilization as a problem of minimizing the amount of jerkiness in motion trajectories, which guarantees convergence with the help of fixed-point theory. Extensive experimental results are reported to demonstrate the superiority of the proposed approach in terms of computational speed and visual quality. The code will be available on GitHub.",
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+    {
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+        "text": "1. Introduction",
+        "text_level": 1,
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+        "type": "text",
+        "text": "With the growing popularity of short videos on social media platforms (e.g., TikTok, Instagram), video has played an increasingly important role in our daily life. However, casually captured videos are often shaky and wobbly due to",
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+        "type": "text",
+        "text": "amateur shooting. Although it is possible to alleviate those problems by resorting to professional equipment (e.g., dollies and steadicams), the cost of hardware-based solutions is often expensive, making it impractical for real-world applications. By contrast, software-based or computational solutions such as video stabilization algorithms [12] have become an attractive alternative to improve the visual quality of shaky video by eliminating undesirable jitter.",
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+        "text": "Existing video stabilization techniques can be classified into two categories: optimization-based and learning-based. Traditional optimization-based algorithms [11,18,21,27,39] have been widely studied due to their speed and robustness. The challenges of them are the occlusion caused by changes in depth of field and the interference caused by foreground objects on camera pose regression. Furthermore, their results often contain large missing regions at frame borders, particularly when videos with a large camera motion. In recent years, learning-based video stabilization algorithms [6, 23, 45, 50] have shown their superiority by achieving higher visual quality compared to traditional methods. However, their stabilization model is too complex for rapid computation, and its generalization property is unknown due to the scarcity of training datasets.",
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+        "text": "To overcome those limitations, we present an iterative optimization-based learning approach that is efficient and robust, capable of achieving high-quality stabilization results with full-frame rendering, as shown in Fig. 1. The probabilistic stabilized network addresses the issues of occlusion and interfering objects, and achieves fast pose estimation. Then the full-frame outpainting module retains the original field of view (FoV) without aggressive cropping. An important new insight brought by our approach is that the objective of video stabilization is to suppress the implicitly embedded noise in the video frames rather than the explicit noise in the pixel intensity values. This inspired us to adopt an expectation-maximization (EM)-like approach for video stabilization. Importantly, considering the strong redundancy of video in the temporal domain, we ingeniously",
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+        "type": "header",
+        "text": "CVF",
+        "bbox": [
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+    },
+    {
+        "type": "header",
+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
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+    {
+        "type": "page_footnote",
+        "text": "*Corresponding author https://github.com/zwyking/Fast-Stab",
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+        "type": "page_number",
+        "text": "23534",
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+    },
+    {
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+        "img_path": "images/46ce70956dbb13bf03558f871ba573814a0f25c71411f32d1433c1e16de2fa95.jpg",
+        "image_caption": [
+            "Figure 1. Overview of our video stabilization framework. It consists of motion trajectory smoothing (in Sec. 3 and Sec. 4) and full-frame outpainting modules (in Sec. 5). The former adopts the two-level (coarse-to-fine) stabilizing algorithm to obtain a stabilized video. The latter further render a full-frame video with strategies of flow outpainting and multiframe fusion."
+        ],
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+        "img_path": "images/6a9817f6acb7f629182e51ff59b460f3fdc999660c43d844aeee05ce28645268.jpg",
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+        "type": "text",
+        "text": "consider stable video (the target of video stabilization) as the fixed point of nonlinear mapping. Such a fixed-point perspective allows us to formulate an optimization problem of the optical flow field in commonly shared regions. Unlike most methods that resort to the ad hoc video dataset [56] or the deblurred dataset [35] as stabilized videos, we propose to construct a synthetic training dataset to facilitate joint optimization of model parameters in different network modules.",
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+        "type": "text",
+        "text": "To solve the formulated iterative optimization problem, we take a divide-and-conquer approach by designing two modules: probabilistic stabilization network (for motion trajectory smoothing) and video outpainting network (for full-frame video rendering). For the former, we propose to build on the previous work of PDCNet [37, 38] and extend it using a coarse-to-fine strategy to improve robustness. For a more robust estimate of the uncertainty of the optical flow, we infer masks from the optical flow by bidirectional propagation with a low computational cost (around  $1/5$  of the time Yu et al. [50]). Accordingly, we have developed a two-level (coarse-to-fine) flow smoothing strategy that first aligns adjacent frames by global affine transformation and then refines the result by warping the fields of intermediate frames. For the latter, we propose a two-stage approach (flow and image outpainting) to render full-frame video. Our experimental results have shown highly competent performance against others on three public benchmark datasets. The main contribution of this work is threefold:",
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+        "sub_type": "text",
+        "list_items": [
+            "- We propose a formulation of video stabilization as a fixed-point problem of the optical flow field and propose a novel procedure to generate a model-based synthetic dataset.",
+            "- We construct a probabilistic stabilization network based on PDCNet and propose an effective coarse-to-fine strategy for robust and efficient smoothing of optical flow fields.",
+            "- We propose a novel video outpainting network to render stabilized full-frame video by exploiting the spatial"
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "coherence in the flow field.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Related Work",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1. Video Stabilization",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Most video stabilization methods can typically be summarized as a three-step procedure: motion estimation, smoothing the trajectory, and generating stable frames. Traditional methods focus primarily on 2D features or image alignment [29] when it comes to motion estimation. These methods are different in modeling approaches to motion, including the trajectory matrix [19], epipolar geometry [10, 21, 47, 53], and the optical flow field [22, 54]. Regarding the smoothing trajectory, particle filter tracking [46], space-time path smoothing [19, 40], and L1 optimization [11] have been proposed. Existing methods for generating stable frames rely mainly on 2D transformations [26], grid warping [19, 21], and dense flow field warping [22].",
+        "bbox": [
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+        "type": "text",
+        "text": "Compared to the 2D method, some approaches turn to 3D reconstruction [17]. However, specialized hardware such as depth camera [20] and light field camera [34] are necessary for these methods based on 3D reconstruction. Some methods [30, 43, 44, 49] tackle video stabilization from the perspective of deep learning. In [49], the optimization of video stabilization was formulated in the CNN weight space. Recent work [50] represents motion by flow field and attempts to learn a stable optical flow to warp frames. Another method [6] aims to learn stable frames by interpolation. These deep-learning methods generate stable videos with less distortion.",
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+    {
+        "type": "text",
+        "text": "2.2. Large FOV Video",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unlike the early work (e.g., traffic video stabilization [16]), large field of view (Field Of View) video stabilization has been attracting more researchers' attention. For most video stabilization methods, cropping is inevitable, which is why the FOV is reduced. Several approaches have been proposed to maintain a high FOV ratio. OVS [45]",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "23535",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "proposed to improve FOV by extrapolation. DIFRINT [6] choose iterative interpolation to generate high-FOV frames. FuSta [23] used neural rendering to synthesize high-FOV frames from feature space. To a great extent, the performance of interpolation-based video stabilization [6,45] depends on the selected frames. If selected frames have little correspondence with each other, performance will deteriorate dramatically. Neural rendering [23] synthesizes the image by weighted summing, causing blur. Most recently, a deep neural network [33] (DNN) has jointly exploited sensor data and optical flow to stabilize videos.",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. Stabilization via Iterative Optimization",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Motivation. Despite rapid advances in video stabilization [30], existing methods still suffer from several notable limitations [12]. First, a systematic treatment of various uncertainty factors (e.g., low-texture regions in the background and moving objects in the foreground) in the problem formulation is still lacking. These uncertainty factors often cause occlusion-related problems and interfere with the motion estimation process. Second, the high computational cost has remained a technical barrier to real-time video processing. The motivation behind our approach is two-fold. On the one hand, we advocate for finding commonly shared regions among successive frames to address various uncertainty factors in handheld video. On the other hand, in contrast to these prestabilization algorithms [21, 23, 49, 50] based on traditional approaches, we proposed a novel high-efficiency prestabilization algorithm based on probabilistic optical flow. Flow-based methods are generally more accurate in motion estimation and deserve a high time cost. Accuracy and efficiency, we have both.",
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+        "type": "text",
+        "text": "Approach. We propose to formulate video stabilization as a problem of minimizing the amount of jerkiness in motion trajectories. It is enlightening to think of video stabilization as a special kind of \"video denoising\" where noise contamination is not associated with pixel intensity values, but embedded into the motion trajectories of foreground and background objects. Conceptually, for video restoration in which unknown motion is the hidden variable, we can treat video stabilization as a chicken-and-egg problem [13] - i.e., the objective of smoothing motion trajectories is intertwined with that of video frame interpolation. Note that an improved estimation of motion information can facilitate the task of frame interpolation and vice versa. Such an observation naturally inspires us to tackle the problem of video stabilization by iterative optimization.",
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+    {
+        "type": "text",
+        "text": "Through divide-and-conquer, we propose to formulate video stabilization as the following optimization problem. Given a sequence of  $n$  frames along with the set of optical flows  $\\mathcal{V}$ , we first define the confidence map, which originally indicates the reliability and accuracy of the optical flow prediction at each pixel. Here, we have thresholded the",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "confidence map as a binary image, which represents accurate matches (as shown in the 4-th column of Fig. 2). Then we can denote  $n$  frames and the corresponding set of confidence maps  $\\mathcal{M}$  by:",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {Y} = \\left\\{Y _ {1}, Y _ {2}, \\dots , Y _ {q} \\right\\}, \\mathcal {M} = \\left\\{M _ {1}, M _ {2}, \\dots , M _ {q} \\right\\}. \\quad (1)\n$$\n",
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+    {
+        "type": "text",
+        "text": "4. Probabilistic Stabilization Network",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The problem of video stabilization can be formulated as finding a nonlinear mapping  $f: \\mathcal{V} \\to \\hat{\\mathcal{V}}$  where  $\\hat{\\mathcal{V}}$  denotes the optical flow set of the stabilized video. We hypothesize that a desirable objective to pursue  $f$  is the well-known fixed-point property, i.e.,  $\\hat{\\mathcal{V}} = f(\\hat{\\mathcal{V}})$ . To achieve this objective, we aim to minimize an objective function  $\\mathbb{F}$  characterized by the magnitude of optical flows between commonly shared regions, as represented by  $\\mathcal{M}$ . Note that  $\\mathcal{M}$  is the hidden variable in our problem formulation (that is, we need to estimate  $\\mathcal{M}$  from  $\\hat{\\mathcal{V}}$ ). A popular strategy to address this chicken-and-egg problem is to alternatively solve the two subproblems of unknown  $\\mathcal{M}$  and  $\\hat{\\mathcal{V}}$ . More specifically, the optimization of  $\\mathbb{F}$  is decomposed into the following two subproblems:",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\left(\\hat {M} _ {1}, \\dots , \\hat {M} _ {q}\\right) = \\underset {\\mathcal {M}} {\\arg \\min } \\mathbb {F} \\left(\\hat {Y} _ {1} \\odot M _ {1}, \\dots , \\hat {Y} _ {q} \\odot M _ {q}\\right), \\tag {2}\n$$\n",
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+    {
+        "type": "equation",
+        "text": "\n$$\n(\\hat {Y} _ {1}, \\dots , \\hat {Y} _ {q}) = \\underset {\\mathcal {Y}} {\\arg \\min } \\mathbb {F} (Y _ {1} \\odot \\hat {M} _ {1}, \\dots , Y _ {q} \\odot \\hat {M} _ {q}),\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $Y / \\hat{Y}$  denotes the magnitude of the optical flow values before and after stabilization and  $\\odot$  is the Hadamard product. Instead of analyzing the estimated motion in the frequency domain, we hypothesize that stabilized videos are the fixed points of video stabilization algorithms, which minimize the above objective function. A fixed point [1] is a mathematical object that does not change under a given transformation. Numerically, fixed-point iteration is a method of computing fixed points of a function. It has been widely applied in data science [7], and image restoration [5, 25]. Here, we denote  $\\mathbb{F}$  in Eq. (2) as the function to be optimized, and the fixed point of  $\\mathbb{F}$  is defined as the stabilized video. Next, we solve these two subproblems by constructing the module of stabilization.",
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+    {
+        "type": "text",
+        "text": "4.1. Probabilistic Flow Field",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "First, we start with an interesting observation. When playing an unstable video at a slower speed (e.g., from 50fps to 30fps), the video tends to appear less wobbly. It follows from the observation that the fundamental cause of video instability is the fast frequency and the large amplitude of the still objects' motion speed. Therefore, the core task of motion smoothing is to identify the region that needs to be stabilized. As Yu et al. [50] pointed out, mismatches, moving objects, and inconsistent motion areas lead to variation of the estimated motion from the true motion trajectories and",
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+        "text": "23536",
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+    {
+        "type": "image",
+        "img_path": "images/a6c0690151f1ab2bb86ee086185771599464ba2e7cde3e86ede584ed8ec55c82.jpg",
+        "image_caption": [
+            "Figure 2. Visualization of confidence map back-propagation results. The flow field and confidence map are predicted by PDCNet between source and target images. The obtained masks in the last column represent the shared regions among these frames."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "should be masked. An important new insight brought about by this work is that these regions of inconsistency in the optical flow fields tend to show greater variability as the frame interval increases. Therefore, if these inconsistent regions are excluded, the video stabilization task can be simplified to the first sub-problem in Eq. (2) - i.e., minimizing  $\\mathbb{F}$  over  $\\mathcal{M}$  for a fixed  $\\mathcal{V}$ .",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To detect unreliable matches and inconsistent regions, we have adopted the probabilistic flow network - PDCNet [37, 38], that jointly tackles the problem of dense correspondence and uncertainty estimation, as our building block. Suppose that PDCNet estimates the optical flow  $Y_{k}$  from frame  $k + d$  to frame  $k$  with the resulting confidence map  $C_k$ . Although  $C_k$  denotes the inaccuracy of the optical flow estimate in  $Y_{k}$ , it is often sensitive to the frame interval  $d$ . For example, it is difficult to identify the inconsistent region when  $d$  is small, while the common area is less controllable when  $d$  is large. Therefore, simply using an optical flow field to estimate inconsistent regions is not sufficient.",
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+    {
+        "type": "text",
+        "text": "We have designed a more robust solution to the joint estimation of dense flow and confidence map based on a coarse-to-fine strategy. The basic idea is to first obtain the probabilistic flow field at a coarse scale (e.g., with the down-sampled video sequence by a factor of  $d$  along the temporal axis) and then fill in the rest (i.e., the frames between the adjacent frames in the down-sampled video) at the fine scale. Such a two-level estimation strategy effectively overcomes the limitations of PDCNet by propagating the estimation results of probabilistic flow fields bidirectionally, as we will elaborate next.",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Coarse-scale strategy As shown by the second-to-last column in Fig. 2, the confidence map estimated by PDCNet can identify the mismatched region, but fails to locate the objects with small motions (e.g., people and the sky). To",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4b7a72e0ff1aa18faa8873256305d6ac94511f316b848b6dd35c65a18a39f689.jpg",
+        "image_caption": [
+            "Figure 3. Architecture of the camera pose regression network. Given the flow field and mask, our network predicts the corresponding affine transformation parameters by closed-loop iterations."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "overcome this difficulty, we introduce the binary mask as a warped and aggregated confidence map (refer to the last column of Fig. 2). Specifically, we first propose to obtain the confidence map  $\\hat{C}_{k + (n - 1)d}$  with down-sampled video (the last row of Fig. 2). In the forward direction, we estimate dense flow and confidence map using PDCNet; then  $\\hat{C}_{k + (n - 1)d}$  is backpropagated to update the binary mask  $\\hat{M}$  by thresholding and setting intersection operators. Through bidirectional propagation, the region covered by  $\\mathcal{M}$  is the shared content from frame  $k$  to frame  $k + (n - 1)d$ . The complete procedure can be found in Supp. S1.",
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+    {
+        "type": "text",
+        "text": "Fine-scale strategy Based on the coarse-scale estimation result for downsampled video (i.e.,  $\\mathcal{M} = \\{\\hat{M}_k, \\hat{M}_{k+d}, \\dots, \\hat{M}_{k+(n-1)d}\\}$ ), we fill in the missing  $d-1$  frames at the fine scale. Specifically, considering the sequential frames from  $k$  to  $k+d$ , we can obtain two sets similar to Eq. (1), which are  $\\mathcal{V} = \\{Y_k, Y_{k+1}, \\dots, Y_{k+d-1}, \\hat{Y}_{k+d}\\}$  and their corresponding confidence map set  $\\mathcal{C}$ . Note that  $\\hat{M}_{k+d} \\in \\mathcal{M}$  has been calculated in coarse stage. Setting  $d=1$ , we can call the algorithm again to obtain the set of output masks  $\\mathcal{M}$  for the rest of  $d-1$  frames.",
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+    },
+    {
+        "type": "text",
+        "text": "4.2. Coarse-scale Stabilizer",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To coarsely stabilize the video, we first propose aligning the adjacent frames with a global affine transformation [52]. The optimization function  $\\mathbb{F}$  in Eq. (2) is represented as",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {T} ^ {*} = \\underset {\\mathbf {T}} {\\arg \\min } \\mathbf {T} (Y \\odot \\hat {M}), \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{T}(\\cdot)$  denotes the image transformation applied to the shared region  $\\hat{M}$  (the result of Sec. 4.1) of the optical flow field  $Y$ . Most conventional methods [23,44,50] adopt image matching to obtain  $\\mathbf{T}(\\cdot)$ -e.g., keypoint detection [2,24,31], feature matching [2,3,51]; and camera pose estimation is implemented by OpenCV-python. However, these methods are often time-consuming and computationally expensive. For example, two adjacent frames of unstable videos usually share a large area and are free from perspective transformation. Thus, an affine transformation, including translation, rotation, and scaling, is sufficient. More importantly, within the optimization-based learning framework, we can",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "23537",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "regress these linear parameters of  $\\mathbf{T}(\\cdot)$  from the optical flow field, which characterize the relative coordinate transformation of the matched features.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel camera pose regression network, as shown in Fig. 3. Given an optical flow field  $Y$  and the corresponding mask field  $\\hat{M}$ , our network  $\\Phi(\\cdot)$  can directly estimate the unknown parameters  $\\mathbf{T}(\\cdot) \\propto \\{\\theta, s, d_x, d_y\\} = \\Phi(Y, \\hat{M})$ . To solve the optimization problem of Eq. (3), we use the estimated parameters to iteratively compute the corresponding residual optical flow fields such that",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {Y} = Y - (S \\cdot R \\cdot V + T), \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $T$ ,  $S$ , and  $R$ , respectively, denote the translation, scaling, and rotation matrix, and  $V \\in \\mathbb{R}^{2 \\times H \\times W}$  represents an image coordination grid. Then  $\\{\\Delta \\theta, \\Delta s, \\Delta x, \\delta y\\} = \\Phi(\\bar{Y}, \\tilde{M})$  is calculated iteratively to produce the updated parameters  $\\{\\theta + \\Delta \\theta, s \\cdot \\Delta s, d_x + \\Delta x, d_y + \\Delta y\\}$ . The finally estimated affine transformation is smoothed by a moving Gaussian filter with a window size of 20 pixels.",
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+            388
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Loss functions Our loss functions include robust loss  $\\ell_1$  and grid loss commonly applied in consistency filtering [51]. We directly calculate the  $\\ell_1$  loss between the predictions and their ground truth  $\\{\\hat{\\theta},\\hat{s},\\hat{d}_x,\\hat{d}_y\\}$",
+        "bbox": [
+            75,
+            395,
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+            455
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} L _ {g t} = \\lambda_ {\\theta} \\| | \\theta - \\hat {\\theta} \\| | _ {1} + \\lambda_ {s} \\| | 1 - \\frac {s}{\\hat {s}} \\| | _ {1} \\tag {5} \\\\ + \\lambda_ {t} \\| | d _ {x} - \\hat {d} _ {x} | + | d _ {y} - \\hat {d} _ {y} | \\| _ {1}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            129,
+            465,
+            468,
+            513
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For better supervision of the estimated pose, we calculate the loss  $L_{grid}$  with the grid  $V \\in \\mathbb{R}^{2 \\times h \\times h}$  in Eq. (4), i.e.,",
+        "bbox": [
+            75,
+            523,
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+            555
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {g r i d} = \\left\\| (\\hat {S} \\cdot \\hat {R} \\cdot V + \\hat {T}) - (S \\cdot R \\cdot V + T) + \\epsilon \\right\\| _ {1} \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            86,
+            565,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\hat{\\cdot}$  denotes the ground truth and  $\\epsilon$  is a small value for stability. The final loss  $L_{stab}$  consists of  $L_{gt}$  and  $L_{grid}$ ,",
+        "bbox": [
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+            594,
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+            626
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {s t a b} = L _ {g t} + \\lambda_ {g r i d} L _ {g r i d}. \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            637,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3. Fine-scale Stabilizer",
+        "text_level": 1,
+        "bbox": [
+            76,
+            662,
+            269,
+            679
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The assumption with an affine transformation of the coarse-scale stabilizer could cause structural discontinuity and local distortion. Therefore, our objective is to refine the coarsely stabilized video by optical flow smoothing. Unlike Eq. 3 which applies an image transformation matrix to optimize the optical flow field, we optimize it at the pixel level by a flow warping field  $\\mathbf{W}$ . Thus, the function  $\\mathbb{F}$  in Eq. (2) is given by",
+        "bbox": [
+            75,
+            686,
+            468,
+            808
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {W} ^ {*} = \\underset {\\mathbf {W}} {\\arg \\min } \\sum_ {i = 0} ^ {N - 1} \\mathbf {W} _ {i} \\left(Y _ {i} \\odot \\hat {M} _ {i}\\right). \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            151,
+            819,
+            468,
+            859
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The flow smoothing network follows the U-Net architecture in [56]. We use  $N$  frames of optical flow fields  $\\mathbf{F}$  and mask",
+        "bbox": [
+            75,
+            869,
+            468,
+            900
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/6fe7a98576fa7769bc42ea3ca51803b12244536840d7b80a8ef6546f35c8b5f6.jpg",
+        "image_caption": [
+            "Figure 4. Illustration of flow fields and warped images under different FOVs. The target image with large FOV presents the smoother flow field and better warped result."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            511,
+            89,
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+            188
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "fields  $\\hat{M}$  as input and obtain  $(N - 1)$  frame warp fields  $\\mathbf{W}$  of intermediate frames. Specifically, for the optical flow field  $Y_{k}$  (from frame  $k + 1$  to frame  $k$ ), we denote the aligned matrices of each frame in Sec. 4.2 as  $H_{k} \\in \\mathbb{R}^{2 \\times 3}$  and  $H_{k + 1} \\in \\mathbb{R}^{2 \\times 3}$ , respectively. Then the input optical flow  $\\mathbf{F}_{k} \\in \\mathbb{R}^{2 \\times HW}$  can be represented by",
+        "bbox": [
+            498,
+            257,
+            890,
+            349
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {F} _ {k} = H _ {k + 1} \\cdot \\left[ V + Y _ {k} \\mid 1 \\right] - H _ {k} \\cdot \\left[ V \\mid 1 \\right], \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            553,
+            359,
+            890,
+            378
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $V \\in \\mathbb{R}^{2 \\times HW}$  and  $[\\cdot|1]$  denote the normalized coordinate representation. Furthermore, to better adapt the flow smoothing network to the mask field  $\\hat{M}$ , we fine-tune it using our synthetic dataset (as we will elaborate in Sec. 6.1). The loss function follows the motion loss [49]",
+        "bbox": [
+            498,
+            387,
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+            464
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {\\text {s m o o t h}} = \\sum_ {k = 0} ^ {N - 1} \\left(\\mathbf {F} _ {k} + \\mathbf {W} _ {k} - \\mathbf {F} _ {k} \\left(\\mathbf {W} _ {k + 1}\\right)\\right) \\odot \\hat {M}, \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            514,
+            474,
+            890,
+            516
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{W}_0 = \\mathbf{W}_N = 0$",
+        "bbox": [
+            500,
+            526,
+            665,
+            542
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5. Video Outpainting Network",
+        "text_level": 1,
+        "bbox": [
+            500,
+            555,
+            756,
+            573
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Most video stabilization methods crop an input video with a small field of view (FOV), excluding missing margin pixels due to frame warping. In contrast, full-frame video stabilization [6, 23, 26] proposes to generate a video that maintains the same FOV as the input video without cropping. They directly generate stabilized video frames with large FOV by fusing the information from neighboring frames. An important limitation of existing fusion strategies is the unequal importance of different frames (i.e., the current and neighboring frames are weighted equally), which would lead to unpleasant distortions in fast-moving situations. To overcome this weakness, we propose a two-stage framework to combine flow and image outpainting strategies [32]. In the first stage, we used flow-outpainting to iteratively align the neighboring frames with the target frame. In the second stage, we fill the target frame with adjacent aligned frame pixels by image outpainting.",
+        "bbox": [
+            496,
+            580,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1. Flow Outpainting Network",
+        "text_level": 1,
+        "bbox": [
+            500,
+            847,
+            743,
+            863
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let  $I^t$  denote the outpainted target image and  $I^s$  the neighboring source image. We aim to fill the missing pixel",
+        "bbox": [
+            500,
+            869,
+            890,
+            900
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "23538",
+        "bbox": [
+            478,
+            944,
+            517,
+            955
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/7b857824a8c24cae006118bc7c1c0e035fb96460916c2e667be345621f1776cd.jpg",
+        "image_caption": [
+            "Figure 5. Architecture of flow outpainting network. Given a small FOV flow field and valid mask, the network predicts the large FOV flow field."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            86,
+            88,
+            460,
+            196
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "region  $M_{\\varnothing}^{t}$  of  $I^{t}$  with the pixels of  $I^{s}$ . As shown in Fig. 4, we take two different FOVs of  $I^{t}$  as input and obtain the corresponding optical flow fields and warped results  $I_{warp}^{s}$ . The small FOV  $I^{t}$  cannot guide  $I^{s}$  well to fill the regions in  $M_{\\varnothing}^{t}$ . Since the predicted optical flow field in  $M_{\\varnothing}^{t}$  is unreliable due to the lack of pixel guidance of  $I^{t}$ , the out-of-view region of  $M_{\\varnothing}^{t}$  has artifacts (marked in the last column of Fig. 4). We observe that the optical flow field of the large FOV  $I^{t}$  is continuous in  $M_{\\varnothing}^{t}$ , which inspired us to extrapolate the flow of  $M_{\\varnothing}^{t}$  using the reliable flow region (the 3rd column of Fig. 4).",
+        "bbox": [
+            75,
+            266,
+            468,
+            433
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel flow outpainting network (Fig. 5), which extrapolates the large FOV flow field using a small FOV flow field and the corresponding valid mask. Specifically, we adopt a U-Net architecture and apply a sequence of gated convolution layers with downsampling / upsampling to obtain the large flow field of FOV  $Y_{large}$ . Note that the input flow field  $Y_{small}$  and the valid mask  $M_{valid}$  have been estimated by PDCNet (see Sec. 4.1).",
+        "bbox": [
+            75,
+            433,
+            468,
+            554
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Loss functions Our loss functions include robust loss  $\\ell_1$  and loss in the frequency domain [4]. We directly calculate the  $\\ell_1$  loss between  $Y_{large}$  and its ground truth  $\\hat{Y}_{large}$ ,",
+        "bbox": [
+            75,
+            560,
+            468,
+            607
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} L _ {Y} = \\lambda_ {i n} \\cdot \\| | Y _ {\\text {l a r g e}} - Y _ {\\text {s m a l l}} | \\odot M _ {\\text {v a l i d}} \\| _ {1} \\tag {11} \\\\ + \\lambda_ {o u t} \\cdot \\| | Y _ {l a r g e} - \\dot {Y} _ {l a r g e} | \\odot (\\sim M _ {v a l i d}) \\| _ {1}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            98,
+            616,
+            468,
+            651
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To encourage low frequency and smoothing  $Y_{large}$ , we add the loss in the frequency domain  $L_{F} = \\| \\hat{\\mathbf{G}}\\cdot \\mathcal{F}Y_{large}\\|_{2}$  where the normalized Gaussian map  $\\hat{\\mathbf{G}}$  with  $\\mu = 0$  and  $\\sigma = 3$  is inverted by its maximum value and  $\\mathcal{F}Y_{large}$  denotes the Fourier spectrum of  $Y_{large}$ . The final loss consists of  $L_{Y}$  and  $L_{F}$",
+        "bbox": [
+            75,
+            660,
+            468,
+            753
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {\\text {o u t p a i n t}} = L _ {Y} + \\lambda_ {F} L _ {F}. \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            753,
+            468,
+            770
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2. Image Margin Outpainting",
+        "text_level": 1,
+        "bbox": [
+            76,
+            786,
+            321,
+            803
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Based on our proposed flow-outpainting network, we design a margin-outpainting method by iteratively aligning frame pairs (see Fig. 6). As discussed in Sec. 5.1, we can obtain a large FOV flow field  $Y_{large}$ . The neighboring reference frame  $I^{s}$  is warped as  $I^{warp} = Y_{large}(I^{s})$  as shown in Fig. 6. In theory, the outpainted frame  $I^{result} =$",
+        "bbox": [
+            75,
+            810,
+            468,
+            900
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/05918f014562ceec451629fc9b4ace86011898cec5ae90fa9887c6c10599b66a.jpg",
+        "image_caption": [
+            "Figure 6. Overview image margin outpainting. Given the target frame  $I^{t}$ , the reference frame  $I^{s}$  is coarsely aligned to  $I^{warp}$  by the predicted large-FOV flow field  $Y_{large}$ . Then, we adopt a margin fusion approach to obtain the result frame  $I^{result}$ , by carefully aggregating  $I^{t}$  and  $I^{warp}$ ."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            511,
+            88,
+            883,
+            244
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "$I^t \\cdot M_{valid} + I^{warp} \\cdot (\\sim M_{valid})$ . However, we notice that there are obvious distortions at the image border. To further align the margins, we take a margin fusion approach (the detailed algorithm can be found in Supp. S1) We crop  $I^{warp}$  and  $I^t$  to  $I_c^s$  and  $I_c^t$ . Then, we can obtain a new warped frame  $I_c^{warp}$  by flow outpainting. In particular, we did not choose to add  $I_c^{warp}$  and  $I^t$  directly. To identify the misaligned region, we propose to outpaint the mask  $M_{I^t}$  by extending the watershed outward from the center. Instead of a preset threshold, we adaptively choose between the target image  $I^t$  and the warped image  $I_c^{warp}$ . Then, the final frame  $I^{result}$  consists of  $I^t$  and  $I_c^{warp}$ :  $I^{result} = I^t \\cdot M_{I^t} + I_c^{warp} \\cdot (\\sim M_{I^t})$ . Compared to the two results  $I^{result}$ , our strategy successfully mitigates misalignment and distortions at the boundary of video frames.",
+        "bbox": [
+            496,
+            343,
+            890,
+            570
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Multi-frame fusion During the final stage of rendering, we use a sequence of neighboring frames to outpaint the target frame, while they may have filled duplicate regions. It is important to find out which frame and which region should be selected. We proposed the selection strategy for multi-frame fusion (the details can be found in Supp. S1). By weighing the metric parameters of each frame, we finally obtain the target frame with large FOV. Note that each frame has an added margin in the stabilization process, so we need to crop them to the original resolution. Although we have outpainted the target frame, some missing pixel holes may still exit at boundaries. Here, we apply the state-of-the-art LaMa image inpainting method [36] to fill these holes using nearest-neighbor interpolation.",
+        "bbox": [
+            496,
+            578,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6. Experiments",
+        "text_level": 1,
+        "bbox": [
+            500,
+            805,
+            632,
+            821
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6.1. Synthetic Datasets for Supervised Learning",
+        "text_level": 1,
+        "bbox": [
+            500,
+            830,
+            867,
+            847
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Due to the limited amount of paired training data, we propose a novel model-based data generation method by carefully designing synthetic datasets for video stabiliza",
+        "bbox": [
+            496,
+            854,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "23539",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/7cfe027c59935e040de8a2046fc98ab869833848eae4235df964d19c18b81bb1.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1. Quantitative results on the NUS dataset [21], the DeepStab dataset [39] and the Selfie Dataset [48]. We evaluate the following metrics: Cropping Ratio(C.), Distortion Value(D.), Stability Score(S). * indicates the results obtained from original paper. We highlight the best method in bold and underline the second-best."
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">NUS dataset [21]</td><td colspan=\"3\">DeepStab dataset [39]</td><td colspan=\"3\">Selfie dataset [48]</td></tr><tr><td>C.↑</td><td>D.↑</td><td>S.↑</td><td>C.↑</td><td>D.↑</td><td>S.↑</td><td>C.↑</td><td>D.↑</td><td>S.↑</td></tr><tr><td>Grundmann et al. [11]</td><td>0.71</td><td>0.76</td><td>0.62</td><td>0.77</td><td>0.87</td><td>0.80</td><td>0.75</td><td>0.81</td><td>0.83</td></tr><tr><td>Liu et al. [21]</td><td>0.81</td><td>0.78</td><td>0.82</td><td>0.80</td><td>0.90</td><td>0.85</td><td>0.74</td><td>0.89</td><td>0.8</td></tr><tr><td>Wang et al. [39]</td><td>0.67</td><td>0.72</td><td>0.41</td><td>-</td><td>-</td><td>-</td><td>0.68</td><td>0.71</td><td>0.82</td></tr><tr><td>Yu and Ramamoorthi [49]</td><td>0.78</td><td>0.77</td><td>0.82</td><td>0.85</td><td>0.89</td><td>0.76</td><td>0.79</td><td>0.77</td><td>0.84</td></tr><tr><td>Yu and Ramamoorthi [50]</td><td>0.85</td><td>0.81</td><td>0.86</td><td>0.87</td><td>0.92</td><td>0.82</td><td>0.83</td><td>0.87</td><td>0.86</td></tr><tr><td>Yu [50]+OVS* [45]</td><td>0.92</td><td>0.78</td><td>0.83</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DUT [44]</td><td>0.71</td><td>0.81</td><td>0.83</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DIFRINT [6]</td><td>1.00</td><td>0.85</td><td>0.84</td><td>1.00</td><td>0.91</td><td>0.78</td><td>1.00</td><td>0.78</td><td>0.84</td></tr><tr><td>FuSta [23]</td><td>1.00</td><td>0.87</td><td>0.86</td><td>1.00</td><td>0.92</td><td>0.82</td><td>1.00</td><td>0.83</td><td>0.87</td></tr><tr><td>Ours</td><td>1.00</td><td>0.91</td><td>0.86</td><td>1.00</td><td>0.94</td><td>0.84</td><td>1.00</td><td>0.87</td><td>0.87</td></tr></table>",
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "tion. For our base synthetic dataset, we used a collection of images from the DPED [14], CityScapes [8] and ADE20K [55] datasets. To generate a stable video, we randomly generate the homography parameters for each image, including rotation angle  $\\theta$ , scaling  $s$ , translations  $(d_x,d_y)$  and perspective factors  $(p_x,p_y)$ . Then we divide these transformation parameters into  $N$  bins equally and obtain a video of  $N$  frames by homography transformations. To simulate the presence of moving objects in real scenarios, the stable video is further augmented with additional independently moving random objects. To do so, the objects are sampled from the COCO dataset [15] and inserted on top of the synthetic video frames using their segmentation masks. Specifically, we randomly choose  $m$  objects (no more than 5), and generate randomly affine transformation parameters for each independent of the background transformation. Finally, we cropped each frame to  $720\\times 480$  around its center. For different training requirements, we apply various combinations of synthetic dataset. The implementation and training details can be found in Supp. S2, S3.",
+        "bbox": [
+            75,
+            340,
+            472,
+            643
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.2.Quantitative Evaluation",
+        "text_level": 1,
+        "bbox": [
+            76,
+            657,
+            299,
+            672
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We compare the results of our method with various video stabilization methods, including Grundmann et al. [11], Liu et al. [21], Wang et al. [39], Yu and Ramamoorthi [49, 50], DUT [44], OVS [45], DIFRINT [6], and FuSta [23]. We obtain the results of the compared methods from the videos released by the authors or generated from the publicly available official implementation with default parameters or pre-trained models. Note that OVS [45] does not honor their promise to provide code, thus we only report the results from their paper.",
+        "bbox": [
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+            681,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We evaluate all approaches on the NUS dataset [21], DeepStab dataset [39], and Selfie dataset [48]. The NUS dataset consists of 144 videos and the corresponding ground truths in 6 scenes. The DeepStab dataset con",
+        "bbox": [
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+            902
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "tains 61 videos and the Selfie dataset consist of 33 videos.",
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+        "type": "text",
+        "text": "Metrics. We introduce three metrics widely used in many methods [6, 23, 50] to evaluate our model: 1) Cropping ratio measures the remaining frame area after cropping off the invalid boundaries. 2) Distortion value evaluates the anisotropic scaling of the homography between the input and output frames. 3) Stability score measures the stability of the output video. We calculate the metrics using the evaluation code provided by DIFRINT.",
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+        "text": "Quantitative comparison. The results of the NUS dataset [21] are summarized in Table 1 (Per-category result can be found in Supp. S5). Overall, our method achieves the best distortion value compared to the state-of-the-art method, FuSta [23]. Especially in the quick-rotation and zoom categories, our method outperforms pure image generation methods [6,23]. We suspect that the reduction of the shared information between frames causes the image generation methods to prefer artifacts. However, our method can ensure local structural integrity when outpainting the margin region. Furthermore, our method achieves an average cropping ratio of 1.0 and stability scores comparable to recent approaches [6, 23, 49, 50]. Since FuSta [23] uses Yu et al. [50] to obtain stabilized input videos, they have the same stability scores. It is important to note that although the stability scores are competitive, our method runs 5 times faster than [50] in the video stabilization stage. Moreover, the comparison results on DeepStab dataset [39] and Selfie dataset [48] are also reported in Table 1. Our method still shows effectiveness in different datasets, proving the generalizability of the proposed method. Note that, due to that Wang et al. [39] is trained on the DeepStab dataset, we do not report its results on the DeepStab dataset for a fair comparison. Qualitative comparisons can be found in Supp. S4 and supplementary video.",
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+        "text": "23540",
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+        "type": "table",
+        "img_path": "images/b07c4ab947c05b21ce1278b311353f1ee2eafd66aeaa658f922129bb98a151c7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Stabilizer</td><td></td><td>Stability↑</td><td>Distortion↑</td></tr><tr><td rowspan=\"2\">Image-level</td><td>w.o. mask</td><td>0.76</td><td>0.88</td></tr><tr><td>with mask</td><td>0.83</td><td>0.91</td></tr><tr><td rowspan=\"2\">Pixel-level</td><td>w.o. mask</td><td>0.81</td><td>0.84</td></tr><tr><td>with mask</td><td>0.83</td><td>0.91</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 2. The importance of mask to video stabilization. We validate on the Crowd category of NUS dataset [21]."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td><td>Distortion↑</td></tr><tr><td>PCAFlow [42]</td><td>11.34</td><td>0.50</td><td>0.72</td></tr><tr><td>Our flow outpainting</td><td>19.65</td><td>0.77</td><td>0.93</td></tr><tr><td>w.o. margin outpainting</td><td>18.04</td><td>0.64</td><td>0.81</td></tr><tr><td>w.o. mask \\( M_{I^t} \\)</td><td>20.35</td><td>0.78</td><td>0.86</td></tr><tr><td>w.o. multi-frame selection</td><td>21.99</td><td>0.84</td><td>0.90</td></tr><tr><td>Ours</td><td>23.17</td><td>0.86</td><td>0.91</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "6.3. Ablation Study",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Importance of mask generation. We investigate the influence of the mask on video stabilization at different stages. To better demonstrate the necessity of mask in complex scenarios, we choose the Crowd category of NUS dataset [21] which includes a display of moving pedestrians and occlusions. Stability and distortion at different settings are shown in Table 2. It can be seen that the performance with mask increases significantly in both stabilizers. Specifically, the mask can both improve stability globally and alleviate image warping distortion locally. This result demonstrates the importance of mask generation for video stabilization.",
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+    {
+        "type": "text",
+        "text": "Flow outpainting. We compare our flow outpainting method with the traditional flow inpainting method PCAFlow [42]. Following [50] we fit the first 5 principal components proposed by PCAFlow to the  $M_{valid} = 1$  regions of the optical flow field, and then outpaint the flow vectors in the  $M_{valid} = 0$  regions with reasonable values of the PCA Flow fitted. The result is obtained by warping the source image with the outpainted optical flow field. We perform this comparison on our synthetic validation set and evaluate it with the corresponding ground truth. Additionally, we use PSNR and SSIM [41] to evaluate the quality of the results. As shown in Table 3, ours dramatically outperforms PCAFlow [42] in all objective metrics.",
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+    {
+        "type": "text",
+        "text": "Importance of image filling strategies. We explore the following proposed strategies for image filling: margin outpainting, mask generation  $M_{It}$ , and multiframe selection. We isolate them from our method to compare their results with the complete version. The results are shown in Table 3. The proposed strategies are generally helpful in improving image quality. Especially, margin outpainting and mask  $M_{It}$  are crucial to the results.",
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+    {
+        "type": "table",
+        "img_path": "images/92723f000f91ab01b485f5125e46ec924552e68a01937aec40fc7708b2bfdc03.jpg",
+        "table_caption": [
+            "Table 3. Ablation study of image filling. The evaluation is conducted on our model-based synthetic validation set."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Runtime</td></tr><tr><td>Traditional pose regression</td><td>216ms</td></tr><tr><td>Our pose regression network</td><td>21ms</td></tr><tr><td>Grundmann et al. [11]</td><td>480ms</td></tr><tr><td>Liu et al. [21]</td><td>1360ms</td></tr><tr><td>Wang et al. [39]</td><td>460ms</td></tr><tr><td>Yu and Ramamoorthi [49]</td><td>1610ms</td></tr><tr><td>Yu and Ramamoorthi [50]</td><td>570ms</td></tr><tr><td>DIFRINT [6]</td><td>1530ms</td></tr><tr><td>FuSta [23]</td><td>9820ms</td></tr><tr><td>Ours</td><td>97ms</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 4. Per-frame runtime comparison of camera pose regression and video stabilization.",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/84dca42383bbf2742110eb444613db34b55c7f59cc19713198a14d13c738d0d4.jpg",
+        "image_caption": [
+            "(a) shaky frames"
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+    },
+    {
+        "type": "image",
+        "img_path": "images/8d25420f1065a21d7e02a10dd900842d696a5b5a32b7113e1c20e6d9b86cf614.jpg",
+        "image_caption": [
+            "(b) stable frames",
+            "Figure 7. Experiment of fixed-point iteration on shaky and stable frames. (a) Our fixed-point iteration helps the shaky frames converge to a steady state. (b) For stable frames, the fixed-point iteration guarantees the stability of results."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "6.4. Runtime Comparison",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our network and pipeline are implemented with PyTorch and Python. Table 4 is a summary of the runtime per frame for various competing methods. All timing results are obtained on a desktop with an RTX3090Ti GPU and an Intel(R) Xeon(R) Gold 6226R CPU. First, we compare the run-time of pose regression. Traditional pose regression is time consuming in feature matching [2, 9], homography estimation, and SVD decomposition [28]. Although our learning-based pose regression network runs 10 times faster than the traditional framework. Then, we report the average run time of different methods, including optimization-based [21, 49] and learning-based [6, 11, 23, 39, 50]. Our method takes  $97\\mathrm{ms}$  which gives  $\\sim 5\\mathrm{x}$  speed-up. This is because our method computes the optical flow field only once and without the help of other upstream task methods and manual optimization.",
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+    {
+        "type": "text",
+        "text": "6.5. Fixed-point Experiment",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To demonstrate the stability of our fixed-point optimization solution, we performed an interesting toy experiment. We input a sequence of shaky frames into our coarse-to-fine stabilizers, and the stabilized result will be iteratively restabilized by our stabilizers. For each iteration, we calculate the average magnitude of optical flow filed with global transformation and flow warping for each frame. Specifi-",
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+        "text": "23541",
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+    {
+        "type": "text",
+        "text": "cally, the regions where we calculate are marked by  $\\hat{M}$ . As shown in Fig. 7(a), the deviation of the shaky frames decreases rapidly with each iteration. Furthermore, we have pointed out that plugging a stabilized video into the stabilization system should not have an impact on the input. Thus, we plug stable frames into the coarse-to-fine stabilizers and iteratively stabilize them. The result is shown in Fig. 7(b). The deviation of each frame is perturbed around the value of zero. Obviously, our method has no effect on stable frames, which shows that stabilized video is indeed the fixed point of our developed stabilization system.",
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+    {
+        "type": "text",
+        "text": "7. Conclusion",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have presented a fast full-frame video stabilization technique based on the iterative optimization strategy. Our approach can be interpreted as the combination of probabilistic stabilization network (coarse-to-fine extension of PDC-Net) and video outpainting outwork (flow-based image outpainting). When trained on synthetic data constructed within the optimization-based learning framework, our method achieves state-of-the-art performance at a fraction of the computational cost of other competing techniques. It is also empirically verified that stabilized video is the fixed point of the stabilization network.",
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+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            76,
+            470,
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+        ],
+        "page_idx": 8
+    },
+    {
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+    },
+    {
+        "type": "ref_text",
+        "text": "[56] Tinghui Zhou, Richard Tucker, John Flynn, Graham Fyffe, and Noah Snavely. Stereo magnification: Learning view synthesis using multiplane images. arXiv preprint arXiv:1805.09817, 2018.",
+        "bbox": [
+            78,
+            90,
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+            147
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "page_number",
+        "text": "23544",
+        "bbox": [
+            478,
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+            955
+        ],
+        "page_idx": 10
+    }
+]

2023/Fast Full-frame Video Stabilization with Iterative Optimization/ad2d1b40-bcdd-45b6-94bd-a7d9f53481b5_origin.pdf

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2023/Fast Full-frame Video Stabilization with Iterative Optimization/full.md

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+# Fast Full-frame Video Stabilization with Iterative Optimization
+
+Weiyue Zhao $^{1}$  Xin Li $^{2}$  Zhan Peng $^{1}$  Xianrui Luo $^{1}$  Xinyi Ye $^{1}$  Hao Lu $^{1}$  Zhiguo Cao $^{1*}$ $^{1}$ Key Laboratory of Image Processing and Intelligent Control, Ministry of Education; School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China  
+ $^{2}$ Department of Computer Science, University of Albany, Albany NY 12222 {zhaoweiyue, peng-zhan, xianruiluo, xinyiye, hlu, zgcao}@hust.edu.cn xli48@albany.edu
+
+# Abstract
+
+Video stabilization refers to the problem of transforming a shaky video into a visually pleasing one. The question of how to strike a good trade-off between visual quality and computational speed has remained one of the open challenges in video stabilization. Inspired by the analogy between wobbly frames and jigsaw puzzles, we propose an iterative optimization-based learning approach using synthetic datasets for video stabilization, which consists of two interacting submodules: motion trajectory smoothing and full-frame outpainting. First, we develop a two-level (coarse-to-fine) stabilizing algorithm based on the probabilistic flow field. The confidence map associated with the estimated optical flow is exploited to guide the search for shared regions through backpropagation. Second, we take a divide-and-conquer approach and propose a novel multiframe fusion strategy to render full-frame stabilized views. An important new insight brought about by our iterative optimization approach is that the target video can be interpreted as the fixed point of nonlinear mapping for video stabilization. We formulate video stabilization as a problem of minimizing the amount of jerkiness in motion trajectories, which guarantees convergence with the help of fixed-point theory. Extensive experimental results are reported to demonstrate the superiority of the proposed approach in terms of computational speed and visual quality. The code will be available on GitHub.
+
+# 1. Introduction
+
+With the growing popularity of short videos on social media platforms (e.g., TikTok, Instagram), video has played an increasingly important role in our daily life. However, casually captured videos are often shaky and wobbly due to
+
+amateur shooting. Although it is possible to alleviate those problems by resorting to professional equipment (e.g., dollies and steadicams), the cost of hardware-based solutions is often expensive, making it impractical for real-world applications. By contrast, software-based or computational solutions such as video stabilization algorithms [12] have become an attractive alternative to improve the visual quality of shaky video by eliminating undesirable jitter.
+
+Existing video stabilization techniques can be classified into two categories: optimization-based and learning-based. Traditional optimization-based algorithms [11,18,21,27,39] have been widely studied due to their speed and robustness. The challenges of them are the occlusion caused by changes in depth of field and the interference caused by foreground objects on camera pose regression. Furthermore, their results often contain large missing regions at frame borders, particularly when videos with a large camera motion. In recent years, learning-based video stabilization algorithms [6, 23, 45, 50] have shown their superiority by achieving higher visual quality compared to traditional methods. However, their stabilization model is too complex for rapid computation, and its generalization property is unknown due to the scarcity of training datasets.
+
+To overcome those limitations, we present an iterative optimization-based learning approach that is efficient and robust, capable of achieving high-quality stabilization results with full-frame rendering, as shown in Fig. 1. The probabilistic stabilized network addresses the issues of occlusion and interfering objects, and achieves fast pose estimation. Then the full-frame outpainting module retains the original field of view (FoV) without aggressive cropping. An important new insight brought by our approach is that the objective of video stabilization is to suppress the implicitly embedded noise in the video frames rather than the explicit noise in the pixel intensity values. This inspired us to adopt an expectation-maximization (EM)-like approach for video stabilization. Importantly, considering the strong redundancy of video in the temporal domain, we ingeniously
+
+![](images/46ce70956dbb13bf03558f871ba573814a0f25c71411f32d1433c1e16de2fa95.jpg)  
+Figure 1. Overview of our video stabilization framework. It consists of motion trajectory smoothing (in Sec. 3 and Sec. 4) and full-frame outpainting modules (in Sec. 5). The former adopts the two-level (coarse-to-fine) stabilizing algorithm to obtain a stabilized video. The latter further render a full-frame video with strategies of flow outpainting and multiframe fusion.
+
+![](images/6a9817f6acb7f629182e51ff59b460f3fdc999660c43d844aeee05ce28645268.jpg)
+
+consider stable video (the target of video stabilization) as the fixed point of nonlinear mapping. Such a fixed-point perspective allows us to formulate an optimization problem of the optical flow field in commonly shared regions. Unlike most methods that resort to the ad hoc video dataset [56] or the deblurred dataset [35] as stabilized videos, we propose to construct a synthetic training dataset to facilitate joint optimization of model parameters in different network modules.
+
+To solve the formulated iterative optimization problem, we take a divide-and-conquer approach by designing two modules: probabilistic stabilization network (for motion trajectory smoothing) and video outpainting network (for full-frame video rendering). For the former, we propose to build on the previous work of PDCNet [37, 38] and extend it using a coarse-to-fine strategy to improve robustness. For a more robust estimate of the uncertainty of the optical flow, we infer masks from the optical flow by bidirectional propagation with a low computational cost (around  $1/5$  of the time Yu et al. [50]). Accordingly, we have developed a two-level (coarse-to-fine) flow smoothing strategy that first aligns adjacent frames by global affine transformation and then refines the result by warping the fields of intermediate frames. For the latter, we propose a two-stage approach (flow and image outpainting) to render full-frame video. Our experimental results have shown highly competent performance against others on three public benchmark datasets. The main contribution of this work is threefold:
+
+- We propose a formulation of video stabilization as a fixed-point problem of the optical flow field and propose a novel procedure to generate a model-based synthetic dataset.  
+- We construct a probabilistic stabilization network based on PDCNet and propose an effective coarse-to-fine strategy for robust and efficient smoothing of optical flow fields.  
+- We propose a novel video outpainting network to render stabilized full-frame video by exploiting the spatial
+
+coherence in the flow field.
+
+# 2. Related Work
+
+# 2.1. Video Stabilization
+
+Most video stabilization methods can typically be summarized as a three-step procedure: motion estimation, smoothing the trajectory, and generating stable frames. Traditional methods focus primarily on 2D features or image alignment [29] when it comes to motion estimation. These methods are different in modeling approaches to motion, including the trajectory matrix [19], epipolar geometry [10, 21, 47, 53], and the optical flow field [22, 54]. Regarding the smoothing trajectory, particle filter tracking [46], space-time path smoothing [19, 40], and L1 optimization [11] have been proposed. Existing methods for generating stable frames rely mainly on 2D transformations [26], grid warping [19, 21], and dense flow field warping [22].
+
+Compared to the 2D method, some approaches turn to 3D reconstruction [17]. However, specialized hardware such as depth camera [20] and light field camera [34] are necessary for these methods based on 3D reconstruction. Some methods [30, 43, 44, 49] tackle video stabilization from the perspective of deep learning. In [49], the optimization of video stabilization was formulated in the CNN weight space. Recent work [50] represents motion by flow field and attempts to learn a stable optical flow to warp frames. Another method [6] aims to learn stable frames by interpolation. These deep-learning methods generate stable videos with less distortion.
+
+# 2.2. Large FOV Video
+
+Unlike the early work (e.g., traffic video stabilization [16]), large field of view (Field Of View) video stabilization has been attracting more researchers' attention. For most video stabilization methods, cropping is inevitable, which is why the FOV is reduced. Several approaches have been proposed to maintain a high FOV ratio. OVS [45]
+
+proposed to improve FOV by extrapolation. DIFRINT [6] choose iterative interpolation to generate high-FOV frames. FuSta [23] used neural rendering to synthesize high-FOV frames from feature space. To a great extent, the performance of interpolation-based video stabilization [6,45] depends on the selected frames. If selected frames have little correspondence with each other, performance will deteriorate dramatically. Neural rendering [23] synthesizes the image by weighted summing, causing blur. Most recently, a deep neural network [33] (DNN) has jointly exploited sensor data and optical flow to stabilize videos.
+
+# 3. Stabilization via Iterative Optimization
+
+Motivation. Despite rapid advances in video stabilization [30], existing methods still suffer from several notable limitations [12]. First, a systematic treatment of various uncertainty factors (e.g., low-texture regions in the background and moving objects in the foreground) in the problem formulation is still lacking. These uncertainty factors often cause occlusion-related problems and interfere with the motion estimation process. Second, the high computational cost has remained a technical barrier to real-time video processing. The motivation behind our approach is two-fold. On the one hand, we advocate for finding commonly shared regions among successive frames to address various uncertainty factors in handheld video. On the other hand, in contrast to these prestabilization algorithms [21, 23, 49, 50] based on traditional approaches, we proposed a novel high-efficiency prestabilization algorithm based on probabilistic optical flow. Flow-based methods are generally more accurate in motion estimation and deserve a high time cost. Accuracy and efficiency, we have both.
+
+Approach. We propose to formulate video stabilization as a problem of minimizing the amount of jerkiness in motion trajectories. It is enlightening to think of video stabilization as a special kind of "video denoising" where noise contamination is not associated with pixel intensity values, but embedded into the motion trajectories of foreground and background objects. Conceptually, for video restoration in which unknown motion is the hidden variable, we can treat video stabilization as a chicken-and-egg problem [13] - i.e., the objective of smoothing motion trajectories is intertwined with that of video frame interpolation. Note that an improved estimation of motion information can facilitate the task of frame interpolation and vice versa. Such an observation naturally inspires us to tackle the problem of video stabilization by iterative optimization.
+
+Through divide-and-conquer, we propose to formulate video stabilization as the following optimization problem. Given a sequence of  $n$  frames along with the set of optical flows  $\mathcal{V}$ , we first define the confidence map, which originally indicates the reliability and accuracy of the optical flow prediction at each pixel. Here, we have thresholded the
+
+confidence map as a binary image, which represents accurate matches (as shown in the 4-th column of Fig. 2). Then we can denote  $n$  frames and the corresponding set of confidence maps  $\mathcal{M}$  by:
+
+$$
+\mathcal {Y} = \left\{Y _ {1}, Y _ {2}, \dots , Y _ {q} \right\}, \mathcal {M} = \left\{M _ {1}, M _ {2}, \dots , M _ {q} \right\}. \quad (1)
+$$
+
+# 4. Probabilistic Stabilization Network
+
+The problem of video stabilization can be formulated as finding a nonlinear mapping  $f: \mathcal{V} \to \hat{\mathcal{V}}$  where  $\hat{\mathcal{V}}$  denotes the optical flow set of the stabilized video. We hypothesize that a desirable objective to pursue  $f$  is the well-known fixed-point property, i.e.,  $\hat{\mathcal{V}} = f(\hat{\mathcal{V}})$ . To achieve this objective, we aim to minimize an objective function  $\mathbb{F}$  characterized by the magnitude of optical flows between commonly shared regions, as represented by  $\mathcal{M}$ . Note that  $\mathcal{M}$  is the hidden variable in our problem formulation (that is, we need to estimate  $\mathcal{M}$  from  $\hat{\mathcal{V}}$ ). A popular strategy to address this chicken-and-egg problem is to alternatively solve the two subproblems of unknown  $\mathcal{M}$  and  $\hat{\mathcal{V}}$ . More specifically, the optimization of  $\mathbb{F}$  is decomposed into the following two subproblems:
+
+$$
+\left(\hat {M} _ {1}, \dots , \hat {M} _ {q}\right) = \underset {\mathcal {M}} {\arg \min } \mathbb {F} \left(\hat {Y} _ {1} \odot M _ {1}, \dots , \hat {Y} _ {q} \odot M _ {q}\right), \tag {2}
+$$
+
+$$
+(\hat {Y} _ {1}, \dots , \hat {Y} _ {q}) = \underset {\mathcal {Y}} {\arg \min } \mathbb {F} (Y _ {1} \odot \hat {M} _ {1}, \dots , Y _ {q} \odot \hat {M} _ {q}),
+$$
+
+where  $Y / \hat{Y}$  denotes the magnitude of the optical flow values before and after stabilization and  $\odot$  is the Hadamard product. Instead of analyzing the estimated motion in the frequency domain, we hypothesize that stabilized videos are the fixed points of video stabilization algorithms, which minimize the above objective function. A fixed point [1] is a mathematical object that does not change under a given transformation. Numerically, fixed-point iteration is a method of computing fixed points of a function. It has been widely applied in data science [7], and image restoration [5, 25]. Here, we denote  $\mathbb{F}$  in Eq. (2) as the function to be optimized, and the fixed point of  $\mathbb{F}$  is defined as the stabilized video. Next, we solve these two subproblems by constructing the module of stabilization.
+
+# 4.1. Probabilistic Flow Field
+
+First, we start with an interesting observation. When playing an unstable video at a slower speed (e.g., from 50fps to 30fps), the video tends to appear less wobbly. It follows from the observation that the fundamental cause of video instability is the fast frequency and the large amplitude of the still objects' motion speed. Therefore, the core task of motion smoothing is to identify the region that needs to be stabilized. As Yu et al. [50] pointed out, mismatches, moving objects, and inconsistent motion areas lead to variation of the estimated motion from the true motion trajectories and
+
+![](images/a6c0690151f1ab2bb86ee086185771599464ba2e7cde3e86ede584ed8ec55c82.jpg)  
+Figure 2. Visualization of confidence map back-propagation results. The flow field and confidence map are predicted by PDCNet between source and target images. The obtained masks in the last column represent the shared regions among these frames.
+
+should be masked. An important new insight brought about by this work is that these regions of inconsistency in the optical flow fields tend to show greater variability as the frame interval increases. Therefore, if these inconsistent regions are excluded, the video stabilization task can be simplified to the first sub-problem in Eq. (2) - i.e., minimizing  $\mathbb{F}$  over  $\mathcal{M}$  for a fixed  $\mathcal{V}$ .
+
+To detect unreliable matches and inconsistent regions, we have adopted the probabilistic flow network - PDCNet [37, 38], that jointly tackles the problem of dense correspondence and uncertainty estimation, as our building block. Suppose that PDCNet estimates the optical flow  $Y_{k}$  from frame  $k + d$  to frame  $k$  with the resulting confidence map  $C_k$ . Although  $C_k$  denotes the inaccuracy of the optical flow estimate in  $Y_{k}$ , it is often sensitive to the frame interval  $d$ . For example, it is difficult to identify the inconsistent region when  $d$  is small, while the common area is less controllable when  $d$  is large. Therefore, simply using an optical flow field to estimate inconsistent regions is not sufficient.
+
+We have designed a more robust solution to the joint estimation of dense flow and confidence map based on a coarse-to-fine strategy. The basic idea is to first obtain the probabilistic flow field at a coarse scale (e.g., with the down-sampled video sequence by a factor of  $d$  along the temporal axis) and then fill in the rest (i.e., the frames between the adjacent frames in the down-sampled video) at the fine scale. Such a two-level estimation strategy effectively overcomes the limitations of PDCNet by propagating the estimation results of probabilistic flow fields bidirectionally, as we will elaborate next.
+
+Coarse-scale strategy As shown by the second-to-last column in Fig. 2, the confidence map estimated by PDCNet can identify the mismatched region, but fails to locate the objects with small motions (e.g., people and the sky). To
+
+![](images/4b7a72e0ff1aa18faa8873256305d6ac94511f316b848b6dd35c65a18a39f689.jpg)  
+Figure 3. Architecture of the camera pose regression network. Given the flow field and mask, our network predicts the corresponding affine transformation parameters by closed-loop iterations.
+
+overcome this difficulty, we introduce the binary mask as a warped and aggregated confidence map (refer to the last column of Fig. 2). Specifically, we first propose to obtain the confidence map  $\hat{C}_{k + (n - 1)d}$  with down-sampled video (the last row of Fig. 2). In the forward direction, we estimate dense flow and confidence map using PDCNet; then  $\hat{C}_{k + (n - 1)d}$  is backpropagated to update the binary mask  $\hat{M}$  by thresholding and setting intersection operators. Through bidirectional propagation, the region covered by  $\mathcal{M}$  is the shared content from frame  $k$  to frame  $k + (n - 1)d$ . The complete procedure can be found in Supp. S1.
+
+Fine-scale strategy Based on the coarse-scale estimation result for downsampled video (i.e.,  $\mathcal{M} = \{\hat{M}_k, \hat{M}_{k+d}, \dots, \hat{M}_{k+(n-1)d}\}$ ), we fill in the missing  $d-1$  frames at the fine scale. Specifically, considering the sequential frames from  $k$  to  $k+d$ , we can obtain two sets similar to Eq. (1), which are  $\mathcal{V} = \{Y_k, Y_{k+1}, \dots, Y_{k+d-1}, \hat{Y}_{k+d}\}$  and their corresponding confidence map set  $\mathcal{C}$ . Note that  $\hat{M}_{k+d} \in \mathcal{M}$  has been calculated in coarse stage. Setting  $d=1$ , we can call the algorithm again to obtain the set of output masks  $\mathcal{M}$  for the rest of  $d-1$  frames.
+
+# 4.2. Coarse-scale Stabilizer
+
+To coarsely stabilize the video, we first propose aligning the adjacent frames with a global affine transformation [52]. The optimization function  $\mathbb{F}$  in Eq. (2) is represented as
+
+$$
+\mathbf {T} ^ {*} = \underset {\mathbf {T}} {\arg \min } \mathbf {T} (Y \odot \hat {M}), \tag {3}
+$$
+
+where  $\mathbf{T}(\cdot)$  denotes the image transformation applied to the shared region  $\hat{M}$  (the result of Sec. 4.1) of the optical flow field  $Y$ . Most conventional methods [23,44,50] adopt image matching to obtain  $\mathbf{T}(\cdot)$ -e.g., keypoint detection [2,24,31], feature matching [2,3,51]; and camera pose estimation is implemented by OpenCV-python. However, these methods are often time-consuming and computationally expensive. For example, two adjacent frames of unstable videos usually share a large area and are free from perspective transformation. Thus, an affine transformation, including translation, rotation, and scaling, is sufficient. More importantly, within the optimization-based learning framework, we can
+
+regress these linear parameters of  $\mathbf{T}(\cdot)$  from the optical flow field, which characterize the relative coordinate transformation of the matched features.
+
+We propose a novel camera pose regression network, as shown in Fig. 3. Given an optical flow field  $Y$  and the corresponding mask field  $\hat{M}$ , our network  $\Phi(\cdot)$  can directly estimate the unknown parameters  $\mathbf{T}(\cdot) \propto \{\theta, s, d_x, d_y\} = \Phi(Y, \hat{M})$ . To solve the optimization problem of Eq. (3), we use the estimated parameters to iteratively compute the corresponding residual optical flow fields such that
+
+$$
+\bar {Y} = Y - (S \cdot R \cdot V + T), \tag {4}
+$$
+
+where  $T$ ,  $S$ , and  $R$ , respectively, denote the translation, scaling, and rotation matrix, and  $V \in \mathbb{R}^{2 \times H \times W}$  represents an image coordination grid. Then  $\{\Delta \theta, \Delta s, \Delta x, \delta y\} = \Phi(\bar{Y}, \tilde{M})$  is calculated iteratively to produce the updated parameters  $\{\theta + \Delta \theta, s \cdot \Delta s, d_x + \Delta x, d_y + \Delta y\}$ . The finally estimated affine transformation is smoothed by a moving Gaussian filter with a window size of 20 pixels.
+
+Loss functions Our loss functions include robust loss  $\ell_1$  and grid loss commonly applied in consistency filtering [51]. We directly calculate the  $\ell_1$  loss between the predictions and their ground truth  $\{\hat{\theta},\hat{s},\hat{d}_x,\hat{d}_y\}$
+
+$$
+\begin{array}{l} L _ {g t} = \lambda_ {\theta} \| | \theta - \hat {\theta} \| | _ {1} + \lambda_ {s} \| | 1 - \frac {s}{\hat {s}} \| | _ {1} \tag {5} \\ + \lambda_ {t} \| | d _ {x} - \hat {d} _ {x} | + | d _ {y} - \hat {d} _ {y} | \| _ {1}. \\ \end{array}
+$$
+
+For better supervision of the estimated pose, we calculate the loss  $L_{grid}$  with the grid  $V \in \mathbb{R}^{2 \times h \times h}$  in Eq. (4), i.e.,
+
+$$
+L _ {g r i d} = \left\| (\hat {S} \cdot \hat {R} \cdot V + \hat {T}) - (S \cdot R \cdot V + T) + \epsilon \right\| _ {1} \tag {6}
+$$
+
+where  $\hat{\cdot}$  denotes the ground truth and  $\epsilon$  is a small value for stability. The final loss  $L_{stab}$  consists of  $L_{gt}$  and  $L_{grid}$ ,
+
+$$
+L _ {s t a b} = L _ {g t} + \lambda_ {g r i d} L _ {g r i d}. \tag {7}
+$$
+
+# 4.3. Fine-scale Stabilizer
+
+The assumption with an affine transformation of the coarse-scale stabilizer could cause structural discontinuity and local distortion. Therefore, our objective is to refine the coarsely stabilized video by optical flow smoothing. Unlike Eq. 3 which applies an image transformation matrix to optimize the optical flow field, we optimize it at the pixel level by a flow warping field  $\mathbf{W}$ . Thus, the function  $\mathbb{F}$  in Eq. (2) is given by
+
+$$
+\mathbf {W} ^ {*} = \underset {\mathbf {W}} {\arg \min } \sum_ {i = 0} ^ {N - 1} \mathbf {W} _ {i} \left(Y _ {i} \odot \hat {M} _ {i}\right). \tag {8}
+$$
+
+The flow smoothing network follows the U-Net architecture in [56]. We use  $N$  frames of optical flow fields  $\mathbf{F}$  and mask
+
+![](images/6fe7a98576fa7769bc42ea3ca51803b12244536840d7b80a8ef6546f35c8b5f6.jpg)  
+Figure 4. Illustration of flow fields and warped images under different FOVs. The target image with large FOV presents the smoother flow field and better warped result.
+
+fields  $\hat{M}$  as input and obtain  $(N - 1)$  frame warp fields  $\mathbf{W}$  of intermediate frames. Specifically, for the optical flow field  $Y_{k}$  (from frame  $k + 1$  to frame  $k$ ), we denote the aligned matrices of each frame in Sec. 4.2 as  $H_{k} \in \mathbb{R}^{2 \times 3}$  and  $H_{k + 1} \in \mathbb{R}^{2 \times 3}$ , respectively. Then the input optical flow  $\mathbf{F}_{k} \in \mathbb{R}^{2 \times HW}$  can be represented by
+
+$$
+\mathbf {F} _ {k} = H _ {k + 1} \cdot \left[ V + Y _ {k} \mid 1 \right] - H _ {k} \cdot \left[ V \mid 1 \right], \tag {9}
+$$
+
+where  $V \in \mathbb{R}^{2 \times HW}$  and  $[\cdot|1]$  denote the normalized coordinate representation. Furthermore, to better adapt the flow smoothing network to the mask field  $\hat{M}$ , we fine-tune it using our synthetic dataset (as we will elaborate in Sec. 6.1). The loss function follows the motion loss [49]
+
+$$
+L _ {\text {s m o o t h}} = \sum_ {k = 0} ^ {N - 1} \left(\mathbf {F} _ {k} + \mathbf {W} _ {k} - \mathbf {F} _ {k} \left(\mathbf {W} _ {k + 1}\right)\right) \odot \hat {M}, \tag {10}
+$$
+
+where  $\mathbf{W}_0 = \mathbf{W}_N = 0$
+
+# 5. Video Outpainting Network
+
+Most video stabilization methods crop an input video with a small field of view (FOV), excluding missing margin pixels due to frame warping. In contrast, full-frame video stabilization [6, 23, 26] proposes to generate a video that maintains the same FOV as the input video without cropping. They directly generate stabilized video frames with large FOV by fusing the information from neighboring frames. An important limitation of existing fusion strategies is the unequal importance of different frames (i.e., the current and neighboring frames are weighted equally), which would lead to unpleasant distortions in fast-moving situations. To overcome this weakness, we propose a two-stage framework to combine flow and image outpainting strategies [32]. In the first stage, we used flow-outpainting to iteratively align the neighboring frames with the target frame. In the second stage, we fill the target frame with adjacent aligned frame pixels by image outpainting.
+
+# 5.1. Flow Outpainting Network
+
+Let  $I^t$  denote the outpainted target image and  $I^s$  the neighboring source image. We aim to fill the missing pixel
+
+![](images/7b857824a8c24cae006118bc7c1c0e035fb96460916c2e667be345621f1776cd.jpg)  
+Figure 5. Architecture of flow outpainting network. Given a small FOV flow field and valid mask, the network predicts the large FOV flow field.
+
+region  $M_{\varnothing}^{t}$  of  $I^{t}$  with the pixels of  $I^{s}$ . As shown in Fig. 4, we take two different FOVs of  $I^{t}$  as input and obtain the corresponding optical flow fields and warped results  $I_{warp}^{s}$ . The small FOV  $I^{t}$  cannot guide  $I^{s}$  well to fill the regions in  $M_{\varnothing}^{t}$ . Since the predicted optical flow field in  $M_{\varnothing}^{t}$  is unreliable due to the lack of pixel guidance of  $I^{t}$ , the out-of-view region of  $M_{\varnothing}^{t}$  has artifacts (marked in the last column of Fig. 4). We observe that the optical flow field of the large FOV  $I^{t}$  is continuous in  $M_{\varnothing}^{t}$ , which inspired us to extrapolate the flow of  $M_{\varnothing}^{t}$  using the reliable flow region (the 3rd column of Fig. 4).
+
+We propose a novel flow outpainting network (Fig. 5), which extrapolates the large FOV flow field using a small FOV flow field and the corresponding valid mask. Specifically, we adopt a U-Net architecture and apply a sequence of gated convolution layers with downsampling / upsampling to obtain the large flow field of FOV  $Y_{large}$ . Note that the input flow field  $Y_{small}$  and the valid mask  $M_{valid}$  have been estimated by PDCNet (see Sec. 4.1).
+
+Loss functions Our loss functions include robust loss  $\ell_1$  and loss in the frequency domain [4]. We directly calculate the  $\ell_1$  loss between  $Y_{large}$  and its ground truth  $\hat{Y}_{large}$ ,
+
+$$
+\begin{array}{l} L _ {Y} = \lambda_ {i n} \cdot \| | Y _ {\text {l a r g e}} - Y _ {\text {s m a l l}} | \odot M _ {\text {v a l i d}} \| _ {1} \tag {11} \\ + \lambda_ {o u t} \cdot \| | Y _ {l a r g e} - \dot {Y} _ {l a r g e} | \odot (\sim M _ {v a l i d}) \| _ {1}. \\ \end{array}
+$$
+
+To encourage low frequency and smoothing  $Y_{large}$ , we add the loss in the frequency domain  $L_{F} = \| \hat{\mathbf{G}}\cdot \mathcal{F}Y_{large}\|_{2}$  where the normalized Gaussian map  $\hat{\mathbf{G}}$  with  $\mu = 0$  and  $\sigma = 3$  is inverted by its maximum value and  $\mathcal{F}Y_{large}$  denotes the Fourier spectrum of  $Y_{large}$ . The final loss consists of  $L_{Y}$  and  $L_{F}$
+
+$$
+L _ {\text {o u t p a i n t}} = L _ {Y} + \lambda_ {F} L _ {F}. \tag {12}
+$$
+
+# 5.2. Image Margin Outpainting
+
+Based on our proposed flow-outpainting network, we design a margin-outpainting method by iteratively aligning frame pairs (see Fig. 6). As discussed in Sec. 5.1, we can obtain a large FOV flow field  $Y_{large}$ . The neighboring reference frame  $I^{s}$  is warped as  $I^{warp} = Y_{large}(I^{s})$  as shown in Fig. 6. In theory, the outpainted frame  $I^{result} =$
+
+![](images/05918f014562ceec451629fc9b4ace86011898cec5ae90fa9887c6c10599b66a.jpg)  
+Figure 6. Overview image margin outpainting. Given the target frame  $I^{t}$ , the reference frame  $I^{s}$  is coarsely aligned to  $I^{warp}$  by the predicted large-FOV flow field  $Y_{large}$ . Then, we adopt a margin fusion approach to obtain the result frame  $I^{result}$ , by carefully aggregating  $I^{t}$  and  $I^{warp}$ .
+
+$I^t \cdot M_{valid} + I^{warp} \cdot (\sim M_{valid})$ . However, we notice that there are obvious distortions at the image border. To further align the margins, we take a margin fusion approach (the detailed algorithm can be found in Supp. S1) We crop  $I^{warp}$  and  $I^t$  to  $I_c^s$  and  $I_c^t$ . Then, we can obtain a new warped frame  $I_c^{warp}$  by flow outpainting. In particular, we did not choose to add  $I_c^{warp}$  and  $I^t$  directly. To identify the misaligned region, we propose to outpaint the mask  $M_{I^t}$  by extending the watershed outward from the center. Instead of a preset threshold, we adaptively choose between the target image  $I^t$  and the warped image  $I_c^{warp}$ . Then, the final frame  $I^{result}$  consists of  $I^t$  and  $I_c^{warp}$ :  $I^{result} = I^t \cdot M_{I^t} + I_c^{warp} \cdot (\sim M_{I^t})$ . Compared to the two results  $I^{result}$ , our strategy successfully mitigates misalignment and distortions at the boundary of video frames.
+
+Multi-frame fusion During the final stage of rendering, we use a sequence of neighboring frames to outpaint the target frame, while they may have filled duplicate regions. It is important to find out which frame and which region should be selected. We proposed the selection strategy for multi-frame fusion (the details can be found in Supp. S1). By weighing the metric parameters of each frame, we finally obtain the target frame with large FOV. Note that each frame has an added margin in the stabilization process, so we need to crop them to the original resolution. Although we have outpainted the target frame, some missing pixel holes may still exit at boundaries. Here, we apply the state-of-the-art LaMa image inpainting method [36] to fill these holes using nearest-neighbor interpolation.
+
+# 6. Experiments
+
+# 6.1. Synthetic Datasets for Supervised Learning
+
+Due to the limited amount of paired training data, we propose a novel model-based data generation method by carefully designing synthetic datasets for video stabiliza
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">NUS dataset [21]</td><td colspan="3">DeepStab dataset [39]</td><td colspan="3">Selfie dataset [48]</td></tr><tr><td>C.↑</td><td>D.↑</td><td>S.↑</td><td>C.↑</td><td>D.↑</td><td>S.↑</td><td>C.↑</td><td>D.↑</td><td>S.↑</td></tr><tr><td>Grundmann et al. [11]</td><td>0.71</td><td>0.76</td><td>0.62</td><td>0.77</td><td>0.87</td><td>0.80</td><td>0.75</td><td>0.81</td><td>0.83</td></tr><tr><td>Liu et al. [21]</td><td>0.81</td><td>0.78</td><td>0.82</td><td>0.80</td><td>0.90</td><td>0.85</td><td>0.74</td><td>0.89</td><td>0.8</td></tr><tr><td>Wang et al. [39]</td><td>0.67</td><td>0.72</td><td>0.41</td><td>-</td><td>-</td><td>-</td><td>0.68</td><td>0.71</td><td>0.82</td></tr><tr><td>Yu and Ramamoorthi [49]</td><td>0.78</td><td>0.77</td><td>0.82</td><td>0.85</td><td>0.89</td><td>0.76</td><td>0.79</td><td>0.77</td><td>0.84</td></tr><tr><td>Yu and Ramamoorthi [50]</td><td>0.85</td><td>0.81</td><td>0.86</td><td>0.87</td><td>0.92</td><td>0.82</td><td>0.83</td><td>0.87</td><td>0.86</td></tr><tr><td>Yu [50]+OVS* [45]</td><td>0.92</td><td>0.78</td><td>0.83</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DUT [44]</td><td>0.71</td><td>0.81</td><td>0.83</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DIFRINT [6]</td><td>1.00</td><td>0.85</td><td>0.84</td><td>1.00</td><td>0.91</td><td>0.78</td><td>1.00</td><td>0.78</td><td>0.84</td></tr><tr><td>FuSta [23]</td><td>1.00</td><td>0.87</td><td>0.86</td><td>1.00</td><td>0.92</td><td>0.82</td><td>1.00</td><td>0.83</td><td>0.87</td></tr><tr><td>Ours</td><td>1.00</td><td>0.91</td><td>0.86</td><td>1.00</td><td>0.94</td><td>0.84</td><td>1.00</td><td>0.87</td><td>0.87</td></tr></table>
+
+Table 1. Quantitative results on the NUS dataset [21], the DeepStab dataset [39] and the Selfie Dataset [48]. We evaluate the following metrics: Cropping Ratio(C.), Distortion Value(D.), Stability Score(S). * indicates the results obtained from original paper. We highlight the best method in bold and underline the second-best.
+
+tion. For our base synthetic dataset, we used a collection of images from the DPED [14], CityScapes [8] and ADE20K [55] datasets. To generate a stable video, we randomly generate the homography parameters for each image, including rotation angle  $\theta$ , scaling  $s$ , translations  $(d_x,d_y)$  and perspective factors  $(p_x,p_y)$ . Then we divide these transformation parameters into  $N$  bins equally and obtain a video of  $N$  frames by homography transformations. To simulate the presence of moving objects in real scenarios, the stable video is further augmented with additional independently moving random objects. To do so, the objects are sampled from the COCO dataset [15] and inserted on top of the synthetic video frames using their segmentation masks. Specifically, we randomly choose  $m$  objects (no more than 5), and generate randomly affine transformation parameters for each independent of the background transformation. Finally, we cropped each frame to  $720\times 480$  around its center. For different training requirements, we apply various combinations of synthetic dataset. The implementation and training details can be found in Supp. S2, S3.
+
+# 6.2.Quantitative Evaluation
+
+We compare the results of our method with various video stabilization methods, including Grundmann et al. [11], Liu et al. [21], Wang et al. [39], Yu and Ramamoorthi [49, 50], DUT [44], OVS [45], DIFRINT [6], and FuSta [23]. We obtain the results of the compared methods from the videos released by the authors or generated from the publicly available official implementation with default parameters or pre-trained models. Note that OVS [45] does not honor their promise to provide code, thus we only report the results from their paper.
+
+Datasets. We evaluate all approaches on the NUS dataset [21], DeepStab dataset [39], and Selfie dataset [48]. The NUS dataset consists of 144 videos and the corresponding ground truths in 6 scenes. The DeepStab dataset con
+
+tains 61 videos and the Selfie dataset consist of 33 videos.
+
+Metrics. We introduce three metrics widely used in many methods [6, 23, 50] to evaluate our model: 1) Cropping ratio measures the remaining frame area after cropping off the invalid boundaries. 2) Distortion value evaluates the anisotropic scaling of the homography between the input and output frames. 3) Stability score measures the stability of the output video. We calculate the metrics using the evaluation code provided by DIFRINT.
+
+Quantitative comparison. The results of the NUS dataset [21] are summarized in Table 1 (Per-category result can be found in Supp. S5). Overall, our method achieves the best distortion value compared to the state-of-the-art method, FuSta [23]. Especially in the quick-rotation and zoom categories, our method outperforms pure image generation methods [6,23]. We suspect that the reduction of the shared information between frames causes the image generation methods to prefer artifacts. However, our method can ensure local structural integrity when outpainting the margin region. Furthermore, our method achieves an average cropping ratio of 1.0 and stability scores comparable to recent approaches [6, 23, 49, 50]. Since FuSta [23] uses Yu et al. [50] to obtain stabilized input videos, they have the same stability scores. It is important to note that although the stability scores are competitive, our method runs 5 times faster than [50] in the video stabilization stage. Moreover, the comparison results on DeepStab dataset [39] and Selfie dataset [48] are also reported in Table 1. Our method still shows effectiveness in different datasets, proving the generalizability of the proposed method. Note that, due to that Wang et al. [39] is trained on the DeepStab dataset, we do not report its results on the DeepStab dataset for a fair comparison. Qualitative comparisons can be found in Supp. S4 and supplementary video.
+
+<table><tr><td>Stabilizer</td><td></td><td>Stability↑</td><td>Distortion↑</td></tr><tr><td rowspan="2">Image-level</td><td>w.o. mask</td><td>0.76</td><td>0.88</td></tr><tr><td>with mask</td><td>0.83</td><td>0.91</td></tr><tr><td rowspan="2">Pixel-level</td><td>w.o. mask</td><td>0.81</td><td>0.84</td></tr><tr><td>with mask</td><td>0.83</td><td>0.91</td></tr></table>
+
+Table 2. The importance of mask to video stabilization. We validate on the Crowd category of NUS dataset [21].  
+
+<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td><td>Distortion↑</td></tr><tr><td>PCAFlow [42]</td><td>11.34</td><td>0.50</td><td>0.72</td></tr><tr><td>Our flow outpainting</td><td>19.65</td><td>0.77</td><td>0.93</td></tr><tr><td>w.o. margin outpainting</td><td>18.04</td><td>0.64</td><td>0.81</td></tr><tr><td>w.o. mask \( M_{I^t} \)</td><td>20.35</td><td>0.78</td><td>0.86</td></tr><tr><td>w.o. multi-frame selection</td><td>21.99</td><td>0.84</td><td>0.90</td></tr><tr><td>Ours</td><td>23.17</td><td>0.86</td><td>0.91</td></tr></table>
+
+# 6.3. Ablation Study
+
+Importance of mask generation. We investigate the influence of the mask on video stabilization at different stages. To better demonstrate the necessity of mask in complex scenarios, we choose the Crowd category of NUS dataset [21] which includes a display of moving pedestrians and occlusions. Stability and distortion at different settings are shown in Table 2. It can be seen that the performance with mask increases significantly in both stabilizers. Specifically, the mask can both improve stability globally and alleviate image warping distortion locally. This result demonstrates the importance of mask generation for video stabilization.
+
+Flow outpainting. We compare our flow outpainting method with the traditional flow inpainting method PCAFlow [42]. Following [50] we fit the first 5 principal components proposed by PCAFlow to the  $M_{valid} = 1$  regions of the optical flow field, and then outpaint the flow vectors in the  $M_{valid} = 0$  regions with reasonable values of the PCA Flow fitted. The result is obtained by warping the source image with the outpainted optical flow field. We perform this comparison on our synthetic validation set and evaluate it with the corresponding ground truth. Additionally, we use PSNR and SSIM [41] to evaluate the quality of the results. As shown in Table 3, ours dramatically outperforms PCAFlow [42] in all objective metrics.
+
+Importance of image filling strategies. We explore the following proposed strategies for image filling: margin outpainting, mask generation  $M_{It}$ , and multiframe selection. We isolate them from our method to compare their results with the complete version. The results are shown in Table 3. The proposed strategies are generally helpful in improving image quality. Especially, margin outpainting and mask  $M_{It}$  are crucial to the results.
+
+Table 3. Ablation study of image filling. The evaluation is conducted on our model-based synthetic validation set.  
+
+<table><tr><td>Method</td><td>Runtime</td></tr><tr><td>Traditional pose regression</td><td>216ms</td></tr><tr><td>Our pose regression network</td><td>21ms</td></tr><tr><td>Grundmann et al. [11]</td><td>480ms</td></tr><tr><td>Liu et al. [21]</td><td>1360ms</td></tr><tr><td>Wang et al. [39]</td><td>460ms</td></tr><tr><td>Yu and Ramamoorthi [49]</td><td>1610ms</td></tr><tr><td>Yu and Ramamoorthi [50]</td><td>570ms</td></tr><tr><td>DIFRINT [6]</td><td>1530ms</td></tr><tr><td>FuSta [23]</td><td>9820ms</td></tr><tr><td>Ours</td><td>97ms</td></tr></table>
+
+Table 4. Per-frame runtime comparison of camera pose regression and video stabilization.
+
+![](images/84dca42383bbf2742110eb444613db34b55c7f59cc19713198a14d13c738d0d4.jpg)  
+(a) shaky frames
+
+![](images/8d25420f1065a21d7e02a10dd900842d696a5b5a32b7113e1c20e6d9b86cf614.jpg)  
+(b) stable frames  
+Figure 7. Experiment of fixed-point iteration on shaky and stable frames. (a) Our fixed-point iteration helps the shaky frames converge to a steady state. (b) For stable frames, the fixed-point iteration guarantees the stability of results.
+
+# 6.4. Runtime Comparison
+
+Our network and pipeline are implemented with PyTorch and Python. Table 4 is a summary of the runtime per frame for various competing methods. All timing results are obtained on a desktop with an RTX3090Ti GPU and an Intel(R) Xeon(R) Gold 6226R CPU. First, we compare the run-time of pose regression. Traditional pose regression is time consuming in feature matching [2, 9], homography estimation, and SVD decomposition [28]. Although our learning-based pose regression network runs 10 times faster than the traditional framework. Then, we report the average run time of different methods, including optimization-based [21, 49] and learning-based [6, 11, 23, 39, 50]. Our method takes  $97\mathrm{ms}$  which gives  $\sim 5\mathrm{x}$  speed-up. This is because our method computes the optical flow field only once and without the help of other upstream task methods and manual optimization.
+
+# 6.5. Fixed-point Experiment
+
+To demonstrate the stability of our fixed-point optimization solution, we performed an interesting toy experiment. We input a sequence of shaky frames into our coarse-to-fine stabilizers, and the stabilized result will be iteratively restabilized by our stabilizers. For each iteration, we calculate the average magnitude of optical flow filed with global transformation and flow warping for each frame. Specifi-
+
+cally, the regions where we calculate are marked by  $\hat{M}$ . As shown in Fig. 7(a), the deviation of the shaky frames decreases rapidly with each iteration. Furthermore, we have pointed out that plugging a stabilized video into the stabilization system should not have an impact on the input. Thus, we plug stable frames into the coarse-to-fine stabilizers and iteratively stabilize them. The result is shown in Fig. 7(b). The deviation of each frame is perturbed around the value of zero. Obviously, our method has no effect on stable frames, which shows that stabilized video is indeed the fixed point of our developed stabilization system.
+
+# 7. Conclusion
+
+In this paper, we have presented a fast full-frame video stabilization technique based on the iterative optimization strategy. Our approach can be interpreted as the combination of probabilistic stabilization network (coarse-to-fine extension of PDC-Net) and video outpainting outwork (flow-based image outpainting). When trained on synthetic data constructed within the optimization-based learning framework, our method achieves state-of-the-art performance at a fraction of the computational cost of other competing techniques. It is also empirically verified that stabilized video is the fixed point of the stabilization network.
+
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2023/Fast Globally Optimal Surface Normal Estimation from an Affine Correspondence/ecc07b62-9596-44a0-9745-a517e83aa84a_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "Fast Globally Optimal Surface Normal from an Affine Correspondence",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Levente Hajder, Lajos Lóczi  \nGeometric Computer Vision Group  \nEötvös Loránd University, Budapest, Hungary  \n{hajder, lloczi}@inf.elte.hu",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Daniel Barath  \nComputer Vision and Geometry Group  \nETH Zurich, Switzerland  \ndbarath@inf.ethz.ch",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We present a new solver for estimating a surface normal from a single affine correspondence in two calibrated views. The proposed approach provides a new globally optimal solution for this over-determined problem and proves that it reduces to a linear system that can be solved extremely efficiently. This allows for performing significantly faster than other recent methods, solving the same problem and obtaining the same globally optimal solution. We demonstrate on  $15k$  image pairs from standard benchmarks that the proposed approach leads to the same results as other optimal algorithms while being, on average, five times faster than the fastest alternative. Besides its theoretical value, we demonstrate that such an approach has clear benefits, e.g., in image-based visual localization, due to not requiring a dense point cloud to recover the surface normal. We show on the Cambridge Landmarks dataset that leveraging the proposed surface normal estimation further improves localization accuracy. Matlab and  $C++$  implementations are also published in the supplementary material.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Acquiring oriented point clouds is a fundamental problem both in computer vision and in robotics with a number of important real-world applications benefiting from the extra information it provides. Such applications include 3D reconstruction [1, 8, 31, 62, 72, 65, 56] and simultaneous localization and mapping [19, 23, 50], where the 3D point orientations provide useful information about the underlying structure. Image-based visual localization approaches [43, 53, 60, 61] can also benefit from known surface normals [68]. The widely used Poisson reconstruction [37, 38] is based on both the point coordinates and their orientations. The visual odometry [51, 52] of an autonomous vehicle can also benefit from understanding surface orientation to, e.g., segment ground or facades. Also, having an oriented point cloud greatly simplifies geometric model and multi-model estimation [32, 4, 5] due to allowing to design solvers that",
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+    {
+        "type": "image",
+        "img_path": "images/0c48ef7f051b42a370dc08fa7ade8ffa2f4139ea4d5f11d69b72a1c707315bda.jpg",
+        "image_caption": [
+            "Figure 1. Geometric configuration: oriented 3D point  $\\mathbf{P} \\in \\mathbb{E}^3$  with normal  $\\mathbf{n} \\in \\mathbb{E}^3$  projected to image  $\\Pi^1$  as  $\\mathbf{p}_1 \\in \\mathbb{E}^2$  and to  $\\Pi^2$  as  $\\mathbf{p}_2 \\in \\mathbb{E}^2$ . The local patch centered at points  $\\mathbf{p}_1$  and  $\\mathbf{p}_2$  are related by the local affine transformation  $\\mathbf{A} \\in \\mathbb{R}^{2 \\times 2}$ ."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "require fewer data points than their point-based counterparts [64]. Moreover, normals can also be leveraged in the numerical refinement of the reconstructed oriented 3D point cloud [27]. In this paper, we focus on a practical problem, i.e., estimating the normal at corresponding features in two images in a solely photometric manner. This can provide a lightweight approach of obtaining surface normals without requiring a dense point cloud to be known. Next, we will discuss ways of obtaining oriented points.",
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+        "text": "Normals from Point Clouds. After applying state-of-the-art sparse (e.g., COLMAP [63]) or dense [36, 16] reconstruction pipelines, the resulting point cloud can be straightforwardly equipped with point orientations. In this case, typically, the normals are fitted per-point, applying the widely used Principal Component Analysis (PCA) [35] to estimate tangent planes or higher-order surfaces [14, 33] from the local neighborhood centered at the current point. This procedure, while being frequently used, has two major drawbacks. First, the setting of the neighborhood size greatly affects the accuracy and is often unclear without having a metric reconstruction. Instead, using the  $k$ -nearest-neighbors [17] algorithm to find local neighborhoods is a viable solution, but the estimation becomes largely affected by the sparsity of the point cloud. Second, fitting local surfaces, independently, to the neighborhood of each 3D point",
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+    {
+        "type": "header",
+        "text": "CVF",
+        "bbox": [
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+    },
+    {
+        "type": "header",
+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "3390",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "in the reconstruction is an extremely expensive operation.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Normals from Homographies. Another method for estimating surface normals from images, without requiring the underlying 3D structure to be known, is via homographies. Homographies are projective transformations between images of co-planar 3D points that depends on the intrinsic and extrinsic camera parameters and the parameters of the underlying 3D plane [24]. A homography can be estimated from a minimum of four point correspondences in two images [30]. While this method allows the estimation of dominant planes supported by a number of correspondences in the images, it does not provide means for normal estimation for each local feature correspondence independently.",
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+    {
+        "type": "text",
+        "text": "Normals from Local Mappings. In recent years, affine correspondences (AC) have emerged as a popular topic in computer vision, offering a valuable source of information about the underlying 3D geometry of the surrounding environment. An AC consists of a corresponding pair of points in two images and a linear transformation that maps the infinitesimal region around the point in the first image to the point in the second one. They have been used to solve various geometric problems, including relative pose estimation in the general case [15, 55, 3, 22, 10], for known vertical direction [25], planar motion [25, 26], or known feature depths [21] and homography estimation [2].",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Traditionally, finding affine features involve four main steps: (scale-covariant) keypoint detection, orientation and affine shape estimation, and descriptor extraction. Under such paradigm, keypoint detection is typically performed on the scale pyramid with the help of a handcrafted response function, such as Hessian [13, 45], Harris [28, 45], Difference of Gaussians (DoG [42]), or learned ones like FAST [57] or Key.Net [11]. Keypoint detection obtains a triplet  $(x,y,\\mathrm{scale})$  which defines a square or circular image patch. Then the patch orientation is estimated with the help of handcrafted - dominant gradient orientation [42], center of the mass [58] - or learned methods [71, 48, 41]. Next, the affine-covariant shape [12, 48] is estimated. Finally, the patch is geometrically rectified and fed into a local patch descriptor, such as SIFT, HardNet [47], SOSNet [66], or others. In summary, all feature detectors, even recent learning-based ones like SuperPoint [20], can be made affine covariant by applying, e.g., AffNet [48] as a post-processing step on the found features.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The problem of surface normal estimation from ACs has been addressed in several works. The first algorithm for normal estimation from ACs was proposed by Kevin Koeser [40] assuming that the camera location is known with respect to the observed 3D point. Megyesi et al. [44] proposed a special surface normal estimator for standard (rectified) stereo, where the degree of freedom of the affine transformations decreases to two. In 2014, Molnár and Chetverikov [49] derived a formula connecting the surface",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "normals, camera parameters, affine transformations, and patch locations into a single expression for arbitrary projective functions, like the catadioptric camera model. Barath et al. [9] proposed a globally optimal estimator in the least squares sense as the solution of a quartic polynomial. Recently, Le Minh and Hajder [46] improved on this by reducing the degree and showed that the optimal solution can be found as a real root of a cubic polynomial.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The main goal of this paper is to propose a rapid normal estimator based on affine correspondences. Our algorithm can be inserted into a 3D reconstruction system, such as Structure-from-Motion [27] (SfM) or Simultaneous Localization and Mapping (SLAM), where the normal estimation is performed many times, therefore its running time influences the time demand of the whole 3D pipeline.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Contribution. In this paper, we propose a new linear and globally optimal solution for surface normal estimation from a single affine correspondence. The proposed solver requires only the four basic arithmetic operations and contains 19 parameters, consisting of the coordinates of the five input vectors and the four affine parameters. This approach completely avoids root extractions, solving high-degree polynomial equations, or employing numerical or iterative methods, resulting in a particularly fast algorithm that runs for only  $0.6\\mu s$  in our experiments. The solver is five times faster than the cubic solver from [46]. Additionally, we propose a way of exploiting AC-wise normals in image-based visual localization pipelines. The algorithms are tested on 15k image pairs from standard benchmarks, and our results show superior accuracy and faster wall-clock times compared to the baselines.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2. Proposed Method",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The setting is visualized in Fig 1. We are given two images, denoted by  $\\Pi^1$  and  $\\Pi^2$ . The 2D and 3D point locations are visualized as black dots. For standard PC-based stereo vision, the 3D location can be obtained by the so-called triangulation method. The local affine transformation, denoted by  $\\mathbf{A} \\in \\mathbb{R}^{2 \\times 2}$ , is defined as the first-order Taylor-approximation of the imaging function at points  $\\mathbf{p}_1$  and  $\\mathbf{p}_2$ . It maps the infinitesimal patch around the observed points from the first to the second image. If  $\\mathbf{A}$  is known, the surface normal  $\\mathbf{n}$  can be directly estimated without knowing the actual 3D structure of the underlying scene [49].",
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+        "type": "text",
+        "text": "Parameterization. A surface normal  $\\mathbf{n} \\in \\mathbb{R}^3$  is a 3D direction, therefore it can be represented by two parameters. Usually, it is parameterized by three coordinates  $\\mathbf{n} = [n_x, n_y, n_z]^\\top$  and its length being set to 1. However, as it turns out below, some other scaling will be more convenient in the present situation.",
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+    {
+        "type": "text",
+        "text": "Problem Statement. Let us assume that we are given a general camera that projects the spatial location  $\\mathbf{P} = [X, Y, Z]^\\top$  to image (pixel) coordinates  $[u v]^\\top$ , and let us",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "3391",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "denote the projective coordinate functions by  $\\Pi_u: \\mathbb{E}^3 \\to \\mathbb{R}$  and  $\\Pi_v: \\mathbb{E}^3 \\to \\mathbb{R}$ . Function  $\\Pi_u$  calculates coordinate  $u$  of the projected point, and  $\\Pi_v$  calculates  $v$ . As Molnár and Chetverikov showed [49], the affine transformation can be defined via the spatial gradients of the projective coordinate functions  $\\Pi_u$  and  $\\Pi_v$ .",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "An affine correspondence is represented by 2D image coordinates  $[u_1 v_1]^\\top$  and  $[u_2 v_2]^\\top$  in the stereo image pair, in conjunction with the affine transformation  $\\mathbf{A}$  that maps the surrounding area of the locations from first image to the second one. The spatial location of the corresponding spatial point is also required. It can be estimated from the image coordinates and camera parameters by standard triangulation techniques [29].",
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+    {
+        "type": "text",
+        "text": "The normal estimation problem is written [9] as an argument minimization to determine the optimal  $\\mathbf{n}^*$  such that",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {n} ^ {*} = \\operatorname * {a r g   m i n} _ {\\mathbf {n}} f (\\mathbf {n}),\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "with",
+        "bbox": [
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+    {
+        "type": "equation",
+        "text": "\n$$\nf (\\mathbf {n}) := \\sum_ {i = 1} ^ {2} \\sum_ {j = 1} ^ {2} \\left(\\frac {\\mathbf {n} ^ {\\top} \\mathbf {w} _ {i j}}{\\mathbf {n} ^ {\\top} \\mathbf {w} _ {c}} - a _ {i j}\\right) ^ {2}. \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, the affine transformation yields the four affine parameters  $a_{ij}$  ( $i,j\\in [1,2]$ ) according to",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {A} = \\left[ \\begin{array}{c c} a _ {1 1} & a _ {1 2} \\\\ a _ {2 1} & a _ {2 2} \\end{array} \\right],\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            200,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and the vectors  $\\mathbf{w}_{ij}$  and  $\\mathbf{w}_c$  are obtained as cross products of the spatial gradient vectors as follows:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {w} _ {1 1} = \\nabla \\Pi_ {v} ^ {1} \\times \\nabla \\Pi_ {u} ^ {2}, \\quad \\mathbf {w} _ {1 2} = \\nabla \\Pi_ {u} ^ {2} \\times \\nabla \\Pi_ {u} ^ {1}, \\\\ \\mathbf {w} _ {2 1} = \\nabla \\Pi_ {v} ^ {1} \\times \\nabla \\Pi_ {v} ^ {2}, \\quad \\mathbf {w} _ {2 2} = \\nabla \\Pi_ {v} ^ {2} \\times \\nabla \\Pi_ {u} ^ {1}, \\\\ \\mathbf {w} _ {c} = \\nabla \\Pi_ {u} ^ {1} \\times \\nabla \\Pi_ {v} ^ {1}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the superscripts refer to the corresponding image index. The coordinates of these vectors will be denoted by  $\\mathbf{w}_{ij,x},\\mathbf{w}_{ij,y},\\mathbf{w}_{ij,z}$ $(i,j\\in \\{1,2\\})$  and  $\\mathbf{w}_{c,x},\\mathbf{w}_{c,y},\\mathbf{w}_{c,z}$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Pin-hole (perspective) projection model. For a pin-hole camera, the projection model is as follows:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{c} u \\\\ v \\\\ 1 \\end{array} \\right] \\sim \\left[ \\begin{array}{c c c c} P _ {1 1} & P _ {1 2} & P _ {1 3} & P _ {1 4} \\\\ P _ {2 1} & P _ {2 2} & P _ {2 3} & P _ {2 4} \\\\ P _ {3 1} & P _ {3 2} & P _ {3 3} & P _ {3 4} \\end{array} \\right] \\left[ \\begin{array}{c} X \\\\ Y \\\\ Z \\\\ 1 \\end{array} \\right],\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            114,
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+            787
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\sim$  denotes the equality up to scale operation. Therefore, image coordinates  $u$  and  $v$  are obtained as",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\Pi_ {u} = \\frac {P _ {1 1} X + P _ {1 2} Y + P _ {1 3} Z + P _ {1 4}}{P _ {3 1} X + P _ {3 2} Y + P _ {3 3} Z + P _ {3 4}}, \\\\ \\Pi_ {v} = \\frac {P _ {2 1} X + P _ {2 2} Y + P _ {2 3} Z + P _ {2 4}}{P _ {3 1} X + P _ {3 2} Y + P _ {3 3} Z + P _ {3 4}}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Spatial gradients, that are  $\\nabla \\Pi = \\left[ \\begin{array}{ccc}\\frac{\\partial\\Pi}{\\partial X} & \\frac{\\partial\\Pi}{\\partial Y} & \\frac{\\partial\\Pi}{\\partial Z} \\end{array} \\right]^{\\top}$ , can be written for the pin-hole model in compact forms as",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\begin{array}{r l} \\nabla \\Pi_ {u} & = \\frac {1}{s} \\left[ \\begin{array}{c} P _ {1 1} \\\\ P _ {1 2} \\\\ P _ {1 3} \\end{array} \\right] ^ {\\top} - \\frac {u}{s} \\left[ \\begin{array}{c} P _ {3 1} \\\\ P _ {3 2} \\\\ P _ {3 3} \\end{array} \\right] ^ {\\top}, \\end{array} \\\\ \\nabla \\Pi_ {v}   =   \\frac {1}{s} \\left[ \\begin{array}{c} P _ {2 1} \\\\ P _ {2 2} \\\\ P _ {2 3} \\end{array} \\right] ^ {\\top} - \\frac {v}{s} \\left[ \\begin{array}{c} P _ {3 1} \\\\ P _ {3 2} \\\\ P _ {3 3} \\end{array} \\right] ^ {\\top}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where  $s = P_{31}X + P_{32}Y + P_{33}Z + P_{34}$  is the projective depth.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Solution. For the problem to be meaningful, we can assume that  $\\mathbf{w}_c \\neq \\mathbf{0}$ . Let us set",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nI := \\inf  _ {\\| \\mathbf {n} \\| = 1, \\mathbf {n} ^ {\\top} \\mathbf {w} _ {c} \\neq 0} f (\\mathbf {n}),\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $\\| \\mathbf{n}\\| \\coloneqq \\sqrt{\\mathbf{n}^\\top\\mathbf{n}}$  denotes the standard Euclidean norm. Clearly,  $f\\geq 0$  is undefined along the \"equator\" of the unit sphere  $\\| \\mathbf{n}\\| = 1$  with  $\\mathbf{n}^{\\top}\\mathbf{w}_{c} = 0$ . Let us consider two cases.",
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+    {
+        "type": "text",
+        "text": "If all  $\\mathbf{w}_{ij}$  and  $\\mathbf{w}_c$  are equal, then trivially  $f(\\mathbf{n}) = \\sum_{i,j=1}^{2}(a_{ij} - 1)^2$  is a constant function.",
+        "bbox": [
+            500,
+            410,
+            890,
+            444
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Otherwise, at least one of the expressions  $\\frac{\\mathbf{n}^{\\top}\\mathbf{w}_{ij}}{\\mathbf{n}^{\\top}\\mathbf{w}_{c}}$  becomes arbitrarily large in modulus when  $\\mathbf{n}$  is chosen such that  $\\mathbf{n}^{\\top}\\mathbf{w}_c$  gets sufficiently close to 0; that is,  $|f(\\mathbf{n})|\\to +\\infty$  as  $\\mathbf{n}^{\\top}\\mathbf{w}_c\\rightarrow 0$  with  $\\| \\mathbf{n}\\| = 1$ . Therefore, by taking into account the continuity of  $f$  as well, the infimum  $I$  is in fact a minimum, and an optimal unit vector  $\\mathbf{n}^{*}$  with  $f(\\mathbf{n}^{*}) = I$  exists. In what follows, we will explicitly determine such an  $\\mathbf{n}^{*}$  as a function of the 19 parameters<sup>1</sup> by using only the four basic arithmetic operations.",
+        "bbox": [
+            498,
+            445,
+            890,
+            584
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A key observation is that  $f(\\lambda \\mathbf{n}) = f(\\mathbf{n})$  for any  $\\lambda \\neq 0$ , so (after a possible rescaling) we can assume that the last component of  $\\mathbf{n}$  is either 1 or 0. These cases, Cases A and B, respectively, will be discussed separately below. To simplify the notation, let us define two auxiliary functions",
+        "bbox": [
+            498,
+            585,
+            890,
+            660
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ng \\left(n _ {x}, n _ {y}\\right) := f \\left(n _ {x}, n _ {y}, 1\\right) \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            606,
+            674,
+            890,
+            690
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and",
+        "bbox": [
+            500,
+            703,
+            527,
+            715
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh \\left(n _ {x}, n _ {y}\\right) := f \\left(n _ {x}, n _ {y}, 0\\right). \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            604,
+            719,
+            890,
+            736
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For the objective function  $g$ , we will obtain a unique candidate extremum  $\\mathbf{n}_g^* \\in \\mathbb{R}^2$ ; whereas for the function  $h$ , a one-parameter family of candidate extrema  $\\mathbf{n}_h^* \\in \\mathbb{R}^2$  will be found but with the same function value  $h(\\mathbf{n}_h^*)$ . The optimal  $\\mathbf{n}^*$  is then",
+        "bbox": [
+            498,
+            744,
+            890,
+            818
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {n} ^ {*} = \\left\\{ \\begin{array}{l l} \\left[ \\mathbf {n} _ {g} ^ {*}, 1 \\right] ^ {\\top}, & \\text {i f} g \\left(\\mathbf {n} _ {g} ^ {*}\\right) \\leq h \\left(\\mathbf {n} _ {h} ^ {*}\\right), \\\\ \\left[ \\mathbf {n} _ {h} ^ {*}, 0 \\right] ^ {\\top}, & \\text {i f} g \\left(\\mathbf {n} _ {g} ^ {*}\\right) > h \\left(\\mathbf {n} _ {h} ^ {*}\\right). \\end{array} \\right. \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            557,
+            829,
+            890,
+            869
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>They are the four affine parameters  $a_{ij}$ , and the 15 components of the vectors  $\\mathbf{w}_{ij}$  and  $\\mathbf{w}_c$ .",
+        "bbox": [
+            500,
+            875,
+            890,
+            900
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3392",
+        "bbox": [
+            482,
+            944,
+            514,
+            955
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Remark 1 In other words, we will be utilizing the simple constraints  $n_z = 1$  or  $n_z = 0$ . We have found that other linear or non-linear constraints, such as  $\\| \\mathbf{n} \\| = 1$ , would render the symbolic calculations to minimize the objective function  $f$  infeasible.",
+        "bbox": [
+            73,
+            90,
+            472,
+            167
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Case A:  $n_z = 1$",
+        "bbox": [
+            76,
+            186,
+            189,
+            202
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use the standard technique and solve the system",
+        "bbox": [
+            76,
+            210,
+            419,
+            226
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\{ \\begin{array}{l} \\partial_ {1} g \\left(n _ {x}, n _ {y}\\right) = 0, \\\\ \\partial_ {2} g \\left(n _ {x}, n _ {y}\\right) = 0 \\end{array} \\right. \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            197,
+            238,
+            468,
+            280
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "to find the optimal vector  $\\mathbf{n}_g^* = [n_x, n_y]^\\top$  as follows.",
+        "bbox": [
+            76,
+            290,
+            426,
+            308
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "By computing and putting the partial derivative",
+        "bbox": [
+            96,
+            308,
+            410,
+            321
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\partial_ {1} g (n _ {x}, n _ {y})\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            228,
+            335,
+            315,
+            353
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "over a common denominator, we see that its numerator depends on  $n_x$  linearly; and it depends on  $n_y$  quadratically; so one can easily express  $n_x$  from the equation  $\\partial_1 g(n_x, n_y) = 0$  as follows:",
+        "bbox": [
+            76,
+            364,
+            468,
+            421
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nn _ {x} = \\frac {\\operatorname {n u m e r} _ {1}}{\\operatorname {d e n o m} _ {1}}, \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            218,
+            420,
+            468,
+            449
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we present the expressions for  $\\mathrm{numer}_1$  and  $\\mathrm{denom}_1$  in the supplementary material. Then (6) is substituted into the other partial derivative  $\\partial_2 g(n_x, n_y)$ . The resulting rational function of  $n_y$ , also containing the other 19 parameters, can be simplified and factorized by Wolfram Mathematica's Factor command<sup>2</sup>. In this way, we obtain a fraction whose structure is the following:",
+        "bbox": [
+            76,
+            455,
+            468,
+            560
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {\\operatorname {l i n e a r} _ {1} \\cdot \\operatorname {l i n e a r} _ {2}}{(\\text {q u a d r a t i c}) ^ {2}}, \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            571,
+            468,
+            604
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the expressions \"linear\" and \"quadratic\" denote polynomials in  $n_y$  having the corresponding degrees.",
+        "bbox": [
+            76,
+            614,
+            468,
+            645
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The optimum of  $g$  can occur when  $n_y$  is chosen such that the fraction (7), that is, its numerator is equal to 0. It can be shown that the assumption  $\\mathrm{linear}_1 = 0$  leads to a contradiction - indeed, by solving  $\\mathrm{linear}_1 = 0$  for  $n_y$ , the resulting formula would make  $\\mathrm{denom}_1$  in (6) identically equal to 0. Hence, we are only left with the equation  $\\mathrm{linear}_2 = 0$ , whose solution is",
+        "bbox": [
+            76,
+            645,
+            468,
+            750
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nn _ {y} = \\frac {\\operatorname {n u m e r} _ {2}}{\\operatorname {d e n o m} _ {2}}. \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            218,
+            758,
+            468,
+            787
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The actual formulas for  $\\mathrm{numer}_2$  and  $\\mathrm{denom}_2$  are found again in the supplementary material.",
+        "bbox": [
+            76,
+            797,
+            468,
+            827
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Thus, the second component of  $\\mathbf{n}_g^*$  is given by (8) (expressed in terms of the 19 parameters). Then, by using this together with (6), we obtain  $n_x$ , the first component of  $\\mathbf{n}_g^*$ .",
+        "bbox": [
+            76,
+            828,
+            468,
+            876
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark 2 The essential feature of these computations is that the solution of system (5) is given in a fully explicit symbolic form by using only elementary operations. Without the clever intermediate symbolic simplifications and factorizations (most notably (7)), one would need to solve a higher-degree polynomial equation to obtain  $n_y$ , requiring either taking roots or using an iterative scheme. For example, if one starts solving (5) with  $n_x$  and  $n_y$  as symbolic parameters but all the coefficients of  $g$  as floating-point numbers, then, in general, no intermediate factorizations or simplifications are possible, so in order to determine  $n_y$ , one would need to solve a quintic equation.",
+        "bbox": [
+            500,
+            90,
+            893,
+            273
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Case B:  $n_z = 0$",
+        "bbox": [
+            500,
+            290,
+            609,
+            304
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We proceed similarly and solve the system",
+        "bbox": [
+            500,
+            314,
+            782,
+            329
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\{ \\begin{array}{l} \\partial_ {1} h \\left(n _ {x}, n _ {y}\\right) = 0, \\\\ \\partial_ {2} h \\left(n _ {x}, n _ {y}\\right) = 0. \\end{array} \\right. \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            620,
+            337,
+            890,
+            378
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "By expressing  $n_x$  from the first equation of (9) and substituting the resulting expression into the second equation, we find that this time the second equation is automatically satisfied. Therefore, (9) possesses a one-parameter family of solutions given by",
+        "bbox": [
+            498,
+            387,
+            892,
+            462
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nn _ {x} = \\frac {\\operatorname {n u m e r} _ {3}}{\\operatorname {d e n o m} _ {3}}, \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            642,
+            469,
+            890,
+            500
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "see the supplementary material. By substituting this into  $h$ , the function value  $h(\\mathbf{n}_h^*)$  appearing in (4) is obtained as",
+        "bbox": [
+            498,
+            507,
+            890,
+            539
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nh \\left(\\mathbf {n} _ {h} ^ {*}\\right) = \\frac {\\operatorname {n u m e r} _ {4}}{\\operatorname {d e n o m} _ {4}}. \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            630,
+            546,
+            890,
+            575
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "See the supplementary material for details.",
+        "bbox": [
+            500,
+            584,
+            782,
+            599
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark 3 One directly checks that the value  $h(\\mathbf{n}_h^*) = h\\left(\\frac{\\mathrm{numer}_3}{\\mathrm{denom}_3}, n_y\\right)$  is indeed independent of  $n_y$ , as claimed.",
+        "bbox": [
+            498,
+            609,
+            890,
+            648
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.1. Summary of the Algorithm",
+        "text_level": 1,
+        "bbox": [
+            500,
+            657,
+            743,
+            672
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given the elements  $a_{ij} \\in \\mathbb{R}$  in the affine transformation and 3D vectors  $\\mathbf{w}_{ij}$  and  $\\mathbf{w}_c \\neq \\mathbf{0}$ , let us define the function  $f$  by (1) for  $\\mathbf{n}$  with  $\\mathbf{n}^\\top \\mathbf{w}_c \\neq \\mathbf{0}$ , and also define the functions  $g$  and  $h$  by (2)-(3). If  $\\mathbf{w}_{11} = \\mathbf{w}_{12} = \\mathbf{w}_{21} = \\mathbf{w}_{22} = \\mathbf{w}_c$ , then any unit vector  $\\mathbf{n}^*$  can be chosen as optimal normal vector.",
+        "bbox": [
+            498,
+            681,
+            890,
+            755
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Otherwise, we compute  $\\mathbf{n}_g^* \\coloneqq [n_x, n_y]^\\top$  from (8) and (6). Then we compute  $h(\\mathbf{n}_h^*)$  from (11). Finally, the optimal  $\\mathbf{n}^*$  is determined by using (4).",
+        "bbox": [
+            498,
+            756,
+            890,
+            801
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3. Experiments",
+        "text_level": 1,
+        "bbox": [
+            500,
+            815,
+            632,
+            830
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We focus on the accuracy and time demand of the algorithms when testing. Both synthetic testing data, including configurations for a number of important camera motions, and standard real-world benchmarks are considered here.",
+        "bbox": [
+            498,
+            840,
+            890,
+            898
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{2}$ https://www.wolfram.com/mathematica/",
+        "bbox": [
+            94,
+            886,
+            316,
+            900
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "3393",
+        "bbox": [
+            482,
+            944,
+            514,
+            955
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/3361adbf5f991a037b270f6cca617713817d6b0f9a68b30df32547514bbdf2f7.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            80,
+            88,
+            271,
+            188
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/9443b008b2f21844cc9e34eed51e5c36849a35ae5c047f53f1ef7a7de01f9ca6.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            88,
+            465,
+            188
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/7aa5f0a54ff8a3fb5ede75cbbad692e3c95bd5b99bdad1a20caee658846f7d38.jpg",
+        "image_caption": [
+            "Figure 2. Images and the reference reconstruction used from the Cambridge Landmarks dataset [39] in localization experiments."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            98,
+            202,
+            250,
+            287
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/52da8b4b8b4e061f0e13354f93a23bc7e07fd79f0a4315df454e46bb22dc7e4d.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            277,
+            195,
+            464,
+            280
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Compared Algorithms. In this section, we discuss the algorithms that can be used to estimate the surface normal from a single affine correspondence.",
+        "bbox": [
+            76,
+            342,
+            468,
+            388
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. The provably most accurate algorithms are the ones that are optimal in the least squares sense. Method  $\\mathrm{OPT}_{\\mathrm{QUARTIC}}$  refers to [9] which solves the problem as a real root of a quartic polynomial equation. The cubic solution  $\\mathrm{OPT}_{\\mathrm{CUBIC}}$  is from [46]. The proposed optimal solver, as it is solved via linear equations, is called  $\\mathrm{OPT}_{\\mathrm{LINEAR}}$  in the rest of the paper.",
+            "2. The simplest and, thus, the fastest method is the Fast Normal Estimation (FNE) algorithm of [9]. A more complex variant, when all possible linear equations are considered, is also constructed here, it is denoted as  $\\mathrm{FNE}_{\\mathrm{SUPER}}$ . The modified equations can be seen in the source codes in the supplementary material.",
+            "3. Raposo and Barreto [56] proposed a solution that estimates the normal via homographies. We have not found an implementation or an exact definition of the algorithm, so we implemented it based on the description. The solution is computed via an over-determined inhomogeneous linear system of equations. The formulas are generated by the MATLAB Symbolic Toolbox. Details are given in the supplementary material. The method is called RAPOSO in the charts."
+        ],
+        "bbox": [
+            89,
+            396,
+            468,
+            738
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The synthetic tests are performed in MATLAB. In the real-world experiments, optimized  $\\mathrm{C}++$  codes run on an Intel(R) Core(TM) i9-10900K CPU @ 3.70GHz.",
+        "bbox": [
+            76,
+            747,
+            468,
+            792
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1. Synthetic Experiments",
+        "text_level": 1,
+        "bbox": [
+            76,
+            801,
+            287,
+            816
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the synthetic tests, a spherical object is generated, thus straightforwardly allowing to sample oriented points from its surface. Two cameras are randomly generated with focal lengths from the range [500, 1500] and with image size  $2000 \\times 2000$ . Their intrinsic matrices are  $\\mathbf{K}_1$  and  $\\mathbf{K}_2$ .",
+        "bbox": [
+            76,
+            825,
+            467,
+            900
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The 3D point was projected into both cameras. The ground truth (GT) affine parameters were calculated directly from the camera parameters and the surface normals as shown in [9]. Several camera configurations are considered to simulate various real-world scenarios. They are as follows:",
+        "bbox": [
+            498,
+            90,
+            890,
+            167
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- For general stereo pose, the locations of the cameras are randomly generated inside a unit-radius sphere. The camera projection matrices are constructed as  $\\mathbf{P}_1 = \\mathbf{K}_1[\\mathbf{I} \\mid \\mathbf{0}]$  and  $\\mathbf{P}_2 = \\mathbf{K}_2[\\mathbf{R} \\mid \\mathbf{t}]$ .",
+            "- For rectified stereo, the cameras are located next to each other with identical orientation. In this case, the projection matrices are  $\\mathbf{P}_1 = \\mathbf{K}_1[\\mathbf{I} \\mid \\mathbf{0}]$  and  $\\mathbf{P}_2 = \\mathbf{K}_2[\\mathbf{I} \\mid \\mathbf{t}]$ , where  $\\mathbf{t} = [\\Delta X 0 0]^\\top$ .",
+            "- Planar motion simulates the case when a camera is mounted on a vehicle that moves on a planar road with the image plane orthogonal to the ground. Such a motion can be represented by two angles around the vertical axis, one for the rotation and one for the translation. In this case, the projection matrices can be written as  $\\mathbf{P}_1 = \\mathbf{K}_1[\\mathbf{I} \\mid 0]$  and  $\\mathbf{P}_2 = \\mathbf{K}_2[\\mathbf{R}_{\\mathrm{Y}} \\mid \\mathbf{t}]$ , where  $\\mathbf{t} = \\rho [\\cos \\beta, 0, \\sin \\beta]^\\top$  and"
+        ],
+        "bbox": [
+            517,
+            174,
+            893,
+            426
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {R} _ {\\mathrm {Y}} = \\left[ \\begin{array}{c c c} \\cos \\alpha & 0 & \\sin \\alpha \\\\ 0 & 1 & 0 \\\\ - \\sin \\alpha & 0 & \\cos \\alpha \\end{array} \\right].\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Moto-motion. This test case simulates a camera that is mounted on a motorcycle. If the bike leans, another rotation, the roll is involved in the problem. In this case, the camera parameters are  $\\mathbf{P}_1 = \\mathbf{K}_1[\\mathbf{I} \\mid \\mathbf{0}]$  and  $\\mathbf{P}_2 = \\mathbf{K}_2[\\mathbf{R}_{\\mathrm{YZ}} \\mid \\mathbf{t}]$ , where  $\\mathbf{t} = \\rho [\\cos \\beta 0 \\sin \\beta]^\\top$  and  $\\mathbf{R}_{YZ} = \\mathbf{R}_Y \\mathbf{R}_Z$ .",
+            "- Drone-motion is another generalization of the planar motion. A camera mounted on a flying drone. Its location is arbitrary, but the orientation is limited to a vertical rotation. Therefore,  $\\mathbf{P}_1 = \\mathbf{K}_1[\\mathbf{I} \\mid \\mathbf{0}]$  and  $\\mathbf{P}_2 = \\mathbf{K}_2[\\mathbf{R}_Y \\mid \\mathbf{t}]$ , where  $\\mathbf{t} = [X Y Z]^\\top$ ."
+        ],
+        "bbox": [
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+            494,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Noise. We have added zero-mean Gaussian noise to the 2D coordinates, therefore, the point noise is given in pixels. The deviation of the noise is the parameter in the charts.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The  $2 \\times 2$  affine transformations are decomposed by SVD into two 2D rotations and another two scaling parameters. Then zero-mean Gaussian noise is added to the angle of rotations, in radians, and the scaling parameters. The standard deviation of the Gaussian noise distribution is equally set for all the four parameters.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3D Reconstruction Pipeline. In order to simulate real scenarios, first, we triangulate each generated noisy point correspondence by the optimal triangulation method proposed by Hartley and Sturm [29]. The noisy affine parameters are corrected by the method proposed in [7] to be consistent with the camera parameters.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "3394",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/e29247ae079a115614ff62377b1ca561bfb6a47a6f259aefd29da6c614187ebc.jpg",
+        "image_caption": [
+            "Figure 3. The frequencies (in 10000 runs) of the  $\\log_{10}$  errors in the  $\\log_{10}$  angular errors in degrees in the noise-free case."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Stability test. We generated 10000 random problem instances considering general camera motion and ran the normal estimators on noiseless affine correspondences. Fig. 3 shows histograms of  $\\log_{10}$  angular errors in degrees. Note that for better visualization, we replace the exact zero values by  $10^{-16}$  as it is the precision of the floating point numbers in MATLAB. The plots show that all our solvers are very stable – there is no peak close to  $10^0$ . For all methods, the majority of the errors are distributed around  $10^{-16}$  which actually is the precision of MATLAB. This means that in most of the cases, all methods are particularly stable returning almost exactly 0 error. A small peak is over  $10^{-6}$  for all methods similarly. Having  $10^{-6}$  degree error is acceptable in most of the scenarios considered in computer vision. There is no peak close to  $10^0$  implying that all solvers are stable. Note that these stability tests usually are for highlighting potential degeneracies where the error exceeds  $10^0$ .",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We also conducted tests using single-point precision in MATLAB. The OPTQUARTIC approach failed entirely under these conditions. For all other methods, we noticed an increase in error, which suggests that double precision is necessary for reliable normal estimation. Interestingly, our proposed OPTLINEAR method appears to be slightly more sensitive to precision than OPTCUBIC is. This is likely due to the fact that OPTLINEAR employs more complex formulas and, consequently, performs more stably with double precision.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Accuracy test. The accuracy of the normals is examined for all the considered camera poses independently. The results are shown in Fig 9. The effect of noise in point coordinates and affine parameters are separately evaluated. Both the mean and the median of the errors are plotted.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The observations we can make from the plots are as follows. First, the three optimal methods  $\\mathrm{OPT}_{\\mathrm{QUARTIC}}$ ,  $\\mathrm{OPT}_{\\mathrm{CUBIC}}$  and  $\\mathrm{OPT}_{\\mathrm{LINEAR}}$  always lead exactly to the same solution, thus their curves coincide in all plots. Second, the optimal methods always lead to the most accurate results if the affine parameters are contaminated by noise. This is not",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "surprising as these methods optimize the normal parameters such that the implied affine transformation is the closest to the measured one. In the case of point-noise only, the accuracy of the optimal algorithm is slightly below that of the competitors. However, it is an unlikely situation in real-world images to have noisy points but perfect local affine frames. It is usually the other way around. Third, as the FNE method uses less information, it is clearly the worst one in terms of accuracy.  $\\mathrm{FNE}_{\\mathrm{SUPER}}$  performs significantly better, however, it requires more time as the right singular vector of a  $6\\times 3$  matrix, corresponding to the smallest singular value, should be computed. Another benefit of  $\\mathrm{FNE}_{\\mathrm{SUPER}}$  is, contrary to FNE and Raposo [56], that it works in all test cases. It can cope with both planar motion and rectified stereo, similarly to the optimal algorithms.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Problematic methods. Methods FNE and Raposo [56] do not work if the cameras undergo special translations. FNE is a special method, several variants can be formed based on the selection of the affine transformations. For rectified stereo, the third and fourth elements are fixed [44], they cannot be used for the estimation.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The Raposo [56] method is the most problematic one as the homography can be decomposed into the camera rotation, translation and surface normal. If the translation has zero elements, the retrieved equations are linearly dependent, and the estimation does not yield a correct solution. Due to this reason, Raposo [56] does not work for rectified stereo and planar motion. We removed these methods from the plots of the respective camera configurations.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Combined noise. In the previous experiments, noise is added either to the point coordinates or affine parameters. Here, we contaminate both types of input data. The mean and median errors of the proposed linear method are visualized in Fig. 10. As it is expected, the curves show similar trends, but the case when both the points and affines are noisy leads to higher errors.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Interpretable affine noise. In the previous experiments, the noise is added directly to the elements of the affine matrices. In Fig. 11, we consider a different way of contaminating the input to show that the proposed method acts reasonably with different noise models as well. The results of curves Aff and Comb are obtained by using the noise model from the previous sections. Comb refers to the case when both the points and the affine matrices are noisy. For AffElem and Comb-Elem, we decompose A by SVD into two orthogonal and a diagonal matrix. Fig. 11 demonstrates that the noise generation does not strongly influence the results.",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4. Real-world Experiments",
+        "text_level": 1,
+        "bbox": [
+            500,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To test normal estimators, we use the Cambridge Landmarks [39] and ScanNet datasets [18]. We chose Cambridge Landmarks, see Fig. 2 for example images, since it provides a large number of images for testing directly the",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "3395",
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+        "img_path": "images/e00cb504bc546457895ce0aa123a2fe0c8d9fb9be95d72a65326bff9fb5c44e0.jpg",
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+        "img_path": "images/b05dd400a77b1142754bc5e9cc19aa3b8a2e009f68fc685032cd21281077175e.jpg",
+        "image_caption": [
+            "Figure 4. General Motion."
+        ],
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+        "image_caption": [
+            "Figure 5. Rectified Stereo."
+        ],
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+        "img_path": "images/7a3d482fe64ec4cac62e80206b50e66a1356cbbd350843476ad055bb216694d7.jpg",
+        "image_caption": [
+            "Figure 6. Planar Motion."
+        ],
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+        "image_caption": [
+            "Figure 7. Moto Motion."
+        ],
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+        ],
+        "page_idx": 6
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+        "img_path": "images/44a2e7f549b351c8d796252cecccb50f32c7086737d70ced94c94b5755d43200.jpg",
+        "image_caption": [],
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/cdab81ac4f53dee3550ff07f2cb78219f422692eec1625340859fb715ff3ac2e.jpg",
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+        "img_path": "images/2f63d8ff61554da8b97f8bab4cd258b676cb22ba9d136c3c45a1ee820c178455.jpg",
+        "image_caption": [
+            "Figure 9. Average and median angular errors (in degrees) of surface normal estimators on synthetic data. Five camera configurations are considered. From left to right: (1) Median error; noise added to affine parameters. (2) Average error; noise added to affine parameters. (3) Median error; noise added to point coordinates. (4) Average error; noise added to point coordinates."
+        ],
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+        "img_path": "images/71f07ae3ccc694ec59f324292b47b7e713b21625dd14fc2e66ba00eb6d1f459a.jpg",
+        "image_caption": [
+            "Figure 8. Drone Motion."
+        ],
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+    {
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+        "img_path": "images/438bf52d4be6ad617d635fd593c4a4d9169606aedd16012c75b9c15a72f0c273.jpg",
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+    {
+        "type": "image",
+        "img_path": "images/125656e0a522fe3bbf44d20cfcd3921be0458d1b5d078f40f22ac2a806541cbd.jpg",
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+    },
+    {
+        "type": "text",
+        "text": "normal accuracy and, also, we can evaluate localization as an application of the proposed procedure. Also, we chose ScanNet since the reference normals can be calculated directly from the dense depth maps which provides a complementary comparison to the tests on Cambridge Landmarks,",
+        "bbox": [
+            75,
+            823,
+            470,
+            898
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where the normals come from a 3D reconstruction.",
+        "bbox": [
+            500,
+            823,
+            836,
+            835
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Cambridge Landmarks consists of five scenes with varying number of images, ranging from 231 to 3015. The reference pose of each image was reconstructed using the VisualSFM structure-from-motion software [70, 69] and the",
+        "bbox": [
+            498,
+            840,
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+            900
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "3396",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/afe3cccb487e6fa90b540974d96e7f2bfb61553b1921b007398e949eb17c0d4b.jpg",
+        "image_caption": [
+            "Figure 10. Mean and median errors of the proposed OPT LINEAR normal estimator with noise added to point coordinates (Point), affine parameters (Affine) or both (Combined)."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b9507a16a4962393dd58f28b472a2706f878510d3e77bd11f2d4779683469989.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/cfa87fde61ea568549103ddeb3b6007b3743ace39c2b1b98cf661245e154abdd.jpg",
+        "image_caption": [
+            "Figure 11. Mean and median errors of the proposed OPT LINEAR normal estimator with noise added to only the affine  $(A\\text{ff})$  elements and, also, to the point coordinates  $(Comb)$ . For Aff and Comb, zero-mean Gaussian noise is added directly to the affine elements. For Aff-Elem and Comb-Elem, the affine matrix is decomposed by SVD to two rotations and a diagonal matrix. Zero-mean Gaussian noise is added to the rotation angles and diagonal elements."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            99,
+            268,
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+            369
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/166482e29d8bfca0ca6612feb0cd377d7bca63f7c3742e99f5ecb687fe59ba8c.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "images are divided into training and testing sets. To have more accurate reference data, we ran COLMAP [63] obtaining a dense point cloud. Finally, we applied the normal estimator implemented in MeshLab<sup>3</sup> to obtain an oriented point cloud to be considered reference. The normals are fitted to the 200 nearest neighbors. For each query image, we ran DenseVLAD [67] to get 20 potential database images. In our experiments, the algorithms are tested on each obtained image pair, i.e., 20 pairs for each query image. For ScanNet, we use the 1500 image pairs used in [59].",
+        "bbox": [
+            75,
+            508,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "On Cambridge Landmarks, we detect affine correspondences by detecting DoG features [42], finding the affine shape by AffNet [48], and extracting HardNet [47] descriptors. This approach is among the leaders in the IMC 2020 benchmark [34]. On ScanNet, we detect SuperPoint [20] features, match them by SuperGlue [59], and find the affine shape by, first, estimating scale and orientation by SelfScaleOri [41] and, then, finding the affine shapes by AffNet.",
+        "bbox": [
+            75,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Normal Estimation. To test normal estimators, we ran all of them on the data described in the previous section. For each image pair, we iterate through all detected correspondences that are consistent with the camera pose - they have less than 5 pixels of Sampson distance. For each of",
+        "bbox": [
+            75,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/1bb67d9598776c047ef3944af4a4be4982d425f647c69f790568eade56b61920.jpg",
+        "image_caption": [
+            "Figure 12. The median angular error in degrees (left) and the AUC@  $10^{\\circ}$  score (right) of the estimated normals plotted as a function of the error in the detected affine correspondences on the Cambridge Landmarks dataset."
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4b414b70d67b0597142c805eaa4d8ea1d857e02ceaa495e357e61aa2c07afe43.jpg",
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+    {
+        "type": "text",
+        "text": "them, we ran multiple normal estimators, measure their angular error and run-time. To understand how the estimators proceed with respect to the error in the detected affine transformations  $\\mathbf{A}$ , we calculate the reference transformation  $\\mathbf{A}^*$  from the camera parameters and the surface normal as shown in [9]. We define the affine error as the Frobenius norm of the difference matrix  $\\| \\mathbf{A} - \\mathbf{A}^* \\|_{\\mathrm{F}}$ .",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The Area Under the recall Curve (AUC) at  $5^{\\circ}$ ,  $10^{\\circ}$  and  $20^{\\circ}$ , the avg. and med. angular errors with the standard deviation in degrees and the run-time in nanoseconds on the Cambridge Landmarks dataset are reported in Table 1. As expected, the best results are achieved by the optimal algorithms, all of them leading to exactly the same results. Our proposed linear solution of the optimal problem,  $\\mathrm{OPT}_{\\mathrm{LINEAR}}$ , is five times faster than the cubic solution  $\\mathrm{OPT}_{\\mathrm{CUBIC}}$ . While  $\\mathrm{OPT}_{\\mathrm{LINEAR}}$  is slightly slower than the fastest FNE method, it is significantly more accurate. Interestingly, while the method of Raposo et al. [56] worked well in the synthetic experiments, we did not manage to achieve good accuracy with it in the real-world experiments. Figure 12 plots the AUC@ $10^{\\circ}$  score and the median angular error as a function of the noise in the detected affine parameters. As expected, normal estimation is highly sensitive to the quality of the affine parameters. The optimal algorithms are the most stable ones.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The results on the ScanNet dataset are reported in Table 3. Similarly as on Cambridge Landmarks, all optimal solvers lead to the same accuracy while the proposed one being the fastest by a large margin.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Visual Localization. To test the impact of surface normal estimation on localization, on the Cambridge Landmarks dataset, we applied the following procedure. We iterate through all query images, each having 10 database images retrieved by DenseVLAD. First, we estimate the relative pose between the query and a random database image by GC-RANSAC [6] with P3P [54]. Using the obtained pose, we estimate the surface normal of each inlier affine correspondence in the coordinate system of the query image. Next, we iterate through all unused database images and use",
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_footnote",
+        "text": "3Meshlab is an open source tool for 3D point cloud processing and visualization.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "3397",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/45c1e3e795772d2512d0efa644949931b2a50682507161b861e38e33bdd7e804.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Estimator</td><td>AUC@5°↑</td><td>@10°↑</td><td>@20°↑</td><td>Avg. err. (°) ↓</td><td>Med. err. (°) ↓</td><td>Std. dev. (°) ↓</td><td>Time (ns) ↓</td></tr><tr><td>FNE [49]</td><td>6.0</td><td>16.0</td><td>33.7</td><td>21.0</td><td>14.3</td><td>21.3</td><td>172</td></tr><tr><td>FNESUPER</td><td>8.9</td><td>22.1</td><td>42.0</td><td>10.9</td><td>16.8</td><td>15.6</td><td>1721</td></tr><tr><td>Raposo [56]</td><td>0.4</td><td>1.5</td><td>5.7</td><td>38.5</td><td>36.5</td><td>18.2</td><td>883</td></tr><tr><td>OPTQUARTIC [9]</td><td>11.7</td><td>27.3</td><td>48.7</td><td>8.8</td><td>13.6</td><td>12.9</td><td>2528</td></tr><tr><td>OPTCUBIC [46]</td><td>11.7</td><td>27.3</td><td>48.7</td><td>8.8</td><td>13.6</td><td>12.9</td><td>2484</td></tr><tr><td>Proposed OPTLINEAR</td><td>11.7</td><td>27.3</td><td>48.7</td><td>8.8</td><td>13.6</td><td>12.9</td><td>596</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/409f1644eb0b6a4ba44a6d75210ee1b851d9740d437615c5d1072035b03d04fe.jpg",
+        "table_caption": [
+            "Table 1. The AUC score at  $5^{\\circ}$ ,  $10^{\\circ}$ , and  $20^{\\circ}$  of the estimated surface normals, their average and median errors in degrees, the standard deviation, and the average run-time, in nanoseconds, of a single normal estimation on the Cambridge Landmarks dataset."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Solver</td><td rowspan=\"2\">Normal estimator</td><td colspan=\"3\">Pose accuracy ↑</td><td rowspan=\"2\">Med. rot. err. (°) ↓</td><td rowspan=\"2\">Med. pos. err. (cm) ↓</td></tr><tr><td>AUC@1°, 10cm</td><td>@2.5°, 25cm</td><td>@5°, 50cm</td></tr><tr><td>P3P</td><td>-</td><td>44.3</td><td>59.1</td><td>67.5</td><td>10.1</td><td>13.5</td></tr><tr><td>P1AC</td><td>FNESUPER</td><td>47.8</td><td>62.0</td><td>69.7</td><td>6.2</td><td>12.8</td></tr><tr><td>P1AC</td><td>Proposed OPTLINEAR</td><td>48.0</td><td>62.3</td><td>69.9</td><td>5.2</td><td>1.8</td></tr><tr><td>P1AC</td><td>GT</td><td>48.4</td><td>62.9</td><td>70.4</td><td>4.6</td><td>2.1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/a8f46f017481fc6419eba75c72b1d0cd8120a2ac64ab9d492c127ad9e343b269.jpg",
+        "table_caption": [
+            "Table 2. Localization AUC scores at  $(1^{\\circ}, 10\\mathrm{cm})$ ,  $(2.5^{\\circ}, 25\\mathrm{cm})$ , and  $(5^{\\circ}, 50\\mathrm{cm})$ , the median rotation (in degrees) and translation errors (in centimeters) on the Cambridge Landmarks dataset [39] when using either the P3P solver or the normal-based P1AC solver with FNEsuper or the proposed surface normal estimator OPTLINEAR. We also show the results with the reference normals. The minimal solvers are used within Graph-Cut RANSAC [6] that takes care of the robust estimation."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>AUC@5↑</td><td>AUC@10↑</td><td>AUC@20↑</td><td>Time (ns)↓</td></tr><tr><td>FNE</td><td>14.8</td><td>15.4</td><td>17.0</td><td>207</td></tr><tr><td>FNESUPER</td><td>34.9</td><td>35.2</td><td>36.6</td><td>2142</td></tr><tr><td>OPTQUARTIC</td><td>35.0</td><td>35.5</td><td>37.3</td><td>18745</td></tr><tr><td>OPTCUBIC</td><td>35.0</td><td>35.5</td><td>37.3</td><td>3146</td></tr><tr><td>OPTLINEAR</td><td>35.0</td><td>35.5</td><td>37.3</td><td>747</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "the P1AC [68] solver inside GC-RANSAC to estimate the pose. P1AC requires a single AC and the surface normal. We only test the proposed solver and  $\\mathrm{FNE}_{\\mathrm{SUPER}}$ . All other optimal solvers lead exactly to the same accuracy as ours.  $\\mathrm{FNE}_{\\mathrm{SUPER}}$  is the most accurate sub-optimal algorithm.",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The accuracy of the estimated pose for the database images with respect to their GT pose is reported in Table 2. All solvers, benefiting from normals and affine correspondences, are more accurate than using P3P. We also report the accuracy when using the reference normals to see the upper bound achievable using normals. The proposed optimal solver OPT LINEAR is only slightly (i.e.  $< 1$  AUC point) behind the accuracy of the GT normal, while being more accurate than the P3P-based solution. The AUC@  $10^{\\circ}$  scores on each scene are reported in Table 4. The proposed OPT LINEAR leads to the best accuracy on all but two scenes.",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5. Conclusions",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we have presented a novel solver for estimating surface normals from affine correspondences when having calibrated cameras. The proposed method yields the",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/5dafa51b1ca9af1e1eab02f97d156dbeeca9a3d0a97c8f5f42a93489b1ea6b7f.jpg",
+        "table_caption": [
+            "Table 3. The AUC scores at  $5^{\\circ}$ ,  $10^{\\circ}$  and  $20^{\\circ}$ , and the avg. time (in nanoseconds) of normal estimators on the ScanNet datasets [18]."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">P3P</td><td colspan=\"3\">P1AC</td></tr><tr><td>GT</td><td>FNESUPER</td><td>Prop. OPTLINEAR</td></tr><tr><td>St. Mary&#x27;s Church</td><td>44.2</td><td>47.7</td><td>46.1</td><td>46.5</td></tr><tr><td>Old Hospital</td><td>44.1</td><td>57.4</td><td>56.8</td><td>57.3</td></tr><tr><td>King&#x27;s College</td><td>52.1</td><td>54.2</td><td>54.1</td><td>54.0</td></tr><tr><td>Shop Facade</td><td>65.5</td><td>65.0</td><td>63.8</td><td>64.1</td></tr><tr><td>Great Court</td><td>62.7</td><td>63.2</td><td>63.0</td><td>63.5</td></tr><tr><td>Street</td><td>17.4</td><td>22.8</td><td>23.1</td><td>23.0</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4. The area under the recall curve (AUC) at  $1^{\\circ}$  rotation and 10cm position errors on the Cambridge Landmarks dataset. We combine the P1AC solver with  $\\mathrm{FNE}_{\\mathrm{SUPER}}$  and the proposed normal estimator. Also, we test it with the reference normals.",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "optimal solution, in the least squares sense, with respect to the affine error, and is particularly fast as the optimal normal is obtained via linear equations. We compared our method to all existing alternatives that estimate normals from a single correspondence on both simulated and real-world data. Additionally, we provide tests that highlight the benefits of using surface normals in image-based localization. The proposed estimator has demonstrated superior accuracy compared to the fastest existing method while being the second fastest overall. Our results suggest that the proposed algorithm is a promising alternative for surface normal estimation in computer vision applications.",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement. The authors warmly thank the reviewers for their very valuable comments and suggestions. The work was supported by the ETH Postdoc fellowship.",
+        "bbox": [
+            498,
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+    },
+    {
+        "type": "page_number",
+        "text": "3398",
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+    },
+    {
+        "type": "text",
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+]

2023/Fast Globally Optimal Surface Normal Estimation from an Affine Correspondence/ecc07b62-9596-44a0-9745-a517e83aa84a_origin.pdf

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2023/Fast Globally Optimal Surface Normal Estimation from an Affine Correspondence/full.md

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+# Fast Globally Optimal Surface Normal from an Affine Correspondence
+
+Levente Hajder, Lajos Lóczi  
+Geometric Computer Vision Group  
+Eötvös Loránd University, Budapest, Hungary  
+{hajder, lloczi}@inf.elte.hu
+
+Daniel Barath  
+Computer Vision and Geometry Group  
+ETH Zurich, Switzerland  
+dbarath@inf.ethz.ch
+
+# Abstract
+
+We present a new solver for estimating a surface normal from a single affine correspondence in two calibrated views. The proposed approach provides a new globally optimal solution for this over-determined problem and proves that it reduces to a linear system that can be solved extremely efficiently. This allows for performing significantly faster than other recent methods, solving the same problem and obtaining the same globally optimal solution. We demonstrate on  $15k$  image pairs from standard benchmarks that the proposed approach leads to the same results as other optimal algorithms while being, on average, five times faster than the fastest alternative. Besides its theoretical value, we demonstrate that such an approach has clear benefits, e.g., in image-based visual localization, due to not requiring a dense point cloud to recover the surface normal. We show on the Cambridge Landmarks dataset that leveraging the proposed surface normal estimation further improves localization accuracy. Matlab and  $C++$  implementations are also published in the supplementary material.
+
+# 1. Introduction
+
+Acquiring oriented point clouds is a fundamental problem both in computer vision and in robotics with a number of important real-world applications benefiting from the extra information it provides. Such applications include 3D reconstruction [1, 8, 31, 62, 72, 65, 56] and simultaneous localization and mapping [19, 23, 50], where the 3D point orientations provide useful information about the underlying structure. Image-based visual localization approaches [43, 53, 60, 61] can also benefit from known surface normals [68]. The widely used Poisson reconstruction [37, 38] is based on both the point coordinates and their orientations. The visual odometry [51, 52] of an autonomous vehicle can also benefit from understanding surface orientation to, e.g., segment ground or facades. Also, having an oriented point cloud greatly simplifies geometric model and multi-model estimation [32, 4, 5] due to allowing to design solvers that
+
+![](images/0c48ef7f051b42a370dc08fa7ade8ffa2f4139ea4d5f11d69b72a1c707315bda.jpg)  
+Figure 1. Geometric configuration: oriented 3D point  $\mathbf{P} \in \mathbb{E}^3$  with normal  $\mathbf{n} \in \mathbb{E}^3$  projected to image  $\Pi^1$  as  $\mathbf{p}_1 \in \mathbb{E}^2$  and to  $\Pi^2$  as  $\mathbf{p}_2 \in \mathbb{E}^2$ . The local patch centered at points  $\mathbf{p}_1$  and  $\mathbf{p}_2$  are related by the local affine transformation  $\mathbf{A} \in \mathbb{R}^{2 \times 2}$ .
+
+require fewer data points than their point-based counterparts [64]. Moreover, normals can also be leveraged in the numerical refinement of the reconstructed oriented 3D point cloud [27]. In this paper, we focus on a practical problem, i.e., estimating the normal at corresponding features in two images in a solely photometric manner. This can provide a lightweight approach of obtaining surface normals without requiring a dense point cloud to be known. Next, we will discuss ways of obtaining oriented points.
+
+Normals from Point Clouds. After applying state-of-the-art sparse (e.g., COLMAP [63]) or dense [36, 16] reconstruction pipelines, the resulting point cloud can be straightforwardly equipped with point orientations. In this case, typically, the normals are fitted per-point, applying the widely used Principal Component Analysis (PCA) [35] to estimate tangent planes or higher-order surfaces [14, 33] from the local neighborhood centered at the current point. This procedure, while being frequently used, has two major drawbacks. First, the setting of the neighborhood size greatly affects the accuracy and is often unclear without having a metric reconstruction. Instead, using the  $k$ -nearest-neighbors [17] algorithm to find local neighborhoods is a viable solution, but the estimation becomes largely affected by the sparsity of the point cloud. Second, fitting local surfaces, independently, to the neighborhood of each 3D point
+
+in the reconstruction is an extremely expensive operation.
+
+Normals from Homographies. Another method for estimating surface normals from images, without requiring the underlying 3D structure to be known, is via homographies. Homographies are projective transformations between images of co-planar 3D points that depends on the intrinsic and extrinsic camera parameters and the parameters of the underlying 3D plane [24]. A homography can be estimated from a minimum of four point correspondences in two images [30]. While this method allows the estimation of dominant planes supported by a number of correspondences in the images, it does not provide means for normal estimation for each local feature correspondence independently.
+
+Normals from Local Mappings. In recent years, affine correspondences (AC) have emerged as a popular topic in computer vision, offering a valuable source of information about the underlying 3D geometry of the surrounding environment. An AC consists of a corresponding pair of points in two images and a linear transformation that maps the infinitesimal region around the point in the first image to the point in the second one. They have been used to solve various geometric problems, including relative pose estimation in the general case [15, 55, 3, 22, 10], for known vertical direction [25], planar motion [25, 26], or known feature depths [21] and homography estimation [2].
+
+Traditionally, finding affine features involve four main steps: (scale-covariant) keypoint detection, orientation and affine shape estimation, and descriptor extraction. Under such paradigm, keypoint detection is typically performed on the scale pyramid with the help of a handcrafted response function, such as Hessian [13, 45], Harris [28, 45], Difference of Gaussians (DoG [42]), or learned ones like FAST [57] or Key.Net [11]. Keypoint detection obtains a triplet  $(x,y,\mathrm{scale})$  which defines a square or circular image patch. Then the patch orientation is estimated with the help of handcrafted - dominant gradient orientation [42], center of the mass [58] - or learned methods [71, 48, 41]. Next, the affine-covariant shape [12, 48] is estimated. Finally, the patch is geometrically rectified and fed into a local patch descriptor, such as SIFT, HardNet [47], SOSNet [66], or others. In summary, all feature detectors, even recent learning-based ones like SuperPoint [20], can be made affine covariant by applying, e.g., AffNet [48] as a post-processing step on the found features.
+
+The problem of surface normal estimation from ACs has been addressed in several works. The first algorithm for normal estimation from ACs was proposed by Kevin Koeser [40] assuming that the camera location is known with respect to the observed 3D point. Megyesi et al. [44] proposed a special surface normal estimator for standard (rectified) stereo, where the degree of freedom of the affine transformations decreases to two. In 2014, Molnár and Chetverikov [49] derived a formula connecting the surface
+
+normals, camera parameters, affine transformations, and patch locations into a single expression for arbitrary projective functions, like the catadioptric camera model. Barath et al. [9] proposed a globally optimal estimator in the least squares sense as the solution of a quartic polynomial. Recently, Le Minh and Hajder [46] improved on this by reducing the degree and showed that the optimal solution can be found as a real root of a cubic polynomial.
+
+The main goal of this paper is to propose a rapid normal estimator based on affine correspondences. Our algorithm can be inserted into a 3D reconstruction system, such as Structure-from-Motion [27] (SfM) or Simultaneous Localization and Mapping (SLAM), where the normal estimation is performed many times, therefore its running time influences the time demand of the whole 3D pipeline.
+
+Contribution. In this paper, we propose a new linear and globally optimal solution for surface normal estimation from a single affine correspondence. The proposed solver requires only the four basic arithmetic operations and contains 19 parameters, consisting of the coordinates of the five input vectors and the four affine parameters. This approach completely avoids root extractions, solving high-degree polynomial equations, or employing numerical or iterative methods, resulting in a particularly fast algorithm that runs for only  $0.6\mu s$  in our experiments. The solver is five times faster than the cubic solver from [46]. Additionally, we propose a way of exploiting AC-wise normals in image-based visual localization pipelines. The algorithms are tested on 15k image pairs from standard benchmarks, and our results show superior accuracy and faster wall-clock times compared to the baselines.
+
+# 2. Proposed Method
+
+The setting is visualized in Fig 1. We are given two images, denoted by  $\Pi^1$  and  $\Pi^2$ . The 2D and 3D point locations are visualized as black dots. For standard PC-based stereo vision, the 3D location can be obtained by the so-called triangulation method. The local affine transformation, denoted by  $\mathbf{A} \in \mathbb{R}^{2 \times 2}$ , is defined as the first-order Taylor-approximation of the imaging function at points  $\mathbf{p}_1$  and  $\mathbf{p}_2$ . It maps the infinitesimal patch around the observed points from the first to the second image. If  $\mathbf{A}$  is known, the surface normal  $\mathbf{n}$  can be directly estimated without knowing the actual 3D structure of the underlying scene [49].
+
+Parameterization. A surface normal  $\mathbf{n} \in \mathbb{R}^3$  is a 3D direction, therefore it can be represented by two parameters. Usually, it is parameterized by three coordinates  $\mathbf{n} = [n_x, n_y, n_z]^\top$  and its length being set to 1. However, as it turns out below, some other scaling will be more convenient in the present situation.
+
+Problem Statement. Let us assume that we are given a general camera that projects the spatial location  $\mathbf{P} = [X, Y, Z]^\top$  to image (pixel) coordinates  $[u v]^\top$ , and let us
+
+denote the projective coordinate functions by  $\Pi_u: \mathbb{E}^3 \to \mathbb{R}$  and  $\Pi_v: \mathbb{E}^3 \to \mathbb{R}$ . Function  $\Pi_u$  calculates coordinate  $u$  of the projected point, and  $\Pi_v$  calculates  $v$ . As Molnár and Chetverikov showed [49], the affine transformation can be defined via the spatial gradients of the projective coordinate functions  $\Pi_u$  and  $\Pi_v$ .
+
+An affine correspondence is represented by 2D image coordinates  $[u_1 v_1]^\top$  and  $[u_2 v_2]^\top$  in the stereo image pair, in conjunction with the affine transformation  $\mathbf{A}$  that maps the surrounding area of the locations from first image to the second one. The spatial location of the corresponding spatial point is also required. It can be estimated from the image coordinates and camera parameters by standard triangulation techniques [29].
+
+The normal estimation problem is written [9] as an argument minimization to determine the optimal  $\mathbf{n}^*$  such that
+
+$$
+\mathbf {n} ^ {*} = \operatorname * {a r g   m i n} _ {\mathbf {n}} f (\mathbf {n}),
+$$
+
+with
+
+$$
+f (\mathbf {n}) := \sum_ {i = 1} ^ {2} \sum_ {j = 1} ^ {2} \left(\frac {\mathbf {n} ^ {\top} \mathbf {w} _ {i j}}{\mathbf {n} ^ {\top} \mathbf {w} _ {c}} - a _ {i j}\right) ^ {2}. \tag {1}
+$$
+
+Here, the affine transformation yields the four affine parameters  $a_{ij}$  ( $i,j\in [1,2]$ ) according to
+
+$$
+\mathbf {A} = \left[ \begin{array}{c c} a _ {1 1} & a _ {1 2} \\ a _ {2 1} & a _ {2 2} \end{array} \right],
+$$
+
+and the vectors  $\mathbf{w}_{ij}$  and  $\mathbf{w}_c$  are obtained as cross products of the spatial gradient vectors as follows:
+
+$$
+\begin{array}{l} \mathbf {w} _ {1 1} = \nabla \Pi_ {v} ^ {1} \times \nabla \Pi_ {u} ^ {2}, \quad \mathbf {w} _ {1 2} = \nabla \Pi_ {u} ^ {2} \times \nabla \Pi_ {u} ^ {1}, \\ \mathbf {w} _ {2 1} = \nabla \Pi_ {v} ^ {1} \times \nabla \Pi_ {v} ^ {2}, \quad \mathbf {w} _ {2 2} = \nabla \Pi_ {v} ^ {2} \times \nabla \Pi_ {u} ^ {1}, \\ \mathbf {w} _ {c} = \nabla \Pi_ {u} ^ {1} \times \nabla \Pi_ {v} ^ {1}, \\ \end{array}
+$$
+
+where the superscripts refer to the corresponding image index. The coordinates of these vectors will be denoted by  $\mathbf{w}_{ij,x},\mathbf{w}_{ij,y},\mathbf{w}_{ij,z}$ $(i,j\in \{1,2\})$  and  $\mathbf{w}_{c,x},\mathbf{w}_{c,y},\mathbf{w}_{c,z}$ .
+
+Pin-hole (perspective) projection model. For a pin-hole camera, the projection model is as follows:
+
+$$
+\left[ \begin{array}{c} u \\ v \\ 1 \end{array} \right] \sim \left[ \begin{array}{c c c c} P _ {1 1} & P _ {1 2} & P _ {1 3} & P _ {1 4} \\ P _ {2 1} & P _ {2 2} & P _ {2 3} & P _ {2 4} \\ P _ {3 1} & P _ {3 2} & P _ {3 3} & P _ {3 4} \end{array} \right] \left[ \begin{array}{c} X \\ Y \\ Z \\ 1 \end{array} \right],
+$$
+
+where  $\sim$  denotes the equality up to scale operation. Therefore, image coordinates  $u$  and  $v$  are obtained as
+
+$$
+\begin{array}{l} \Pi_ {u} = \frac {P _ {1 1} X + P _ {1 2} Y + P _ {1 3} Z + P _ {1 4}}{P _ {3 1} X + P _ {3 2} Y + P _ {3 3} Z + P _ {3 4}}, \\ \Pi_ {v} = \frac {P _ {2 1} X + P _ {2 2} Y + P _ {2 3} Z + P _ {2 4}}{P _ {3 1} X + P _ {3 2} Y + P _ {3 3} Z + P _ {3 4}}. \\ \end{array}
+$$
+
+Spatial gradients, that are  $\nabla \Pi = \left[ \begin{array}{ccc}\frac{\partial\Pi}{\partial X} & \frac{\partial\Pi}{\partial Y} & \frac{\partial\Pi}{\partial Z} \end{array} \right]^{\top}$ , can be written for the pin-hole model in compact forms as
+
+$$
+\begin{array}{l} \begin{array}{r l} \nabla \Pi_ {u} & = \frac {1}{s} \left[ \begin{array}{c} P _ {1 1} \\ P _ {1 2} \\ P _ {1 3} \end{array} \right] ^ {\top} - \frac {u}{s} \left[ \begin{array}{c} P _ {3 1} \\ P _ {3 2} \\ P _ {3 3} \end{array} \right] ^ {\top}, \end{array} \\ \nabla \Pi_ {v}   =   \frac {1}{s} \left[ \begin{array}{c} P _ {2 1} \\ P _ {2 2} \\ P _ {2 3} \end{array} \right] ^ {\top} - \frac {v}{s} \left[ \begin{array}{c} P _ {3 1} \\ P _ {3 2} \\ P _ {3 3} \end{array} \right] ^ {\top}, \\ \end{array}
+$$
+
+where  $s = P_{31}X + P_{32}Y + P_{33}Z + P_{34}$  is the projective depth.
+
+Solution. For the problem to be meaningful, we can assume that  $\mathbf{w}_c \neq \mathbf{0}$ . Let us set
+
+$$
+I := \inf  _ {\| \mathbf {n} \| = 1, \mathbf {n} ^ {\top} \mathbf {w} _ {c} \neq 0} f (\mathbf {n}),
+$$
+
+where  $\| \mathbf{n}\| \coloneqq \sqrt{\mathbf{n}^\top\mathbf{n}}$  denotes the standard Euclidean norm. Clearly,  $f\geq 0$  is undefined along the "equator" of the unit sphere  $\| \mathbf{n}\| = 1$  with  $\mathbf{n}^{\top}\mathbf{w}_{c} = 0$ . Let us consider two cases.
+
+If all  $\mathbf{w}_{ij}$  and  $\mathbf{w}_c$  are equal, then trivially  $f(\mathbf{n}) = \sum_{i,j=1}^{2}(a_{ij} - 1)^2$  is a constant function.
+
+Otherwise, at least one of the expressions  $\frac{\mathbf{n}^{\top}\mathbf{w}_{ij}}{\mathbf{n}^{\top}\mathbf{w}_{c}}$  becomes arbitrarily large in modulus when  $\mathbf{n}$  is chosen such that  $\mathbf{n}^{\top}\mathbf{w}_c$  gets sufficiently close to 0; that is,  $|f(\mathbf{n})|\to +\infty$  as  $\mathbf{n}^{\top}\mathbf{w}_c\rightarrow 0$  with  $\| \mathbf{n}\| = 1$ . Therefore, by taking into account the continuity of  $f$  as well, the infimum  $I$  is in fact a minimum, and an optimal unit vector  $\mathbf{n}^{*}$  with  $f(\mathbf{n}^{*}) = I$  exists. In what follows, we will explicitly determine such an  $\mathbf{n}^{*}$  as a function of the 19 parameters<sup>1</sup> by using only the four basic arithmetic operations.
+
+A key observation is that  $f(\lambda \mathbf{n}) = f(\mathbf{n})$  for any  $\lambda \neq 0$ , so (after a possible rescaling) we can assume that the last component of  $\mathbf{n}$  is either 1 or 0. These cases, Cases A and B, respectively, will be discussed separately below. To simplify the notation, let us define two auxiliary functions
+
+$$
+g \left(n _ {x}, n _ {y}\right) := f \left(n _ {x}, n _ {y}, 1\right) \tag {2}
+$$
+
+and
+
+$$
+h \left(n _ {x}, n _ {y}\right) := f \left(n _ {x}, n _ {y}, 0\right). \tag {3}
+$$
+
+For the objective function  $g$ , we will obtain a unique candidate extremum  $\mathbf{n}_g^* \in \mathbb{R}^2$ ; whereas for the function  $h$ , a one-parameter family of candidate extrema  $\mathbf{n}_h^* \in \mathbb{R}^2$  will be found but with the same function value  $h(\mathbf{n}_h^*)$ . The optimal  $\mathbf{n}^*$  is then
+
+$$
+\mathbf {n} ^ {*} = \left\{ \begin{array}{l l} \left[ \mathbf {n} _ {g} ^ {*}, 1 \right] ^ {\top}, & \text {i f} g \left(\mathbf {n} _ {g} ^ {*}\right) \leq h \left(\mathbf {n} _ {h} ^ {*}\right), \\ \left[ \mathbf {n} _ {h} ^ {*}, 0 \right] ^ {\top}, & \text {i f} g \left(\mathbf {n} _ {g} ^ {*}\right) > h \left(\mathbf {n} _ {h} ^ {*}\right). \end{array} \right. \tag {4}
+$$
+
+Remark 1 In other words, we will be utilizing the simple constraints  $n_z = 1$  or  $n_z = 0$ . We have found that other linear or non-linear constraints, such as  $\| \mathbf{n} \| = 1$ , would render the symbolic calculations to minimize the objective function  $f$  infeasible.
+
+Case A:  $n_z = 1$
+
+We use the standard technique and solve the system
+
+$$
+\left\{ \begin{array}{l} \partial_ {1} g \left(n _ {x}, n _ {y}\right) = 0, \\ \partial_ {2} g \left(n _ {x}, n _ {y}\right) = 0 \end{array} \right. \tag {5}
+$$
+
+to find the optimal vector  $\mathbf{n}_g^* = [n_x, n_y]^\top$  as follows.
+
+By computing and putting the partial derivative
+
+$$
+\partial_ {1} g (n _ {x}, n _ {y})
+$$
+
+over a common denominator, we see that its numerator depends on  $n_x$  linearly; and it depends on  $n_y$  quadratically; so one can easily express  $n_x$  from the equation  $\partial_1 g(n_x, n_y) = 0$  as follows:
+
+$$
+n _ {x} = \frac {\operatorname {n u m e r} _ {1}}{\operatorname {d e n o m} _ {1}}, \tag {6}
+$$
+
+where we present the expressions for  $\mathrm{numer}_1$  and  $\mathrm{denom}_1$  in the supplementary material. Then (6) is substituted into the other partial derivative  $\partial_2 g(n_x, n_y)$ . The resulting rational function of  $n_y$ , also containing the other 19 parameters, can be simplified and factorized by Wolfram Mathematica's Factor command<sup>2</sup>. In this way, we obtain a fraction whose structure is the following:
+
+$$
+\frac {\operatorname {l i n e a r} _ {1} \cdot \operatorname {l i n e a r} _ {2}}{(\text {q u a d r a t i c}) ^ {2}}, \tag {7}
+$$
+
+where the expressions "linear" and "quadratic" denote polynomials in  $n_y$  having the corresponding degrees.
+
+The optimum of  $g$  can occur when  $n_y$  is chosen such that the fraction (7), that is, its numerator is equal to 0. It can be shown that the assumption  $\mathrm{linear}_1 = 0$  leads to a contradiction - indeed, by solving  $\mathrm{linear}_1 = 0$  for  $n_y$ , the resulting formula would make  $\mathrm{denom}_1$  in (6) identically equal to 0. Hence, we are only left with the equation  $\mathrm{linear}_2 = 0$ , whose solution is
+
+$$
+n _ {y} = \frac {\operatorname {n u m e r} _ {2}}{\operatorname {d e n o m} _ {2}}. \tag {8}
+$$
+
+The actual formulas for  $\mathrm{numer}_2$  and  $\mathrm{denom}_2$  are found again in the supplementary material.
+
+Thus, the second component of  $\mathbf{n}_g^*$  is given by (8) (expressed in terms of the 19 parameters). Then, by using this together with (6), we obtain  $n_x$ , the first component of  $\mathbf{n}_g^*$ .
+
+Remark 2 The essential feature of these computations is that the solution of system (5) is given in a fully explicit symbolic form by using only elementary operations. Without the clever intermediate symbolic simplifications and factorizations (most notably (7)), one would need to solve a higher-degree polynomial equation to obtain  $n_y$ , requiring either taking roots or using an iterative scheme. For example, if one starts solving (5) with  $n_x$  and  $n_y$  as symbolic parameters but all the coefficients of  $g$  as floating-point numbers, then, in general, no intermediate factorizations or simplifications are possible, so in order to determine  $n_y$ , one would need to solve a quintic equation.
+
+Case B:  $n_z = 0$
+
+We proceed similarly and solve the system
+
+$$
+\left\{ \begin{array}{l} \partial_ {1} h \left(n _ {x}, n _ {y}\right) = 0, \\ \partial_ {2} h \left(n _ {x}, n _ {y}\right) = 0. \end{array} \right. \tag {9}
+$$
+
+By expressing  $n_x$  from the first equation of (9) and substituting the resulting expression into the second equation, we find that this time the second equation is automatically satisfied. Therefore, (9) possesses a one-parameter family of solutions given by
+
+$$
+n _ {x} = \frac {\operatorname {n u m e r} _ {3}}{\operatorname {d e n o m} _ {3}}, \tag {10}
+$$
+
+see the supplementary material. By substituting this into  $h$ , the function value  $h(\mathbf{n}_h^*)$  appearing in (4) is obtained as
+
+$$
+h \left(\mathbf {n} _ {h} ^ {*}\right) = \frac {\operatorname {n u m e r} _ {4}}{\operatorname {d e n o m} _ {4}}. \tag {11}
+$$
+
+See the supplementary material for details.
+
+Remark 3 One directly checks that the value  $h(\mathbf{n}_h^*) = h\left(\frac{\mathrm{numer}_3}{\mathrm{denom}_3}, n_y\right)$  is indeed independent of  $n_y$ , as claimed.
+
+# 2.1. Summary of the Algorithm
+
+Given the elements  $a_{ij} \in \mathbb{R}$  in the affine transformation and 3D vectors  $\mathbf{w}_{ij}$  and  $\mathbf{w}_c \neq \mathbf{0}$ , let us define the function  $f$  by (1) for  $\mathbf{n}$  with  $\mathbf{n}^\top \mathbf{w}_c \neq \mathbf{0}$ , and also define the functions  $g$  and  $h$  by (2)-(3). If  $\mathbf{w}_{11} = \mathbf{w}_{12} = \mathbf{w}_{21} = \mathbf{w}_{22} = \mathbf{w}_c$ , then any unit vector  $\mathbf{n}^*$  can be chosen as optimal normal vector.
+
+Otherwise, we compute  $\mathbf{n}_g^* \coloneqq [n_x, n_y]^\top$  from (8) and (6). Then we compute  $h(\mathbf{n}_h^*)$  from (11). Finally, the optimal  $\mathbf{n}^*$  is determined by using (4).
+
+# 3. Experiments
+
+We focus on the accuracy and time demand of the algorithms when testing. Both synthetic testing data, including configurations for a number of important camera motions, and standard real-world benchmarks are considered here.
+
+![](images/3361adbf5f991a037b270f6cca617713817d6b0f9a68b30df32547514bbdf2f7.jpg)
+
+![](images/9443b008b2f21844cc9e34eed51e5c36849a35ae5c047f53f1ef7a7de01f9ca6.jpg)
+
+![](images/7aa5f0a54ff8a3fb5ede75cbbad692e3c95bd5b99bdad1a20caee658846f7d38.jpg)  
+Figure 2. Images and the reference reconstruction used from the Cambridge Landmarks dataset [39] in localization experiments.
+
+![](images/52da8b4b8b4e061f0e13354f93a23bc7e07fd79f0a4315df454e46bb22dc7e4d.jpg)
+
+Compared Algorithms. In this section, we discuss the algorithms that can be used to estimate the surface normal from a single affine correspondence.
+
+1. The provably most accurate algorithms are the ones that are optimal in the least squares sense. Method  $\mathrm{OPT}_{\mathrm{QUARTIC}}$  refers to [9] which solves the problem as a real root of a quartic polynomial equation. The cubic solution  $\mathrm{OPT}_{\mathrm{CUBIC}}$  is from [46]. The proposed optimal solver, as it is solved via linear equations, is called  $\mathrm{OPT}_{\mathrm{LINEAR}}$  in the rest of the paper.  
+2. The simplest and, thus, the fastest method is the Fast Normal Estimation (FNE) algorithm of [9]. A more complex variant, when all possible linear equations are considered, is also constructed here, it is denoted as  $\mathrm{FNE}_{\mathrm{SUPER}}$ . The modified equations can be seen in the source codes in the supplementary material.  
+3. Raposo and Barreto [56] proposed a solution that estimates the normal via homographies. We have not found an implementation or an exact definition of the algorithm, so we implemented it based on the description. The solution is computed via an over-determined inhomogeneous linear system of equations. The formulas are generated by the MATLAB Symbolic Toolbox. Details are given in the supplementary material. The method is called RAPOSO in the charts.
+
+The synthetic tests are performed in MATLAB. In the real-world experiments, optimized  $\mathrm{C}++$  codes run on an Intel(R) Core(TM) i9-10900K CPU @ 3.70GHz.
+
+# 3.1. Synthetic Experiments
+
+For the synthetic tests, a spherical object is generated, thus straightforwardly allowing to sample oriented points from its surface. Two cameras are randomly generated with focal lengths from the range [500, 1500] and with image size  $2000 \times 2000$ . Their intrinsic matrices are  $\mathbf{K}_1$  and  $\mathbf{K}_2$ .
+
+The 3D point was projected into both cameras. The ground truth (GT) affine parameters were calculated directly from the camera parameters and the surface normals as shown in [9]. Several camera configurations are considered to simulate various real-world scenarios. They are as follows:
+
+- For general stereo pose, the locations of the cameras are randomly generated inside a unit-radius sphere. The camera projection matrices are constructed as  $\mathbf{P}_1 = \mathbf{K}_1[\mathbf{I} \mid \mathbf{0}]$  and  $\mathbf{P}_2 = \mathbf{K}_2[\mathbf{R} \mid \mathbf{t}]$ .  
+- For rectified stereo, the cameras are located next to each other with identical orientation. In this case, the projection matrices are  $\mathbf{P}_1 = \mathbf{K}_1[\mathbf{I} \mid \mathbf{0}]$  and  $\mathbf{P}_2 = \mathbf{K}_2[\mathbf{I} \mid \mathbf{t}]$ , where  $\mathbf{t} = [\Delta X 0 0]^\top$ .  
+- Planar motion simulates the case when a camera is mounted on a vehicle that moves on a planar road with the image plane orthogonal to the ground. Such a motion can be represented by two angles around the vertical axis, one for the rotation and one for the translation. In this case, the projection matrices can be written as  $\mathbf{P}_1 = \mathbf{K}_1[\mathbf{I} \mid 0]$  and  $\mathbf{P}_2 = \mathbf{K}_2[\mathbf{R}_{\mathrm{Y}} \mid \mathbf{t}]$ , where  $\mathbf{t} = \rho [\cos \beta, 0, \sin \beta]^\top$  and
+
+$$
+\mathbf {R} _ {\mathrm {Y}} = \left[ \begin{array}{c c c} \cos \alpha & 0 & \sin \alpha \\ 0 & 1 & 0 \\ - \sin \alpha & 0 & \cos \alpha \end{array} \right].
+$$
+
+- Moto-motion. This test case simulates a camera that is mounted on a motorcycle. If the bike leans, another rotation, the roll is involved in the problem. In this case, the camera parameters are  $\mathbf{P}_1 = \mathbf{K}_1[\mathbf{I} \mid \mathbf{0}]$  and  $\mathbf{P}_2 = \mathbf{K}_2[\mathbf{R}_{\mathrm{YZ}} \mid \mathbf{t}]$ , where  $\mathbf{t} = \rho [\cos \beta 0 \sin \beta]^\top$  and  $\mathbf{R}_{YZ} = \mathbf{R}_Y \mathbf{R}_Z$ .  
+- Drone-motion is another generalization of the planar motion. A camera mounted on a flying drone. Its location is arbitrary, but the orientation is limited to a vertical rotation. Therefore,  $\mathbf{P}_1 = \mathbf{K}_1[\mathbf{I} \mid \mathbf{0}]$  and  $\mathbf{P}_2 = \mathbf{K}_2[\mathbf{R}_Y \mid \mathbf{t}]$ , where  $\mathbf{t} = [X Y Z]^\top$ .
+
+Noise. We have added zero-mean Gaussian noise to the 2D coordinates, therefore, the point noise is given in pixels. The deviation of the noise is the parameter in the charts.
+
+The  $2 \times 2$  affine transformations are decomposed by SVD into two 2D rotations and another two scaling parameters. Then zero-mean Gaussian noise is added to the angle of rotations, in radians, and the scaling parameters. The standard deviation of the Gaussian noise distribution is equally set for all the four parameters.
+
+3D Reconstruction Pipeline. In order to simulate real scenarios, first, we triangulate each generated noisy point correspondence by the optimal triangulation method proposed by Hartley and Sturm [29]. The noisy affine parameters are corrected by the method proposed in [7] to be consistent with the camera parameters.
+
+![](images/e29247ae079a115614ff62377b1ca561bfb6a47a6f259aefd29da6c614187ebc.jpg)  
+Figure 3. The frequencies (in 10000 runs) of the  $\log_{10}$  errors in the  $\log_{10}$  angular errors in degrees in the noise-free case.
+
+Stability test. We generated 10000 random problem instances considering general camera motion and ran the normal estimators on noiseless affine correspondences. Fig. 3 shows histograms of  $\log_{10}$  angular errors in degrees. Note that for better visualization, we replace the exact zero values by  $10^{-16}$  as it is the precision of the floating point numbers in MATLAB. The plots show that all our solvers are very stable – there is no peak close to  $10^0$ . For all methods, the majority of the errors are distributed around  $10^{-16}$  which actually is the precision of MATLAB. This means that in most of the cases, all methods are particularly stable returning almost exactly 0 error. A small peak is over  $10^{-6}$  for all methods similarly. Having  $10^{-6}$  degree error is acceptable in most of the scenarios considered in computer vision. There is no peak close to  $10^0$  implying that all solvers are stable. Note that these stability tests usually are for highlighting potential degeneracies where the error exceeds  $10^0$ .
+
+We also conducted tests using single-point precision in MATLAB. The OPTQUARTIC approach failed entirely under these conditions. For all other methods, we noticed an increase in error, which suggests that double precision is necessary for reliable normal estimation. Interestingly, our proposed OPTLINEAR method appears to be slightly more sensitive to precision than OPTCUBIC is. This is likely due to the fact that OPTLINEAR employs more complex formulas and, consequently, performs more stably with double precision.
+
+Accuracy test. The accuracy of the normals is examined for all the considered camera poses independently. The results are shown in Fig 9. The effect of noise in point coordinates and affine parameters are separately evaluated. Both the mean and the median of the errors are plotted.
+
+The observations we can make from the plots are as follows. First, the three optimal methods  $\mathrm{OPT}_{\mathrm{QUARTIC}}$ ,  $\mathrm{OPT}_{\mathrm{CUBIC}}$  and  $\mathrm{OPT}_{\mathrm{LINEAR}}$  always lead exactly to the same solution, thus their curves coincide in all plots. Second, the optimal methods always lead to the most accurate results if the affine parameters are contaminated by noise. This is not
+
+surprising as these methods optimize the normal parameters such that the implied affine transformation is the closest to the measured one. In the case of point-noise only, the accuracy of the optimal algorithm is slightly below that of the competitors. However, it is an unlikely situation in real-world images to have noisy points but perfect local affine frames. It is usually the other way around. Third, as the FNE method uses less information, it is clearly the worst one in terms of accuracy.  $\mathrm{FNE}_{\mathrm{SUPER}}$  performs significantly better, however, it requires more time as the right singular vector of a  $6\times 3$  matrix, corresponding to the smallest singular value, should be computed. Another benefit of  $\mathrm{FNE}_{\mathrm{SUPER}}$  is, contrary to FNE and Raposo [56], that it works in all test cases. It can cope with both planar motion and rectified stereo, similarly to the optimal algorithms.
+
+Problematic methods. Methods FNE and Raposo [56] do not work if the cameras undergo special translations. FNE is a special method, several variants can be formed based on the selection of the affine transformations. For rectified stereo, the third and fourth elements are fixed [44], they cannot be used for the estimation.
+
+The Raposo [56] method is the most problematic one as the homography can be decomposed into the camera rotation, translation and surface normal. If the translation has zero elements, the retrieved equations are linearly dependent, and the estimation does not yield a correct solution. Due to this reason, Raposo [56] does not work for rectified stereo and planar motion. We removed these methods from the plots of the respective camera configurations.
+
+Combined noise. In the previous experiments, noise is added either to the point coordinates or affine parameters. Here, we contaminate both types of input data. The mean and median errors of the proposed linear method are visualized in Fig. 10. As it is expected, the curves show similar trends, but the case when both the points and affines are noisy leads to higher errors.
+
+Interpretable affine noise. In the previous experiments, the noise is added directly to the elements of the affine matrices. In Fig. 11, we consider a different way of contaminating the input to show that the proposed method acts reasonably with different noise models as well. The results of curves Aff and Comb are obtained by using the noise model from the previous sections. Comb refers to the case when both the points and the affine matrices are noisy. For AffElem and Comb-Elem, we decompose A by SVD into two orthogonal and a diagonal matrix. Fig. 11 demonstrates that the noise generation does not strongly influence the results.
+
+# 4. Real-world Experiments
+
+To test normal estimators, we use the Cambridge Landmarks [39] and ScanNet datasets [18]. We chose Cambridge Landmarks, see Fig. 2 for example images, since it provides a large number of images for testing directly the
+
+![](images/e00cb504bc546457895ce0aa123a2fe0c8d9fb9be95d72a65326bff9fb5c44e0.jpg)
+
+![](images/eeacceaf62dc6d144c117dab0fba5bc60a93abeff01bb0403dca86e754fa110e.jpg)
+
+![](images/b05dd400a77b1142754bc5e9cc19aa3b8a2e009f68fc685032cd21281077175e.jpg)  
+Figure 4. General Motion.
+
+![](images/7cba10c32b4f2843d7ec8d2599b7d2f90625f6359189829e2cd5aa48f5831ad2.jpg)
+
+![](images/948ac9746a92375e8283b0ab8e0331b152b1941855a96f3683ce6ac30384690e.jpg)
+
+![](images/f5823d95f2ef91a088d04317aee25aa099e38d425f75a7c472ec120c8677a40e.jpg)
+
+![](images/0b8be2ec24e87b1682ce1ab3c497291b1d077f74ae3e305862a028e3d1829f52.jpg)  
+Figure 5. Rectified Stereo.
+
+![](images/ebb91366dee31e11c32a664398fefc899af2b32abee7350f70399503818cd040.jpg)
+
+![](images/f2f5cd5f73b8b32db8338da54395dec2b2afa99d1db88d314d5821edbb439577.jpg)
+
+![](images/d028ae3a9cc7fe72a13b88d912d1d0ed8b79c242c95d9fb6fb23d80e6c08bab8.jpg)
+
+![](images/7a3d482fe64ec4cac62e80206b50e66a1356cbbd350843476ad055bb216694d7.jpg)  
+Figure 6. Planar Motion.
+
+![](images/dd12fafc323c83b4e17753fd5ebea86cc28761dc8a09eaf89112364871bd90a7.jpg)
+
+![](images/2c9d020dc92e5aedee75bc7d4d2afafdf5d6218a4bd665f9fcce69da0ceed6a6.jpg)
+
+![](images/e0eb25234c40f4ac1dfb29acafb4ced8150fd4e857288dccf9a403dba5a08020.jpg)
+
+![](images/d4285f04584e2e24052763f147b6078dc863bcfec0635a86a9193d6f936b4ae9.jpg)  
+Figure 7. Moto Motion.
+
+![](images/44a2e7f549b351c8d796252cecccb50f32c7086737d70ced94c94b5755d43200.jpg)
+
+![](images/cdab81ac4f53dee3550ff07f2cb78219f422692eec1625340859fb715ff3ac2e.jpg)
+
+![](images/2f63d8ff61554da8b97f8bab4cd258b676cb22ba9d136c3c45a1ee820c178455.jpg)  
+Figure 9. Average and median angular errors (in degrees) of surface normal estimators on synthetic data. Five camera configurations are considered. From left to right: (1) Median error; noise added to affine parameters. (2) Average error; noise added to affine parameters. (3) Median error; noise added to point coordinates. (4) Average error; noise added to point coordinates.
+
+![](images/71f07ae3ccc694ec59f324292b47b7e713b21625dd14fc2e66ba00eb6d1f459a.jpg)  
+Figure 8. Drone Motion.
+
+![](images/438bf52d4be6ad617d635fd593c4a4d9169606aedd16012c75b9c15a72f0c273.jpg)
+
+![](images/125656e0a522fe3bbf44d20cfcd3921be0458d1b5d078f40f22ac2a806541cbd.jpg)
+
+normal accuracy and, also, we can evaluate localization as an application of the proposed procedure. Also, we chose ScanNet since the reference normals can be calculated directly from the dense depth maps which provides a complementary comparison to the tests on Cambridge Landmarks,
+
+where the normals come from a 3D reconstruction.
+
+Cambridge Landmarks consists of five scenes with varying number of images, ranging from 231 to 3015. The reference pose of each image was reconstructed using the VisualSFM structure-from-motion software [70, 69] and the
+
+![](images/afe3cccb487e6fa90b540974d96e7f2bfb61553b1921b007398e949eb17c0d4b.jpg)  
+Figure 10. Mean and median errors of the proposed OPT LINEAR normal estimator with noise added to point coordinates (Point), affine parameters (Affine) or both (Combined).
+
+![](images/b9507a16a4962393dd58f28b472a2706f878510d3e77bd11f2d4779683469989.jpg)
+
+![](images/cfa87fde61ea568549103ddeb3b6007b3743ace39c2b1b98cf661245e154abdd.jpg)  
+Figure 11. Mean and median errors of the proposed OPT LINEAR normal estimator with noise added to only the affine  $(A\text{ff})$  elements and, also, to the point coordinates  $(Comb)$ . For Aff and Comb, zero-mean Gaussian noise is added directly to the affine elements. For Aff-Elem and Comb-Elem, the affine matrix is decomposed by SVD to two rotations and a diagonal matrix. Zero-mean Gaussian noise is added to the rotation angles and diagonal elements.
+
+![](images/166482e29d8bfca0ca6612feb0cd377d7bca63f7c3742e99f5ecb687fe59ba8c.jpg)
+
+images are divided into training and testing sets. To have more accurate reference data, we ran COLMAP [63] obtaining a dense point cloud. Finally, we applied the normal estimator implemented in MeshLab<sup>3</sup> to obtain an oriented point cloud to be considered reference. The normals are fitted to the 200 nearest neighbors. For each query image, we ran DenseVLAD [67] to get 20 potential database images. In our experiments, the algorithms are tested on each obtained image pair, i.e., 20 pairs for each query image. For ScanNet, we use the 1500 image pairs used in [59].
+
+On Cambridge Landmarks, we detect affine correspondences by detecting DoG features [42], finding the affine shape by AffNet [48], and extracting HardNet [47] descriptors. This approach is among the leaders in the IMC 2020 benchmark [34]. On ScanNet, we detect SuperPoint [20] features, match them by SuperGlue [59], and find the affine shape by, first, estimating scale and orientation by SelfScaleOri [41] and, then, finding the affine shapes by AffNet.
+
+Normal Estimation. To test normal estimators, we ran all of them on the data described in the previous section. For each image pair, we iterate through all detected correspondences that are consistent with the camera pose - they have less than 5 pixels of Sampson distance. For each of
+
+![](images/1bb67d9598776c047ef3944af4a4be4982d425f647c69f790568eade56b61920.jpg)  
+Figure 12. The median angular error in degrees (left) and the AUC@  $10^{\circ}$  score (right) of the estimated normals plotted as a function of the error in the detected affine correspondences on the Cambridge Landmarks dataset.
+
+![](images/4b414b70d67b0597142c805eaa4d8ea1d857e02ceaa495e357e61aa2c07afe43.jpg)
+
+them, we ran multiple normal estimators, measure their angular error and run-time. To understand how the estimators proceed with respect to the error in the detected affine transformations  $\mathbf{A}$ , we calculate the reference transformation  $\mathbf{A}^*$  from the camera parameters and the surface normal as shown in [9]. We define the affine error as the Frobenius norm of the difference matrix  $\| \mathbf{A} - \mathbf{A}^* \|_{\mathrm{F}}$ .
+
+The Area Under the recall Curve (AUC) at  $5^{\circ}$ ,  $10^{\circ}$  and  $20^{\circ}$ , the avg. and med. angular errors with the standard deviation in degrees and the run-time in nanoseconds on the Cambridge Landmarks dataset are reported in Table 1. As expected, the best results are achieved by the optimal algorithms, all of them leading to exactly the same results. Our proposed linear solution of the optimal problem,  $\mathrm{OPT}_{\mathrm{LINEAR}}$ , is five times faster than the cubic solution  $\mathrm{OPT}_{\mathrm{CUBIC}}$ . While  $\mathrm{OPT}_{\mathrm{LINEAR}}$  is slightly slower than the fastest FNE method, it is significantly more accurate. Interestingly, while the method of Raposo et al. [56] worked well in the synthetic experiments, we did not manage to achieve good accuracy with it in the real-world experiments. Figure 12 plots the AUC@ $10^{\circ}$  score and the median angular error as a function of the noise in the detected affine parameters. As expected, normal estimation is highly sensitive to the quality of the affine parameters. The optimal algorithms are the most stable ones.
+
+The results on the ScanNet dataset are reported in Table 3. Similarly as on Cambridge Landmarks, all optimal solvers lead to the same accuracy while the proposed one being the fastest by a large margin.
+
+Visual Localization. To test the impact of surface normal estimation on localization, on the Cambridge Landmarks dataset, we applied the following procedure. We iterate through all query images, each having 10 database images retrieved by DenseVLAD. First, we estimate the relative pose between the query and a random database image by GC-RANSAC [6] with P3P [54]. Using the obtained pose, we estimate the surface normal of each inlier affine correspondence in the coordinate system of the query image. Next, we iterate through all unused database images and use
+
+<table><tr><td>Estimator</td><td>AUC@5°↑</td><td>@10°↑</td><td>@20°↑</td><td>Avg. err. (°) ↓</td><td>Med. err. (°) ↓</td><td>Std. dev. (°) ↓</td><td>Time (ns) ↓</td></tr><tr><td>FNE [49]</td><td>6.0</td><td>16.0</td><td>33.7</td><td>21.0</td><td>14.3</td><td>21.3</td><td>172</td></tr><tr><td>FNESUPER</td><td>8.9</td><td>22.1</td><td>42.0</td><td>10.9</td><td>16.8</td><td>15.6</td><td>1721</td></tr><tr><td>Raposo [56]</td><td>0.4</td><td>1.5</td><td>5.7</td><td>38.5</td><td>36.5</td><td>18.2</td><td>883</td></tr><tr><td>OPTQUARTIC [9]</td><td>11.7</td><td>27.3</td><td>48.7</td><td>8.8</td><td>13.6</td><td>12.9</td><td>2528</td></tr><tr><td>OPTCUBIC [46]</td><td>11.7</td><td>27.3</td><td>48.7</td><td>8.8</td><td>13.6</td><td>12.9</td><td>2484</td></tr><tr><td>Proposed OPTLINEAR</td><td>11.7</td><td>27.3</td><td>48.7</td><td>8.8</td><td>13.6</td><td>12.9</td><td>596</td></tr></table>
+
+Table 1. The AUC score at  $5^{\circ}$ ,  $10^{\circ}$ , and  $20^{\circ}$  of the estimated surface normals, their average and median errors in degrees, the standard deviation, and the average run-time, in nanoseconds, of a single normal estimation on the Cambridge Landmarks dataset.  
+
+<table><tr><td rowspan="2">Solver</td><td rowspan="2">Normal estimator</td><td colspan="3">Pose accuracy ↑</td><td rowspan="2">Med. rot. err. (°) ↓</td><td rowspan="2">Med. pos. err. (cm) ↓</td></tr><tr><td>AUC@1°, 10cm</td><td>@2.5°, 25cm</td><td>@5°, 50cm</td></tr><tr><td>P3P</td><td>-</td><td>44.3</td><td>59.1</td><td>67.5</td><td>10.1</td><td>13.5</td></tr><tr><td>P1AC</td><td>FNESUPER</td><td>47.8</td><td>62.0</td><td>69.7</td><td>6.2</td><td>12.8</td></tr><tr><td>P1AC</td><td>Proposed OPTLINEAR</td><td>48.0</td><td>62.3</td><td>69.9</td><td>5.2</td><td>1.8</td></tr><tr><td>P1AC</td><td>GT</td><td>48.4</td><td>62.9</td><td>70.4</td><td>4.6</td><td>2.1</td></tr></table>
+
+Table 2. Localization AUC scores at  $(1^{\circ}, 10\mathrm{cm})$ ,  $(2.5^{\circ}, 25\mathrm{cm})$ , and  $(5^{\circ}, 50\mathrm{cm})$ , the median rotation (in degrees) and translation errors (in centimeters) on the Cambridge Landmarks dataset [39] when using either the P3P solver or the normal-based P1AC solver with FNEsuper or the proposed surface normal estimator OPTLINEAR. We also show the results with the reference normals. The minimal solvers are used within Graph-Cut RANSAC [6] that takes care of the robust estimation.  
+
+<table><tr><td></td><td>AUC@5↑</td><td>AUC@10↑</td><td>AUC@20↑</td><td>Time (ns)↓</td></tr><tr><td>FNE</td><td>14.8</td><td>15.4</td><td>17.0</td><td>207</td></tr><tr><td>FNESUPER</td><td>34.9</td><td>35.2</td><td>36.6</td><td>2142</td></tr><tr><td>OPTQUARTIC</td><td>35.0</td><td>35.5</td><td>37.3</td><td>18745</td></tr><tr><td>OPTCUBIC</td><td>35.0</td><td>35.5</td><td>37.3</td><td>3146</td></tr><tr><td>OPTLINEAR</td><td>35.0</td><td>35.5</td><td>37.3</td><td>747</td></tr></table>
+
+the P1AC [68] solver inside GC-RANSAC to estimate the pose. P1AC requires a single AC and the surface normal. We only test the proposed solver and  $\mathrm{FNE}_{\mathrm{SUPER}}$ . All other optimal solvers lead exactly to the same accuracy as ours.  $\mathrm{FNE}_{\mathrm{SUPER}}$  is the most accurate sub-optimal algorithm.
+
+The accuracy of the estimated pose for the database images with respect to their GT pose is reported in Table 2. All solvers, benefiting from normals and affine correspondences, are more accurate than using P3P. We also report the accuracy when using the reference normals to see the upper bound achievable using normals. The proposed optimal solver OPT LINEAR is only slightly (i.e.  $< 1$  AUC point) behind the accuracy of the GT normal, while being more accurate than the P3P-based solution. The AUC@  $10^{\circ}$  scores on each scene are reported in Table 4. The proposed OPT LINEAR leads to the best accuracy on all but two scenes.
+
+# 5. Conclusions
+
+In this work, we have presented a novel solver for estimating surface normals from affine correspondences when having calibrated cameras. The proposed method yields the
+
+Table 3. The AUC scores at  $5^{\circ}$ ,  $10^{\circ}$  and  $20^{\circ}$ , and the avg. time (in nanoseconds) of normal estimators on the ScanNet datasets [18].  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">P3P</td><td colspan="3">P1AC</td></tr><tr><td>GT</td><td>FNESUPER</td><td>Prop. OPTLINEAR</td></tr><tr><td>St. Mary&#x27;s Church</td><td>44.2</td><td>47.7</td><td>46.1</td><td>46.5</td></tr><tr><td>Old Hospital</td><td>44.1</td><td>57.4</td><td>56.8</td><td>57.3</td></tr><tr><td>King&#x27;s College</td><td>52.1</td><td>54.2</td><td>54.1</td><td>54.0</td></tr><tr><td>Shop Facade</td><td>65.5</td><td>65.0</td><td>63.8</td><td>64.1</td></tr><tr><td>Great Court</td><td>62.7</td><td>63.2</td><td>63.0</td><td>63.5</td></tr><tr><td>Street</td><td>17.4</td><td>22.8</td><td>23.1</td><td>23.0</td></tr></table>
+
+Table 4. The area under the recall curve (AUC) at  $1^{\circ}$  rotation and 10cm position errors on the Cambridge Landmarks dataset. We combine the P1AC solver with  $\mathrm{FNE}_{\mathrm{SUPER}}$  and the proposed normal estimator. Also, we test it with the reference normals.
+
+optimal solution, in the least squares sense, with respect to the affine error, and is particularly fast as the optimal normal is obtained via linear equations. We compared our method to all existing alternatives that estimate normals from a single correspondence on both simulated and real-world data. Additionally, we provide tests that highlight the benefits of using surface normals in image-based localization. The proposed estimator has demonstrated superior accuracy compared to the fastest existing method while being the second fastest overall. Our results suggest that the proposed algorithm is a promising alternative for surface normal estimation in computer vision applications.
+
+Acknowledgement. The authors warmly thank the reviewers for their very valuable comments and suggestions. The work was supported by the ETH Postdoc fellowship.
+
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+        "type": "text",
+        "text": "Fast Inference and Update of Probabilistic Density Estimation on Trajectory Prediction",
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+        "text": "Takahiro Maeda Norimichi Ukita",
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+        "text": "Toyota Technological Institute, Japan",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Safety-critical applications such as autonomous vehicles and social robots require fast computation and accurate probability density estimation on trajectory prediction. To address both requirements, this paper presents a new normalizing flow-based trajectory prediction model named FlowChain. FlowChain is a stack of conditional continuously-indexed flows (CIFs) that are expressive and allow analytical probability density computation. This analytical computation is faster than the generative models that need additional approximations such as kernel density estimation. Moreover, FlowChain is more accurate than the Gaussian mixture-based models due to fewer assumptions on the estimated density. FlowChain also allows a rapid update of estimated probability densities. This update is achieved by adopting the newest observed position and reusing the flow transformations and its log-det-jacobians that represent the motion trend. This update is completed in less than one millisecond because this reuse greatly omits the computational cost. Experimental results showed our FlowChain achieved state-of-the-art trajectory prediction accuracy compared to previous methods. Furthermore, our FlowChain demonstrated superiority in the accuracy and speed of density estimation. Our code is available at https://github.com/meaten/FlowChain-ICCV2023.",
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+        "text": "1. Introduction",
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+        "text": "Human trajectory prediction is a challenging problem because human movements are not deterministic, unlike the billiard balls on a pool table. Starting from deterministic approaches [26, 49], to tackle its indeterministic nature, many stochastic approaches are proposed, such as Gaussian distribution-based [1, 42, 66], generative adversarial networks (GANs)-based [9, 10, 25, 57], variational autoencoder (VAE)-based [58, 71, 73], diffusion-based [24], and normalizing flow-based [59] methods. These methods",
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+        "image_caption": [
+            "Figure 1. Effects of our FlowChain. Our FlowChain can rapidly estimate accurate density than VAE-based methods as shown in green. Furthermore, we propose an update of estimated density that only requires an instant of time, as shown in blue."
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "can be applied to human-interactive systems, including autonomous vehicles [8,11,38,55,69] and social robots [4,39].",
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+        "text": "In these safety-critical applications, autonomous systems must infer the existence probability density distribution (\"density\" for brevity) of other moving objects such as pedestrians around on all prediction steps for risk assessment (i.e., collision avoidance). However, most of the indeterministic approaches mentioned above merely generate possible future trajectories and cannot estimate the density alone. We need to further apply kernel density estimation (KDE) to these future trajectories for density estimation.",
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+        "type": "text",
+        "text": "Thus, trajectory prediction with density estimation usually takes a larger computational cost than only generating possible trajectories. This computational cost accumulates upon every single prediction update along with highly-frequent observations. This is unacceptable to autonomous systems that work in rapidly changing environments.",
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+        "text": "To address both computational cost and density estimation, we propose a new normalizing flow-based model named FlowChain (Fig. 1). This speed and accuracy are because normalizing flows need no additional approximation such as Gaussian mixture modeling or KDE with limited samples. FlowChain can estimate a density on each prediction time step thanks to our stacked normalizing flow archi",
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+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
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+    {
+        "type": "text",
+        "text": "tecture unlike Flomo [59], which is closest to FlowChain. While Flomo uses one flow module that only estimates the spatial density of a trajectory, FlowChain further predicts its spatio-temporal density so that the spatial density of an agent position at each time step is also predicted. This stacking architecture also allows a rapid update of an estimated density in an instant of time e.g. less than one millisecond. This update is achieved by reusing the flow transformations and its log-det-jacobians that represent the motion trend such as going straight or turning right. This update with the reuse significantly omits the computational cost while maintaining the accuracy of density estimation because the motion trend doesn't change for several time steps.",
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+        "text": "Our contributions are three-fold as follows:",
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+            "- We propose a normalizing flow-based trajectory prediction network named FlowChain that rapidly estimates the accurate density of future positions on each prediction time step.",
+            "- Our update procedure for the density requires less than one millisecond by adopting new observations and reusing the motion trend as the flow transformations.",
+            "- Our FlowChain achieved comparable prediction accuracy to other computationally-intense state-of-the-art methods on several benchmarks."
+        ],
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+        "text": "2. Related Work",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Human Trajectory Prediction. The earliest works on human trajectory prediction [26, 49] are deterministic regression models using attractive and repulsive forces between humans and goal positions. However, human movements are indeterministic in nature. Therefore, many approaches are proposed to model this uncertainty. Social-LSTM [1] and Social-STGCNN [42] regress the parameters of the bivariate Gaussian distributions to model the uncertainty of future trajectories. These approaches [1, 42] are simple and require low computational cost but cannot catch the future multi-modality we often see at road intersections.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To address this multi-modality, many stochastic prediction models are proposed. Trajectory++ [58], AgentFormer [71], and ExpertTraj [73] combined the aforementioned Gaussian regression with VAE [32]. These models predict a bivariate Gaussian mixture based on stochastically sampled latent variables [28, 52]. We can obtain the density by summing up each estimated Gaussian density map. However, these Gaussian mixture methods have limited expressive power and thus predict inaccurate distributions.",
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+    {
+        "type": "text",
+        "text": "On the other hand, GAN [18]-based approaches implicitly handle this uncertainty. Social-GAN [25], SoPhie [57], Goal-GAN [10], and MG-GAN [9] use a GAN architecture to predict diverse trajectories conditioned on random",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "noises. While they cannot estimate the density alone, we can estimate the density by applying KDE [48, 54] on sampled trajectories",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "The denoising diffusion probabilistic model [27] can also implicitly handle uncertainty by generating multiple trajectories. Recently, denoising diffusion attracts much attention for its state-of-the-art performances in several generative tasks, including image synthesis [12] and audio generation [33]. MID [24] is a trajectory prediction model incorporating denoising diffusion, which achieves state-of-the-art performances. We can also estimate the density of MID by applying KDE. However, GAN-based and diffusion-based approaches cannot estimate accurate densities because KDE is performed on limited samples.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "On the contrary, normalizing flow [51, 64, 65]-based trajectory prediction method named Flomo [59] can rapidly generate future trajectories and their accurate probabilities thanks to the analytical computation of normalizing flows. However, this approach only estimates the probability of each future trajectory and thus cannot estimate the density of positions on each prediction step alone.",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Normalizing Flows. Thanks to its bijective process, normalizing flow can compute the density analytically. This analytical computation allows us fast and accurate density estimation because it does not need additional approximation such as KDE with limited samples. For more expressive power, several bijective families were proposed such as affine coupling layers [13, 14], autoregressive maps [17, 47], invertible linear transformations [31], and invertible ResNet block [2, 6]. All flows mentioned above have a topology constraint between the base and estimated densities. This topology constraint limits the expressive power of normalizing flows. To address this constraint, some approaches incorporated stochasticity into flows [68]. However, this stochasticity makes analytical density calculation impossible. Therefore, further density approximation such as KDE is required for this stochastic normalizing flow. Other approaches [5, 7, 44] use surjective layers, which relax the topology constraint but make analytical density computation with a little looseness. In summary, the optimal normalizing flow model architecture for future trajectory density estimation is still unknown.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Density Estimation. Kernel density estimation [48, 54] is the most representative non-parametric method that estimates a continuous density from a finite set of data. Suppose that we have  $A$  input data and  $B$  points evaluated, the computational cost is  $\\mathcal{O}(AB)$ . This is a huge cost for autonomous systems. Therefore, several accelerations are proposed such as data binning [19, 60, 62, 67], fast sum updating [15, 16, 35, 61], fast Gauss transform [22, 23, 34], and dual-tree method [20, 21, 36]. However, we confirmed that the KDE implementation utilizing GPU is faster than these accelerations including [3, 45, 46]. We use the KDE utilizing",
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "9796",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "GPU for comparison in Sec. 4.",
+        "bbox": [
+            76,
+            90,
+            282,
+            104
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. Proposed Approach",
+        "text_level": 1,
+        "bbox": [
+            76,
+            119,
+            267,
+            137
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present our normalizing flow-based trajectory prediction named FlowChain and its update of estimated density. First, we formulate the trajectory prediction task and conditional CIFs in Sec. 3.1. and Sec. 3.2 respectively. Then, we present our FlowChain model architecture in Sec. 3.3. After that, we describe its fast update procedure of estimated density in Sec.3.4. Finally, we explain how to obtain the density map in Sec. 3.5.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1. Problem Formulation",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We aim to estimate agents' plausible future trajectory densities from their past trajectories. The trajectory of each agent is defined as a sequence  $(\\mathbf{x}_0, \\mathbf{x}_1, \\ldots, \\mathbf{x}_T)$  of positions  $\\mathbf{x}_t = (x_t, y_t)$  over discrete timesteps  $t \\in \\{0, \\ldots, T\\}$ . Given the current time step  $t$ , we predict future  $T_{\\mathrm{f}}$ -step trajectory densities  $\\{p(\\tilde{\\mathbf{x}}_{t+1}^i | \\mathbf{o}_t), \\ldots, p(\\tilde{\\mathbf{x}}_{t+T_{\\mathrm{f}}}^i | \\mathbf{o}_t)\\}$  given  $T_{\\mathrm{o}}$ -step observed trajectory  $\\mathbf{o}_t = \\{(x_{t-T_{\\mathrm{o}}}^i - 1, \\ldots, x_t^i) | i \\in \\{1, 2, \\ldots, N_a\\}\\}$ , where  $N_a$  and  $i$  denote the number of agents and the index of each agent, respectively. This index will be omitted for brevity in later use. We denote  $\\mathbf{x}$  as an observed position,  $\\tilde{\\mathbf{x}}$  as a predicted future position, and  $\\hat{\\mathbf{x}}$  as a ground truth future position.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2. Conditional Continuously-indexed Flow",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Conditional normalizing flow is a density estimation method that uses a parameterized bijective transformation  $f$  conditioned on additional information  $\\mathbf{c}$  such as an encoded vector from observed trajectories, i.e.,  $\\mathbf{o}_t$ , for trajectory prediction. The bijection  $f$  transforms a base density  $p(\\mathbf{z})$  to an expressive density  $p(\\mathbf{y})$  by the analytical computation of the change-of-variables formula as follows:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {y} = f (\\mathbf {z}) \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\np (\\mathbf {y}) = p (\\mathbf {z}) \\left| \\det  \\nabla_ {\\mathbf {y}} f ^ {- 1} (\\mathbf {y}, \\mathbf {c}) \\right| \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The parameters of bijective transformation  $f$  can be learned by maximizing the likelihood of samples  $\\hat{\\mathbf{y}}$  from datasets, or minimizing the negative log-likelihood as follows:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nN L L = - \\log p (\\hat {\\mathbf {y}} | \\mathbf {c}) \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            200,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A normalizing flow with only bijective transformations has a constraint that the topology of a base density is preserved. For example, if the base density is a simple unimodal Gaussian distribution, the normalizing flow fails to fit a distinct multi-modal density. To avoid this problem of bijective transformation, we use continuously-indexed flows (CIFs) [7], akin to an infinite mixture of normalizing flows. While CIFs potentially avoid the topology constraint, CIFs still enable analytical density computation with a little",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "looseness  $\\mathcal{E}$  because CIFs incorporate inference and generative surjections according to the survey [44]. We investigated the tradeoff between the expressive power and exactness of estimated density on CIFs. We concluded this looseness of estimated density can be ignored in trajectory prediction, as we show in the supplemental materials. Based on the above discussion, We use conditional CIFs as the base normalizing flow model of our FlowChain.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.3. FlowChain model",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The overview of the FlowChain model is shown in Fig. 2. Given the current time step  $t$ , we estimate future trajectory densities  $\\{p(\\tilde{\\mathbf{x}}_{t + 1}|\\mathbf{o}_t),\\dots ,p(\\tilde{\\mathbf{x}}_{t + T_{\\mathrm{f}}}|\\mathbf{o}_t)\\}$  by chaining the conditional CIFs, each of which transforms the density of the former time step  $p(\\tilde{\\mathbf{x}}_{t + n - 1}|\\mathbf{o}_t)$  to the density of the next time step  $p(\\tilde{\\mathbf{x}}_{t + n}|\\mathbf{o}_t)$ , shown in the upper side of Fig. 2. Our formulation consists of two equations: the initial condition for the first step prediction (Eq.(4)) and the recurrence relation between the latter steps (Eq.(5)) as follows.",
+        "bbox": [
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+            244,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} p \\left(\\tilde {\\mathbf {x}} _ {t + 1} \\mid \\mathbf {o} _ {t}\\right) = \\mathcal {N} (\\mathbf {z}; \\mathbf {x} _ {t}, \\sigma) | \\det  \\nabla_ {\\tilde {\\mathbf {x}} _ {t + 1}} f _ {1} ^ {- 1} \\left(\\tilde {\\mathbf {x}} _ {t + 1}, \\mathbf {o} _ {t}\\right) | (4) \\\\ p \\left(\\tilde {\\mathbf {x}} _ {t + n} \\mid \\mathbf {o} _ {t}\\right) = p \\left(\\tilde {\\mathbf {x}} _ {t + n - 1} \\mid \\mathbf {o} _ {t}\\right) | \\det  \\nabla_ {\\tilde {\\mathbf {x}} _ {t + n}} f _ {n} ^ {- 1} \\left(\\tilde {\\mathbf {x}} _ {t + n}, \\mathbf {o} _ {t}\\right) | (5) \\\\ = \\exp \\left(\\log \\mathcal {N} (\\mathbf {z}; \\mathbf {x} _ {t}, \\sigma) \\right. \\\\ \\left. + \\sum_ {n ^ {\\prime} = 1} ^ {n} \\log | \\det  \\nabla_ {\\tilde {\\mathbf {x}} _ {t + n ^ {\\prime}}} f _ {n ^ {\\prime}} ^ {- 1} \\left(\\tilde {\\mathbf {x}} _ {t + n ^ {\\prime}}, \\mathbf {o} _ {t}\\right) |\\right) (6) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            509,
+            393,
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+            537
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $n \\in \\{2,3,\\dots,T_{\\mathrm{f}}\\}$ .  $f_{n}$  is a conditional CIF that transforms the sample  $\\tilde{\\mathbf{x}}_{t + n - 1}$  from a former density  $p(\\tilde{\\mathbf{x}}_{t + n - 1}|\\mathbf{o}_t)$  to the sample  $\\tilde{\\mathbf{x}}_{t + n}$  from a next density  $p(\\tilde{\\mathbf{x}}_{t + n}|\\mathbf{o}_t)$  as  $\\tilde{\\mathbf{x}}_{t + n} = f_n(\\tilde{\\mathbf{x}}_{t + n - 1},\\mathbf{o}_t)$ .  $f_{1}$  transforms the sample  $\\mathbf{z}$  from a base Gaussian density  $\\mathcal{N}(\\cdot ;\\mathbf{x}_t,\\sigma)$ .  $\\mathcal{N}(\\cdot ;\\mathbf{x}_t,\\sigma)$  is created from the newest observed position  $\\mathbf{x}_t$  as a mean position. The standard deviation  $\\sigma$  is a trainable parameter of FlowChain model. All of  $f_{1}$  and  $f_{n}$  are independently trained with the loss function (Eq.(3)) in our method.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "From  $f_{n}$ , we can sample multiple trajectories for best-of-N metrics [25] by sampling multiple  $\\mathbf{z}$  as follows:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\{\\tilde {\\mathbf {x}} _ {t + 1}, \\dots , \\tilde {\\mathbf {x}} _ {t + n} \\} = \\left\\{f _ {1} (\\mathbf {z}, \\mathbf {o} _ {t}), \\dots , f _ {n} \\circ \\dots \\circ f _ {2} \\circ f _ {1} (\\mathbf {z}, \\mathbf {o} _ {t}) \\right\\} \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.4. Update of estimated density",
+        "text_level": 1,
+        "bbox": [
+            500,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For accurate prediction by reusing the previous predictions, we focused on two features of the observed trajectory: the newest observed position and motion trend, shown in Fig. 2. Our FlowChain generates future trajectories by transforming the newest observed position  $\\mathbf{x}_t$  based on the motion trend e.g. go straight or turning left/right, modeled",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "9797",
+        "bbox": [
+            480,
+            944,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/bbb7e99b80d26d3aae1e45a2e55c9596c832c06ab6fc2acebc38dd25d322e822.jpg",
+        "image_caption": [
+            "Figure 2. Overview of our FlowChain model. FlowChain consists of sequential conditional CIFs denoted by  $f_{n}$ .  $f_{n}$  is conditioned on the feature vector denoted by Motion Trend from the Temporal-Social Encoder, which encodes the observed trajectories  $\\mathbf{o}_t$ . Since  $\\mathbf{o}_t$  includes the trajectories of other agents as well as an agent of interest (i.e., the target of prediction), the future trajectory is predicted while taking into account social interaction with the others.  $f_{n}$  transforms the density of the former time step  $p(\\tilde{\\mathbf{x}}_{t + n - 1}|\\mathbf{o}_t)$  to the density of the next time step  $p(\\tilde{\\mathbf{x}}_{t + n}|\\mathbf{o}_t)$  as shown in the upper side. However, for the first step, the Gaussian distribution  $\\mathcal{N}(\\cdot ;\\mathbf{x}_t,\\sigma)$ , which is created from the newest observed position  $\\mathbf{x}_t$ , is transformed. The boxes of past, current (i.e., newest), and future densities are colored by red, blue, and purple, respectively. In the update procedure shown on the bottom side, the Gaussian distribution  $\\mathcal{N}(\\cdot ;\\mathbf{x}_{n + 1},\\sigma)$ , created from newest observed position  $\\mathbf{x}_{t + 1}$  at  $t + 1$ , replace the estimated density from the former time step. The other calculations are reused as shown by gray arrows. We can properly update the density by using the newest observed position at  $t + 1$ ,  $\\mathbf{x}_{t + 1}$ , instead of  $p(\\tilde{\\mathbf{x}}_{t + 1}|\\mathbf{o}_t)$ . In a toy example shown in this figure, another incorrect mode observed in the predicted densities,  $p(\\tilde{\\mathbf{x}}_{t + 2}|\\mathbf{o}_t)$  and  $p(\\tilde{\\mathbf{x}}_{t + 3}|\\mathbf{o}_t)$ , is successfully suppressed in the updated predictions  $p^{\\prime}(\\tilde{\\mathbf{x}}_{t + 2}|\\mathbf{o}_{t + 1})$  and  $p^{\\prime}(\\tilde{\\mathbf{x}}_{t + 3}|\\mathbf{o}_{t + 1})$ . As shown in Sec. 4, our experiments validate this advantage of our proposed fast update procedure is greater than its disadvantage (i.e., a decrease in the prediction accuracy due to the reuse of the motion trend) so that the updated predictions are better than the initial predictions (e.g.,  $p^{\\prime}(\\tilde{\\mathbf{x}}_{t + 3}|\\mathbf{o}_{t + 1})$  is better than  $p(\\tilde{\\mathbf{x}}_{t + 3}|\\mathbf{o}_t)$  in the figure)."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            84,
+            89,
+            883,
+            356
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "by each flow  $f_{n}(\\mathbf{x}_{t + n},\\mathbf{o}_{t})$ . We cannot reuse the newest observed position  $\\mathbf{x}_t$  at  $t$  for the next time step because this position is a starting point of the future trajectories and thus greatly affect the accuracy of probability density estimation. Therefore, FlowChain adopt the newest observed position  $\\mathbf{x}_{t + n}$  at  $t + n$  by replacing the estimated density  $p(\\tilde{\\mathbf{x}}_{t + n}|\\mathbf{o}_t)$  with the Gaussian density  $\\mathcal{N}(\\cdot ;\\mathbf{x}_{t + n},\\sigma)$ , as shown in the bottom of Fig. 2. This replacement propagates the information of the newest observed position properly to the latter time steps.",
+        "bbox": [
+            75,
+            564,
+            472,
+            717
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, the motion trend can be reused because it doesn't change for several time steps unless a sudden strong external force is applied to a prediction target. Therefore, our FlowChain model reuses its trend in each flow transformation  $f_{n}(\\tilde{\\mathbf{x}}_{t + n},\\mathbf{o}_{t})$ . Specifically, we use the previously generated trajectories  $\\{\\tilde{\\mathbf{x}}_{t + 1},\\dots ,\\tilde{\\mathbf{x}}_{t + n}\\}$ . Thus, we can reuse the corresponding log-det-jacobians  $\\log |\\operatorname *{det}\\nabla_{\\tilde{\\mathbf{x}}_{t + n}}f_n^{-1}(\\tilde{\\mathbf{x}}_{t + n + 1},\\mathbf{o}_t)|$  which is the variation of the estimated densities, as shown by the gray arrows in Fig. 2. The equation of FlowChain's update is expressed",
+        "bbox": [
+            75,
+            750,
+            472,
+            902
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "similar to Eq.(4) and Eq.(5) as follows:",
+        "bbox": [
+            500,
+            564,
+            759,
+            579
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\np ^ {\\prime} \\left(\\tilde {\\mathbf {x}} _ {t + 2} \\mid \\mathbf {o} _ {t + 1}\\right) = \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {t + 1}; \\mathbf {x} _ {t + 1}, \\sigma\\right) | \\det  \\nabla_ {\\tilde {\\mathbf {x}} _ {t + 2}} f _ {2} ^ {- 1} \\left(\\tilde {\\mathbf {x}} _ {t + 2}, \\mathbf {o} _ {t}\\right) | \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            500,
+            590,
+            906,
+            636
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} p ^ {\\prime} \\left(\\tilde {\\mathbf {x}} _ {t + n} \\mid \\mathbf {o} _ {t + 1}\\right) = p ^ {\\prime} \\left(\\tilde {\\mathbf {x}} _ {t + n - 1} \\mid \\mathbf {o} _ {t + 1}\\right) | \\det  \\nabla_ {\\tilde {\\mathbf {x}} _ {t + n}} f _ {n} ^ {- 1} \\left(\\tilde {\\mathbf {x}} _ {t + n}\\right) | (9) \\\\ = \\exp \\left(\\log \\mathcal {N} (\\tilde {\\mathbf {x}} _ {t + 1}; \\mathbf {x} _ {t + 1}, \\sigma) \\right. \\\\ \\left. + \\sum_ {n ^ {\\prime} = 2} ^ {n} \\log | \\det  \\nabla_ {\\tilde {\\mathbf {x}} _ {t + n ^ {\\prime}}} f _ {n ^ {\\prime}} ^ {- 1} \\left(\\tilde {\\mathbf {x}} _ {t + n ^ {\\prime}}, \\mathbf {o} _ {t}\\right) |\\right) (10) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            500,
+            638,
+            893,
+            765
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where,  $n \\in \\{3, 4, \\ldots, T_{\\mathrm{f}}\\}$ . Thanks to reusing each flow transformation  $f_{n}(\\mathbf{x}_{t + n}, \\mathbf{o}_{t})$ , we don't need to evaluate log-det-jacobians  $\\log |\\operatorname*{det}\\nabla_{\\tilde{\\mathbf{x}}_{t + n}}f_n^{-1}(\\tilde{\\mathbf{x}}_{t + n})|$  again. This will greatly omit the computational cost. As a result, we can update the probability density by only calculating the Gaussian base distribution, summing up the log-det-jacobians, and taking exponentials, which take less than  $1ms$  in total. This rapid update is beneficial, especially for autonomous",
+        "bbox": [
+            496,
+            779,
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+            902
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "9798",
+        "bbox": [
+            480,
+            944,
+            514,
+            955
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/52ac5db57636058d0d1358939e84c2ce76c08ffe6801a183030550912d708e22.jpg",
+        "image_caption": [
+            "Trajectories",
+            "Time steps",
+            "Figure 3. Overview of Simfork dataset. The trajectories shown in the left figure are examples of the observed and ground truth trajectories for training. The position of each step is sampled from the distributions of the middle figures. The sum of each distribution is shown in the right figure. The boxes of past, current (i.e., newest), and future densities are colored by red, blue, and purple, respectively."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            108,
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+            233
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/49a2a5f9a20d09fb5532c5217e056cf0bdd0a2dd983c76ad9ea17624e40d3e02.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            488,
+            95,
+            771,
+            233
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/272646ed6adfb81e2ef5a3be6a67db80d94dd65d0b015974e7b254183515b3f3.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            774,
+            95,
+            867,
+            218
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "systems where computational resources are limited and often become busy by executing other computationally-intense tasks such as object detection, mapping, and path planning.",
+        "bbox": [
+            75,
+            310,
+            470,
+            371
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.5. How to obtain the density map",
+        "text_level": 1,
+        "bbox": [
+            76,
+            378,
+            349,
+            395
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The density map, not like the mere one-point estimate, is needed when autonomous systems perform visualization of density or successive tasks such as navigation. We can correct the possible future points with estimated density by sampling multiple trajectories from Eq.(7) and its density from Eq.(6). The density map is obtained by collecting these points with density at the same time step.",
+        "bbox": [
+            75,
+            401,
+            468,
+            508
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments and Results Analysis",
+        "text_level": 1,
+        "bbox": [
+            76,
+            520,
+            385,
+            537
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1. Experimental Setup",
+        "text_level": 1,
+        "bbox": [
+            76,
+            545,
+            267,
+            561
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets. Experiments were conducted on various datasets including our developed dataset with synthesized data Simfork, and real-image benchmarks including ETH/UCY [37, 50] and SDD [53]. On all the datasets, the numbers of past observed and future predicted steps are  $T_{o} = 8$  and  $T_{f} = 12$ , respectively. All steps have an interval of 0.4 seconds in accordance with the literature such as [25, 58].",
+        "bbox": [
+            75,
+            568,
+            468,
+            672
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Simfork is a synthesized dataset for comparison on the accuracy of predicted densities, which are produced as explained in Sec. 3.5. In all trajectories in this dataset, either of the two types of traveling directions (i.e., turning left or right) is observed after a straight trajectory, as shown in Fig. 3. Positions before and after the fork (colored by green and black, respectively, in the leftmost figure of Fig. 3) are used as past observed and future ground-truth positions, respectively. Each position of a trajectory is independently perturbed by the Gaussian noise, which results in the density at each time step.",
+        "bbox": [
+            75,
+            674,
+            468,
+            839
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Two pedestrian trajectory datasets, ETH [50] and UCY [37], are merged to ETH/UCY. This dataset has five scenes: ETH, HOTEL, UNIV, ZARA1, and ZARA2. For training and testing, we follow the standard leave-one-out",
+        "bbox": [
+            75,
+            839,
+            470,
+            900
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "approach, where we train on 4 scenes and test on the remaining one. We follow the train-validation-test split of Social-GAN [25].",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$SDD$  consists of 20 scenes captured in bird's eye view using aerial drones. This dataset contains various moving agents such as pedestrians, bicycles, and cars, but we focus on pedestrians for comparison, as with Trajnet [56]. We also follow the train-test split of Trajnet [56].",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Metrics Evaluation of accuracy on trajectory prediction is done with the widely-used metrics, Average Displacement Error (ADE) and Final Displacement Error (FDE). ADE is the average L2 distance between the predicted and ground truth trajectories over time. FDE is the L2 distance between the endpoints of the predicted and ground truth trajectories. We follow the Best-of-N procedure [25] with  $N = 20$ .",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For evaluation of the accuracy of density estimation, we use earth mover's distance (EMD) and log-probability. EMD is a measure of the distance between two densities by the minimum cost to move one density pile to the other. EMD is calculated between the estimated density and ground truth density on each prediction step. Thus, we use EMD on Simfork dataset, where the ground truth densities are available. The advantage of EMD over the widely-used KL divergence is that EMD works properly even if the two density is disjoint while KL divergence returns a constant value. We use log-probability on the pedestrian trajectory prediction datasets, where the ground truth density is unavailable. The log-probability is calculated on the ground truth trajectory instead of density.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details. Each conditional CIFs in our FlowChain used three-layer RealNVP [14] with multilayer perceptions (MLPs) of three hidden layers and 128 hidden units. All the conditional CIFs are with independent weights conditioned on the feature vectors from Trajectory++ [58] encoder as the temporal-social encoder. We used a batch size of 128 and ADAM optimizer [30] with a learning rate of  $10^{-4}$ . All experiments were conducted on a single NVIDIA A100 GPU. We use a KDE implementation utilizing GPU as mentioned in Sec. 2.",
+        "bbox": [
+            496,
+            750,
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+            900
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "9799",
+        "bbox": [
+            480,
+            944,
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+            955
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/5d818df1050df31267b2edd0fb12bfd2a196ad86d95bd387946b1ce33363768f.jpg",
+        "table_caption": [
+            "Table 1. Quantitative results of trajectory prediction on ETH/UCY dataset with Best-of-20 metrics. T + I denotes its method takes a trajectory and an image as inputs while T denotes its method takes only a trajectory. Scores are in meters. Red and blue scores denote the best and the second-best in methods with T. Lower is better. The results of Trajectron++ [58] and MID [24] are updated according to an implementation issue #53 on the Trajectron++ GitHub page."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Input</td><td colspan=\"2\">ETH</td><td colspan=\"2\">HOTEL</td><td colspan=\"2\">UNIV</td><td colspan=\"2\">ZARA1</td><td colspan=\"2\">ZARA2</td><td colspan=\"2\">Mean</td></tr><tr><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td></tr><tr><td>SoPhie [57]</td><td>T + I</td><td>0.70</td><td>1.43</td><td>0.76</td><td>1.67</td><td>0.54</td><td>1.24</td><td>0.30</td><td>0.63</td><td>0.38</td><td>0.78</td><td>0.54</td><td>1.15</td></tr><tr><td>Goal-GAN [10]</td><td>T + I</td><td>0.59</td><td>1.18</td><td>0.19</td><td>0.35</td><td>0.60</td><td>1.19</td><td>0.43</td><td>0.87</td><td>0.32</td><td>0.65</td><td>0.43</td><td>0.85</td></tr><tr><td>MG-GAN [9]</td><td>T + I</td><td>0.47</td><td>0.91</td><td>0.14</td><td>0.24</td><td>0.54</td><td>1.07</td><td>0.36</td><td>0.73</td><td>0.29</td><td>0.60</td><td>0.36</td><td>0.71</td></tr><tr><td>Social-LSTM [1]</td><td>T</td><td>1.09</td><td>2.35</td><td>0.79</td><td>1.76</td><td>0.67</td><td>1.40</td><td>0.47</td><td>1.00</td><td>0.56</td><td>1.17</td><td>0.72</td><td>1.54</td></tr><tr><td>Social-GAN [25]</td><td>T</td><td>0.87</td><td>1.62</td><td>0.67</td><td>1.37</td><td>0.76</td><td>1.52</td><td>0.35</td><td>0.68</td><td>0.42</td><td>0.84</td><td>0.61</td><td>1.21</td></tr><tr><td>STGAT [29]</td><td>T</td><td>0.65</td><td>1.12</td><td>0.35</td><td>0.66</td><td>0.52</td><td>1.10</td><td>0.34</td><td>0.69</td><td>0.29</td><td>0.60</td><td>0.43</td><td>0.83</td></tr><tr><td>Social-STGCNN [42]</td><td>T</td><td>0.64</td><td>1.11</td><td>0.49</td><td>0.85</td><td>0.44</td><td>0.79</td><td>0.34</td><td>0.53</td><td>0.30</td><td>0.48</td><td>0.44</td><td>0.75</td></tr><tr><td>PECNet [41]</td><td>T</td><td>0.61</td><td>1.07</td><td>0.22</td><td>0.39</td><td>0.34</td><td>0.56</td><td>0.25</td><td>0.45</td><td>0.19</td><td>0.33</td><td>0.32</td><td>0.56</td></tr><tr><td>Trajectory++ [58]</td><td>T</td><td>0.61</td><td>1.03</td><td>0.20</td><td>0.28</td><td>0.30</td><td>0.55</td><td>0.24</td><td>0.41</td><td>0.18</td><td>0.32</td><td>0.31</td><td>0.52</td></tr><tr><td>MID [24]</td><td>T</td><td>0.55</td><td>0.88</td><td>0.20</td><td>0.35</td><td>0.30</td><td>0.55</td><td>0.29</td><td>0.51</td><td>0.20</td><td>0.38</td><td>0.31</td><td>0.53</td></tr><tr><td>Social Implicit [43]</td><td>T</td><td>0.66</td><td>1.44</td><td>0.20</td><td>0.36</td><td>0.31</td><td>0.60</td><td>0.25</td><td>0.50</td><td>0.22</td><td>0.43</td><td>0.33</td><td>0.67</td></tr><tr><td>FlowChain(Ours)</td><td>T</td><td>0.55</td><td>0.99</td><td>0.20</td><td>0.35</td><td>0.29</td><td>0.54</td><td>0.22</td><td>0.40</td><td>0.20</td><td>0.34</td><td>0.29</td><td>0.52</td></tr></table>",
+        "bbox": [
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+            142,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/ddbf6351dfbee96c7272e71ca3b90ff42612cfffb64ff0afe295efafb05a877a.jpg",
+        "table_caption": [
+            "Table 2. Quantitative results of trajectory prediction on SDD dataset with Best-of-20 metrics. Scores are in pixels. See the caption of Tab. 1 also for details."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Input</td><td>ADE</td><td>FDE</td></tr><tr><td>SoPhie [57]</td><td>T + I</td><td>16.27</td><td>29.38</td></tr><tr><td>Goal-GAN [10]</td><td>T + I</td><td>12.20</td><td>22.10</td></tr><tr><td>MG-GAN [9]</td><td>T + I</td><td>13.60</td><td>25.80</td></tr><tr><td>Social-LSTM [1]</td><td>T</td><td>31.19</td><td>56.97</td></tr><tr><td>Social-GAN [25]</td><td>T</td><td>27.25</td><td>41.44</td></tr><tr><td>STGAT [29]</td><td>T</td><td>14.85</td><td>28.17</td></tr><tr><td>Social-STGCNN [42]</td><td>T</td><td>20.76</td><td>33.18</td></tr><tr><td>PECNet [41]</td><td>T</td><td>9.97</td><td>15.89</td></tr><tr><td>Trajectory++ [58]</td><td>T</td><td>11.40</td><td>20.12</td></tr><tr><td>MID [24]</td><td>T</td><td>10.31</td><td>17.37</td></tr><tr><td>FlowChain (Ours)</td><td>T</td><td>9.93</td><td>17.17</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2. Prediction Accuracy",
+        "text_level": 1,
+        "bbox": [
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+            632,
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+            648
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "A wide range of previous trajectory prediction methods is quantitatively compared with our FlowChain on ETH/UCY and SDD datasets, shown in Tab. 1 and Tab. 2. We didn't include some methods [40, 63, 70-72] due to a different train-validation-test split.",
+        "bbox": [
+            75,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For ETH/UCY dataset, FlowChain achieved mean ADE/FDE of 0.29/0.52 in meters, which are the best among all trajectory prediction methods. On SDD dataset, FlowChain achieved average ADE/FDE of 9.93/17.17 in pixels, which are the best and second-best among all methods taking only trajectories as inputs [1,24,25,41,58,63].",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Although our focus is on density estimation with a minimal computational cost, FlowChain is comparable with the state-of-the-art methods also in terms of errors. We didn't evaluate the accuracy of future trajectories from FlowChain's update because the trajectories do not change",
+        "bbox": [
+            75,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/f530bdaa1cf5a2d18a228a98f412c3ee79b75f5eabac471307052eeca901d5f5.jpg",
+        "image_caption": [
+            "Figure 4. Temporal estimated densities on Simfork dataset."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            383,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "in the update as mentioned in Sec. 3.4.",
+        "bbox": [
+            500,
+            729,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3. Analysis on Density Estimation",
+        "text_level": 1,
+        "bbox": [
+            500,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The results of density estimation are compared between FlowChain and previous methods based on VAE, GAN, and diffusion approaches. Trajectory++ is the representative VAE-based trajectory prediction model with density estimation by GMM. Social-GAN is a minimal model of GAN-based trajectory prediction. MID is the diffusion-based trajectory prediction model with state-of-the-art prediction accuracy. We applied KDE to Social-GAN and MID since",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "9800",
+        "bbox": [
+            482,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/839b4e1985cd61c6514f7c80757822fc96ed14f964bc3cbf215c9e00143da48b.jpg",
+        "image_caption": [
+            "Figure 5. Temporal estimated densities on ETH/UCY dataset."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            84,
+            90,
+            455,
+            366
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "they cannot estimate the density alone. For a fair comparison, FlowChain, Social-GAN, and MID generated the same number of samples for future trajectories;  $10^{5}$  samples.",
+        "bbox": [
+            76,
+            411,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluated the computational time and the accuracy of density estimation of each method, as shown in Tab. 3. Social-GAN and MID take much longer time because KDE is slow in spite of running on GPU. Trajectory++ is a bit faster than our FlowChain due to its simple Gaussian mixture model but suffers from the poor quality of estimated density, as shown by the worst EMD, 2.545, on Simfork dataset and worst log-probability, -281.86, on ETH/UCY dataset. On the other hand, our FlowChain achieved the best EMD, 1.408, and the best log-probability, -0.26, in spite of the fast computation 37.0ms. Thus, our method successfully addresses both the computational cost and accurate density estimation.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The estimated densities on Simfork are visualized in Fig. 4. The densities estimated by Trajectory++ are too large compared to the ground truth densities. While the densities of Social-GAN and MID are stretched to the vertical axis, our FlowChain successfully estimated the tighter round-shape densities closest to the ground truth densities.",
+        "bbox": [
+            75,
+            656,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The estimated densities on  $ETH/UCY$  are shown in Fig. 5. The size of each density represents the positional uncertainty of a predicted position. As within Simfork, in  $ETH/UCY$ , an agent cannot change its traveling direction significantly in a short period (i.e.,  $T_{f} = 12$  is  $0.4 \\times 12 = 4.8$  seconds), the size of the density expected to be tight. The densities estimated by Social-GAN are tight but deviated from the ground truth trajectory. Trajectory++ generated the densities with several disconnected modes due to the approximation as a Gaussian mixture. While the densi",
+        "bbox": [
+            75,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/ba883f7cf9f760f6123000d999e38e9eeed29de25e15f81a06112c1804114af8.jpg",
+        "image_caption": [
+            "Figure 6. Temporal densities estimated by our update procedure on Simfork dataset."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            506,
+            90,
+            883,
+            281
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/14a0b33d57f3e80ef5505842a4b22a14ade06801bc05c2e66f6bfbaf37b077b4.jpg",
+        "image_caption": [
+            "Figure 7. Temporal densities estimated by our update procedure on ETH/UCY dataset."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            506,
+            320,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ties from MID are close to the ground truth trajectory, MID could not suggest tight densities and their modes deviated from the ground truth trajectories. On the contrary, the density from our FlowChain covers the ground truth trajectory with tight distributions. thanks to the explicit density estimation of conditional CIFs and our expressive model architecture. We should note that the estimated density of FlowChain has some fluctuation due to the looseness of the density computation of CIFs. However, we did not observe any potential problem or failure of density estimation related to this in our experiments.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4. Proposed Update of Density Estimation",
+        "text_level": 1,
+        "bbox": [
+            500,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As shown in Tab. 3, the accuracy of density estimation degraded along with the number of updates. However, this update is remarkably fast  $0.6ms$ , also shown in Tab. 3. Thus, autonomous systems can select the accurate prediction from scratch or the super rapid update by reusing predictions depending on the utilization of their computational resources.",
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The estimated densities of our update procedure are shown in Fig. 6 and Fig. 7, which show the densities up",
+        "bbox": [
+            500,
+            869,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "9801",
+        "bbox": [
+            482,
+            944,
+            513,
+            955
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/c58c0473e9af68b3d9290cfe44242a85d1a1c22d2e8709e1187f1c8283ed923e.jpg",
+        "table_caption": [
+            "Table 3. Accuracy and computational time of density estimation. Scores are in milliseconds for computational time. While lower is better for EMD on Simfork dataset, higher is better for log-probability on ETH/UCY dataset. We report the averaged EMD and log-probability over the 12 prediction steps."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td></td><td colspan=\"2\">Comp. Time</td><td colspan=\"2\">Accuracy of Density Estimation</td></tr><tr><td></td><td>All</td><td>w/o KDE</td><td>EMD ↓</td><td>log-probability ↑</td></tr><tr><td>Social-GAN [25]</td><td></td><td>11498.8</td><td>9.8</td><td>1.544</td><td>-3.40</td></tr><tr><td>Trajectory++ [58]</td><td></td><td>33.7</td><td>-</td><td>2.545</td><td>-281.86</td></tr><tr><td>MID [24]</td><td></td><td>23904.1</td><td>12219.1</td><td>2.135</td><td>-0.90</td></tr><tr><td rowspan=\"3\">FlowChain (Ours)</td><td>Pred at t</td><td>37.0</td><td>-</td><td>1.408</td><td>-0.26</td></tr><tr><td>Update from t - 1</td><td>0.6</td><td>-</td><td>2.539</td><td>-0.45</td></tr><tr><td>Update from t - 5</td><td>0.5</td><td>-</td><td>2.899</td><td>-0.53</td></tr></table>",
+        "bbox": [
+            142,
+            128,
+            823,
+            253
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/76d1a9de8abd9ffa3060dc4aa6c5873422784d1b9f4a0d21f49818c11241d778.jpg",
+        "table_caption": [
+            "Table 4. Accuracy of density estimation with the update."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\">Method</td><td colspan=\"8\">log-probability on each step</td></tr><tr><td>t+6</td><td>t+7</td><td>t+8</td><td>t+9</td><td>t+10</td><td>t+11</td><td>t+12</td><td>mean</td></tr><tr><td colspan=\"2\">Social-GAN [25]</td><td>-2.65</td><td>-2.86</td><td>-3.07</td><td>-3.26</td><td>-3.44</td><td>-3.62</td><td>-3.80</td><td>-3.24</td></tr><tr><td colspan=\"2\">Trajectory++ [58]</td><td>-121.19</td><td>-93.61</td><td>-71.62</td><td>-51.36</td><td>-36.85</td><td>-27.92</td><td>-22.72</td><td>-97.79</td></tr><tr><td colspan=\"2\">MID [24]</td><td>-1.06</td><td>-1.43</td><td>-1.75</td><td>-2.02</td><td>-2.26</td><td>-2.46</td><td>-2.65</td><td>-1.95</td></tr><tr><td rowspan=\"2\">FlowChain</td><td>Without Update</td><td>0.21</td><td>0.36</td><td>0.44</td><td>-0.08</td><td>-0.78</td><td>-0.75</td><td>-1.19</td><td>-0.26</td></tr><tr><td>With Update</td><td>0.20</td><td>0.50</td><td>0.39</td><td>0.44</td><td>0.20</td><td>0.26</td><td>0.16</td><td>0.31</td></tr></table>",
+        "bbox": [
+            120,
+            273,
+            844,
+            383
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a76851cb4fc158b0af1c58891e65eb3faa4e67d0e242ea4c520878e06ff7c78.jpg",
+        "image_caption": [
+            "Figure 8. Comparison of standard methods and FlowChain with updates."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            102,
+            406,
+            434,
+            593
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "dated at  $t + 1$  and  $t + 2$  from the prediction at  $t$ . On Simfork dataset, the wrong mode is successfully suppressed by exploiting the information of the new observation. On ETH/UCY dataset, the initial prediction deviated from the ground truth because the observed trajectory observed until  $t$  does not have the information on turning left. However, the updated densities covered the ground truth well by adopting the newest observed position despite the minimal computational cost.",
+        "bbox": [
+            75,
+            655,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We further perform a comparison with a practical setting. Models usually make predictions with intervals due to the large computational cost. Therefore, we usually cannot obtain the prediction from the latest observation, as shown by the upper side of Fig. 8. For example, assume we need the density of six time-step ahead such as  $p_{t+6}$  from  $\\mathbf{x}_t$ , we cannot obtain  $p_{t+7}$  from  $\\mathbf{x}_{t+1}$  or  $p_{t+8}$  from  $\\mathbf{x}_{t+2}$  due to the",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "computational overhead. However, with FlowChain's rapid update, we can obtain the update  $(p_{t + 7}, p_{t + 8})$  from the latest observation  $(\\mathbf{x}_{t + 1}, \\mathbf{x}_{t + 2})$  on every time step thanks to the minimal computational cost, as shown by the bottom side of Fig. 8. We conducted the comparison with the frame of interest as six time-step ahead  $t + 6$  and the log-probability metric. As shown by Tab. 4, our FlowChain with update achieves the highest log-probability. Therefore, we can conclude that FlowChain's update can improve the density estimation with minimal computational cost.",
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+        "page_idx": 7
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+        "text": "5. Concluding Remarks",
+        "text_level": 1,
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+        "text": "In this paper, we proposed a new normalizing flow-based trajectory prediction network named FlowChain that achieves fast and accurate probability density estimation. Furthermore, our update procedure reliably generates the density estimation of the next step based on the density estimation of the previous step in an instant. Experimental results demonstrated that FlowChain achieved comparable accuracy to the state-of-the-art trajectory prediction models and superiority in the accuracy of density estimation.",
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+        "type": "text",
+        "text": "Our FlowChain has one limitation. Our update procedure can reliably estimate the density for several steps but the performance will degrade along with the time steps because the reused flow transformations do not reflect the new observations. Our future work is determining when should we stop to use the update procedure and make predictions from scratch. Furthermore, FlowChain is the domain-agnostic model that can be applied to a wide range of prediction tasks such as stock price prediction, human motion prediction, and video prediction. We will investigate the applicability of our FlowChain on these other domains.",
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+    },
+    {
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+        "text": "9802",
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+    {
+        "type": "text",
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2023/Fast Inference and Update of Probabilistic Density Estimation on Trajectory Prediction/19d05057-5302-4e5f-a291-21f3188ae791_origin.pdf

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+# Fast Inference and Update of Probabilistic Density Estimation on Trajectory Prediction
+
+Takahiro Maeda Norimichi Ukita
+
+Toyota Technological Institute, Japan
+
+{sd21601,ukita}@toyota-ti.ac.jp
+
+# Abstract
+
+Safety-critical applications such as autonomous vehicles and social robots require fast computation and accurate probability density estimation on trajectory prediction. To address both requirements, this paper presents a new normalizing flow-based trajectory prediction model named FlowChain. FlowChain is a stack of conditional continuously-indexed flows (CIFs) that are expressive and allow analytical probability density computation. This analytical computation is faster than the generative models that need additional approximations such as kernel density estimation. Moreover, FlowChain is more accurate than the Gaussian mixture-based models due to fewer assumptions on the estimated density. FlowChain also allows a rapid update of estimated probability densities. This update is achieved by adopting the newest observed position and reusing the flow transformations and its log-det-jacobians that represent the motion trend. This update is completed in less than one millisecond because this reuse greatly omits the computational cost. Experimental results showed our FlowChain achieved state-of-the-art trajectory prediction accuracy compared to previous methods. Furthermore, our FlowChain demonstrated superiority in the accuracy and speed of density estimation. Our code is available at https://github.com/meaten/FlowChain-ICCV2023.
+
+# 1. Introduction
+
+Human trajectory prediction is a challenging problem because human movements are not deterministic, unlike the billiard balls on a pool table. Starting from deterministic approaches [26, 49], to tackle its indeterministic nature, many stochastic approaches are proposed, such as Gaussian distribution-based [1, 42, 66], generative adversarial networks (GANs)-based [9, 10, 25, 57], variational autoencoder (VAE)-based [58, 71, 73], diffusion-based [24], and normalizing flow-based [59] methods. These methods
+
+![](images/61587bff2e78dc4ebe9603181c4add73529e15eebc64fa2b03fd0994a5fb6b63.jpg)  
+Figure 1. Effects of our FlowChain. Our FlowChain can rapidly estimate accurate density than VAE-based methods as shown in green. Furthermore, we propose an update of estimated density that only requires an instant of time, as shown in blue.
+
+can be applied to human-interactive systems, including autonomous vehicles [8,11,38,55,69] and social robots [4,39].
+
+In these safety-critical applications, autonomous systems must infer the existence probability density distribution ("density" for brevity) of other moving objects such as pedestrians around on all prediction steps for risk assessment (i.e., collision avoidance). However, most of the indeterministic approaches mentioned above merely generate possible future trajectories and cannot estimate the density alone. We need to further apply kernel density estimation (KDE) to these future trajectories for density estimation.
+
+Thus, trajectory prediction with density estimation usually takes a larger computational cost than only generating possible trajectories. This computational cost accumulates upon every single prediction update along with highly-frequent observations. This is unacceptable to autonomous systems that work in rapidly changing environments.
+
+To address both computational cost and density estimation, we propose a new normalizing flow-based model named FlowChain (Fig. 1). This speed and accuracy are because normalizing flows need no additional approximation such as Gaussian mixture modeling or KDE with limited samples. FlowChain can estimate a density on each prediction time step thanks to our stacked normalizing flow archi
+
+tecture unlike Flomo [59], which is closest to FlowChain. While Flomo uses one flow module that only estimates the spatial density of a trajectory, FlowChain further predicts its spatio-temporal density so that the spatial density of an agent position at each time step is also predicted. This stacking architecture also allows a rapid update of an estimated density in an instant of time e.g. less than one millisecond. This update is achieved by reusing the flow transformations and its log-det-jacobians that represent the motion trend such as going straight or turning right. This update with the reuse significantly omits the computational cost while maintaining the accuracy of density estimation because the motion trend doesn't change for several time steps.
+
+Our contributions are three-fold as follows:
+
+- We propose a normalizing flow-based trajectory prediction network named FlowChain that rapidly estimates the accurate density of future positions on each prediction time step.  
+- Our update procedure for the density requires less than one millisecond by adopting new observations and reusing the motion trend as the flow transformations.  
+- Our FlowChain achieved comparable prediction accuracy to other computationally-intense state-of-the-art methods on several benchmarks.
+
+# 2. Related Work
+
+Human Trajectory Prediction. The earliest works on human trajectory prediction [26, 49] are deterministic regression models using attractive and repulsive forces between humans and goal positions. However, human movements are indeterministic in nature. Therefore, many approaches are proposed to model this uncertainty. Social-LSTM [1] and Social-STGCNN [42] regress the parameters of the bivariate Gaussian distributions to model the uncertainty of future trajectories. These approaches [1, 42] are simple and require low computational cost but cannot catch the future multi-modality we often see at road intersections.
+
+To address this multi-modality, many stochastic prediction models are proposed. Trajectory++ [58], AgentFormer [71], and ExpertTraj [73] combined the aforementioned Gaussian regression with VAE [32]. These models predict a bivariate Gaussian mixture based on stochastically sampled latent variables [28, 52]. We can obtain the density by summing up each estimated Gaussian density map. However, these Gaussian mixture methods have limited expressive power and thus predict inaccurate distributions.
+
+On the other hand, GAN [18]-based approaches implicitly handle this uncertainty. Social-GAN [25], SoPhie [57], Goal-GAN [10], and MG-GAN [9] use a GAN architecture to predict diverse trajectories conditioned on random
+
+noises. While they cannot estimate the density alone, we can estimate the density by applying KDE [48, 54] on sampled trajectories
+
+The denoising diffusion probabilistic model [27] can also implicitly handle uncertainty by generating multiple trajectories. Recently, denoising diffusion attracts much attention for its state-of-the-art performances in several generative tasks, including image synthesis [12] and audio generation [33]. MID [24] is a trajectory prediction model incorporating denoising diffusion, which achieves state-of-the-art performances. We can also estimate the density of MID by applying KDE. However, GAN-based and diffusion-based approaches cannot estimate accurate densities because KDE is performed on limited samples.
+
+On the contrary, normalizing flow [51, 64, 65]-based trajectory prediction method named Flomo [59] can rapidly generate future trajectories and their accurate probabilities thanks to the analytical computation of normalizing flows. However, this approach only estimates the probability of each future trajectory and thus cannot estimate the density of positions on each prediction step alone.
+
+Normalizing Flows. Thanks to its bijective process, normalizing flow can compute the density analytically. This analytical computation allows us fast and accurate density estimation because it does not need additional approximation such as KDE with limited samples. For more expressive power, several bijective families were proposed such as affine coupling layers [13, 14], autoregressive maps [17, 47], invertible linear transformations [31], and invertible ResNet block [2, 6]. All flows mentioned above have a topology constraint between the base and estimated densities. This topology constraint limits the expressive power of normalizing flows. To address this constraint, some approaches incorporated stochasticity into flows [68]. However, this stochasticity makes analytical density calculation impossible. Therefore, further density approximation such as KDE is required for this stochastic normalizing flow. Other approaches [5, 7, 44] use surjective layers, which relax the topology constraint but make analytical density computation with a little looseness. In summary, the optimal normalizing flow model architecture for future trajectory density estimation is still unknown.
+
+Density Estimation. Kernel density estimation [48, 54] is the most representative non-parametric method that estimates a continuous density from a finite set of data. Suppose that we have  $A$  input data and  $B$  points evaluated, the computational cost is  $\mathcal{O}(AB)$ . This is a huge cost for autonomous systems. Therefore, several accelerations are proposed such as data binning [19, 60, 62, 67], fast sum updating [15, 16, 35, 61], fast Gauss transform [22, 23, 34], and dual-tree method [20, 21, 36]. However, we confirmed that the KDE implementation utilizing GPU is faster than these accelerations including [3, 45, 46]. We use the KDE utilizing
+
+GPU for comparison in Sec. 4.
+
+# 3. Proposed Approach
+
+In this section, we present our normalizing flow-based trajectory prediction named FlowChain and its update of estimated density. First, we formulate the trajectory prediction task and conditional CIFs in Sec. 3.1. and Sec. 3.2 respectively. Then, we present our FlowChain model architecture in Sec. 3.3. After that, we describe its fast update procedure of estimated density in Sec.3.4. Finally, we explain how to obtain the density map in Sec. 3.5.
+
+# 3.1. Problem Formulation
+
+We aim to estimate agents' plausible future trajectory densities from their past trajectories. The trajectory of each agent is defined as a sequence  $(\mathbf{x}_0, \mathbf{x}_1, \ldots, \mathbf{x}_T)$  of positions  $\mathbf{x}_t = (x_t, y_t)$  over discrete timesteps  $t \in \{0, \ldots, T\}$ . Given the current time step  $t$ , we predict future  $T_{\mathrm{f}}$ -step trajectory densities  $\{p(\tilde{\mathbf{x}}_{t+1}^i | \mathbf{o}_t), \ldots, p(\tilde{\mathbf{x}}_{t+T_{\mathrm{f}}}^i | \mathbf{o}_t)\}$  given  $T_{\mathrm{o}}$ -step observed trajectory  $\mathbf{o}_t = \{(x_{t-T_{\mathrm{o}}}^i - 1, \ldots, x_t^i) | i \in \{1, 2, \ldots, N_a\}\}$ , where  $N_a$  and  $i$  denote the number of agents and the index of each agent, respectively. This index will be omitted for brevity in later use. We denote  $\mathbf{x}$  as an observed position,  $\tilde{\mathbf{x}}$  as a predicted future position, and  $\hat{\mathbf{x}}$  as a ground truth future position.
+
+# 3.2. Conditional Continuously-indexed Flow
+
+Conditional normalizing flow is a density estimation method that uses a parameterized bijective transformation  $f$  conditioned on additional information  $\mathbf{c}$  such as an encoded vector from observed trajectories, i.e.,  $\mathbf{o}_t$ , for trajectory prediction. The bijection  $f$  transforms a base density  $p(\mathbf{z})$  to an expressive density  $p(\mathbf{y})$  by the analytical computation of the change-of-variables formula as follows:
+
+$$
+\mathbf {y} = f (\mathbf {z}) \tag {1}
+$$
+
+$$
+p (\mathbf {y}) = p (\mathbf {z}) \left| \det  \nabla_ {\mathbf {y}} f ^ {- 1} (\mathbf {y}, \mathbf {c}) \right| \tag {2}
+$$
+
+The parameters of bijective transformation  $f$  can be learned by maximizing the likelihood of samples  $\hat{\mathbf{y}}$  from datasets, or minimizing the negative log-likelihood as follows:
+
+$$
+N L L = - \log p (\hat {\mathbf {y}} | \mathbf {c}) \tag {3}
+$$
+
+A normalizing flow with only bijective transformations has a constraint that the topology of a base density is preserved. For example, if the base density is a simple unimodal Gaussian distribution, the normalizing flow fails to fit a distinct multi-modal density. To avoid this problem of bijective transformation, we use continuously-indexed flows (CIFs) [7], akin to an infinite mixture of normalizing flows. While CIFs potentially avoid the topology constraint, CIFs still enable analytical density computation with a little
+
+looseness  $\mathcal{E}$  because CIFs incorporate inference and generative surjections according to the survey [44]. We investigated the tradeoff between the expressive power and exactness of estimated density on CIFs. We concluded this looseness of estimated density can be ignored in trajectory prediction, as we show in the supplemental materials. Based on the above discussion, We use conditional CIFs as the base normalizing flow model of our FlowChain.
+
+# 3.3. FlowChain model
+
+The overview of the FlowChain model is shown in Fig. 2. Given the current time step  $t$ , we estimate future trajectory densities  $\{p(\tilde{\mathbf{x}}_{t + 1}|\mathbf{o}_t),\dots ,p(\tilde{\mathbf{x}}_{t + T_{\mathrm{f}}}|\mathbf{o}_t)\}$  by chaining the conditional CIFs, each of which transforms the density of the former time step  $p(\tilde{\mathbf{x}}_{t + n - 1}|\mathbf{o}_t)$  to the density of the next time step  $p(\tilde{\mathbf{x}}_{t + n}|\mathbf{o}_t)$ , shown in the upper side of Fig. 2. Our formulation consists of two equations: the initial condition for the first step prediction (Eq.(4)) and the recurrence relation between the latter steps (Eq.(5)) as follows.
+
+$$
+\begin{array}{l} p \left(\tilde {\mathbf {x}} _ {t + 1} \mid \mathbf {o} _ {t}\right) = \mathcal {N} (\mathbf {z}; \mathbf {x} _ {t}, \sigma) | \det  \nabla_ {\tilde {\mathbf {x}} _ {t + 1}} f _ {1} ^ {- 1} \left(\tilde {\mathbf {x}} _ {t + 1}, \mathbf {o} _ {t}\right) | (4) \\ p \left(\tilde {\mathbf {x}} _ {t + n} \mid \mathbf {o} _ {t}\right) = p \left(\tilde {\mathbf {x}} _ {t + n - 1} \mid \mathbf {o} _ {t}\right) | \det  \nabla_ {\tilde {\mathbf {x}} _ {t + n}} f _ {n} ^ {- 1} \left(\tilde {\mathbf {x}} _ {t + n}, \mathbf {o} _ {t}\right) | (5) \\ = \exp \left(\log \mathcal {N} (\mathbf {z}; \mathbf {x} _ {t}, \sigma) \right. \\ \left. + \sum_ {n ^ {\prime} = 1} ^ {n} \log | \det  \nabla_ {\tilde {\mathbf {x}} _ {t + n ^ {\prime}}} f _ {n ^ {\prime}} ^ {- 1} \left(\tilde {\mathbf {x}} _ {t + n ^ {\prime}}, \mathbf {o} _ {t}\right) |\right) (6) \\ \end{array}
+$$
+
+where  $n \in \{2,3,\dots,T_{\mathrm{f}}\}$ .  $f_{n}$  is a conditional CIF that transforms the sample  $\tilde{\mathbf{x}}_{t + n - 1}$  from a former density  $p(\tilde{\mathbf{x}}_{t + n - 1}|\mathbf{o}_t)$  to the sample  $\tilde{\mathbf{x}}_{t + n}$  from a next density  $p(\tilde{\mathbf{x}}_{t + n}|\mathbf{o}_t)$  as  $\tilde{\mathbf{x}}_{t + n} = f_n(\tilde{\mathbf{x}}_{t + n - 1},\mathbf{o}_t)$ .  $f_{1}$  transforms the sample  $\mathbf{z}$  from a base Gaussian density  $\mathcal{N}(\cdot ;\mathbf{x}_t,\sigma)$ .  $\mathcal{N}(\cdot ;\mathbf{x}_t,\sigma)$  is created from the newest observed position  $\mathbf{x}_t$  as a mean position. The standard deviation  $\sigma$  is a trainable parameter of FlowChain model. All of  $f_{1}$  and  $f_{n}$  are independently trained with the loss function (Eq.(3)) in our method.
+
+From  $f_{n}$ , we can sample multiple trajectories for best-of-N metrics [25] by sampling multiple  $\mathbf{z}$  as follows:
+
+$$
+\{\tilde {\mathbf {x}} _ {t + 1}, \dots , \tilde {\mathbf {x}} _ {t + n} \} = \left\{f _ {1} (\mathbf {z}, \mathbf {o} _ {t}), \dots , f _ {n} \circ \dots \circ f _ {2} \circ f _ {1} (\mathbf {z}, \mathbf {o} _ {t}) \right\} \tag {7}
+$$
+
+# 3.4. Update of estimated density
+
+For accurate prediction by reusing the previous predictions, we focused on two features of the observed trajectory: the newest observed position and motion trend, shown in Fig. 2. Our FlowChain generates future trajectories by transforming the newest observed position  $\mathbf{x}_t$  based on the motion trend e.g. go straight or turning left/right, modeled
+
+![](images/bbb7e99b80d26d3aae1e45a2e55c9596c832c06ab6fc2acebc38dd25d322e822.jpg)  
+Figure 2. Overview of our FlowChain model. FlowChain consists of sequential conditional CIFs denoted by  $f_{n}$ .  $f_{n}$  is conditioned on the feature vector denoted by Motion Trend from the Temporal-Social Encoder, which encodes the observed trajectories  $\mathbf{o}_t$ . Since  $\mathbf{o}_t$  includes the trajectories of other agents as well as an agent of interest (i.e., the target of prediction), the future trajectory is predicted while taking into account social interaction with the others.  $f_{n}$  transforms the density of the former time step  $p(\tilde{\mathbf{x}}_{t + n - 1}|\mathbf{o}_t)$  to the density of the next time step  $p(\tilde{\mathbf{x}}_{t + n}|\mathbf{o}_t)$  as shown in the upper side. However, for the first step, the Gaussian distribution  $\mathcal{N}(\cdot ;\mathbf{x}_t,\sigma)$ , which is created from the newest observed position  $\mathbf{x}_t$ , is transformed. The boxes of past, current (i.e., newest), and future densities are colored by red, blue, and purple, respectively. In the update procedure shown on the bottom side, the Gaussian distribution  $\mathcal{N}(\cdot ;\mathbf{x}_{n + 1},\sigma)$ , created from newest observed position  $\mathbf{x}_{t + 1}$  at  $t + 1$ , replace the estimated density from the former time step. The other calculations are reused as shown by gray arrows. We can properly update the density by using the newest observed position at  $t + 1$ ,  $\mathbf{x}_{t + 1}$ , instead of  $p(\tilde{\mathbf{x}}_{t + 1}|\mathbf{o}_t)$ . In a toy example shown in this figure, another incorrect mode observed in the predicted densities,  $p(\tilde{\mathbf{x}}_{t + 2}|\mathbf{o}_t)$  and  $p(\tilde{\mathbf{x}}_{t + 3}|\mathbf{o}_t)$ , is successfully suppressed in the updated predictions  $p^{\prime}(\tilde{\mathbf{x}}_{t + 2}|\mathbf{o}_{t + 1})$  and  $p^{\prime}(\tilde{\mathbf{x}}_{t + 3}|\mathbf{o}_{t + 1})$ . As shown in Sec. 4, our experiments validate this advantage of our proposed fast update procedure is greater than its disadvantage (i.e., a decrease in the prediction accuracy due to the reuse of the motion trend) so that the updated predictions are better than the initial predictions (e.g.,  $p^{\prime}(\tilde{\mathbf{x}}_{t + 3}|\mathbf{o}_{t + 1})$  is better than  $p(\tilde{\mathbf{x}}_{t + 3}|\mathbf{o}_t)$  in the figure).
+
+by each flow  $f_{n}(\mathbf{x}_{t + n},\mathbf{o}_{t})$ . We cannot reuse the newest observed position  $\mathbf{x}_t$  at  $t$  for the next time step because this position is a starting point of the future trajectories and thus greatly affect the accuracy of probability density estimation. Therefore, FlowChain adopt the newest observed position  $\mathbf{x}_{t + n}$  at  $t + n$  by replacing the estimated density  $p(\tilde{\mathbf{x}}_{t + n}|\mathbf{o}_t)$  with the Gaussian density  $\mathcal{N}(\cdot ;\mathbf{x}_{t + n},\sigma)$ , as shown in the bottom of Fig. 2. This replacement propagates the information of the newest observed position properly to the latter time steps.
+
+On the other hand, the motion trend can be reused because it doesn't change for several time steps unless a sudden strong external force is applied to a prediction target. Therefore, our FlowChain model reuses its trend in each flow transformation  $f_{n}(\tilde{\mathbf{x}}_{t + n},\mathbf{o}_{t})$ . Specifically, we use the previously generated trajectories  $\{\tilde{\mathbf{x}}_{t + 1},\dots ,\tilde{\mathbf{x}}_{t + n}\}$ . Thus, we can reuse the corresponding log-det-jacobians  $\log |\operatorname *{det}\nabla_{\tilde{\mathbf{x}}_{t + n}}f_n^{-1}(\tilde{\mathbf{x}}_{t + n + 1},\mathbf{o}_t)|$  which is the variation of the estimated densities, as shown by the gray arrows in Fig. 2. The equation of FlowChain's update is expressed
+
+similar to Eq.(4) and Eq.(5) as follows:
+
+$$
+p ^ {\prime} \left(\tilde {\mathbf {x}} _ {t + 2} \mid \mathbf {o} _ {t + 1}\right) = \mathcal {N} \left(\tilde {\mathbf {x}} _ {t + 1}; \mathbf {x} _ {t + 1}, \sigma\right) | \det  \nabla_ {\tilde {\mathbf {x}} _ {t + 2}} f _ {2} ^ {- 1} \left(\tilde {\mathbf {x}} _ {t + 2}, \mathbf {o} _ {t}\right) | \tag {8}
+$$
+
+$$
+\begin{array}{l} p ^ {\prime} \left(\tilde {\mathbf {x}} _ {t + n} \mid \mathbf {o} _ {t + 1}\right) = p ^ {\prime} \left(\tilde {\mathbf {x}} _ {t + n - 1} \mid \mathbf {o} _ {t + 1}\right) | \det  \nabla_ {\tilde {\mathbf {x}} _ {t + n}} f _ {n} ^ {- 1} \left(\tilde {\mathbf {x}} _ {t + n}\right) | (9) \\ = \exp \left(\log \mathcal {N} (\tilde {\mathbf {x}} _ {t + 1}; \mathbf {x} _ {t + 1}, \sigma) \right. \\ \left. + \sum_ {n ^ {\prime} = 2} ^ {n} \log | \det  \nabla_ {\tilde {\mathbf {x}} _ {t + n ^ {\prime}}} f _ {n ^ {\prime}} ^ {- 1} \left(\tilde {\mathbf {x}} _ {t + n ^ {\prime}}, \mathbf {o} _ {t}\right) |\right) (10) \\ \end{array}
+$$
+
+where,  $n \in \{3, 4, \ldots, T_{\mathrm{f}}\}$ . Thanks to reusing each flow transformation  $f_{n}(\mathbf{x}_{t + n}, \mathbf{o}_{t})$ , we don't need to evaluate log-det-jacobians  $\log |\operatorname*{det}\nabla_{\tilde{\mathbf{x}}_{t + n}}f_n^{-1}(\tilde{\mathbf{x}}_{t + n})|$  again. This will greatly omit the computational cost. As a result, we can update the probability density by only calculating the Gaussian base distribution, summing up the log-det-jacobians, and taking exponentials, which take less than  $1ms$  in total. This rapid update is beneficial, especially for autonomous
+
+![](images/52ac5db57636058d0d1358939e84c2ce76c08ffe6801a183030550912d708e22.jpg)  
+Trajectories  
+Time steps  
+Figure 3. Overview of Simfork dataset. The trajectories shown in the left figure are examples of the observed and ground truth trajectories for training. The position of each step is sampled from the distributions of the middle figures. The sum of each distribution is shown in the right figure. The boxes of past, current (i.e., newest), and future densities are colored by red, blue, and purple, respectively.
+
+![](images/49a2a5f9a20d09fb5532c5217e056cf0bdd0a2dd983c76ad9ea17624e40d3e02.jpg)
+
+![](images/272646ed6adfb81e2ef5a3be6a67db80d94dd65d0b015974e7b254183515b3f3.jpg)
+
+systems where computational resources are limited and often become busy by executing other computationally-intense tasks such as object detection, mapping, and path planning.
+
+# 3.5. How to obtain the density map
+
+The density map, not like the mere one-point estimate, is needed when autonomous systems perform visualization of density or successive tasks such as navigation. We can correct the possible future points with estimated density by sampling multiple trajectories from Eq.(7) and its density from Eq.(6). The density map is obtained by collecting these points with density at the same time step.
+
+# 4. Experiments and Results Analysis
+
+# 4.1. Experimental Setup
+
+Datasets. Experiments were conducted on various datasets including our developed dataset with synthesized data Simfork, and real-image benchmarks including ETH/UCY [37, 50] and SDD [53]. On all the datasets, the numbers of past observed and future predicted steps are  $T_{o} = 8$  and  $T_{f} = 12$ , respectively. All steps have an interval of 0.4 seconds in accordance with the literature such as [25, 58].
+
+Simfork is a synthesized dataset for comparison on the accuracy of predicted densities, which are produced as explained in Sec. 3.5. In all trajectories in this dataset, either of the two types of traveling directions (i.e., turning left or right) is observed after a straight trajectory, as shown in Fig. 3. Positions before and after the fork (colored by green and black, respectively, in the leftmost figure of Fig. 3) are used as past observed and future ground-truth positions, respectively. Each position of a trajectory is independently perturbed by the Gaussian noise, which results in the density at each time step.
+
+Two pedestrian trajectory datasets, ETH [50] and UCY [37], are merged to ETH/UCY. This dataset has five scenes: ETH, HOTEL, UNIV, ZARA1, and ZARA2. For training and testing, we follow the standard leave-one-out
+
+approach, where we train on 4 scenes and test on the remaining one. We follow the train-validation-test split of Social-GAN [25].
+
+$SDD$  consists of 20 scenes captured in bird's eye view using aerial drones. This dataset contains various moving agents such as pedestrians, bicycles, and cars, but we focus on pedestrians for comparison, as with Trajnet [56]. We also follow the train-test split of Trajnet [56].
+
+Metrics Evaluation of accuracy on trajectory prediction is done with the widely-used metrics, Average Displacement Error (ADE) and Final Displacement Error (FDE). ADE is the average L2 distance between the predicted and ground truth trajectories over time. FDE is the L2 distance between the endpoints of the predicted and ground truth trajectories. We follow the Best-of-N procedure [25] with  $N = 20$ .
+
+For evaluation of the accuracy of density estimation, we use earth mover's distance (EMD) and log-probability. EMD is a measure of the distance between two densities by the minimum cost to move one density pile to the other. EMD is calculated between the estimated density and ground truth density on each prediction step. Thus, we use EMD on Simfork dataset, where the ground truth densities are available. The advantage of EMD over the widely-used KL divergence is that EMD works properly even if the two density is disjoint while KL divergence returns a constant value. We use log-probability on the pedestrian trajectory prediction datasets, where the ground truth density is unavailable. The log-probability is calculated on the ground truth trajectory instead of density.
+
+Implementation Details. Each conditional CIFs in our FlowChain used three-layer RealNVP [14] with multilayer perceptions (MLPs) of three hidden layers and 128 hidden units. All the conditional CIFs are with independent weights conditioned on the feature vectors from Trajectory++ [58] encoder as the temporal-social encoder. We used a batch size of 128 and ADAM optimizer [30] with a learning rate of  $10^{-4}$ . All experiments were conducted on a single NVIDIA A100 GPU. We use a KDE implementation utilizing GPU as mentioned in Sec. 2.
+
+Table 1. Quantitative results of trajectory prediction on ETH/UCY dataset with Best-of-20 metrics. T + I denotes its method takes a trajectory and an image as inputs while T denotes its method takes only a trajectory. Scores are in meters. Red and blue scores denote the best and the second-best in methods with T. Lower is better. The results of Trajectron++ [58] and MID [24] are updated according to an implementation issue #53 on the Trajectron++ GitHub page.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Input</td><td colspan="2">ETH</td><td colspan="2">HOTEL</td><td colspan="2">UNIV</td><td colspan="2">ZARA1</td><td colspan="2">ZARA2</td><td colspan="2">Mean</td></tr><tr><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td><td>ADE</td><td>FDE</td></tr><tr><td>SoPhie [57]</td><td>T + I</td><td>0.70</td><td>1.43</td><td>0.76</td><td>1.67</td><td>0.54</td><td>1.24</td><td>0.30</td><td>0.63</td><td>0.38</td><td>0.78</td><td>0.54</td><td>1.15</td></tr><tr><td>Goal-GAN [10]</td><td>T + I</td><td>0.59</td><td>1.18</td><td>0.19</td><td>0.35</td><td>0.60</td><td>1.19</td><td>0.43</td><td>0.87</td><td>0.32</td><td>0.65</td><td>0.43</td><td>0.85</td></tr><tr><td>MG-GAN [9]</td><td>T + I</td><td>0.47</td><td>0.91</td><td>0.14</td><td>0.24</td><td>0.54</td><td>1.07</td><td>0.36</td><td>0.73</td><td>0.29</td><td>0.60</td><td>0.36</td><td>0.71</td></tr><tr><td>Social-LSTM [1]</td><td>T</td><td>1.09</td><td>2.35</td><td>0.79</td><td>1.76</td><td>0.67</td><td>1.40</td><td>0.47</td><td>1.00</td><td>0.56</td><td>1.17</td><td>0.72</td><td>1.54</td></tr><tr><td>Social-GAN [25]</td><td>T</td><td>0.87</td><td>1.62</td><td>0.67</td><td>1.37</td><td>0.76</td><td>1.52</td><td>0.35</td><td>0.68</td><td>0.42</td><td>0.84</td><td>0.61</td><td>1.21</td></tr><tr><td>STGAT [29]</td><td>T</td><td>0.65</td><td>1.12</td><td>0.35</td><td>0.66</td><td>0.52</td><td>1.10</td><td>0.34</td><td>0.69</td><td>0.29</td><td>0.60</td><td>0.43</td><td>0.83</td></tr><tr><td>Social-STGCNN [42]</td><td>T</td><td>0.64</td><td>1.11</td><td>0.49</td><td>0.85</td><td>0.44</td><td>0.79</td><td>0.34</td><td>0.53</td><td>0.30</td><td>0.48</td><td>0.44</td><td>0.75</td></tr><tr><td>PECNet [41]</td><td>T</td><td>0.61</td><td>1.07</td><td>0.22</td><td>0.39</td><td>0.34</td><td>0.56</td><td>0.25</td><td>0.45</td><td>0.19</td><td>0.33</td><td>0.32</td><td>0.56</td></tr><tr><td>Trajectory++ [58]</td><td>T</td><td>0.61</td><td>1.03</td><td>0.20</td><td>0.28</td><td>0.30</td><td>0.55</td><td>0.24</td><td>0.41</td><td>0.18</td><td>0.32</td><td>0.31</td><td>0.52</td></tr><tr><td>MID [24]</td><td>T</td><td>0.55</td><td>0.88</td><td>0.20</td><td>0.35</td><td>0.30</td><td>0.55</td><td>0.29</td><td>0.51</td><td>0.20</td><td>0.38</td><td>0.31</td><td>0.53</td></tr><tr><td>Social Implicit [43]</td><td>T</td><td>0.66</td><td>1.44</td><td>0.20</td><td>0.36</td><td>0.31</td><td>0.60</td><td>0.25</td><td>0.50</td><td>0.22</td><td>0.43</td><td>0.33</td><td>0.67</td></tr><tr><td>FlowChain(Ours)</td><td>T</td><td>0.55</td><td>0.99</td><td>0.20</td><td>0.35</td><td>0.29</td><td>0.54</td><td>0.22</td><td>0.40</td><td>0.20</td><td>0.34</td><td>0.29</td><td>0.52</td></tr></table>
+
+Table 2. Quantitative results of trajectory prediction on SDD dataset with Best-of-20 metrics. Scores are in pixels. See the caption of Tab. 1 also for details.  
+
+<table><tr><td>Method</td><td>Input</td><td>ADE</td><td>FDE</td></tr><tr><td>SoPhie [57]</td><td>T + I</td><td>16.27</td><td>29.38</td></tr><tr><td>Goal-GAN [10]</td><td>T + I</td><td>12.20</td><td>22.10</td></tr><tr><td>MG-GAN [9]</td><td>T + I</td><td>13.60</td><td>25.80</td></tr><tr><td>Social-LSTM [1]</td><td>T</td><td>31.19</td><td>56.97</td></tr><tr><td>Social-GAN [25]</td><td>T</td><td>27.25</td><td>41.44</td></tr><tr><td>STGAT [29]</td><td>T</td><td>14.85</td><td>28.17</td></tr><tr><td>Social-STGCNN [42]</td><td>T</td><td>20.76</td><td>33.18</td></tr><tr><td>PECNet [41]</td><td>T</td><td>9.97</td><td>15.89</td></tr><tr><td>Trajectory++ [58]</td><td>T</td><td>11.40</td><td>20.12</td></tr><tr><td>MID [24]</td><td>T</td><td>10.31</td><td>17.37</td></tr><tr><td>FlowChain (Ours)</td><td>T</td><td>9.93</td><td>17.17</td></tr></table>
+
+# 4.2. Prediction Accuracy
+
+A wide range of previous trajectory prediction methods is quantitatively compared with our FlowChain on ETH/UCY and SDD datasets, shown in Tab. 1 and Tab. 2. We didn't include some methods [40, 63, 70-72] due to a different train-validation-test split.
+
+For ETH/UCY dataset, FlowChain achieved mean ADE/FDE of 0.29/0.52 in meters, which are the best among all trajectory prediction methods. On SDD dataset, FlowChain achieved average ADE/FDE of 9.93/17.17 in pixels, which are the best and second-best among all methods taking only trajectories as inputs [1,24,25,41,58,63].
+
+Although our focus is on density estimation with a minimal computational cost, FlowChain is comparable with the state-of-the-art methods also in terms of errors. We didn't evaluate the accuracy of future trajectories from FlowChain's update because the trajectories do not change
+
+![](images/f530bdaa1cf5a2d18a228a98f412c3ee79b75f5eabac471307052eeca901d5f5.jpg)  
+Figure 4. Temporal estimated densities on Simfork dataset.
+
+in the update as mentioned in Sec. 3.4.
+
+# 4.3. Analysis on Density Estimation
+
+The results of density estimation are compared between FlowChain and previous methods based on VAE, GAN, and diffusion approaches. Trajectory++ is the representative VAE-based trajectory prediction model with density estimation by GMM. Social-GAN is a minimal model of GAN-based trajectory prediction. MID is the diffusion-based trajectory prediction model with state-of-the-art prediction accuracy. We applied KDE to Social-GAN and MID since
+
+![](images/839b4e1985cd61c6514f7c80757822fc96ed14f964bc3cbf215c9e00143da48b.jpg)  
+Figure 5. Temporal estimated densities on ETH/UCY dataset.
+
+they cannot estimate the density alone. For a fair comparison, FlowChain, Social-GAN, and MID generated the same number of samples for future trajectories;  $10^{5}$  samples.
+
+We evaluated the computational time and the accuracy of density estimation of each method, as shown in Tab. 3. Social-GAN and MID take much longer time because KDE is slow in spite of running on GPU. Trajectory++ is a bit faster than our FlowChain due to its simple Gaussian mixture model but suffers from the poor quality of estimated density, as shown by the worst EMD, 2.545, on Simfork dataset and worst log-probability, -281.86, on ETH/UCY dataset. On the other hand, our FlowChain achieved the best EMD, 1.408, and the best log-probability, -0.26, in spite of the fast computation 37.0ms. Thus, our method successfully addresses both the computational cost and accurate density estimation.
+
+The estimated densities on Simfork are visualized in Fig. 4. The densities estimated by Trajectory++ are too large compared to the ground truth densities. While the densities of Social-GAN and MID are stretched to the vertical axis, our FlowChain successfully estimated the tighter round-shape densities closest to the ground truth densities.
+
+The estimated densities on  $ETH/UCY$  are shown in Fig. 5. The size of each density represents the positional uncertainty of a predicted position. As within Simfork, in  $ETH/UCY$ , an agent cannot change its traveling direction significantly in a short period (i.e.,  $T_{f} = 12$  is  $0.4 \times 12 = 4.8$  seconds), the size of the density expected to be tight. The densities estimated by Social-GAN are tight but deviated from the ground truth trajectory. Trajectory++ generated the densities with several disconnected modes due to the approximation as a Gaussian mixture. While the densi
+
+![](images/ba883f7cf9f760f6123000d999e38e9eeed29de25e15f81a06112c1804114af8.jpg)  
+Figure 6. Temporal densities estimated by our update procedure on Simfork dataset.
+
+![](images/14a0b33d57f3e80ef5505842a4b22a14ade06801bc05c2e66f6bfbaf37b077b4.jpg)  
+Figure 7. Temporal densities estimated by our update procedure on ETH/UCY dataset.
+
+ties from MID are close to the ground truth trajectory, MID could not suggest tight densities and their modes deviated from the ground truth trajectories. On the contrary, the density from our FlowChain covers the ground truth trajectory with tight distributions. thanks to the explicit density estimation of conditional CIFs and our expressive model architecture. We should note that the estimated density of FlowChain has some fluctuation due to the looseness of the density computation of CIFs. However, we did not observe any potential problem or failure of density estimation related to this in our experiments.
+
+# 4.4. Proposed Update of Density Estimation
+
+As shown in Tab. 3, the accuracy of density estimation degraded along with the number of updates. However, this update is remarkably fast  $0.6ms$ , also shown in Tab. 3. Thus, autonomous systems can select the accurate prediction from scratch or the super rapid update by reusing predictions depending on the utilization of their computational resources.
+
+The estimated densities of our update procedure are shown in Fig. 6 and Fig. 7, which show the densities up
+
+Table 3. Accuracy and computational time of density estimation. Scores are in milliseconds for computational time. While lower is better for EMD on Simfork dataset, higher is better for log-probability on ETH/UCY dataset. We report the averaged EMD and log-probability over the 12 prediction steps.  
+
+<table><tr><td rowspan="2">Method</td><td></td><td colspan="2">Comp. Time</td><td colspan="2">Accuracy of Density Estimation</td></tr><tr><td></td><td>All</td><td>w/o KDE</td><td>EMD ↓</td><td>log-probability ↑</td></tr><tr><td>Social-GAN [25]</td><td></td><td>11498.8</td><td>9.8</td><td>1.544</td><td>-3.40</td></tr><tr><td>Trajectory++ [58]</td><td></td><td>33.7</td><td>-</td><td>2.545</td><td>-281.86</td></tr><tr><td>MID [24]</td><td></td><td>23904.1</td><td>12219.1</td><td>2.135</td><td>-0.90</td></tr><tr><td rowspan="3">FlowChain (Ours)</td><td>Pred at t</td><td>37.0</td><td>-</td><td>1.408</td><td>-0.26</td></tr><tr><td>Update from t - 1</td><td>0.6</td><td>-</td><td>2.539</td><td>-0.45</td></tr><tr><td>Update from t - 5</td><td>0.5</td><td>-</td><td>2.899</td><td>-0.53</td></tr></table>
+
+Table 4. Accuracy of density estimation with the update.  
+
+<table><tr><td rowspan="2" colspan="2">Method</td><td colspan="8">log-probability on each step</td></tr><tr><td>t+6</td><td>t+7</td><td>t+8</td><td>t+9</td><td>t+10</td><td>t+11</td><td>t+12</td><td>mean</td></tr><tr><td colspan="2">Social-GAN [25]</td><td>-2.65</td><td>-2.86</td><td>-3.07</td><td>-3.26</td><td>-3.44</td><td>-3.62</td><td>-3.80</td><td>-3.24</td></tr><tr><td colspan="2">Trajectory++ [58]</td><td>-121.19</td><td>-93.61</td><td>-71.62</td><td>-51.36</td><td>-36.85</td><td>-27.92</td><td>-22.72</td><td>-97.79</td></tr><tr><td colspan="2">MID [24]</td><td>-1.06</td><td>-1.43</td><td>-1.75</td><td>-2.02</td><td>-2.26</td><td>-2.46</td><td>-2.65</td><td>-1.95</td></tr><tr><td rowspan="2">FlowChain</td><td>Without Update</td><td>0.21</td><td>0.36</td><td>0.44</td><td>-0.08</td><td>-0.78</td><td>-0.75</td><td>-1.19</td><td>-0.26</td></tr><tr><td>With Update</td><td>0.20</td><td>0.50</td><td>0.39</td><td>0.44</td><td>0.20</td><td>0.26</td><td>0.16</td><td>0.31</td></tr></table>
+
+![](images/0a76851cb4fc158b0af1c58891e65eb3faa4e67d0e242ea4c520878e06ff7c78.jpg)  
+Figure 8. Comparison of standard methods and FlowChain with updates.
+
+dated at  $t + 1$  and  $t + 2$  from the prediction at  $t$ . On Simfork dataset, the wrong mode is successfully suppressed by exploiting the information of the new observation. On ETH/UCY dataset, the initial prediction deviated from the ground truth because the observed trajectory observed until  $t$  does not have the information on turning left. However, the updated densities covered the ground truth well by adopting the newest observed position despite the minimal computational cost.
+
+We further perform a comparison with a practical setting. Models usually make predictions with intervals due to the large computational cost. Therefore, we usually cannot obtain the prediction from the latest observation, as shown by the upper side of Fig. 8. For example, assume we need the density of six time-step ahead such as  $p_{t+6}$  from  $\mathbf{x}_t$ , we cannot obtain  $p_{t+7}$  from  $\mathbf{x}_{t+1}$  or  $p_{t+8}$  from  $\mathbf{x}_{t+2}$  due to the
+
+computational overhead. However, with FlowChain's rapid update, we can obtain the update  $(p_{t + 7}, p_{t + 8})$  from the latest observation  $(\mathbf{x}_{t + 1}, \mathbf{x}_{t + 2})$  on every time step thanks to the minimal computational cost, as shown by the bottom side of Fig. 8. We conducted the comparison with the frame of interest as six time-step ahead  $t + 6$  and the log-probability metric. As shown by Tab. 4, our FlowChain with update achieves the highest log-probability. Therefore, we can conclude that FlowChain's update can improve the density estimation with minimal computational cost.
+
+# 5. Concluding Remarks
+
+In this paper, we proposed a new normalizing flow-based trajectory prediction network named FlowChain that achieves fast and accurate probability density estimation. Furthermore, our update procedure reliably generates the density estimation of the next step based on the density estimation of the previous step in an instant. Experimental results demonstrated that FlowChain achieved comparable accuracy to the state-of-the-art trajectory prediction models and superiority in the accuracy of density estimation.
+
+Our FlowChain has one limitation. Our update procedure can reliably estimate the density for several steps but the performance will degrade along with the time steps because the reused flow transformations do not reflect the new observations. Our future work is determining when should we stop to use the update procedure and make predictions from scratch. Furthermore, FlowChain is the domain-agnostic model that can be applied to a wide range of prediction tasks such as stock price prediction, human motion prediction, and video prediction. We will investigate the applicability of our FlowChain on these other domains.
+
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+        "type": "text",
+        "text": "Fast Neural Scene Flow",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Xueqian Li\\*1 Jianqiao Zheng\\* Francesco Ferroni\\*2 Jhony Kaesemodel Pontes\\*3 Simon Lucey\\*1 1The University of Adelaide 2NVIDIA 3Latitude AI",
+        "bbox": [
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+        "image_caption": [
+            "(a) Lidar depth map densification"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            385
+        ],
+        "page_idx": 0
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+        "img_path": "images/54d9e247bdfa3c4b2fa33496b37c6945f5a17ec88ca028c376d0a7bf0ca96a32.jpg",
+        "image_caption": [
+            "(b) Efficiency of network architectures"
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+        "img_path": "images/6e1324fa6d030cfc56e5cd611dac84ed3d0751e4de311ee5791ddb6be45afc55.jpg",
+        "image_caption": [
+            "(c) Error and computation time",
+            "Figure 1. Scene flow is an important problem as it provides low-level motion cues for many downstream tasks. State-of-the-art learning methods are usually fast and can achieve impressive performance on in-domain data, but usually fail to generalize to out-of-the-distribution (OOD) data or handle dense point clouds. In this paper, we focus on a runtime optimization-based neural scene flow pipeline. In (a) one can see its application in the densification of lidar. However, in (c) one sees that the major drawback is the extensive computation time. We identify that the common speedup strategy in network architectures for coordinate networks has little effect on scene flow acceleration [see green (b)] unlike image reconstruction [see pink (b)]. With the dominant computational burden stemming instead from the Chamfer loss function, we propose to use a distance transform-based loss function to accelerate [see purple (b)], which achieves up to  $30 \\times$  speedup and on-par estimation performance compared to NSFP [see (c)]. When tested on 8k points, it is as efficient [see (c)] as leading learning methods, achieving real-time performance."
+        ],
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+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
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+        "text": "Neural Scene Flow Prior (NSFP) is of significant interest to the vision community due to its inherent robustness to out-of-distribution (OOD) effects and its ability to deal with dense lidar points. The approach utilizes a coordinate neural network to estimate scene flow at runtime, without any training. However, it is up to 100 times slower than current state-of-the-art learning methods. In other applications such as image, video, and radiance function reconstruction innovations in speeding up the runtime performance of coordinate networks have centered upon architectural changes. In this paper, we demonstrate that scene flow is different—with the dominant computational bottleneck stemming from the loss function itself (i.e., Chamfer distance). Further, we rediscover the distance transform (DT) as an efficient, correspondence-free loss function that dramatically speeds up the runtime optimization. Our fast neural scene flow (FNSF) approach reports for the first time real-time performance comparable to learning methods,",
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+        "text": "*Part of the work was done while at Argo AI. Corresponding e-mail: xueqian.li@adelaide.edu.au. Code available at https://github.com/Lilac-Lee/FastNSF.git",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "without any training or OOD bias on two of the largest open autonomous driving (AV) lidar datasets Waymo Open [62] and Argoverse [8].",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
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+        "text": "Neural Scene Flow Prior (NSFP) [37] is considered the dominant method in open-world perception [47] and scene densification [37, 69] using lidar (see Fig. 1 (a)). NSFP achieves state-of-the-art scene flow estimates from dense lidar point clouds (up to  $150\\mathrm{k}+$  points) and works on a variety of sensor setups with little to no refinement or adaptation. Unlike supervised or unsupervised learning-based methods, NSFP does not require learning from large offline datasets and has no limits on point density ( $<8\\mathrm{k}$  for most learning methods). Instead, it leverages the architecture of a neural network to implicitly regularize the flow estimate and employs a runtime optimization that can easily scale to large out-of-distribution (OOD) scenes, which is a challenge for current learning-based methods [15, 31, 37, 47, 51].",
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+        "text": "A fundamental drawback, however, to NSFP is the speed of its runtime optimization which is in some instances of orders of magnitude slower than its learning counterparts",
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+        "type": "header",
+        "text": "CVF",
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+    {
+        "type": "header",
+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
+        "bbox": [
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+        "text": "9878",
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+        "text": "(see Fig. 1). As a result, NSFP has widely been used offline for (i) providing scene flow supervision for efficient learning methods and (ii) as a pre-processing step for training open-world perception systems [47]. However, the considerable computational cost of NSFP limits its current applications only to these offline tasks.",
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+    {
+        "type": "text",
+        "text": "A central narrative of our approach is that the dominant computational burden in runtime scene flow optimization (NSFP) is not the network architecture, but the loss function—specifically Chamfer distance (CD) [17]. This differs considerably from other applications of coordinate networks throughout vision and learning such as neural radiance fields (NeRF [44]) and high-fidelity image reconstruction ([61]) which have gained significant speedups through architectural innovations [83]. A visual depiction of this discrepancy can be found in Fig. 1 (b).",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Key to our approach is the use of correspondence-free loss function—distance transform (DT) [5, 13, 58] as a proxy for the computationally expensive CD loss. Even though DT has been extensively studied by the graphics and vision community over a few decades, its application as an efficient loss function in deep geometry has largely been overlooked up until this point. We believe that the inherent efficiencies of the DT are especially pertinent for runtime network training such as in NSFP. Our approach shares similarities with Plenoxels [81]—a recent approach for efficient radiance field estimation using coordinate networks—as we trade memory consumption for computation time, allowing for significant speedups during runtime, which provides an alternative solution when exploring more efficient loss functions for dense scene flow estimation. We differ from Plenoxels, however, in that our memory consumption stems from our proposed loss function, not the neural architecture itself.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we present for the first time an approximately real-time (for 8k points) runtime optimization method as computationally efficient as leading learning methods whilst preserving the scalability to dense point clouds and state-of-the-art performance on OOD scenes like NSFP. We compare the performance and the computation time of our approach on two of the largest open lidar AV datasets available: Waymo Open [62] and Argoverse [8]. Our fast neural scene flow achieves up to  $\\sim 30$  times speedup than NSFP [37] (our faster implementation) and of comparable speed to leading learning-based methods (see Fig. 1 (c)) with the same number of points (8,192). It opens up the possibility of employing a fast, robust, and generalizable approach for dense scene flow, which is not prone to OOD effects in real-time vision and robotic applications.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2. Related work",
+        "text_level": 1,
+        "bbox": [
+            76,
+            833,
+            215,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Scene flow estimation. Scene flow denotes the motion field in 3D space [67] uplifted from 2D optical flow. To reconstruct 3D flow, traditional image-based methods [2, 24,",
+        "bbox": [
+            75,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "25,27,28,36,53] formulate an optimization problem utilizing RGB or depth information, and learning-based RGB/RGB-D methods [6, 29, 30, 57, 60, 64, 78] rely on single/multiple image features which encode with a large amount of data supervisions. On the other hand, to estimate scene flow directly from 3D, traditional point cloud-based methods [1, 11, 50] solve for a non-rigid registration problem, while recent work prefers full-supervised learning [23, 34, 38, 39, 52, 71, 73, 75] that uses point-based features or self-supervised learning [3, 21, 45, 65, 75] that employs a self-supervised loss. Recent non-learning-based methods [37, 51] draw our attention back to runtime optimization that easily scales to large data. Graph prior [51] explicitly builds a graph on the point cloud and uses a graph Laplacian regularizer. While neural scene flow prior [37] uses the network as an implicit regularizer to smooth motions. In this paper, we explore point cloud-based scene flow using runtime optimization.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Accelerating coordinate networks. There exists a line of work [9, 20, 26, 46, 56, 81, 82] that focuses on accelerating coordinate networks by trading slow, memory efficient, deep network architectures for fast, memory hungry, shallow architectures. Most of these innovations have been applied to the problem of neural radiance fields most notably Plenoxels [81] and TensorRF [9]. Recently, this trend was generalized for arbitrary signals through the introduction of complex positional encoding [83] with shallow linear networks. In this paper, we claim that these architectural innovations have little utility in speeding up neural scene flow without first addressing the computational cost of the Chamfer loss it uses.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Distance transform. DT [4, 5, 13, 41, 58] has played an important role in image processing, especially binary image analysis [48, 66]. Further applications are also found in medical image segmentation [12, 33, 59, 70, 72], robotics motion planning [55, 76], geometric representation [7, 10, 49], and accelerated point cloud registration [19, 79]. Among them, various distance measures have been used, such as city block, chessboard, and Euclidean distance [13, 77]. Naturally, Euclidean distance is preferred in computing point distance but it is also the most difficult metric to compute due to the temporal complexity [22]. Many work attempts to speedup Euclidean DT computation including raster-scan-based algorithms [5, 18, 35, 41, 54], fast marching-based algorithms [16, 40, 68], etc., and has achieved linear time computation. In this paper, we investigate the raster-scan-based algorithm for the 3D point cloud.",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. Approach",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.1. Background",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Scene flow optimization. Suppose we have a moving sensor (e.g., lidar mounted on a car, depth camera tied to a robot, etc.) collecting point cloud in a dynamic environment.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "9879",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "At time  $t-1$ , a point cloud  $S_{1}$  (source) of the scene is sampled. Then given the movements of the sensor and objects in certain directions, another point cloud  $S_{2}$  (target) is sampled at time  $t$ . In order to find out all the motions in the environment, we model the translation of each point cloud  $\\mathbf{p} \\in S_{1}$  from time  $t-1$  to time  $t$  as a flow vector  $\\mathbf{f} \\in \\mathbb{R}^{3}$ , where  $\\mathbf{p}' = \\mathbf{p} + \\mathbf{f}$ . The translational vector set of all 3D points in  $S_{1}$  is defined as the scene flow  $\\mathcal{F} = \\{\\mathbf{f}_i\\}_{i=1}^{|S_1|}$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the optimization of the scene flow is to minimize the point distance between the source  $S_{1}$  and the target  $S_{2}$ . Usually, a regularization C, such as a Laplacian regularizer, is needed due to the highly unconstrained non-rigid flows. The overall optimization becomes",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {F} ^ {*} = \\underset {\\mathcal {F}} {\\arg \\min } \\sum_ {\\mathbf {p} \\in \\mathcal {S} _ {1}} \\mathrm {D} (\\mathbf {p} + \\mathbf {f}, \\mathcal {S} _ {2}) + \\lambda \\mathrm {C}, \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $D$  is a point distance function,  $\\lambda$  is a coefficient of the regularizer  $C$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural scene flow prior. NSFP uses traditional runtime optimization to optimize a neural network. Different from learning-based methods, NSFP does not rely on any prior knowledge of large-scale datasets. And different from traditional scene flow optimization with an explicit regularizer that is mentioned above, neural scene flow prior optimizes parameters of a network which implicitly imposes a regularization by its structure:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {\\Theta} ^ {*} = \\underset {\\boldsymbol {\\Theta}} {\\arg \\min } \\sum_ {\\mathbf {p} \\in \\mathcal {S} _ {1}} D (\\mathbf {p} + g (\\mathbf {p}; \\boldsymbol {\\Theta}), \\mathcal {S} _ {2}). \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            531,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\Theta$  is a parameter set of network  $g$  to be optimized.  $\\mathbf{p}$  is the input source point, and the flow  $\\mathbf{f} = g(\\mathbf{p};\\Theta)$  is the output of the network. The optimization converges at  $\\mathbf{f}^{*} = g(\\mathbf{p};\\Theta^{*})$ . The network  $g$  here is chosen to be a commonly used ReLU-MLP.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Since the points in source  $S_{1}$  and target  $S_{2}$  are not in correspondence, nor having the same number of points, i.e.,  $|S_{1}| \\neq |S_{2}|$ , we use a distance function that handles these problems as",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nD (\\mathbf {p}, \\mathcal {S}) = \\min  _ {\\mathbf {x} \\in \\mathcal {S}} \\| \\mathbf {p} - \\mathbf {x} \\| _ {2} ^ {2}. \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            729,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Practically, a bidirectional point distance is found, yielding the above equation equivalent to the Chamfer loss [17].",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Chamfer distance. In point cloud processing, Chamfer distance (CD) [17] is an important loss function and metric for computing the point distance of two point clouds that do not necessarily have points in correspondence. Chamfer distance loss computes the point distance of both source-to-target and target-to-source directions. In detail, the CD loss",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/ba3b70147a46e9e29a4960e668728d47221fffca4b4b58831470bca793841440.jpg",
+        "image_caption": [
+            " Initialized DT map",
+            "Figure 2. An example of how to build a DT map of 2D points. The initialized DT map is set to all zeros. The rasterized 2D point image is set to zero if a point is presented in the grid, and set to one if the grid contains no points. After computation, the final DT map is shown in the right figure."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/7dd8f1cd9ae17d1b5cd46d33439fd7062dcd1a12de5feafccfb2e358b359ebc4.jpg",
+        "image_caption": [
+            "Rasterized 2D points"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/0790784cd9f52965c075bbf94be5651102e85302f5870c4b294403ca2ebde626.jpg",
+        "image_caption": [
+            "Computed DT map"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            764,
+            90,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "can be written as",
+        "bbox": [
+            500,
+            297,
+            614,
+            311
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} C D \\left(\\mathcal {S} _ {1}, \\mathcal {S} _ {2}\\right) = \\sum_ {\\mathbf {p} \\in \\mathcal {S} _ {2}} \\mathrm {D} \\left(\\mathbf {p}, \\mathcal {S} _ {1}\\right) + \\sum_ {\\mathbf {q} \\in \\mathcal {S} _ {1}} \\mathrm {D} \\left(\\mathbf {q}, \\mathcal {S} _ {2}\\right) \\\\ = \\sum_ {\\mathbf {p} \\in \\mathcal {S} _ {2}} \\min  _ {\\mathbf {x} \\in \\mathcal {S} _ {1}} \\| \\mathbf {p} - \\mathbf {x} \\| _ {2} ^ {2} + \\sum_ {\\mathbf {q} \\in \\mathcal {S} _ {1}} \\min  _ {\\mathbf {y} \\in \\mathcal {S} _ {2}} \\| \\mathbf {q} - \\mathbf {y} \\| _ {2} ^ {2}. \\tag {4} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            321,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To compute the point distance, correspondences from source-to-target and target-to-source are searched among the nearest point neighbors. However, the exhaustive point correspondence search is extremely slow, especially when dealing with dense point clouds that contain more than 10k points (e.g., in Waymo Open and Argoverse datasets, the number of points can be up to  $150\\mathrm{k}+$ ).",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2. Correspondence-free point distance transform",
+        "text_level": 1,
+        "bbox": [
+            500,
+            541,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "DT is widely used in 2D image processing, such as segmentation, boundary detection, pattern matching, skeletonization, etc. However, general usage in irregular and unordered 3D point cloud tasks is not broadly discussed. Given that the image grid is regular and ordered, a DT map is easily obtained by computing the minimum distance of each sampled point  $\\mathbf{x} \\in S$  to the vertex  $\\mathbf{q}$  of a DT map  $\\mathcal{G}$  as",
+        "bbox": [
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+            564,
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+            670
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {D T} (\\mathbf {q}) = \\min  _ {\\mathbf {x} \\in \\mathcal {S}, \\mathbf {q} \\in \\mathcal {G}} \\mathrm {D} (\\mathbf {x}, \\mathbf {q}). \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            598,
+            681,
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+            705
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We extend Eq. (5) to fit in the scene flow task such that  $\\mathbf{D}$  refers to Euclidean distance,  $\\mathbf{x}$  denotes the target point, and  $\\mathbf{q}$  is the regularly spaced point in a voxel (3D).",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Approximation of DT map. However, with a large number of points in a point cloud and the grid map, directly computing Eq. (5) builds high dimensional matrices which will lead to large memory occupation. Even when we presume that each axis of the grid/voxel is separable, and the distance per axis is pre-computed, the memory consumption is still huge that cannot be processed on a single GPU. One strategy is to iteratively aggregate small",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "9880",
+        "bbox": [
+            482,
+            944,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/6a84d27d465517007a9ffc68a9a9e3ec3d25ae3c38ca967d7222e543f43f026f.jpg",
+        "image_caption": [
+            "Figure 3. Distance transform map of a target point cloud in Argoverse dataset. Purple points are the target, and green points are the source. It is important to note that the DT map is constructed from the target points, and the source points are only shown for reference purposes. The colormap denotes the DT values (in meters), with light yellow denoting a smaller distance and dark purple indicating a larger distance."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            89,
+            88,
+            460,
+            329
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "matrix multiplications while at a cost of extensive time consumption. Another strategy is to use k-d tree query to compute the Euclidean distance. Building a k-d tree is one-time computing, but querying from a large point set to an even larger k-d tree is expensive.",
+        "bbox": [
+            75,
+            446,
+            468,
+            521
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Instead, we use a fast distance approximation by rasterizing the irregular points onto the nearest regular cuboid/grid. Specifically, similar to classical efficient binary image-based DT methods [12, 18], we construct two 3D binary images for target points and an initialized DT map. An example of 2D DT map is shown in Fig. 2. Specifically, the DT map is set to be all zeros, and the target binary image pixel is set to one when the grid contains no target points in it. Then a two-pass raster scan-based algorithm [18] is applied to each axis to compute the final DT, which means a 3D image needs six propagation passes (left to right, right to left, front to back, back to front, top to bottom, bottom to top). Although we rasterize target points using a binary 3D image approximation, we still compute the exact Euclidean distance between these two binary images instead of an approximated point distance. We empirically find that such discretization does not hurt performance. One reason is the local rigidity of the scene flow benefits from reasonably small but not necessarily infinitesimal DT grids/voxels.",
+        "bbox": [
+            75,
+            522,
+            472,
+            809
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A visualization of a 2D bird's eye view (BEV) DT map is presented in Fig. 3, where we see that when the point density is large, the distance value is relatively small, and the distance value becomes extremely large when no points exist. By choosing an appropriate grid size, the pattern of DT is distinct among different points while",
+        "bbox": [
+            75,
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+            472,
+            902
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "maintaining a relatively smooth structure in the neighboring area. Evidently, the correspondence-free distance transform can act as an effective and efficient surrogate for point correspondence-based Chamfer loss.",
+        "bbox": [
+            496,
+            90,
+            893,
+            151
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "DT query and loss function. Once a DT map is built for the target point cloud, we can look up the pre-computed DT map to get the distance of the nearest source point in an extremely fast fashion. The loss function becomes the queried Euclidean distance between the target and the nearest deformed source points:",
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} = \\min  _ {\\mathbf {y} \\in \\mathcal {S} ^ {\\prime}, \\mathbf {q} \\in \\mathcal {G}} \\| \\mathbf {y} - \\mathbf {q} \\| _ {2}, \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            612,
+            270,
+            893,
+            295
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $S'$  is the deformed source point cloud. Unlike CD loss, no point correspondence search is required in Eq. (6), making the DT query exceptionally fast. Further ablation studies can be found in Sec. 4.3.",
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments",
+        "text_level": 1,
+        "bbox": [
+            498,
+            386,
+            633,
+            404
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Datasets We are primarily interested in scene flow methods that are well-suited for large-scale, realistic, lidar-based OOD scenes, which are commonly encountered in autonomous driving (AV) applications. To this end, we focus on two AV datasets: Waymo Open [62] and Argoverse [8], which contain numerous challenging dynamic scenes. Unfortunately, no ground truth annotations were provided for the open-world dataset. We pre-processed the Argoverse and Waymo Open pseudo ground truth scene flow datasets following [37, 51] and [31, 80] respectively.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Metrics We follow scene flow metrics used in [37, 38, 45, 51, 75] to evaluate the performance. We also include the computation time breakdown in the table. These metrics are:",
+        "bbox": [
+            496,
+            575,
+            893,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1) 3D end-point error  $\\mathcal{E}(m)$  that measures the mean absolute point distance between estimated and target points;",
+            "2) strict accuracy  $Acc_5(\\%)$ , which is the accuracy of the estimated flow that satisfies the absolute point error  $\\mathcal{E} < 0.05\\mathrm{m}$  or the relative point error  $\\mathcal{E}' < 5\\%$ ;",
+            "3) relaxed accuracy  $Acc_{10}(\\%)$ , which is the accuracy of the estimated flow that satisfies the absolute point error  $\\mathcal{E} < 0.1\\mathrm{m}$  or the relative point error  $\\mathcal{E}' < 10\\%$ ;",
+            "4) angle error  $\\theta_{\\epsilon}(rad)$  that measures the mean angle error (in radiance) of the estimated and the pseudo ground truth translational vectors.",
+            "5) computation time  $t(ms)$  breaks down to four parts. Pre-compute includes data loading and building a DT map. Corr. / DT query counts the time needed to search point correspondences or query point distance within a DT map. We also include computation time for Network forward and backward propagation. Finally, Total time in seconds / milliseconds is measured."
+        ],
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "9881",
+        "bbox": [
+            480,
+            944,
+            513,
+            955
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Implementation details We provide the details of implementation for each algorithm we compared. Further information can be found in the supplementary materials.",
+        "bbox": [
+            76,
+            90,
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+            137
+        ],
+        "page_idx": 4
+    },
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+            "1) NSFP [37]. The original implementation is extremely slow. Based on the official code released by the authors while keeping the same parameter settings specified in the original paper, we implemented a faster version (NSFP) for a fair comparison. Note that to reflect the independence of each pair of point clouds when estimating scene flow, we randomly initialized the network before each optimization.",
+            "2) Baseline. Note that although the backward flow enforces a cycle consistency between the deformed point cloud and the original point cloud, we empirically found that removing cycle consistency does not hurt the overall performance when dealing with dense lidar point clouds, but improves the computational efficiency. Here we removed the backward flow in the original NSFP and implemented a NSFP (baseline) version as our baseline for a fair comparison—all computation times were compared with this model. We further modified the baseline model using a linear model with complex positional encodings (NSFP (baseline, linear)) to demonstrate the effect of network architecture changes. Detailed explanations can be found in Sec. 4.1 and the supplementary material.",
+            "3) Ours. We implemented a DT-based neural scene flow method with 8-layer ReLU-MLPs (Ours). We chose the grid cell size of the DT map to be 0.1 meters. Ablation studies on the choice of grid cell size can be found in Sec. 4.3. We further modified our method using a linear model with complex positional encodings (Ours (linear)) to demonstrate the effect of network architecture changes.",
+            "4) FlowStep3D [34] and FLOT [52] are fully supervised methods trained on synthetic FlyingThings3D [42] datasets. PointPWC-Net [75] can be used as a self-supervised method. We used the official code released by the authors and directly tested the pretrained model (pretrained on FlyingThings3D) on our datasets. However, as full/self-learning-based methods, they performed poorly on OOD datasets, which is also observed in [15, 31, 37, 47, 51]. Here we only chose the method with the best performance and the lowest computation time—FLOT—for comparison in the main table. The comparison of other learning-based methods is included in the supplementary material. R3DSF [21] is a weakly supervised method with no direct supervision of ground truth dynamic flows. We tested the method using the pretrained model (on KITTI [43]) provided by the authors."
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+        "type": "text",
+        "text": "All models were implemented using CUDA 11.6-supported PyTorch. All experiments were run on a computer with a single NVIDIA RTX 3090Ti GPU and a 24 AMD Ryzen 95900X 12-Core CPU @ 4.95GHz.",
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+        "img_path": "images/5a10fe8f3f5c48187796ab4ce9a77c3e4a182690a2b61eb3c1266e88e34bee3b.jpg",
+        "image_caption": [
+            "Figure 4. Illustration of 3D complex PE in scene flow problem. The 3D space of source points (yellow) and target points (green) is divided into small cubes. The number of cubes on each axis differs but the edge length is the same. Encoding of a point in a certain grid is the blending/interpolation of encodings of all 8 grid vertices. These grid points (blue) are encoded by complex PE along each axis. Note the vec and outer product  $(\\times)$  notation is for better visual understanding and it is equivalent to the Kronecker product."
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+        "type": "text",
+        "text": "4.1. Speedup in network architecture",
+        "text_level": 1,
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+        "text": "Plenoxels [81] is a method that accelerates NeRF [44] optimization by replacing deep neural networks with voxel-based spherical harmonics representations, leading to efficient volume rendering. Similar network speedup approaches can also be used in neural scene flow optimization. We incorporate recent innovations in positional encodings, specifically complex positional encoding (complex PE) [83] to represent high-frequency signals and enable a linear reconstruction model which is similar in spirit to [81].",
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+        "type": "text",
+        "text": "Complex positional encodings Positional encodings (PEs) are encodings for input positions—e.g., 2D image grids, 3D voxel grids, etc.—that are usually used in coordinate networks. PEs has been shown to improve the performance and convergence speed of coordinate networks [63]. Simple PE is a simple concatenation of the encoding in each input dimension, while a complex PE is a more complicated encoding that computes the Kronecker product of the per-dimension encoding. An illustration of complex PE of scene flow is shown in Fig. 4.",
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+        "type": "text",
+        "text": "With a complex PE and a linear model parameterized by  $\\mathbf{W} \\in \\mathbb{R}^{W_x W_y W_z \\times 3}$ , the scene flow can be represented as",
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+        "type": "equation",
+        "text": "\n$$\n\\mathbf {f} \\approx \\mathbf {B} (\\mathbf {p}; \\phi) \\operatorname {v e c} \\left(\\phi (\\mathbf {x}) \\mathbf {W} \\phi (\\mathbf {z}) ^ {T} \\phi (\\mathbf {y}) ^ {T}\\right), \\tag {7}\n$$\n",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where  $\\mathbf{B}(\\cdot)$  is the blending function,  $\\phi (\\cdot)$  is the encoder,  $\\phi (\\mathbf{x})\\mathbf{W}\\phi (\\mathbf{z})^T\\phi (\\mathbf{y})^T$  is a simple notation for  $n$  -mode multiplication, which is equivalent to Kronecker product. The blending function interpolate the 3D regular grid encodings to handle the irregular and unordered nature of 3D point clouds. Additional details of complex PE can be found in the supplementary materials.",
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+        "text": "9882",
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+    {
+        "type": "table",
+        "img_path": "images/6753d63ab75f620d755562fd7bfc6205a1695d2aa25db490027a5ce210c3bc2a.jpg",
+        "table_caption": [
+            "Table 1. Computation time and performance on Waymo Open Scene Flow dataset. We generated 202 testing scene flow examples where each point cloud has 8k-144k points. The upper tabular between blue bars are experiments with the full point cloud, and the lower tabular between orange bars are experiments with only 8,192 points. All query time is listed as: time per optimization step [total query time]. The speedups are marked with green  $\\times$  numbers, and the slow-downs are marked with red  $\\times$ . The computation time of using full point cloud is compared to values in the red box, and the computation time of using 8,192 points is compared to values in green box. Bold texts are the best performance, and underlined texts denote the second-best performance.  $\\uparrow$  means larger values are better while  $\\downarrow$  means smaller values are better. Corr. / DT query denotes correspondence search or DT query. Note: performance of learning methods PointPWC-Net [75], FlowStep3D [34], PV-RAFT [74] can be found in the supplementary material."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">ε(m)↓</td><td rowspan=\"2\">Acc5(%)↑</td><td rowspan=\"2\">Acc10(%)↑</td><td rowspan=\"2\">θe(rad)↓</td><td colspan=\"4\">t (ms) ↓</td></tr><tr><td>Pre-compute</td><td>Corr. / DT query</td><td>Network</td><td>Total</td></tr><tr><td>NSFP [37]</td><td>0.100</td><td>76.62</td><td>88.56</td><td>0.286</td><td>—</td><td>114 [30972]</td><td>4.62 [1361]</td><td>35.51 s</td></tr><tr><td>NSFP (baseline)</td><td>0.118</td><td>74.16</td><td>86.70</td><td>0.300</td><td>—</td><td>43.1 [15036]</td><td>2.38 [904]</td><td>18.39 s</td></tr><tr><td>NSFP (baseline, linear)</td><td>0.096</td><td>70.78</td><td>86.31</td><td>0.310</td><td>9.48</td><td>40.38 [8037] | 1.1×</td><td>1.53 [319] | 1.6×</td><td>10.20 s | 1.80×</td></tr><tr><td>Ours</td><td>0.072</td><td>84.73</td><td>92.24</td><td>0.280</td><td>40.85</td><td>0.25 [13] | 172×</td><td>2.89 [149] | 1.2×</td><td>0.58 s | 31.7×</td></tr><tr><td>Ours (linear)</td><td>0.109</td><td>71.27</td><td>85.80</td><td>0.321</td><td>44.88</td><td>0.23 [19] | 187×</td><td>1.62 [138] | 1.5×</td><td>0.49 s | 37.5×</td></tr><tr><td>FLOT [52]</td><td>0.702</td><td>2.46</td><td>11.30</td><td>0.808</td><td>—</td><td>—</td><td>—</td><td>99 ms</td></tr><tr><td>R3DSF [21]</td><td>0.414</td><td>35.47</td><td>44.96</td><td>0.527</td><td>—</td><td>—</td><td>—</td><td>140 ms</td></tr><tr><td>NSFP [37] (8,192 pts)</td><td>0.138</td><td>53.62</td><td>78.57</td><td>0.339</td><td>—</td><td>5.42 [1285] | 21×</td><td>4.44 [1051] | 1.1×</td><td>2459 ms | 17.6×</td></tr><tr><td>Ours (8,192 pts)</td><td>0.106</td><td>77.53</td><td>88.99</td><td>0.329</td><td>35.22</td><td>0.23 [6.5] | 496×</td><td>2.60 [76] | 1.8×</td><td>121 ms | 1.16×</td></tr></table>",
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+        "table_caption": [
+            "Table 2. Computation time and performance on Argoverse Scene Flow dataset. Argoverse has 212 testing scene flow examples where each point cloud has  $30\\mathrm{k - }70\\mathrm{k}$  points. Notations are the same as in the table above."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">ε(m)↓</td><td rowspan=\"2\">Acc5(%)↑</td><td rowspan=\"2\">Acc10(%)↑</td><td rowspan=\"2\">θe(rad)↓</td><td colspan=\"4\">t (ms)↓</td></tr><tr><td>Pre-compute</td><td>Corr. / DT query</td><td>Network</td><td>Total</td></tr><tr><td>NSFP [37]</td><td>0.069</td><td>71.56</td><td>87.80</td><td>0.235</td><td>—</td><td>47 [14310]</td><td>4.66 [1507]</td><td>18.08 s</td></tr><tr><td>NSFP (baseline)</td><td>0.078</td><td>69.46</td><td>86.22</td><td>0.253</td><td>—</td><td>17 [5901]</td><td>2.31 [848]</td><td>8.38 s</td></tr><tr><td>NSFP (baseline, linear)</td><td>0.097</td><td>67.03</td><td>83.20</td><td>0.314</td><td>9.19</td><td>14.7 [2786] | 1.1×</td><td>1.51 [297] | 1.5×</td><td>3.55 s | 2.36×</td></tr><tr><td>Ours</td><td>0.071</td><td>80.05</td><td>90.71</td><td>0.289</td><td>43.61</td><td>0.24 [14] | 71×</td><td>2.57 [149] | 1.1×</td><td>0.51 s | 16.4×</td></tr><tr><td>Ours (linear)</td><td>0.106</td><td>65.00</td><td>82.85</td><td>0.319</td><td>48.59</td><td>0.23 [20] | 74×</td><td>1.69 [149] | 1.4×</td><td>0.43 s | 19.5×</td></tr><tr><td>FLOT [52]</td><td>0.821</td><td>2.00</td><td>8.84</td><td>0.967</td><td>—</td><td>—</td><td>—</td><td>88 ms</td></tr><tr><td>R3DSF [21]</td><td>0.417</td><td>32.52</td><td>42.52</td><td>0.551</td><td>—</td><td>—</td><td>—</td><td>113 ms</td></tr><tr><td>NSFP [37] (8,192 pts)</td><td>0.113</td><td>46.32</td><td>72.68</td><td>0.347</td><td>—</td><td>5.40 [1500] | 8.7×</td><td>4.42 [1233] | 1.1×</td><td>2864 ms | 25.6×</td></tr><tr><td>Ours (8,192 pts)</td><td>0.118</td><td>69.93</td><td>83.55</td><td>0.352</td><td>41.57</td><td>0.22 [6.33] | 214×</td><td>2.51 [72.69] | 1.9×</td><td>124 ms | 1.10×</td></tr></table>",
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+        "type": "text",
+        "text": "Network speedup comparison We would like to point out that although various strategies have been proposed to accelerate network architectures, they do not lead to substantial speedup in neural scene flow estimation. To demonstrate this, we compare the relative efficiency of using 8-layer Relu-MLPs and complex PE with a linear model in both 2D image reconstruction and 3D scene flow tasks, shown in Fig. 1 (b). In detail, following [83], we used a size of  $256 \\times 256$  image dataset for image reconstruction. For a fair comparison, the relative efficiency is obtained by normalizing the computation time of two tasks. The computation time of single image reconstruction is  $17.63\\mathrm{s}$  and  $0.15\\mathrm{s}$  for deep network and linear network respectively, leading to  $118 \\times$  speedup. However, the linear model only results in  $\\sim 2 \\times$  and  $\\sim 1.2 \\times$  speedup for NSFP and our method respectively. Our results show that complex PE-based speedup achieves significant acceleration in 2D image reconstruction compared to deep neural networks, but the",
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+    {
+        "type": "text",
+        "text": "benefits of such speedup are more limited when estimating neural scene flow. More detailed comparisons are in Sec. 4.2.",
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+    {
+        "type": "text",
+        "text": "4.2. Comparison of performance",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We show the performance of our method with different variants compared to NSFP, FLOT, and R3DSF on Waymo Open (Tab. 1) and Argoverse (Tab. 2) scene flow datasets. We denote results on Waymo Open (xx) and Argoverse (yy) as xx/yy. Visual results and applications of densification are shown in Fig. 5 and supplementary materials.",
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+        "type": "text",
+        "text": "Dense scene flow estimation. For dense point clouds, the baseline NSFP took  $18.39 / 8.38$  s to converge. When replacing the 8-layer MLPs with a linear network and applying complex PE, we observed a speedup of only  $1.80 / 2.36 \\times$  in total, primarily due to the network propagation speedup. When we replaced naive CD loss with the DT loss, a significant  $31.7 / 16.4 \\times$  speedup was achieved while",
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+        "type": "page_number",
+        "text": "9883",
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+    {
+        "type": "image",
+        "img_path": "images/2e9861c823f7c6bac92698a5310c18a16c04b8e19ed13482c9b76b4b643903a0.jpg",
+        "image_caption": [
+            "Figure 5. Visual examples of the scene flow prediction using our method on Argoverse dataset. We show 3 different autonomous driving scenarios where the autonomous vehicle (AV) was making a right turn (left), the AV stopped for crossing traffic (middle), and the AV was driving in the city (right). We found that the scene flow predicted by our method is close to the ground truth flows. The upper left corner color wheel indicates the flow magnitude (color intensity) and the flow direction (angle). For example, the yellow vehicles in the middle figure are heading south with a relatively large speed wrt the AV."
+        ],
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+    {
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+    {
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+        "type": "text",
+        "text": "maintaining comparable high accuracy. However, replacing a deep network with a linear model when using DT loss only resulted in an additional  $1.2 / 1.2 \\times$  speedup at the cost of a decrease in flow accuracy. Specifically, the total computation time of our proposed method with DT loss is  $0.58 / 0.51$  s, and with DT loss and a linear model is  $0.49 / 0.43$  s. However, with a marginal decrease in computation time, we observed a relatively large drop in performance such that the strict accuracy of our method with DT is  $84.73 / 80.05\\%$  and  $71.27 / 65.00\\%$  with DT and a linear model. These results strongly support our argument in the previous section, which is the general strategy of simplifying or replacing network architectures to accelerate coordinate networks is not particularly effective when optimizing scene flow through coordinate networks.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Overall, the performance of our method using DT loss has on-par performance compared to NSFP while being orders of magnitude faster. We have noticed that there is an \"improved accuracy\" of our method compared to NSFP in Tab. 1, Tab. 2 based on these facts: 1. The performance of our method and NSFP is similar—our method achieves slightly higher performance on some metrics (e.g., accuracy), and NSFP achieves slightly higher performance on other metrics (e.g., angular error). These results indicate that distance transform (DT) is an effective and efficient surrogate for the Chamfer distance (CD) in the scene flow problem. 2. Compared to the \"exact\" point distance (CD), DT queries the distance based on a voxelized DT map, which naturally smooths the flow estimation. In real-world applications, we believe using a DT-accelerated deep network model will achieve both high-fidelity accuracy and efficient computation.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Computation time breakdown. A detailed time breakdown is provided in both tables, including pre-computation, correspondence search / DT distance query, and network forward / backward propagation. The naive",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Chamfer distance requires per-point correspondence search, which can be extremely slow (43.1/17 ms per iteration), especially when the point cloud is denser. For instance, correspondence search is much slower in the Waymo Open dataset than in the Argoverse dataset since Waymo Open has an average of 90k points while Argoverse has an average of 50k points in a single point cloud. DT query is a pre-defined table lookup that is exceptionally faster than the naive point correspondence search, achieving a remarkable  $172 / 71 \\times$  speedup per optimization step—the main contributor to the overall efficiency of our method. Note that the pre-computing of the DT map ( $\\sim 40$  ms) is acceptable given that it is one-time computing and the total computation time is in the range of hundreds of milliseconds. Additional complex PE and a linear model only provide a modest speedup of  $1.6 / 1.5 \\times$  in the network propagation step. Further optimization of the pre-computation and faster implementation can be achieved with full-CUDA support. Note that although in NSFP, the CD loss was implemented using PyTorch3D with CUDA acceleration, it still requires significant computation time, indicating its inherent limitations that cannot be easily overcome through engineering techniques alone.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "OOD generalizability. We further extend our model using fewer points (8,192 points) to accommodate for a fair comparison against learning-based methods [21, 34, 52, 74, 75]. For these learning models, a fixed number of points is required and they only operate on fewer points, such as 2,048 [38, 52] or 8,192 points [21, 34, 52, 75]. Also, these learning methods need to crop the point cloud to a small range. Therefore, learning models cannot be easily adapted to large-scale dense point clouds. Jund et al. explored to inference dense point cloud [32], but they did not directly process the full point cloud as the input. A simple solution to address this challenge is to iteratively predict scene flow for",
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+        "text": "9884",
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+    {
+        "type": "image",
+        "img_path": "images/af6b5027363274c10f701a2258a6af4bcd00496558e6f9e6d1c6a18196da739b.jpg",
+        "image_caption": [
+            "Figure 6. 2D DT map with different grid sizes. A larger grid size results in extremely coarse DT, as the grid cell size decreases, the DT accuracy increases. Here we zoom in on the original DT map in Fig. 3 to show the detailed distance values."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/15adaa81bf4cde9f36025e1253d956535536de24f27302b4254f5000964c02b6.jpg",
+        "image_caption": [
+            "Figure 7. Performance of distance transform with different grid cell sizes on a 2D scene from Argoverse scene flow dataset. Generally, the larger the grid cell is, the lower the accuracy becomes. The computation time and memory consumption also decrease as the grid cell size increases (the number of grids decreases). The total grid of the DT map is of size  $160 \\times 150$ ."
+        ],
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+    {
+        "type": "text",
+        "text": "small subsets of dense point clouds using learning methods. However, this approach requires intensive computation, and it might cause out-of-memory issues [32].",
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+    {
+        "type": "text",
+        "text": "Fully supervised learning methods FlowStep3D, FLOT, PV-RAFT, and self-supervised method PointPWC-Net do not generalize to new autonomous driving datasets (see Tab. 1, Tab. 2) due to the domain gap between the training and testing datasets [15, 31, 37, 47, 51]. We provide additional performance in the supplementary material. Despite its poor performance on OOD data, FLOT achieves a competitive inference time, making it one of the fastest learning-based methods available. Note that since the weakly supervised learning method R3DSF takes supervision from object segmentation and AV ego motions and was trained on a lidar point cloud-based dataset, it has a relatively high accuracy compared to full/self-supervised learning methods, but its performance is still inferior to non-learning-based methods NSFP and our method. Such evidence strongly suggests that our method—a runtime optimization—is robust to OOD effects, and can be directly applied to applications where no training data is readily available.",
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+        "type": "text",
+        "text": "Towards real-time computation. Our method attains real-time performance (121/124 ms) which is comparable to learning methods and maintains high accuracy, while",
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+        "type": "table",
+        "img_path": "images/6c376d9faf336b5a839bd89f50806e65cc9e1ec0dcd99a0fa8f1c6f1c647cf58.jpg",
+        "table_caption": [
+            "Table 3. Comparison of our method and NSFP++ on Argoverse scene flow dataset with ego-motion compensation."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>E↓(m)</td><td>Acc5↑(%)</td><td>Acc10↑(%)</td><td>θe↓(rad)</td><td>t↓(ms)</td></tr><tr><td>Ours</td><td>0.411</td><td>34.94</td><td>46.82</td><td>0.731</td><td>335</td></tr><tr><td>NSFP++ [47]</td><td>0.295</td><td>31.82</td><td>62.61</td><td>0.343</td><td>16188</td></tr><tr><td>NSFP++ (DT)</td><td>0.272</td><td>30.26</td><td>60.25</td><td>0.365</td><td>2001</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "learning-based methods struggle to generalize on OOD data. It indicates the significant potential of applying robust and accurate runtime optimization-based methods in many vision-based applications.",
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+        "type": "text",
+        "text": "Other learning methods. Learning-based scene flow methods are usually fast and achieve good accuracy when applied to in-domain small-scale data (training and testing are on the same dataset with a specific range: e.g., KITTI data with point clouds within  $35\\mathrm{m}$  of the scene center) with a limited number of points (usually 2,048 / 8,192 points). As discussed in [15,31,37,47,51], the domain gap is a significant challenge for learning methods—a specific dataset with specific configurations—e.g., coordinate systems, viewing directions, etc.—that match the testing data needs to be used during training. However, we are interested in exploring runtime optimization-based methods that are robust to large-scale OOD data that can be employed in many real-world applications, such as autonomous driving scenarios, where no labels are readily available.",
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+        "type": "text",
+        "text": "4.3. Ablation study of DT grid size",
+        "text_level": 1,
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+        "type": "text",
+        "text": "DT splits the space into small grid cells, while the grid size affects the accuracy, the computation time, and the memory accuracy. We provide a performance comparison using different grid cell sizes of DT in Fig. 7. A 2D visualization of DT with different grid sizes is also shown in Fig. 6. We can clearly see that a relatively small grid cell size  $(grid \\leq 0.1m)$  is required to ensure the fidelity of the distance transform map. Note that when rasterized points are closer to the original irregular points, DT is a closer approximation to the exact Euclidean distance between two point clouds. Moreover, with a rasterized point-based DT strategy, the pre-computation of the map is no longer an overhead, and the grid cell size will largely affect the memory instead of the computation time—we trade memory consumption with computation time.",
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+        "img_path": "images/0996e1206160f744f23a64b3f5acd1a7afd1097427d6480dae328ef0df70051d.jpg",
+        "image_caption": [
+            "Figure 8. Failure cases on Argoverse dataset. The left figure shows some noisy motions were predicted in the background. The right figure shows our method failed to predict when the motion of some cars is relatively small to the AV."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.4. Speedup in other methods",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The proposed DT loss is a general loss that can be used in other 3D geometry tasks. Specifically, for optimization-based methods such as RSF [14], DT could replace the nearest neighbor distance loss to speed up the overall computation. For learning methods such as PointPWCNet [75], DT could replace Chamfer loss to speed up optimization during training, especially when dealing with a large number of points. For methods using cycle consistency, when replacing CD  $(\\mathcal{O}(N^2))$  with DT, the forward flow computation will be substantially speeded up (similar speedup to the paper). The backward flow computation will also be speeded up even though the DT map will be computed in every optimization step.",
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+    {
+        "type": "text",
+        "text": "We provide an example of comparing the results of our method to  $\\mathrm{NSFP}++$  [47] in Tab. 3. We used our version of  $\\mathrm{NSFP}++$  due to no publicly available official implementation. Since  $\\mathrm{NSFP}++$  requires ego-motion compensation, we also created an additional Argoverse scene flow dataset that removes the ego motions of autonomous vehicles. These ego motions are provided in the original Argoverse dataset [8].",
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+    {
+        "type": "text",
+        "text": "Following [47], we define dynamic points as points that the norm between the deformed source (i.e., source point cloud deformed by the ground truth flow) and the ego-motion compensated source (i.e., the source point cloud is transformed by the ego-motion) exceeds a threshold of 0.05. All the metrics are computed only on dynamic points. During optimization, for our method, we removed the ego-motion of the point cloud and estimated the scene flow using the full point cloud, and for NSFP++, we only used dynamic points as required.",
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+    {
+        "type": "text",
+        "text": "In Tab. 3, we show that our method achieves worse performance than  $\\mathrm{NSFP}++$  while being  $\\sim 50\\times$  faster. We further replaced the Chamfer distance loss used in  $\\mathrm{NSFP}++$  with our proposed distance transform (DT) loss, and we observed a  $\\sim 8\\times$  speedup. The distance transform loss can be a robust and efficient surrogate to Chamfer distance loss in many deep geometry vision tasks.",
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+    {
+        "type": "text",
+        "text": "4.5. Limitations",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One drawback of our method is that creating a DT map using rasterization can lead to discretization errors, especially when the grid size of the DT map is large. To mitigate these, it is necessary to build a DT map with relatively fine-resolution grids. In this case, the memory consumption will increase especially when dealing with high-dimensional data, such as 3D point clouds. Further engineering efforts of pre-creating an efficient high-resolution DT map are required to maintain a reasonable memory cost. However, we empirically find that in the context of scene flow estimation—specifically for scene flow in autonomous driving scenarios—a relatively smooth representation is preferred for local rigidity assumptions. Nevertheless, the tradeoff between the grid resolution, the memory consumption, and the estimation accuracy should be carefully considered and chosen based on specific real-world applications.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Failure cases. We show two typical failure cases in Fig. 8. The first case (left column) is when dynamic points are much sparser than the background, our prediction can result in noisy non-rigid motions in the background. The second case (right column) is when the dynamic motion is relatively small, our model may fail to recognize the dynamic scene and only predict rigid motions.",
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+    {
+        "type": "text",
+        "text": "5. Conclusion",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we revisit the runtime optimization-based scene flow method NSFP and propose a method that is both efficient and generalizable to large-scale OOD data and dense lidar points. We identify that common strategies for speeding up network architectures do not yield significant time reductions—the major computation overhead is the Chamfer distance loss. Therefore, we propose to use an efficient correspondence-free distance transform loss as a robust surrogate. The rediscovery of DT in scene flow estimation opens up an innovative venue to leverage its efficiency and robustness for various deep geometry tasks. Compared to NSFP, our method maintains comparable accuracy but gains up to  $\\sim 30\\times$  speedups. We report for the first time a real-time performance ( $\\sim 120\\mathrm{ms}$ ) with neural scene flow and runtime optimization when using fewer points (8,192). The efficient runtime optimization-based neural scene flow can be widely applied in lidar scenes to do point cloud densification, open-world object detection, and scene clustering, such as in autonomous driving scenarios, where no ground truth or training data are readily available.",
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+    {
+        "type": "text",
+        "text": "Acknowledgement: We would like to thank Haosen Xing for the careful review of the manuscript and the help throughout the project. We thank Kavisha Vidanapathirana for the initial implementation of NSFP++ [47].",
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+    {
+        "type": "text",
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2023/Fast Neural Scene Flow/ffd5b6d9-5f0d-4f78-9688-f85e6ce5fa64_origin.pdf

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+# Fast Neural Scene Flow
+
+Xueqian Li\*1 Jianqiao Zheng\* Francesco Ferroni\*2 Jhony Kaesemodel Pontes\*3 Simon Lucey\*1 1The University of Adelaide 2NVIDIA 3Latitude AI
+
+![](images/42f33bc54254aecd92f7b2ef0351b6c949acfd95b315cd30d79131286e68f85a.jpg)  
+(a) Lidar depth map densification
+
+![](images/54d9e247bdfa3c4b2fa33496b37c6945f5a17ec88ca028c376d0a7bf0ca96a32.jpg)  
+(b) Efficiency of network architectures
+
+![](images/6e1324fa6d030cfc56e5cd611dac84ed3d0751e4de311ee5791ddb6be45afc55.jpg)  
+(c) Error and computation time  
+Figure 1. Scene flow is an important problem as it provides low-level motion cues for many downstream tasks. State-of-the-art learning methods are usually fast and can achieve impressive performance on in-domain data, but usually fail to generalize to out-of-the-distribution (OOD) data or handle dense point clouds. In this paper, we focus on a runtime optimization-based neural scene flow pipeline. In (a) one can see its application in the densification of lidar. However, in (c) one sees that the major drawback is the extensive computation time. We identify that the common speedup strategy in network architectures for coordinate networks has little effect on scene flow acceleration [see green (b)] unlike image reconstruction [see pink (b)]. With the dominant computational burden stemming instead from the Chamfer loss function, we propose to use a distance transform-based loss function to accelerate [see purple (b)], which achieves up to  $30 \times$  speedup and on-par estimation performance compared to NSFP [see (c)]. When tested on 8k points, it is as efficient [see (c)] as leading learning methods, achieving real-time performance.
+
+# Abstract
+
+Neural Scene Flow Prior (NSFP) is of significant interest to the vision community due to its inherent robustness to out-of-distribution (OOD) effects and its ability to deal with dense lidar points. The approach utilizes a coordinate neural network to estimate scene flow at runtime, without any training. However, it is up to 100 times slower than current state-of-the-art learning methods. In other applications such as image, video, and radiance function reconstruction innovations in speeding up the runtime performance of coordinate networks have centered upon architectural changes. In this paper, we demonstrate that scene flow is different—with the dominant computational bottleneck stemming from the loss function itself (i.e., Chamfer distance). Further, we rediscover the distance transform (DT) as an efficient, correspondence-free loss function that dramatically speeds up the runtime optimization. Our fast neural scene flow (FNSF) approach reports for the first time real-time performance comparable to learning methods,
+
+*Part of the work was done while at Argo AI. Corresponding e-mail: xueqian.li@adelaide.edu.au. Code available at https://github.com/Lilac-Lee/FastNSF.git
+
+without any training or OOD bias on two of the largest open autonomous driving (AV) lidar datasets Waymo Open [62] and Argoverse [8].
+
+# 1. Introduction
+
+Neural Scene Flow Prior (NSFP) [37] is considered the dominant method in open-world perception [47] and scene densification [37, 69] using lidar (see Fig. 1 (a)). NSFP achieves state-of-the-art scene flow estimates from dense lidar point clouds (up to  $150\mathrm{k}+$  points) and works on a variety of sensor setups with little to no refinement or adaptation. Unlike supervised or unsupervised learning-based methods, NSFP does not require learning from large offline datasets and has no limits on point density ( $<8\mathrm{k}$  for most learning methods). Instead, it leverages the architecture of a neural network to implicitly regularize the flow estimate and employs a runtime optimization that can easily scale to large out-of-distribution (OOD) scenes, which is a challenge for current learning-based methods [15, 31, 37, 47, 51].
+
+A fundamental drawback, however, to NSFP is the speed of its runtime optimization which is in some instances of orders of magnitude slower than its learning counterparts
+
+(see Fig. 1). As a result, NSFP has widely been used offline for (i) providing scene flow supervision for efficient learning methods and (ii) as a pre-processing step for training open-world perception systems [47]. However, the considerable computational cost of NSFP limits its current applications only to these offline tasks.
+
+A central narrative of our approach is that the dominant computational burden in runtime scene flow optimization (NSFP) is not the network architecture, but the loss function—specifically Chamfer distance (CD) [17]. This differs considerably from other applications of coordinate networks throughout vision and learning such as neural radiance fields (NeRF [44]) and high-fidelity image reconstruction ([61]) which have gained significant speedups through architectural innovations [83]. A visual depiction of this discrepancy can be found in Fig. 1 (b).
+
+Key to our approach is the use of correspondence-free loss function—distance transform (DT) [5, 13, 58] as a proxy for the computationally expensive CD loss. Even though DT has been extensively studied by the graphics and vision community over a few decades, its application as an efficient loss function in deep geometry has largely been overlooked up until this point. We believe that the inherent efficiencies of the DT are especially pertinent for runtime network training such as in NSFP. Our approach shares similarities with Plenoxels [81]—a recent approach for efficient radiance field estimation using coordinate networks—as we trade memory consumption for computation time, allowing for significant speedups during runtime, which provides an alternative solution when exploring more efficient loss functions for dense scene flow estimation. We differ from Plenoxels, however, in that our memory consumption stems from our proposed loss function, not the neural architecture itself.
+
+In this paper, we present for the first time an approximately real-time (for 8k points) runtime optimization method as computationally efficient as leading learning methods whilst preserving the scalability to dense point clouds and state-of-the-art performance on OOD scenes like NSFP. We compare the performance and the computation time of our approach on two of the largest open lidar AV datasets available: Waymo Open [62] and Argoverse [8]. Our fast neural scene flow achieves up to  $\sim 30$  times speedup than NSFP [37] (our faster implementation) and of comparable speed to leading learning-based methods (see Fig. 1 (c)) with the same number of points (8,192). It opens up the possibility of employing a fast, robust, and generalizable approach for dense scene flow, which is not prone to OOD effects in real-time vision and robotic applications.
+
+# 2. Related work
+
+Scene flow estimation. Scene flow denotes the motion field in 3D space [67] uplifted from 2D optical flow. To reconstruct 3D flow, traditional image-based methods [2, 24,
+
+25,27,28,36,53] formulate an optimization problem utilizing RGB or depth information, and learning-based RGB/RGB-D methods [6, 29, 30, 57, 60, 64, 78] rely on single/multiple image features which encode with a large amount of data supervisions. On the other hand, to estimate scene flow directly from 3D, traditional point cloud-based methods [1, 11, 50] solve for a non-rigid registration problem, while recent work prefers full-supervised learning [23, 34, 38, 39, 52, 71, 73, 75] that uses point-based features or self-supervised learning [3, 21, 45, 65, 75] that employs a self-supervised loss. Recent non-learning-based methods [37, 51] draw our attention back to runtime optimization that easily scales to large data. Graph prior [51] explicitly builds a graph on the point cloud and uses a graph Laplacian regularizer. While neural scene flow prior [37] uses the network as an implicit regularizer to smooth motions. In this paper, we explore point cloud-based scene flow using runtime optimization.
+
+Accelerating coordinate networks. There exists a line of work [9, 20, 26, 46, 56, 81, 82] that focuses on accelerating coordinate networks by trading slow, memory efficient, deep network architectures for fast, memory hungry, shallow architectures. Most of these innovations have been applied to the problem of neural radiance fields most notably Plenoxels [81] and TensorRF [9]. Recently, this trend was generalized for arbitrary signals through the introduction of complex positional encoding [83] with shallow linear networks. In this paper, we claim that these architectural innovations have little utility in speeding up neural scene flow without first addressing the computational cost of the Chamfer loss it uses.
+
+Distance transform. DT [4, 5, 13, 41, 58] has played an important role in image processing, especially binary image analysis [48, 66]. Further applications are also found in medical image segmentation [12, 33, 59, 70, 72], robotics motion planning [55, 76], geometric representation [7, 10, 49], and accelerated point cloud registration [19, 79]. Among them, various distance measures have been used, such as city block, chessboard, and Euclidean distance [13, 77]. Naturally, Euclidean distance is preferred in computing point distance but it is also the most difficult metric to compute due to the temporal complexity [22]. Many work attempts to speedup Euclidean DT computation including raster-scan-based algorithms [5, 18, 35, 41, 54], fast marching-based algorithms [16, 40, 68], etc., and has achieved linear time computation. In this paper, we investigate the raster-scan-based algorithm for the 3D point cloud.
+
+# 3. Approach
+
+# 3.1. Background
+
+Scene flow optimization. Suppose we have a moving sensor (e.g., lidar mounted on a car, depth camera tied to a robot, etc.) collecting point cloud in a dynamic environment.
+
+At time  $t-1$ , a point cloud  $S_{1}$  (source) of the scene is sampled. Then given the movements of the sensor and objects in certain directions, another point cloud  $S_{2}$  (target) is sampled at time  $t$ . In order to find out all the motions in the environment, we model the translation of each point cloud  $\mathbf{p} \in S_{1}$  from time  $t-1$  to time  $t$  as a flow vector  $\mathbf{f} \in \mathbb{R}^{3}$ , where  $\mathbf{p}' = \mathbf{p} + \mathbf{f}$ . The translational vector set of all 3D points in  $S_{1}$  is defined as the scene flow  $\mathcal{F} = \{\mathbf{f}_i\}_{i=1}^{|S_1|}$ .
+
+Therefore, the optimization of the scene flow is to minimize the point distance between the source  $S_{1}$  and the target  $S_{2}$ . Usually, a regularization C, such as a Laplacian regularizer, is needed due to the highly unconstrained non-rigid flows. The overall optimization becomes
+
+$$
+\mathcal {F} ^ {*} = \underset {\mathcal {F}} {\arg \min } \sum_ {\mathbf {p} \in \mathcal {S} _ {1}} \mathrm {D} (\mathbf {p} + \mathbf {f}, \mathcal {S} _ {2}) + \lambda \mathrm {C}, \tag {1}
+$$
+
+where  $D$  is a point distance function,  $\lambda$  is a coefficient of the regularizer  $C$ .
+
+Neural scene flow prior. NSFP uses traditional runtime optimization to optimize a neural network. Different from learning-based methods, NSFP does not rely on any prior knowledge of large-scale datasets. And different from traditional scene flow optimization with an explicit regularizer that is mentioned above, neural scene flow prior optimizes parameters of a network which implicitly imposes a regularization by its structure:
+
+$$
+\boldsymbol {\Theta} ^ {*} = \underset {\boldsymbol {\Theta}} {\arg \min } \sum_ {\mathbf {p} \in \mathcal {S} _ {1}} D (\mathbf {p} + g (\mathbf {p}; \boldsymbol {\Theta}), \mathcal {S} _ {2}). \tag {2}
+$$
+
+where  $\Theta$  is a parameter set of network  $g$  to be optimized.  $\mathbf{p}$  is the input source point, and the flow  $\mathbf{f} = g(\mathbf{p};\Theta)$  is the output of the network. The optimization converges at  $\mathbf{f}^{*} = g(\mathbf{p};\Theta^{*})$ . The network  $g$  here is chosen to be a commonly used ReLU-MLP.
+
+Since the points in source  $S_{1}$  and target  $S_{2}$  are not in correspondence, nor having the same number of points, i.e.,  $|S_{1}| \neq |S_{2}|$ , we use a distance function that handles these problems as
+
+$$
+D (\mathbf {p}, \mathcal {S}) = \min  _ {\mathbf {x} \in \mathcal {S}} \| \mathbf {p} - \mathbf {x} \| _ {2} ^ {2}. \tag {3}
+$$
+
+Practically, a bidirectional point distance is found, yielding the above equation equivalent to the Chamfer loss [17].
+
+Chamfer distance. In point cloud processing, Chamfer distance (CD) [17] is an important loss function and metric for computing the point distance of two point clouds that do not necessarily have points in correspondence. Chamfer distance loss computes the point distance of both source-to-target and target-to-source directions. In detail, the CD loss
+
+![](images/ba3b70147a46e9e29a4960e668728d47221fffca4b4b58831470bca793841440.jpg)  
+ Initialized DT map  
+Figure 2. An example of how to build a DT map of 2D points. The initialized DT map is set to all zeros. The rasterized 2D point image is set to zero if a point is presented in the grid, and set to one if the grid contains no points. After computation, the final DT map is shown in the right figure.
+
+![](images/7dd8f1cd9ae17d1b5cd46d33439fd7062dcd1a12de5feafccfb2e358b359ebc4.jpg)  
+Rasterized 2D points
+
+![](images/0790784cd9f52965c075bbf94be5651102e85302f5870c4b294403ca2ebde626.jpg)  
+Computed DT map
+
+can be written as
+
+$$
+\begin{array}{l} C D \left(\mathcal {S} _ {1}, \mathcal {S} _ {2}\right) = \sum_ {\mathbf {p} \in \mathcal {S} _ {2}} \mathrm {D} \left(\mathbf {p}, \mathcal {S} _ {1}\right) + \sum_ {\mathbf {q} \in \mathcal {S} _ {1}} \mathrm {D} \left(\mathbf {q}, \mathcal {S} _ {2}\right) \\ = \sum_ {\mathbf {p} \in \mathcal {S} _ {2}} \min  _ {\mathbf {x} \in \mathcal {S} _ {1}} \| \mathbf {p} - \mathbf {x} \| _ {2} ^ {2} + \sum_ {\mathbf {q} \in \mathcal {S} _ {1}} \min  _ {\mathbf {y} \in \mathcal {S} _ {2}} \| \mathbf {q} - \mathbf {y} \| _ {2} ^ {2}. \tag {4} \\ \end{array}
+$$
+
+To compute the point distance, correspondences from source-to-target and target-to-source are searched among the nearest point neighbors. However, the exhaustive point correspondence search is extremely slow, especially when dealing with dense point clouds that contain more than 10k points (e.g., in Waymo Open and Argoverse datasets, the number of points can be up to  $150\mathrm{k}+$ ).
+
+# 3.2. Correspondence-free point distance transform
+
+DT is widely used in 2D image processing, such as segmentation, boundary detection, pattern matching, skeletonization, etc. However, general usage in irregular and unordered 3D point cloud tasks is not broadly discussed. Given that the image grid is regular and ordered, a DT map is easily obtained by computing the minimum distance of each sampled point  $\mathbf{x} \in S$  to the vertex  $\mathbf{q}$  of a DT map  $\mathcal{G}$  as
+
+$$
+\mathrm {D T} (\mathbf {q}) = \min  _ {\mathbf {x} \in \mathcal {S}, \mathbf {q} \in \mathcal {G}} \mathrm {D} (\mathbf {x}, \mathbf {q}). \tag {5}
+$$
+
+We extend Eq. (5) to fit in the scene flow task such that  $\mathbf{D}$  refers to Euclidean distance,  $\mathbf{x}$  denotes the target point, and  $\mathbf{q}$  is the regularly spaced point in a voxel (3D).
+
+Approximation of DT map. However, with a large number of points in a point cloud and the grid map, directly computing Eq. (5) builds high dimensional matrices which will lead to large memory occupation. Even when we presume that each axis of the grid/voxel is separable, and the distance per axis is pre-computed, the memory consumption is still huge that cannot be processed on a single GPU. One strategy is to iteratively aggregate small
+
+![](images/6a84d27d465517007a9ffc68a9a9e3ec3d25ae3c38ca967d7222e543f43f026f.jpg)  
+Figure 3. Distance transform map of a target point cloud in Argoverse dataset. Purple points are the target, and green points are the source. It is important to note that the DT map is constructed from the target points, and the source points are only shown for reference purposes. The colormap denotes the DT values (in meters), with light yellow denoting a smaller distance and dark purple indicating a larger distance.
+
+matrix multiplications while at a cost of extensive time consumption. Another strategy is to use k-d tree query to compute the Euclidean distance. Building a k-d tree is one-time computing, but querying from a large point set to an even larger k-d tree is expensive.
+
+Instead, we use a fast distance approximation by rasterizing the irregular points onto the nearest regular cuboid/grid. Specifically, similar to classical efficient binary image-based DT methods [12, 18], we construct two 3D binary images for target points and an initialized DT map. An example of 2D DT map is shown in Fig. 2. Specifically, the DT map is set to be all zeros, and the target binary image pixel is set to one when the grid contains no target points in it. Then a two-pass raster scan-based algorithm [18] is applied to each axis to compute the final DT, which means a 3D image needs six propagation passes (left to right, right to left, front to back, back to front, top to bottom, bottom to top). Although we rasterize target points using a binary 3D image approximation, we still compute the exact Euclidean distance between these two binary images instead of an approximated point distance. We empirically find that such discretization does not hurt performance. One reason is the local rigidity of the scene flow benefits from reasonably small but not necessarily infinitesimal DT grids/voxels.
+
+A visualization of a 2D bird's eye view (BEV) DT map is presented in Fig. 3, where we see that when the point density is large, the distance value is relatively small, and the distance value becomes extremely large when no points exist. By choosing an appropriate grid size, the pattern of DT is distinct among different points while
+
+maintaining a relatively smooth structure in the neighboring area. Evidently, the correspondence-free distance transform can act as an effective and efficient surrogate for point correspondence-based Chamfer loss.
+
+DT query and loss function. Once a DT map is built for the target point cloud, we can look up the pre-computed DT map to get the distance of the nearest source point in an extremely fast fashion. The loss function becomes the queried Euclidean distance between the target and the nearest deformed source points:
+
+$$
+\mathcal {L} = \min  _ {\mathbf {y} \in \mathcal {S} ^ {\prime}, \mathbf {q} \in \mathcal {G}} \| \mathbf {y} - \mathbf {q} \| _ {2}, \tag {6}
+$$
+
+where  $S'$  is the deformed source point cloud. Unlike CD loss, no point correspondence search is required in Eq. (6), making the DT query exceptionally fast. Further ablation studies can be found in Sec. 4.3.
+
+# 4. Experiments
+
+Datasets We are primarily interested in scene flow methods that are well-suited for large-scale, realistic, lidar-based OOD scenes, which are commonly encountered in autonomous driving (AV) applications. To this end, we focus on two AV datasets: Waymo Open [62] and Argoverse [8], which contain numerous challenging dynamic scenes. Unfortunately, no ground truth annotations were provided for the open-world dataset. We pre-processed the Argoverse and Waymo Open pseudo ground truth scene flow datasets following [37, 51] and [31, 80] respectively.
+
+Metrics We follow scene flow metrics used in [37, 38, 45, 51, 75] to evaluate the performance. We also include the computation time breakdown in the table. These metrics are:
+
+1) 3D end-point error  $\mathcal{E}(m)$  that measures the mean absolute point distance between estimated and target points;  
+2) strict accuracy  $Acc_5(\%)$ , which is the accuracy of the estimated flow that satisfies the absolute point error  $\mathcal{E} < 0.05\mathrm{m}$  or the relative point error  $\mathcal{E}' < 5\%$ ;  
+3) relaxed accuracy  $Acc_{10}(\%)$ , which is the accuracy of the estimated flow that satisfies the absolute point error  $\mathcal{E} < 0.1\mathrm{m}$  or the relative point error  $\mathcal{E}' < 10\%$ ;  
+4) angle error  $\theta_{\epsilon}(rad)$  that measures the mean angle error (in radiance) of the estimated and the pseudo ground truth translational vectors.  
+5) computation time  $t(ms)$  breaks down to four parts. Pre-compute includes data loading and building a DT map. Corr. / DT query counts the time needed to search point correspondences or query point distance within a DT map. We also include computation time for Network forward and backward propagation. Finally, Total time in seconds / milliseconds is measured.
+
+Implementation details We provide the details of implementation for each algorithm we compared. Further information can be found in the supplementary materials.
+
+1) NSFP [37]. The original implementation is extremely slow. Based on the official code released by the authors while keeping the same parameter settings specified in the original paper, we implemented a faster version (NSFP) for a fair comparison. Note that to reflect the independence of each pair of point clouds when estimating scene flow, we randomly initialized the network before each optimization.  
+2) Baseline. Note that although the backward flow enforces a cycle consistency between the deformed point cloud and the original point cloud, we empirically found that removing cycle consistency does not hurt the overall performance when dealing with dense lidar point clouds, but improves the computational efficiency. Here we removed the backward flow in the original NSFP and implemented a NSFP (baseline) version as our baseline for a fair comparison—all computation times were compared with this model. We further modified the baseline model using a linear model with complex positional encodings (NSFP (baseline, linear)) to demonstrate the effect of network architecture changes. Detailed explanations can be found in Sec. 4.1 and the supplementary material.  
+3) Ours. We implemented a DT-based neural scene flow method with 8-layer ReLU-MLPs (Ours). We chose the grid cell size of the DT map to be 0.1 meters. Ablation studies on the choice of grid cell size can be found in Sec. 4.3. We further modified our method using a linear model with complex positional encodings (Ours (linear)) to demonstrate the effect of network architecture changes.  
+4) FlowStep3D [34] and FLOT [52] are fully supervised methods trained on synthetic FlyingThings3D [42] datasets. PointPWC-Net [75] can be used as a self-supervised method. We used the official code released by the authors and directly tested the pretrained model (pretrained on FlyingThings3D) on our datasets. However, as full/self-learning-based methods, they performed poorly on OOD datasets, which is also observed in [15, 31, 37, 47, 51]. Here we only chose the method with the best performance and the lowest computation time—FLOT—for comparison in the main table. The comparison of other learning-based methods is included in the supplementary material. R3DSF [21] is a weakly supervised method with no direct supervision of ground truth dynamic flows. We tested the method using the pretrained model (on KITTI [43]) provided by the authors.
+
+All models were implemented using CUDA 11.6-supported PyTorch. All experiments were run on a computer with a single NVIDIA RTX 3090Ti GPU and a 24 AMD Ryzen 95900X 12-Core CPU @ 4.95GHz.
+
+![](images/5a10fe8f3f5c48187796ab4ce9a77c3e4a182690a2b61eb3c1266e88e34bee3b.jpg)  
+Figure 4. Illustration of 3D complex PE in scene flow problem. The 3D space of source points (yellow) and target points (green) is divided into small cubes. The number of cubes on each axis differs but the edge length is the same. Encoding of a point in a certain grid is the blending/interpolation of encodings of all 8 grid vertices. These grid points (blue) are encoded by complex PE along each axis. Note the vec and outer product  $(\times)$  notation is for better visual understanding and it is equivalent to the Kronecker product.
+
+# 4.1. Speedup in network architecture
+
+Plenoxels [81] is a method that accelerates NeRF [44] optimization by replacing deep neural networks with voxel-based spherical harmonics representations, leading to efficient volume rendering. Similar network speedup approaches can also be used in neural scene flow optimization. We incorporate recent innovations in positional encodings, specifically complex positional encoding (complex PE) [83] to represent high-frequency signals and enable a linear reconstruction model which is similar in spirit to [81].
+
+Complex positional encodings Positional encodings (PEs) are encodings for input positions—e.g., 2D image grids, 3D voxel grids, etc.—that are usually used in coordinate networks. PEs has been shown to improve the performance and convergence speed of coordinate networks [63]. Simple PE is a simple concatenation of the encoding in each input dimension, while a complex PE is a more complicated encoding that computes the Kronecker product of the per-dimension encoding. An illustration of complex PE of scene flow is shown in Fig. 4.
+
+With a complex PE and a linear model parameterized by  $\mathbf{W} \in \mathbb{R}^{W_x W_y W_z \times 3}$ , the scene flow can be represented as
+
+$$
+\mathbf {f} \approx \mathbf {B} (\mathbf {p}; \phi) \operatorname {v e c} \left(\phi (\mathbf {x}) \mathbf {W} \phi (\mathbf {z}) ^ {T} \phi (\mathbf {y}) ^ {T}\right), \tag {7}
+$$
+
+where  $\mathbf{B}(\cdot)$  is the blending function,  $\phi (\cdot)$  is the encoder,  $\phi (\mathbf{x})\mathbf{W}\phi (\mathbf{z})^T\phi (\mathbf{y})^T$  is a simple notation for  $n$  -mode multiplication, which is equivalent to Kronecker product. The blending function interpolate the 3D regular grid encodings to handle the irregular and unordered nature of 3D point clouds. Additional details of complex PE can be found in the supplementary materials.
+
+Table 1. Computation time and performance on Waymo Open Scene Flow dataset. We generated 202 testing scene flow examples where each point cloud has 8k-144k points. The upper tabular between blue bars are experiments with the full point cloud, and the lower tabular between orange bars are experiments with only 8,192 points. All query time is listed as: time per optimization step [total query time]. The speedups are marked with green  $\times$  numbers, and the slow-downs are marked with red  $\times$ . The computation time of using full point cloud is compared to values in the red box, and the computation time of using 8,192 points is compared to values in green box. Bold texts are the best performance, and underlined texts denote the second-best performance.  $\uparrow$  means larger values are better while  $\downarrow$  means smaller values are better. Corr. / DT query denotes correspondence search or DT query. Note: performance of learning methods PointPWC-Net [75], FlowStep3D [34], PV-RAFT [74] can be found in the supplementary material.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">ε(m)↓</td><td rowspan="2">Acc5(%)↑</td><td rowspan="2">Acc10(%)↑</td><td rowspan="2">θe(rad)↓</td><td colspan="4">t (ms) ↓</td></tr><tr><td>Pre-compute</td><td>Corr. / DT query</td><td>Network</td><td>Total</td></tr><tr><td>NSFP [37]</td><td>0.100</td><td>76.62</td><td>88.56</td><td>0.286</td><td>—</td><td>114 [30972]</td><td>4.62 [1361]</td><td>35.51 s</td></tr><tr><td>NSFP (baseline)</td><td>0.118</td><td>74.16</td><td>86.70</td><td>0.300</td><td>—</td><td>43.1 [15036]</td><td>2.38 [904]</td><td>18.39 s</td></tr><tr><td>NSFP (baseline, linear)</td><td>0.096</td><td>70.78</td><td>86.31</td><td>0.310</td><td>9.48</td><td>40.38 [8037] | 1.1×</td><td>1.53 [319] | 1.6×</td><td>10.20 s | 1.80×</td></tr><tr><td>Ours</td><td>0.072</td><td>84.73</td><td>92.24</td><td>0.280</td><td>40.85</td><td>0.25 [13] | 172×</td><td>2.89 [149] | 1.2×</td><td>0.58 s | 31.7×</td></tr><tr><td>Ours (linear)</td><td>0.109</td><td>71.27</td><td>85.80</td><td>0.321</td><td>44.88</td><td>0.23 [19] | 187×</td><td>1.62 [138] | 1.5×</td><td>0.49 s | 37.5×</td></tr><tr><td>FLOT [52]</td><td>0.702</td><td>2.46</td><td>11.30</td><td>0.808</td><td>—</td><td>—</td><td>—</td><td>99 ms</td></tr><tr><td>R3DSF [21]</td><td>0.414</td><td>35.47</td><td>44.96</td><td>0.527</td><td>—</td><td>—</td><td>—</td><td>140 ms</td></tr><tr><td>NSFP [37] (8,192 pts)</td><td>0.138</td><td>53.62</td><td>78.57</td><td>0.339</td><td>—</td><td>5.42 [1285] | 21×</td><td>4.44 [1051] | 1.1×</td><td>2459 ms | 17.6×</td></tr><tr><td>Ours (8,192 pts)</td><td>0.106</td><td>77.53</td><td>88.99</td><td>0.329</td><td>35.22</td><td>0.23 [6.5] | 496×</td><td>2.60 [76] | 1.8×</td><td>121 ms | 1.16×</td></tr></table>
+
+Table 2. Computation time and performance on Argoverse Scene Flow dataset. Argoverse has 212 testing scene flow examples where each point cloud has  $30\mathrm{k - }70\mathrm{k}$  points. Notations are the same as in the table above.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">ε(m)↓</td><td rowspan="2">Acc5(%)↑</td><td rowspan="2">Acc10(%)↑</td><td rowspan="2">θe(rad)↓</td><td colspan="4">t (ms)↓</td></tr><tr><td>Pre-compute</td><td>Corr. / DT query</td><td>Network</td><td>Total</td></tr><tr><td>NSFP [37]</td><td>0.069</td><td>71.56</td><td>87.80</td><td>0.235</td><td>—</td><td>47 [14310]</td><td>4.66 [1507]</td><td>18.08 s</td></tr><tr><td>NSFP (baseline)</td><td>0.078</td><td>69.46</td><td>86.22</td><td>0.253</td><td>—</td><td>17 [5901]</td><td>2.31 [848]</td><td>8.38 s</td></tr><tr><td>NSFP (baseline, linear)</td><td>0.097</td><td>67.03</td><td>83.20</td><td>0.314</td><td>9.19</td><td>14.7 [2786] | 1.1×</td><td>1.51 [297] | 1.5×</td><td>3.55 s | 2.36×</td></tr><tr><td>Ours</td><td>0.071</td><td>80.05</td><td>90.71</td><td>0.289</td><td>43.61</td><td>0.24 [14] | 71×</td><td>2.57 [149] | 1.1×</td><td>0.51 s | 16.4×</td></tr><tr><td>Ours (linear)</td><td>0.106</td><td>65.00</td><td>82.85</td><td>0.319</td><td>48.59</td><td>0.23 [20] | 74×</td><td>1.69 [149] | 1.4×</td><td>0.43 s | 19.5×</td></tr><tr><td>FLOT [52]</td><td>0.821</td><td>2.00</td><td>8.84</td><td>0.967</td><td>—</td><td>—</td><td>—</td><td>88 ms</td></tr><tr><td>R3DSF [21]</td><td>0.417</td><td>32.52</td><td>42.52</td><td>0.551</td><td>—</td><td>—</td><td>—</td><td>113 ms</td></tr><tr><td>NSFP [37] (8,192 pts)</td><td>0.113</td><td>46.32</td><td>72.68</td><td>0.347</td><td>—</td><td>5.40 [1500] | 8.7×</td><td>4.42 [1233] | 1.1×</td><td>2864 ms | 25.6×</td></tr><tr><td>Ours (8,192 pts)</td><td>0.118</td><td>69.93</td><td>83.55</td><td>0.352</td><td>41.57</td><td>0.22 [6.33] | 214×</td><td>2.51 [72.69] | 1.9×</td><td>124 ms | 1.10×</td></tr></table>
+
+Network speedup comparison We would like to point out that although various strategies have been proposed to accelerate network architectures, they do not lead to substantial speedup in neural scene flow estimation. To demonstrate this, we compare the relative efficiency of using 8-layer Relu-MLPs and complex PE with a linear model in both 2D image reconstruction and 3D scene flow tasks, shown in Fig. 1 (b). In detail, following [83], we used a size of  $256 \times 256$  image dataset for image reconstruction. For a fair comparison, the relative efficiency is obtained by normalizing the computation time of two tasks. The computation time of single image reconstruction is  $17.63\mathrm{s}$  and  $0.15\mathrm{s}$  for deep network and linear network respectively, leading to  $118 \times$  speedup. However, the linear model only results in  $\sim 2 \times$  and  $\sim 1.2 \times$  speedup for NSFP and our method respectively. Our results show that complex PE-based speedup achieves significant acceleration in 2D image reconstruction compared to deep neural networks, but the
+
+benefits of such speedup are more limited when estimating neural scene flow. More detailed comparisons are in Sec. 4.2.
+
+# 4.2. Comparison of performance
+
+We show the performance of our method with different variants compared to NSFP, FLOT, and R3DSF on Waymo Open (Tab. 1) and Argoverse (Tab. 2) scene flow datasets. We denote results on Waymo Open (xx) and Argoverse (yy) as xx/yy. Visual results and applications of densification are shown in Fig. 5 and supplementary materials.
+
+Dense scene flow estimation. For dense point clouds, the baseline NSFP took  $18.39 / 8.38$  s to converge. When replacing the 8-layer MLPs with a linear network and applying complex PE, we observed a speedup of only  $1.80 / 2.36 \times$  in total, primarily due to the network propagation speedup. When we replaced naive CD loss with the DT loss, a significant  $31.7 / 16.4 \times$  speedup was achieved while
+
+![](images/2e9861c823f7c6bac92698a5310c18a16c04b8e19ed13482c9b76b4b643903a0.jpg)  
+Figure 5. Visual examples of the scene flow prediction using our method on Argoverse dataset. We show 3 different autonomous driving scenarios where the autonomous vehicle (AV) was making a right turn (left), the AV stopped for crossing traffic (middle), and the AV was driving in the city (right). We found that the scene flow predicted by our method is close to the ground truth flows. The upper left corner color wheel indicates the flow magnitude (color intensity) and the flow direction (angle). For example, the yellow vehicles in the middle figure are heading south with a relatively large speed wrt the AV.
+
+![](images/91f49666f983474bc02a473e24bb5441b7e50c4a0a9ff0f768bb2ea35f7eabb7.jpg)
+
+![](images/b7e913dc4fcec0412dfd141d88d574f1fb57e1ba605810710b179b0fb59b6982.jpg)
+
+maintaining comparable high accuracy. However, replacing a deep network with a linear model when using DT loss only resulted in an additional  $1.2 / 1.2 \times$  speedup at the cost of a decrease in flow accuracy. Specifically, the total computation time of our proposed method with DT loss is  $0.58 / 0.51$  s, and with DT loss and a linear model is  $0.49 / 0.43$  s. However, with a marginal decrease in computation time, we observed a relatively large drop in performance such that the strict accuracy of our method with DT is  $84.73 / 80.05\%$  and  $71.27 / 65.00\%$  with DT and a linear model. These results strongly support our argument in the previous section, which is the general strategy of simplifying or replacing network architectures to accelerate coordinate networks is not particularly effective when optimizing scene flow through coordinate networks.
+
+Overall, the performance of our method using DT loss has on-par performance compared to NSFP while being orders of magnitude faster. We have noticed that there is an "improved accuracy" of our method compared to NSFP in Tab. 1, Tab. 2 based on these facts: 1. The performance of our method and NSFP is similar—our method achieves slightly higher performance on some metrics (e.g., accuracy), and NSFP achieves slightly higher performance on other metrics (e.g., angular error). These results indicate that distance transform (DT) is an effective and efficient surrogate for the Chamfer distance (CD) in the scene flow problem. 2. Compared to the "exact" point distance (CD), DT queries the distance based on a voxelized DT map, which naturally smooths the flow estimation. In real-world applications, we believe using a DT-accelerated deep network model will achieve both high-fidelity accuracy and efficient computation.
+
+Computation time breakdown. A detailed time breakdown is provided in both tables, including pre-computation, correspondence search / DT distance query, and network forward / backward propagation. The naive
+
+Chamfer distance requires per-point correspondence search, which can be extremely slow (43.1/17 ms per iteration), especially when the point cloud is denser. For instance, correspondence search is much slower in the Waymo Open dataset than in the Argoverse dataset since Waymo Open has an average of 90k points while Argoverse has an average of 50k points in a single point cloud. DT query is a pre-defined table lookup that is exceptionally faster than the naive point correspondence search, achieving a remarkable  $172 / 71 \times$  speedup per optimization step—the main contributor to the overall efficiency of our method. Note that the pre-computing of the DT map ( $\sim 40$  ms) is acceptable given that it is one-time computing and the total computation time is in the range of hundreds of milliseconds. Additional complex PE and a linear model only provide a modest speedup of  $1.6 / 1.5 \times$  in the network propagation step. Further optimization of the pre-computation and faster implementation can be achieved with full-CUDA support. Note that although in NSFP, the CD loss was implemented using PyTorch3D with CUDA acceleration, it still requires significant computation time, indicating its inherent limitations that cannot be easily overcome through engineering techniques alone.
+
+OOD generalizability. We further extend our model using fewer points (8,192 points) to accommodate for a fair comparison against learning-based methods [21, 34, 52, 74, 75]. For these learning models, a fixed number of points is required and they only operate on fewer points, such as 2,048 [38, 52] or 8,192 points [21, 34, 52, 75]. Also, these learning methods need to crop the point cloud to a small range. Therefore, learning models cannot be easily adapted to large-scale dense point clouds. Jund et al. explored to inference dense point cloud [32], but they did not directly process the full point cloud as the input. A simple solution to address this challenge is to iteratively predict scene flow for
+
+![](images/af6b5027363274c10f701a2258a6af4bcd00496558e6f9e6d1c6a18196da739b.jpg)  
+Figure 6. 2D DT map with different grid sizes. A larger grid size results in extremely coarse DT, as the grid cell size decreases, the DT accuracy increases. Here we zoom in on the original DT map in Fig. 3 to show the detailed distance values.
+
+![](images/15adaa81bf4cde9f36025e1253d956535536de24f27302b4254f5000964c02b6.jpg)  
+Figure 7. Performance of distance transform with different grid cell sizes on a 2D scene from Argoverse scene flow dataset. Generally, the larger the grid cell is, the lower the accuracy becomes. The computation time and memory consumption also decrease as the grid cell size increases (the number of grids decreases). The total grid of the DT map is of size  $160 \times 150$ .
+
+small subsets of dense point clouds using learning methods. However, this approach requires intensive computation, and it might cause out-of-memory issues [32].
+
+Fully supervised learning methods FlowStep3D, FLOT, PV-RAFT, and self-supervised method PointPWC-Net do not generalize to new autonomous driving datasets (see Tab. 1, Tab. 2) due to the domain gap between the training and testing datasets [15, 31, 37, 47, 51]. We provide additional performance in the supplementary material. Despite its poor performance on OOD data, FLOT achieves a competitive inference time, making it one of the fastest learning-based methods available. Note that since the weakly supervised learning method R3DSF takes supervision from object segmentation and AV ego motions and was trained on a lidar point cloud-based dataset, it has a relatively high accuracy compared to full/self-supervised learning methods, but its performance is still inferior to non-learning-based methods NSFP and our method. Such evidence strongly suggests that our method—a runtime optimization—is robust to OOD effects, and can be directly applied to applications where no training data is readily available.
+
+Towards real-time computation. Our method attains real-time performance (121/124 ms) which is comparable to learning methods and maintains high accuracy, while
+
+Table 3. Comparison of our method and NSFP++ on Argoverse scene flow dataset with ego-motion compensation.  
+
+<table><tr><td>Method</td><td>E↓(m)</td><td>Acc5↑(%)</td><td>Acc10↑(%)</td><td>θe↓(rad)</td><td>t↓(ms)</td></tr><tr><td>Ours</td><td>0.411</td><td>34.94</td><td>46.82</td><td>0.731</td><td>335</td></tr><tr><td>NSFP++ [47]</td><td>0.295</td><td>31.82</td><td>62.61</td><td>0.343</td><td>16188</td></tr><tr><td>NSFP++ (DT)</td><td>0.272</td><td>30.26</td><td>60.25</td><td>0.365</td><td>2001</td></tr></table>
+
+learning-based methods struggle to generalize on OOD data. It indicates the significant potential of applying robust and accurate runtime optimization-based methods in many vision-based applications.
+
+Other learning methods. Learning-based scene flow methods are usually fast and achieve good accuracy when applied to in-domain small-scale data (training and testing are on the same dataset with a specific range: e.g., KITTI data with point clouds within  $35\mathrm{m}$  of the scene center) with a limited number of points (usually 2,048 / 8,192 points). As discussed in [15,31,37,47,51], the domain gap is a significant challenge for learning methods—a specific dataset with specific configurations—e.g., coordinate systems, viewing directions, etc.—that match the testing data needs to be used during training. However, we are interested in exploring runtime optimization-based methods that are robust to large-scale OOD data that can be employed in many real-world applications, such as autonomous driving scenarios, where no labels are readily available.
+
+# 4.3. Ablation study of DT grid size
+
+DT splits the space into small grid cells, while the grid size affects the accuracy, the computation time, and the memory accuracy. We provide a performance comparison using different grid cell sizes of DT in Fig. 7. A 2D visualization of DT with different grid sizes is also shown in Fig. 6. We can clearly see that a relatively small grid cell size  $(grid \leq 0.1m)$  is required to ensure the fidelity of the distance transform map. Note that when rasterized points are closer to the original irregular points, DT is a closer approximation to the exact Euclidean distance between two point clouds. Moreover, with a rasterized point-based DT strategy, the pre-computation of the map is no longer an overhead, and the grid cell size will largely affect the memory instead of the computation time—we trade memory consumption with computation time.
+
+![](images/0996e1206160f744f23a64b3f5acd1a7afd1097427d6480dae328ef0df70051d.jpg)  
+Figure 8. Failure cases on Argoverse dataset. The left figure shows some noisy motions were predicted in the background. The right figure shows our method failed to predict when the motion of some cars is relatively small to the AV.
+
+# 4.4. Speedup in other methods
+
+The proposed DT loss is a general loss that can be used in other 3D geometry tasks. Specifically, for optimization-based methods such as RSF [14], DT could replace the nearest neighbor distance loss to speed up the overall computation. For learning methods such as PointPWCNet [75], DT could replace Chamfer loss to speed up optimization during training, especially when dealing with a large number of points. For methods using cycle consistency, when replacing CD  $(\mathcal{O}(N^2))$  with DT, the forward flow computation will be substantially speeded up (similar speedup to the paper). The backward flow computation will also be speeded up even though the DT map will be computed in every optimization step.
+
+We provide an example of comparing the results of our method to  $\mathrm{NSFP}++$  [47] in Tab. 3. We used our version of  $\mathrm{NSFP}++$  due to no publicly available official implementation. Since  $\mathrm{NSFP}++$  requires ego-motion compensation, we also created an additional Argoverse scene flow dataset that removes the ego motions of autonomous vehicles. These ego motions are provided in the original Argoverse dataset [8].
+
+Following [47], we define dynamic points as points that the norm between the deformed source (i.e., source point cloud deformed by the ground truth flow) and the ego-motion compensated source (i.e., the source point cloud is transformed by the ego-motion) exceeds a threshold of 0.05. All the metrics are computed only on dynamic points. During optimization, for our method, we removed the ego-motion of the point cloud and estimated the scene flow using the full point cloud, and for NSFP++, we only used dynamic points as required.
+
+In Tab. 3, we show that our method achieves worse performance than  $\mathrm{NSFP}++$  while being  $\sim 50\times$  faster. We further replaced the Chamfer distance loss used in  $\mathrm{NSFP}++$  with our proposed distance transform (DT) loss, and we observed a  $\sim 8\times$  speedup. The distance transform loss can be a robust and efficient surrogate to Chamfer distance loss in many deep geometry vision tasks.
+
+# 4.5. Limitations
+
+One drawback of our method is that creating a DT map using rasterization can lead to discretization errors, especially when the grid size of the DT map is large. To mitigate these, it is necessary to build a DT map with relatively fine-resolution grids. In this case, the memory consumption will increase especially when dealing with high-dimensional data, such as 3D point clouds. Further engineering efforts of pre-creating an efficient high-resolution DT map are required to maintain a reasonable memory cost. However, we empirically find that in the context of scene flow estimation—specifically for scene flow in autonomous driving scenarios—a relatively smooth representation is preferred for local rigidity assumptions. Nevertheless, the tradeoff between the grid resolution, the memory consumption, and the estimation accuracy should be carefully considered and chosen based on specific real-world applications.
+
+Failure cases. We show two typical failure cases in Fig. 8. The first case (left column) is when dynamic points are much sparser than the background, our prediction can result in noisy non-rigid motions in the background. The second case (right column) is when the dynamic motion is relatively small, our model may fail to recognize the dynamic scene and only predict rigid motions.
+
+# 5. Conclusion
+
+In this work, we revisit the runtime optimization-based scene flow method NSFP and propose a method that is both efficient and generalizable to large-scale OOD data and dense lidar points. We identify that common strategies for speeding up network architectures do not yield significant time reductions—the major computation overhead is the Chamfer distance loss. Therefore, we propose to use an efficient correspondence-free distance transform loss as a robust surrogate. The rediscovery of DT in scene flow estimation opens up an innovative venue to leverage its efficiency and robustness for various deep geometry tasks. Compared to NSFP, our method maintains comparable accuracy but gains up to  $\sim 30\times$  speedups. We report for the first time a real-time performance ( $\sim 120\mathrm{ms}$ ) with neural scene flow and runtime optimization when using fewer points (8,192). The efficient runtime optimization-based neural scene flow can be widely applied in lidar scenes to do point cloud densification, open-world object detection, and scene clustering, such as in autonomous driving scenarios, where no ground truth or training data are readily available.
+
+Acknowledgement: We would like to thank Haosen Xing for the careful review of the manuscript and the help throughout the project. We thank Kavisha Vidanapathirana for the initial implementation of NSFP++ [47].
+
+# References
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+        "text": "Fast and Accurate Transferability Measurement by Evaluating Intra-class Feature Variance",
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+        "text": "Huiwen Xu  \nSeoul National University  \nSeoul, Republic of Korea  \nxuhuiwen33@snu.ac.kr",
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+        "type": "text",
+        "text": "U Kang Seoul National University Seoul, Republic of Korea ukang@snu.ac.kr",
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+        "text": "Abstract",
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+        "text": "Given a set of pre-trained models, how can we quickly and accurately find the most useful pre-trained model for a downstream task? Transferability measurement is to quantify how transferable is a pre-trained model learned on a source task to a target task. It is used for quickly ranking pre-trained models for a given task and thus becomes a crucial step for transfer learning. Existing methods measure transferability as the discrimination ability of a source model for a target data before transfer learning, which cannot accurately estimate the fine-tuning performance. Some of them restrict the application of transferability measurement in selecting the best supervised pre-trained models that have classifiers. It is important to have a general method for measuring transferability that can be applied in a variety of situations, such as selecting the best self-supervised pre-trained models that do not have classifiers, and selecting the best transferring layer for a target task.",
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+        "text": "In this work, we propose TMI (TRANSFERABILITY MEASUREMENT WITH INTRA-CLASS FEATURE VARIANCE), a fast and accurate algorithm to measure transferability. We view transferability as the generalization of a pre-trained model on a target task by measuring intra-class feature variance. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. Compared to previous studies that estimate how discriminative the models are, intra-class variance is more accurate than those as it does not require an optimal feature extractor and classifier. Extensive experiments on real-world datasets show that TMI outperforms competitors for selecting the top-5 best models, and exhibits consistently better correlation in 13 out of 17 cases.",
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+        "text": "1. Introduction",
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+        "text": "Transfer learning is an important concept in the field of machine learning, revolutionizing the way models are",
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+        "image_caption": [
+            "Figure 1: Illustration of selecting the best pre-trained model for a given target task."
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+        "text": "trained and applied to various domains. Transfer learning has proven to be particularly valuable when labeled data in the target domain are scarce or costly to obtain. Within the broader scope of transfer learning, several variants have emerged, such as source-free domain adaptation [10, 27, 23, 7], multi-source transfer learning [16, 14, 7, 9], heterogeneous transfer learning [11, 15], and open-set domain adaptation [6, 21].",
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+        "text": "In transfer learning, fine-tuning a pre-trained model becomes a simple and effective way to improve the performance of a downstream task. A crucial step in transfer learning is to quickly and accurately select the most helpful pre-trained model in a set of given pre-trained models. Transferability measurement is to quantify how transferable is a pre-trained model learned on a source task to a target task. As shown in Figure. 1, transferability measurement is used for ranking pre-trained models for a given task, and is a crucial step for transfer learning.",
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+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
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+        "text": "A desired method for transferability measurement should be faster than running a transfer learning algorithm, accurately select the best pre-trained model, and be general to be implemented in various common scenarios. Taskonomy [26] and Task2Vec [1] are computationally expensive due to their learning with data. NCE [18] and LEEP [13] are free of training but cannot be applied to general cases, such as selecting self-supervised pre-trained models and selecting the best transferring layer. LogMe [25] and H-Score [2] also cannot be applied theoretically to select the best transferring layer. Instead of measuring transferability, they measure how discriminative a pre-trained model is to a target task before implementing transfer learning, which cannot estimate fine-tuning performance. TransRate [5] is simple and general to be applied for transferability measurement, but too slow due to the computation of the entropy of whole hidden representations.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we propose TMI (TRANSFERABILITY MEASUREMENT WITH INTRA-CLASS FEATURE VARIANCE), a simple but effective method for transferability measurement. We deem transferability as the generalization of a pre-trained model on a target task. To quantify the degree of generalization, we measure the intra-class variance by the conditional entropy of target representation given a label. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. Compared to previous studies that estimate how discriminative the models are, intra-class variance is more accurate than those as it does not require an optimal feature extractor and classifier. Large intra-class variance implies the feature is efficient for learning tasks. TMI can be used for selecting not only the best supervised model but also the best self-supervised models and the best transferring layer.",
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+        "text": "Our contributions are summarized as follows:",
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+        "list_items": [
+            "- Algorithm. We propose TMI, a fast, accurate, and general method for measuring transferability. We view transferability as generalization ability of a pre-trained model on a target task. The model generalization is measured by intra-class feature variance.",
+            "- Experiments. We conduct extensive experiments on seventeen datasets with fifty supervised pre-trained models and eleven self-supervised pre-trained models. TMI outperforms competitors for selecting the top-5 pre-trained architecture in 14 out of 17 cases and selecting the best source data in all cases. TMI also shows the best trade-off of the running time and transferability, outperforming competitors in 12 out of 17 cases (see Figure. 2).",
+            "- Case study. We evaluate the transferability in more general cases, such as applying to self-supervised pretrained models and selecting the best transferring layer. Compared to competitors, TMI achieves the highest correlation coefficient in 13 out of 17 for selecting the"
+        ],
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+    {
+        "type": "text",
+        "text": "best self-supervised pre-trained models, and in 16 out of 17 datasets for selecting the best transferring layer.",
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+    {
+        "type": "text",
+        "text": "The rest of this paper is organized as follows. We introduce related works for transferability measurement in Section 2, and propose our method TMI in Section 3. After presenting experimental results in Section 4, we conclude in Section 5.",
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+    {
+        "type": "text",
+        "text": "2. Related Works",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We assume a pre-trained model  $\\theta = (w, h)$  in a source domain where  $w$  is a feature extractor, and  $h$  is a classifier (or head). We transfer the model  $\\theta$  to a target domain and add a new classifier  $k$  at the end of the feature extractor  $w$  before fine-tuning. Since the target task is different from the source task in most cases, the pre-trained classifier  $h$  is inappropriate to solve the target task. Thus, we fine-tune the model by minimizing the cross entropy loss on the target dataset  $D = \\{(x_i, y_i)\\}_{i=1}^n$ :",
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+    {
+        "type": "equation",
+        "text": "\n$$\nw ^ {*}, k ^ {*} = \\underset {w, k} {\\arg \\min } \\mathcal {L} (w, k) \\tag {1}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where a feature extractor  $w$  maps an input  $x_{i}$  to a representation  $r_i = w(x_i)$ , and a classifier  $h$  maps the representation  $r_i$  to a probability distribution  $\\hat{y}_i = \\sigma(h(r_i))$  where  $\\sigma(\\cdot)$  is an activation function. Note that  $\\hat{y}_i$  is a vector of length  $n_c$ , where  $n_c$  is the number of classes, and  $\\mathcal{L}(\\cdot)$  is the cross entropy loss, which is equivalent to the average negative log-likelihood:",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} (w, k) = - \\frac {1}{n} \\sum_ {i = 1} ^ {n} \\log P \\left(\\hat {y} _ {i, c} \\mid x _ {i}; w, k\\right) \\tag {2}\n$$\n",
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+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where  $\\hat{y}_{i,c}$  is the probability of  $i$ -th input instance belonging to its ground truth.",
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+    {
+        "type": "text",
+        "text": "The transferability of a source task  $S$  to a target task  $T$  is defined by the log-likelihood [18]:",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nT r f _ {(S \\rightarrow T)} = \\frac {1}{n} \\sum_ {i = 1} ^ {n} \\log P \\left(\\hat {y} _ {i, c} \\mid x _ {i}; w ^ {*}, k ^ {*}\\right) \\tag {3}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Taskonomy [26] and Task2Vec [1] train the source model using target data to obtain the optimal feature extractor  $w^{*}$  and the optimal classifier  $k^{*}$ . These methods are inconvenient to measure the transferability due to their training with data. There have been works focusing on estimating a bound on the log-likelihood to evaluate the transferability without training. NCE [18] proposes negative conditional entropy, which is drawn from the empirical joint distribution over ground truths and predicted labels from a pretrained model. Similarly, LEEP [13] replaces the ground truth with the average log-likelihood to evaluate the transferability. NCE and LEEP are fast, but are restricted to supervised pre-trained models because the algorithms rely",
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+        "type": "page_number",
+        "text": "11475",
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+            "(a) Caltech-101"
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+            "(c) CIFAR-10"
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+            "(e) MNIST"
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+            "(f) FashionMNIST"
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+            "(g) SVHN"
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+        "image_caption": [
+            "(h)FlowerPhotos"
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+            "(j) Chest X-Ray"
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+        "image_caption": [
+            "(1) FGVC-Aircraft"
+        ],
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+            "(m) CUB-200"
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+            "(n) Cars"
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+        "image_caption": [
+            "(0) DTD"
+        ],
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+        "image_caption": [
+            "(p) Food-101",
+            "Figure 2: Total running time (seconds) vs. Kendall's Correlation between transferability scores and test accuracies for selecting the best model among fifty pre-trained models. TMI provides the best trade-off between running time and correlation."
+        ],
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+        "image_caption": [
+            "(q) Oxford-IIIT Pet"
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+    {
+        "type": "image",
+        "img_path": "images/6f9ba93d8c11955ece7591be98cff4c64123f005d356866d04f60ab5c3d85e8e.jpg",
+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "on the outputs of the pre-trained classifier. To apply measurements in more general cases, LogME [25] calculates the logarithm of maximum evidence given extracted features, and H-Score [2] solves an HGR maximum correlation problem. LogMe [25] and H-Score [2] still cannot be applied to select the best transferred layer since they extract features from only the ultimate layers of the pretrained feature extractors. TransRate [5] is the most recent algorithm and applicable to self-supervised models and transferring layer selection. However, it is time-consuming since it requires to calculate mutual information between extracted feature and ground truth. To simplify the problem, [18, 13, 25, 2, 5] assume that the optimal model can be obtained by re-training the classifier while freezing the feature extractor. They estimate the performance of re-training only the classifier, which is even worse than training a target model from scratch [24]. In this paper, we propose a fast, accurate, and general transferability measurement method that predicts fine-tuned performance of a pre-trained model to a new target task.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3. Proposed Method",
+        "text_level": 1,
+        "bbox": [
+            76,
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+    {
+        "type": "text",
+        "text": "We propose TMI to measure the transferability between a pre-trained model and a target task.",
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+    {
+        "type": "text",
+        "text": "3.1. Overview",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Our goal is to measure the transferability in a fast, accurate, and general manner. The following challenges need to be addressed for the goal.",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "C1 Speed. Training a pre-trained model on the target data requires a high computational cost. How can we measure the transferability without training on the target data?",
+            "C2 Accuracy. Existing methods evaluate the transferability that estimate the performance of the optimal target model obtained by re-training only the classifier, not the feature extractor, which is uncommon in transfer learning. How can we accurately measure the transferability for predicting the fine-tuning performance?",
+            "C3 Generality. How can we measure the transferability such that it is applicable to general settings, including selecting the best supervised pre-trained model, the best self-supervised pre-trained models, and the transferring layer?"
+        ],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We address the aforementioned challenges with the following main ideas:",
+        "bbox": [
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+    },
+    {
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+        "list_items": [
+            "I1 We assess the transferability by evaluating the compactness and sparseness of target representations with only a one-way forward propagation.",
+            "12 We relax the assumption proposed in existing methods"
+        ],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "11476",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "that the pre-trained feature extractor can extract all the information related to the target task. We view transferability as the generalization ability of a pre-trained model on a target task. We estimate the generalization ability using intra-class variance, which implies how efficient the feature is for learning tasks.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "13 We measure the transferability using intra-class variance of latent representations, which is applicable to various settings including self-supervised pre-trained models and transferring layer selection.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2.TMI",
+        "text_level": 1,
+        "bbox": [
+            76,
+            250,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a pre-trained source model and a target dataset with labels, our goal is to measure the transferability between the source model and the target task to predict the fine-tuning performance. Existing methods measure the performance of re-trained classifier while freezing the feature extractor, which is time-consuming. To address the issues, our proposed TMI computes the transferability as the intra-class variance of a pre-trained model on a target task. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. It is unnecessary for TMI to obtain optimal classifier which is a time-consuming process.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "TMI The transferability of a pre-trained feature extractor  $\\omega_{s}$  from a source task  $S$  to a target task  $T$ , denoted by  $Trf_{(S\\rightarrow T)}$ , is measured by TMI with the intra-class feature variance obtained from the pre-trained model  $\\omega_{s}$ .",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nT r f _ {(S \\rightarrow T)} := H \\left(\\omega_ {s} (X) | Y\\right) = \\sum_ {c = 1} ^ {C} \\frac {n _ {c}}{n} H \\left(\\omega_ {s} \\left(X _ {c}\\right)\\right) \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            94,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $X$  and  $Y$  represent the target feature and the target label, respectively.  $C$  represents the number of classes and  $X_{c} = \\{x|Y = c\\}$  is the set of target features in the  $c$ -th class.  $n$  and  $n_{c}$  represent the number of total instances and number of instances in the  $c$ -th class, respectively. We measure the intra-class variance of target representations using conditional entropy. The intra-class variance encourages the pre-trained model to have a generalization ability for learning tasks.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "How to utilize TMI? Given a set of pre-trained models and a target dataset with labels, we first obtain target representations of pre-trained models by one-way forward propagation. Next, we compute the transferability score using Equation (4). We select the pre-trained model with the highest transferability score and then fine-tune the model using the target data. We argue that the larger the TMI, the higher the transfer performance.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments",
+        "text_level": 1,
+        "bbox": [
+            76,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We present experimental results to answer the following questions about TMI:",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "Q1 Evaluation of transferability estimation (Section 4.2). Does TMI accurately select the best pretrained model compared to baselines? Is the transferability score measured by TMI highly correlated to the transfer performance?",
+            "Q2 Comparison of ICV measures (Section 4.3). What is the best measurement to evaluate Intra-Class Variance (ICV)?",
+            "Q3 Evaluation of generalization ability (Section 4.4). Does intra-class variance estimate generalization ability of a pre-trained model on a target task?",
+            "Q4 General applications (Section 4.5). Does TMI accurately estimate the transferability in general cases, such as selecting the best self-supervised pre-trained model and the best transferring layer?",
+            "Q5 Hyperparameter sensitivity (Section 4.6). How sensitive is TMI to hyperparameters?"
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1. Experimental Setup",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We present pre-trained models, datasets, baselines, evaluation, estimator, and hyperparameters for our experiments.",
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Pre-trained models. We use supervised model groups offered by Pytorch: ResNet, DenseNet, MnasNet, ShuffleNet V2, MobileNet V2, MobileNet V3, ResNeXt, Wide ResNet, EfficientNet, EfficientNet V2, and RegNet. We also utilize self-supervised model groups, which have only feature extractors without classifiers. We obtain Swin Transformer and Vision Transformer from Pytorch; we get MoCo v1, MoCo v2, Barlow Twins, SwAV, SimSiam, and Dino from Gitlab repositories. Note that we select fifty supervised pretrained models and eleven self-supervised pre-trained models from the aforementioned model groups.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We use seventeen image datasets summarized in Table 1. We randomly split each dataset into a train set and test set by 8:2 ratio, except for datasets that already have been separated.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Baselines. We compare the proposed method with the following baselines. NCE [18] defines the transferability as negative conditional entropy between ground truths and predicted labels from a pre-trained model. LEEP [13] measures negative conditional entropy between ground truths and predicted average log-likelihood. H-Score [2] estimates the transferability by solving an HGR maximum correlation problem. LogME [25] proposes the logarithm of maximum evidence given extracted features. TransRate [5] measures the transferability by the mutual information between target labels and target representations. We omit the comparison with Taskonomy [26] and Task2Vec [1] due to tremendous time costs.",
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+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "11477",
+        "bbox": [
+            480,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/2261568fa582b7a0bdce772b37610d225b81d533ba018d80ce72c42aa6e069e6.jpg",
+        "table_caption": [
+            "Table 1: Summary of datasets."
+        ],
+        "table_footnote": [
+            "<sup>1</sup>https://data.caltech.edu/records/mzrjq-6wc02",
+            "2 https://data.caltech.edu/records/nyy15-4j048",
+            "<sup>3</sup> https://www.cs.toronto.edu/~kriz/cifar.html",
+            "4 http://yann.learun.com/exdb/mnist/",
+            "5 https://github.com/zalandoresearch/fashion-mnist",
+            "$^{6}$  http://ufldl.stanford.edu/housenumbers/",
+            "$^7$  https://www.tensorflow.org/datasets/catalog/tf_flowers",
+            "8 https://github.com/phelber/eurosat",
+            "9 https://data.mendeley.com/datasets/rscbjbr9sj/2",
+            "10 http://ai.bu.edu/visda-2017/#browse",
+            "11 https://www Robots.ox.ac.uk/~vgg/data/fgvc-aircraft/",
+            "12 http://www.vision.caltech.edu/datasets/cub_200_2011/",
+            "13 https://ai.stanford.edu/~jkrause/cars/car_dataset.html",
+            "14 https://www.robots.ox.ac.uk/~vgg/data/dtd/",
+            "15 https://data.vision.ee.ethz.ch/cvl/datasetsextra/food-101/",
+            "16 https://www Robots.ox.ac.uk/~vgg/data/pets/"
+        ],
+        "table_body": "<table><tr><td>Dataset</td><td># of instances (train / test)</td><td># of classes</td><td>Category</td></tr><tr><td>Caltech-1011</td><td>7,315 / 1,829</td><td>101</td><td>Multiple domains</td></tr><tr><td>Caltech-2562</td><td>24,485 / 6,122</td><td>256</td><td>Multiple domains</td></tr><tr><td>CIFAR-103</td><td>50,000 / 10,000</td><td>10</td><td>Multiple domains</td></tr><tr><td>CIFAR-1003</td><td>50,000 / 10,000</td><td>100</td><td>Multiple domains</td></tr><tr><td>MNIST4</td><td>60,000 / 10,000</td><td>10</td><td>Digit</td></tr><tr><td>FashionMNIST5</td><td>60,000 / 10,000</td><td>10</td><td>Fashion</td></tr><tr><td>SVHN6</td><td>73,257 / 26,032</td><td>10</td><td>Digit</td></tr><tr><td>FlowerPhotos7</td><td>2,936 / 734</td><td>5</td><td>Flower</td></tr><tr><td>EuroSAT8</td><td>21,600 / 5,400</td><td>10</td><td>Landscape</td></tr><tr><td>Chest X-Ray9</td><td>5,216 / 623</td><td>2</td><td>Medicine</td></tr><tr><td>VisDA10</td><td>152,397 / 72,372</td><td>212</td><td>Multiple domains</td></tr><tr><td>FGVC-Aircraft11</td><td>3,334 / 3,333</td><td>100</td><td>Aircraft</td></tr><tr><td>CUB-20012</td><td>9,430 / 2,358</td><td>200</td><td>Animal</td></tr><tr><td>Cars13</td><td>8,144 / 8,041</td><td>196</td><td>Auto</td></tr><tr><td>DTD14</td><td>4,512 / 1,128</td><td>47</td><td>Texture</td></tr><tr><td>Food-10115</td><td>80,800 / 20,200</td><td>101</td><td>Food</td></tr><tr><td>Oxford-IIIT Pet16</td><td>3,680 / 3,669</td><td>37</td><td>Animal</td></tr></table>",
+        "bbox": [
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+            116,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation. We evaluate the performance of transferability measurement methods in two ways. First, do they accurately predict the best model? Second, are the transferability scores correlated to fine-tuned accuracy? We adopt Kendall correlation coefficient to measure the correlations between the transferability and fine-tuned accuracy. Kendall correlation coefficient tests the ordinal association between two measured quantities. Kendall correlation coefficient has values between -1 and 1, and correlations between two variables are high when the values are close to 1.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Estimator. We adopt  $k$ -nearest estimator [8] to evaluate the continuous entropy in Equation (4). Compared to kernel density estimators [12, 17],  $k$ -nearest estimator vastly reduces errors by sacrificing time, especially when data are too big. TMI greatly reduces the size of data to be calculated at once because it computes conditional entropy of representation given label.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameters. We use greedy searching algorithm to find the best hyperparameters for fine-tuning. For transferability measurement, the only hyperparameter in our proposed TMI is the number  $n_k$  of neighbors, which is used in the entropy estimator [8]. We discuss the hyperparameter sensitivity to transferability in Section 4.6.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Ratio of target tasks where their top-5 best source models include the model estimated by each transferability measurement method. The best result is in bold. Note that TMI gives the best ratio, accurately selecting the best pretrained models in 14 out of 17 cases.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/23777e577da089b5b969e9cb617481c0299354687082ad4a3465982aa180eff1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Top-5</td><td>3 / 17</td><td>5 / 17</td><td>4 / 17</td><td>8 / 17</td><td>9 / 17</td><td>14 / 17</td></tr></table>",
+        "bbox": [
+            504,
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+            213
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2. Evaluation of Transferability Estimation",
+        "text_level": 1,
+        "bbox": [
+            498,
+            234,
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+            251
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remind that our goal is to quickly and accurately select the best pre-trained model among a pool of well-trained models. A pre-trained model is determined by two components, source data and model architecture. Thus, we conduct experiments to select the best model architecture and the best source data.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Selecting the best model architecture. We evaluate how the transferability measurement methods accurately select the best source model for a target model. Table 2 shows the ratio of targets tasks where their top-5 best source models include the model estimated by each transferability measurement method. Note that TMI gives the best ratio, accurately selecting the best pre-trained models in 14 out of 17 cases, which is much better than the ratio  $9 / 17$  of the second-best model TransRate.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We also compare the correlations between fine-tuned accuracies and transferability scores in Table 3. Note that our proposed TMI shows the best Kendall correlation in 13 out of 17 datasets, outperforming all the competitors. Some baselines even give negative coefficients, which means that their predictions are far from the correct ranking.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Selecting the best source data. We fine-tune eighteen pre-trained models with the same model architecture ResNet-50, but different source datasets, including ImageNet and seventeen datasets introduced in Section 4.1. We predict the best source data and compare correlations with the ground truths. Table 4 shows that our proposed TMI accurately predicts the best source data in all the 17 cases, outperforming all the competitors. Our proposed TMI gives the best Kendall correlation consistently in 12 out of 17 cases, as shown in Table 5.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3. Comparison of ICV Measures",
+        "text_level": 1,
+        "bbox": [
+            498,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Intra-Class Variance (ICV) is a general concept widely used in metric learning. In addition to the conditional entropy introduced in Section 3, we additionally choose four metric learning methods, Contrast [4], Center [20], SNCA [22], and MS [19]. We use the intra-class variance parts of their objective functions to measure the transferability. We apply the five aforementioned methods to measure",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "11478",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/abfbeb347e3afef8e613d4358e47fe7fd03843fb9e5dd4e4e70679428b92b038.jpg",
+        "table_caption": [
+            "Table 3: Correlations between fine-tuned accuracies and transferability scores when transferring supervised pre-trained models trained on ImageNet. The best result is in bold. Our proposed TMI gives the best Kendall correlation in 13 out of 17 cases, outperforming competitors."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"6\">Kendall Correlation Coefficient</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>0.429</td><td>0.425</td><td>0.316</td><td>0.362</td><td>0.474</td><td>0.524</td></tr><tr><td>Caltech-256</td><td>0.686</td><td>0.455</td><td>0.492</td><td>0.503</td><td>0.509</td><td>0.481</td></tr><tr><td>CIFAR-10</td><td>-0.140</td><td>0.324</td><td>0.275</td><td>0.323</td><td>0.364</td><td>0.527</td></tr><tr><td>CIFAR-100</td><td>-0.202</td><td>0.340</td><td>0.351</td><td>0.398</td><td>0.432</td><td>0.497</td></tr><tr><td>MNIST</td><td>0.057</td><td>0.282</td><td>0.083</td><td>0.137</td><td>0.038</td><td>0.137</td></tr><tr><td>FashionMNIST</td><td>-0.328</td><td>0.127</td><td>-0.038</td><td>0.121</td><td>0.125</td><td>0.467</td></tr><tr><td>SVHN</td><td>-0.023</td><td>0.091</td><td>-0.002</td><td>0.092</td><td>0.206</td><td>0.394</td></tr><tr><td>FlowerPhotos</td><td>0.117</td><td>0.139</td><td>0.224</td><td>0.321</td><td>0.392</td><td>0.499</td></tr><tr><td>EuroSAT</td><td>0.269</td><td>0.082</td><td>-0.005</td><td>0.102</td><td>0.051</td><td>0.085</td></tr><tr><td>Chest X-Ray</td><td>0.071</td><td>0.152</td><td>0.171</td><td>0.333</td><td>0.285</td><td>0.395</td></tr><tr><td>VisDA</td><td>0.298</td><td>0.240</td><td>0.329</td><td>0.334</td><td>0.304</td><td>0.341</td></tr><tr><td>FGVC-Aircraft</td><td>0.366</td><td>-0.011</td><td>0.216</td><td>0.498</td><td>0.421</td><td>0.500</td></tr><tr><td>CUB-200</td><td>0.415</td><td>0.098</td><td>0.325</td><td>0.410</td><td>0.519</td><td>0.523</td></tr><tr><td>Cars</td><td>0.270</td><td>0.506</td><td>0.175</td><td>0.432</td><td>0.455</td><td>0.521</td></tr><tr><td>DTD</td><td>-0.074</td><td>0.501</td><td>0.479</td><td>0.366</td><td>0.499</td><td>0.401</td></tr><tr><td>Food-101</td><td>0.261</td><td>0.515</td><td>0.395</td><td>0.421</td><td>0.448</td><td>0.537</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.536</td><td>0.527</td><td>0.418</td><td>0.493</td><td>0.500</td><td>0.576</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/86ad4169e3bd318844a9f37d12d16f95c936bde71f72ca34ac575fcb76e57f83.jpg",
+        "table_caption": [
+            "Table 4: Ratio of target tasks where the transferability measurement methods accurately find the best source data. The best method is in bold. TMI predicts the best source data in all cases, outperforming competitors."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Top-1</td><td>1 / 17</td><td>2 / 17</td><td>3 / 17</td><td>2 / 17</td><td>13 / 17</td><td>17 / 17</td></tr></table>",
+        "bbox": [
+            81,
+            544,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "the correlations between fine-tuned accuracies and transferability scores. Conditional Entropy (CE) shows the best correlations than the other four metric learning measurements in 15 out of 17 datasets, outperforming all the competitors, as shown in Table 6. M. Boudiaf et al. [3] demonstrate that the conditional entropy is a lower bound of intra-class variance of the four metric learning methods.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.4. Evaluation of Generalization Ability",
+        "text_level": 1,
+        "bbox": [
+            76,
+            724,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We perform experiments on two control groups to demonstrate that transferability is relevant to the generalization ability of a pre-trained model on a target task. In Group A, we use a pre-trained ResNet-50 model trained on ImageNet and fine-tune it using the dataset  $D$ . In Group B, we train a ResNet-50 model trained from scratch on dataset  $D$  and fine-tune it on the same dataset. Dataset  $D$  is one of the datasets introduced in Section 4.1.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 7 shows the difference of fine-tuning performance and that of transferability between two groups. “+” rep",
+        "bbox": [
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+            869,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "resents the difference is larger than 0 while “-” represents that is less than 0. Group A has higher transfer accuracies than Group B in most cases since models in Group A have learned a large amount of information from the ImageNet, which can potentially benefit downstream tasks.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our proposed TMI and TransRate have the same symbols for transfer accuracy in 14 out of 17 cases, while the other four baselines do not provide similar symbols. TransRate also estimates well, but it is computationally expensive since it measures the mutual information between target representation and label. In contrast, TMI is more efficient as it considers only intra-class features.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.5. General Applications",
+        "text_level": 1,
+        "bbox": [
+            500,
+            676,
+            699,
+            694
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Applying to self-supervised pre-trained models. TMI not only measures the transferability between supervised pre-trained models and classification tasks but also can be extended to self-supervised pre-trained models. We prepare eleven self-supervised pre-trained models already trained on ImageNet provided by Github repositories introduced in Section 4.1. NCE [18] and LEEP [13] cannot measure the transferability between a self-supervised pre-trained model and a downstream task because they dig out the relationship between target ground truth and predicted label or probability. Thus, we compare the proposed TMI to LogME [25], H-Score [2], and TransRate [5] in Table 8; TMI outperforms competitors in 13 out of 17 classification datasets.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "11479",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/f9ccf95c3210a5588e9edac8509ea68cf8be26aa939154cc7887356e28551ee6.jpg",
+        "table_caption": [
+            "Table 5: Correlations between fine-tuned accuracies and transferability scores when transferring ResNet-50 models trained on different source data. The best result is in bold. TMI shows the best Kendall correlation in 12 out of 17 cases, outperforming all the competitors."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"6\">Kendall Correlation Coefficient</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>-0.294</td><td>0.542</td><td>0.353</td><td>0.647</td><td>0.425</td><td>0.699</td></tr><tr><td>Caltech-256</td><td>-0.268</td><td>0.542</td><td>0.437</td><td>0.660</td><td>0.477</td><td>0.699</td></tr><tr><td>CIFAR-10</td><td>-0.216</td><td>0.373</td><td>0.383</td><td>0.490</td><td>0.399</td><td>0.630</td></tr><tr><td>CIFAR-100</td><td>-0.098</td><td>0.556</td><td>0.686</td><td>0.345</td><td>0.542</td><td>0.699</td></tr><tr><td>MNIST</td><td>-0.061</td><td>0.439</td><td>0.372</td><td>-0.007</td><td>0.115</td><td>0.162</td></tr><tr><td>FashionMNIST</td><td>-0.086</td><td>0.362</td><td>0.430</td><td>0.475</td><td>0.046</td><td>0.150</td></tr><tr><td>SVHN</td><td>0.039</td><td>0.285</td><td>-0.033</td><td>0.393</td><td>0.170</td><td>0.407</td></tr><tr><td>FlowerPhotos</td><td>-0.359</td><td>0.412</td><td>0.456</td><td>0.307</td><td>0.490</td><td>0.516</td></tr><tr><td>EuroSAT</td><td>-0.388</td><td>0.428</td><td>0.490</td><td>0.322</td><td>0.467</td><td>0.704</td></tr><tr><td>Chest X-Ray</td><td>-0.072</td><td>0.164</td><td>0.359</td><td>-0.111</td><td>0.425</td><td>-0.123</td></tr><tr><td>VisDA</td><td>-0.538</td><td>0.498</td><td>0.094</td><td>-0.039</td><td>0.498</td><td>0.546</td></tr><tr><td>FGVC-Aircraft</td><td>-0.033</td><td>0.373</td><td>0.339</td><td>0.556</td><td>0.399</td><td>0.688</td></tr><tr><td>CUB-200</td><td>-0.503</td><td>0.582</td><td>0.455</td><td>0.464</td><td>0.647</td><td>0.723</td></tr><tr><td>Cars</td><td>-0.294</td><td>0.529</td><td>0.403</td><td>0.752</td><td>0.542</td><td>0.804</td></tr><tr><td>DTD</td><td>-0.438</td><td>0.595</td><td>0.393</td><td>0.647</td><td>0.542</td><td>0.757</td></tr><tr><td>Food-101</td><td>-0.020</td><td>0.281</td><td>0.092</td><td>0.046</td><td>0.229</td><td>0.225</td></tr><tr><td>Oxford-IIIT Pet</td><td>-0.268</td><td>0.569</td><td>0.660</td><td>0.503</td><td>0.516</td><td>0.248</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/4bf5cacc516bd8b3e4945abbf7645da4b39378740f2ba26d29e1d4500de17c44.jpg",
+        "table_caption": [
+            "Table 6: Comparison of intra-class variance measurements. Conditional Entropy (CE) shows the best correlations consistently in 15 out of 17 cases. The best result is in bold."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"5\">Kendall Correlation Coefficient</td></tr><tr><td>Contrast</td><td>Center</td><td>SNCA</td><td>MS</td><td>CE</td></tr><tr><td>Caltech-101</td><td>0.274</td><td>0.283</td><td>0.380</td><td>0.358</td><td>0.442</td></tr><tr><td>Caltech-256</td><td>0.295</td><td>0.347</td><td>0.261</td><td>0.347</td><td>0.354</td></tr><tr><td>CIFAR-10</td><td>0.328</td><td>0.322</td><td>0.275</td><td>0.299</td><td>0.369</td></tr><tr><td>CIFAR-100</td><td>0.374</td><td>0.282</td><td>0.207</td><td>0.200</td><td>0.376</td></tr><tr><td>MNIST</td><td>0.383</td><td>0.267</td><td>0.306</td><td>0.228</td><td>0.374</td></tr><tr><td>FashionMNIST</td><td>0.332</td><td>0.225</td><td>0.260</td><td>0.337</td><td>0.374</td></tr><tr><td>SVHN</td><td>0.303</td><td>0.359</td><td>0.369</td><td>0.360</td><td>0.383</td></tr><tr><td>FlowerPhotos</td><td>0.209</td><td>0.263</td><td>0.291</td><td>0.277</td><td>0.372</td></tr><tr><td>EuroSAT</td><td>0.292</td><td>0.237</td><td>0.367</td><td>0.268</td><td>0.400</td></tr><tr><td>Chest X-Ray</td><td>0.292</td><td>0.289</td><td>0.210</td><td>0.232</td><td>0.357</td></tr><tr><td>VisDA</td><td>0.259</td><td>0.352</td><td>0.200</td><td>0.329</td><td>0.352</td></tr><tr><td>FGVC-Aircraft</td><td>0.220</td><td>0.322</td><td>0.244</td><td>0.315</td><td>0.351</td></tr><tr><td>CUB-200</td><td>0.368</td><td>0.252</td><td>0.379</td><td>0.385</td><td>0.431</td></tr><tr><td>Cars</td><td>0.351</td><td>0.285</td><td>0.398</td><td>0.268</td><td>0.248</td></tr><tr><td>DTD</td><td>0.236</td><td>0.322</td><td>0.228</td><td>0.206</td><td>0.327</td></tr><tr><td>Food-101</td><td>0.322</td><td>0.267</td><td>0.246</td><td>0.269</td><td>0.484</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.301</td><td>0.202</td><td>0.363</td><td>0.274</td><td>0.449</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Selecting the best transferring layer. Given a ResNet-18 trained on ImageNet, we can easily determine the best transferring layer, which produces the best fine-tuning performance for a target task by measuring transferability. We compare the correlations of LogME [25], H-Score [2], and TransRate to select the best transferring layer for fine",
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+        "img_path": "images/fcecd8c634032bc5b009ab8ae35f6b731f2fc16ed4bee9bf3b2fde1570844d74.jpg",
+        "table_caption": [
+            "Table 7: Comparison of the consistency of fine-tuning accuracy and transferability. “+” represents the difference is larger than 0 while “-” represents the difference is less than 0. TMI has consistent signs with fine-tuning accuracy in 14 out of 17 cases, outperforming its competitors."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td rowspan=\"2\">Accuracy (A-B)</td><td colspan=\"6\">Transferability (A-B)</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogME</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Caltech-256</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CIFAR-10</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CIFAR-100</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>MNIST</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>FashionMNIST</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>SVHN</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>FlowerPhotos</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>EuroSAT</td><td>+</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>Chest X-Ray</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>VisDA</td><td>+</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>FGVC-Aircraft</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CUB-200</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Cars</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>DTD</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Food-101</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Oxford-IIIT Pet</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "tuning. Theoretically, LogME [25] and H-Score [2] cannot deal with layer selection because they rely on features from the ultimate layer of the feature extractor. However, we extract features from each layer and directly put them into LogME and H-Score algorithms to calculate the trans",
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+    {
+        "type": "page_number",
+        "text": "11480",
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+    },
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+        "type": "table",
+        "img_path": "images/be0ce93f694e5c1485661c53e9de06f01740fa39c6cd56abb89f8ec397054dc2.jpg",
+        "table_caption": [
+            "Table 8: Correlations between fine-tuned accuracies and transferability scores when transferring self-supervised pretrained models to classification tasks. The best result is in bold. TMI shows the best Kendall correlation consistently in 13 out of 17 cases, outperforming all the competitors."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"4\">Kendall Correlation Coefficient</td></tr><tr><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>-0.278</td><td>0.183</td><td>0.000</td><td>0.367</td></tr><tr><td>Caltech-256</td><td>-0.561</td><td>0.345</td><td>0.164</td><td>0.600</td></tr><tr><td>CIFAR-10</td><td>-0.337</td><td>0.018</td><td>0.164</td><td>0.624</td></tr><tr><td>CIFAR-100</td><td>-0.561</td><td>0.673</td><td>0.236</td><td>0.697</td></tr><tr><td>MNIST</td><td>-0.574</td><td>-0.262</td><td>-0.426</td><td>0.246</td></tr><tr><td>FashionMNIST</td><td>-0.330</td><td>-0.220</td><td>-0.330</td><td>0.094</td></tr><tr><td>SVHN</td><td>0.055</td><td>0.309</td><td>-0.345</td><td>0.075</td></tr><tr><td>FlowerPhotos</td><td>0.389</td><td>-0.241</td><td>0.130</td><td>0.075</td></tr><tr><td>EuroSAT</td><td>-0.073</td><td>-0.037</td><td>0.037</td><td>-0.208</td></tr><tr><td>Chest X-Ray</td><td>-0.037</td><td>-0.037</td><td>-0.257</td><td>0.175</td></tr><tr><td>VisDA</td><td>0.550</td><td>0.055</td><td>0.309</td><td>-0.597</td></tr><tr><td>FGVC-Aircraft</td><td>-0.345</td><td>0.091</td><td>-0.309</td><td>0.150</td></tr><tr><td>CUB-200</td><td>-0.112</td><td>-0.018</td><td>0.091</td><td>0.564</td></tr><tr><td>Cars</td><td>0.000</td><td>0.382</td><td>-0.200</td><td>0.262</td></tr><tr><td>DTD</td><td>0.382</td><td>0.018</td><td>0.224</td><td>0.455</td></tr><tr><td>Food-101</td><td>-0.411</td><td>-0.018</td><td>-0.018</td><td>0.204</td></tr><tr><td>Oxford-IIIT Pet</td><td>-0.585</td><td>0.294</td><td>0.073</td><td>0.807</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c98a0030dc307929921556676a29d661d875515f1787899a5231af92281004ef.jpg",
+        "image_caption": [
+            "Figure 3: Sensitivity to the hyperparameter  $n_k$  in entropy estimator."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ferability. The size of extracted features is quite large in the early layers, resulting in much higher computational cost for transferability measurement. To reduce this cost, we add an avgpool2d layer to the end of transferring layer to reduce the dimension of extracted features. Table 9 shows that TMI achieves the highest correlation coefficients in 16 out of 17 datasets. Correlation coefficients of MNIST, SVHN, and Chest X-Ray are much lower than those of the other datasets because their ranges of minimum and maximum accuracies obtained by layer transfer are smaller than  $0.5\\%$ , making it challenging to rank these dataset accurately.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.6. Hyperparameter Sensitivity",
+        "text_level": 1,
+        "bbox": [
+            76,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "An advantage of TMI is that it does not train models on target datasets because it is time-consuming and sensitive to hyperparameters which may cause different results. The only hyperparameter in TMI is  $n_k$  used in the entropy estimator. In Figure. 3, the correlation between pre-trained",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/132aac5f08356d6efc99d7f4c6332f736ce396e9c4a10bb4c143a652bf9b80be.jpg",
+        "table_caption": [
+            "Table 9: Correlations between fine-tuned accuracies and transferability scores when transferring specific layers of pre-trained models. The best result is in bold. TMI shows the best correlations consistently in 16 out of 17 cases."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"4\">Kendall Correlation Coefficient</td></tr><tr><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>0.346</td><td>0.935</td><td>0.752</td><td>0.941</td></tr><tr><td>Caltech-256</td><td>0.973</td><td>0.317</td><td>0.800</td><td>0.984</td></tr><tr><td>CIFAR-10</td><td>0.360</td><td>0.743</td><td>0.704</td><td>0.746</td></tr><tr><td>CIFAR-100</td><td>0.460</td><td>0.774</td><td>0.774</td><td>0.801</td></tr><tr><td>MNIST</td><td>0.180</td><td>0.055</td><td>0.179</td><td>0.184</td></tr><tr><td>FashionMNIST</td><td>0.294</td><td>0.241</td><td>0.298</td><td>0.308</td></tr><tr><td>SVHN</td><td>0.087</td><td>0.106</td><td>0.040</td><td>0.113</td></tr><tr><td>FlowerPhotos</td><td>0.779</td><td>0.198</td><td>0.687</td><td>0.792</td></tr><tr><td>EuroSAT</td><td>0.691</td><td>0.709</td><td>0.495</td><td>0.735</td></tr><tr><td>Chest X-Ray</td><td>0.113</td><td>0.154</td><td>0.100</td><td>0.054</td></tr><tr><td>VisDA</td><td>0.665</td><td>0.866</td><td>0.826</td><td>0.904</td></tr><tr><td>FGVC-Aircraft</td><td>0.791</td><td>0.302</td><td>0.595</td><td>0.817</td></tr><tr><td>CUB-200</td><td>0.317</td><td>0.970</td><td>0.787</td><td>0.954</td></tr><tr><td>Cars</td><td>0.330</td><td>0.869</td><td>0.752</td><td>0.882</td></tr><tr><td>DTD</td><td>0.924</td><td>0.401</td><td>0.761</td><td>0.931</td></tr><tr><td>Food-101</td><td>0.409</td><td>0.948</td><td>0.869</td><td>0.970</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.349</td><td>0.970</td><td>0.826</td><td>0.990</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "models and TMI shows consistency in most ranges for  $n_k$ .",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5. Conclusion",
+        "text_level": 1,
+        "bbox": [
+            500,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We propose TMI, a fast, accurate, and general algorithm to measure the transferability. TMI quantifies the degree of generalization of a pre-trained model to a new task by estimating the intra-class feature variance. We conduct experiments on seventeen datasets with fifty supervised pretrained models and eleven self-supervised pre-trained models. TMI selects the top-5 pre-trained architecture in 14 out of 17 cases and selects the best source data in all cases, outperforming its competitors. We measure the correlation between the fine-tuning performance and transferability, and our proposed TMI outperforms competitors in 13 out of 17 cases.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments",
+        "text_level": 1,
+        "bbox": [
+            500,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This work was supported by Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) [No.2020-0-00894, Flexible and Efficient Model Compression Method for Various Applications and Environments], [No.2021-0-01343, Artificial Intelligence Graduate School Program (Seoul National University)], and [NO.2021-0-02068, Artificial Intelligence Innovation Hub (Artificial Intelligence Institute, Seoul National University)]. The Institute of Engineering Research at Seoul National University provided research facilities for this work. The ICT at Seoul National University provides research facilities for this study. U Kang is the corresponding author.",
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+    {
+        "type": "page_number",
+        "text": "11481",
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+    },
+    {
+        "type": "text",
+        "text": "References",
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2023/Fast and Accurate Transferability Measurement by Evaluating Intra-class Feature Variance/16d96c9b-c625-4fb0-aaab-b679d45a05de_model.json

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+        {
+            "type": "header",
+            "bbox": [
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+            "angle": 0,
+            "content": "CVF"
+        },
+        {
+            "type": "header",
+            "bbox": [
+                0.239,
+                0.001,
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+            ],
+            "angle": 0,
+            "content": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.112,
+                0.13,
+                0.861,
+                0.175
+            ],
+            "angle": 0,
+            "content": "Fast and Accurate Transferability Measurement by Evaluating Intra-class Feature Variance"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.204,
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+            "angle": 0,
+            "content": "Huiwen Xu  \nSeoul National University  \nSeoul, Republic of Korea  \nxuhuiwen33@snu.ac.kr"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.516,
+                0.204,
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+            ],
+            "angle": 0,
+            "content": "U Kang Seoul National University Seoul, Republic of Korea ukang@snu.ac.kr"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.235,
+                0.309,
+                0.314,
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+            "angle": 0,
+            "content": "Abstract"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.342,
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+            ],
+            "angle": 0,
+            "content": "Given a set of pre-trained models, how can we quickly and accurately find the most useful pre-trained model for a downstream task? Transferability measurement is to quantify how transferable is a pre-trained model learned on a source task to a target task. It is used for quickly ranking pre-trained models for a given task and thus becomes a crucial step for transfer learning. Existing methods measure transferability as the discrimination ability of a source model for a target data before transfer learning, which cannot accurately estimate the fine-tuning performance. Some of them restrict the application of transferability measurement in selecting the best supervised pre-trained models that have classifiers. It is important to have a general method for measuring transferability that can be applied in a variety of situations, such as selecting the best self-supervised pre-trained models that do not have classifiers, and selecting the best transferring layer for a target task."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.601,
+                0.473,
+                0.813
+            ],
+            "angle": 0,
+            "content": "In this work, we propose TMI (TRANSFERABILITY MEASUREMENT WITH INTRA-CLASS FEATURE VARIANCE), a fast and accurate algorithm to measure transferability. We view transferability as the generalization of a pre-trained model on a target task by measuring intra-class feature variance. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. Compared to previous studies that estimate how discriminative the models are, intra-class variance is more accurate than those as it does not require an optimal feature extractor and classifier. Extensive experiments on real-world datasets show that TMI outperforms competitors for selecting the top-5 best models, and exhibits consistently better correlation in 13 out of 17 cases."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.078,
+                0.844,
+                0.21,
+                0.86
+            ],
+            "angle": 0,
+            "content": "1. Introduction"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.871,
+                0.472,
+                0.903
+            ],
+            "angle": 0,
+            "content": "Transfer learning is an important concept in the field of machine learning, revolutionizing the way models are"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.508,
+                0.322,
+                0.891,
+                0.523
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.499,
+                0.538,
+                0.894,
+                0.571
+            ],
+            "angle": 0,
+            "content": "Figure 1: Illustration of selecting the best pre-trained model for a given target task."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.626,
+                0.895,
+                0.747
+            ],
+            "angle": 0,
+            "content": "trained and applied to various domains. Transfer learning has proven to be particularly valuable when labeled data in the target domain are scarce or costly to obtain. Within the broader scope of transfer learning, several variants have emerged, such as source-free domain adaptation [10, 27, 23, 7], multi-source transfer learning [16, 14, 7, 9], heterogeneous transfer learning [11, 15], and open-set domain adaptation [6, 21]."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.75,
+                0.896,
+                0.903
+            ],
+            "angle": 0,
+            "content": "In transfer learning, fine-tuning a pre-trained model becomes a simple and effective way to improve the performance of a downstream task. A crucial step in transfer learning is to quickly and accurately select the most helpful pre-trained model in a set of given pre-trained models. Transferability measurement is to quantify how transferable is a pre-trained model learned on a source task to a target task. As shown in Figure. 1, transferability measurement is used for ranking pre-trained models for a given task, and is a crucial step for transfer learning."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.481,
+                0.945,
+                0.52,
+                0.957
+            ],
+            "angle": 0,
+            "content": "11474"
+        }
+    ],
+    [
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.092,
+                0.473,
+                0.348
+            ],
+            "angle": 0,
+            "content": "A desired method for transferability measurement should be faster than running a transfer learning algorithm, accurately select the best pre-trained model, and be general to be implemented in various common scenarios. Taskonomy [26] and Task2Vec [1] are computationally expensive due to their learning with data. NCE [18] and LEEP [13] are free of training but cannot be applied to general cases, such as selecting self-supervised pre-trained models and selecting the best transferring layer. LogMe [25] and H-Score [2] also cannot be applied theoretically to select the best transferring layer. Instead of measuring transferability, they measure how discriminative a pre-trained model is to a target task before implementing transfer learning, which cannot estimate fine-tuning performance. TransRate [5] is simple and general to be applied for transferability measurement, but too slow due to the computation of the entropy of whole hidden representations."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.348,
+                0.473,
+                0.59
+            ],
+            "angle": 0,
+            "content": "In this paper, we propose TMI (TRANSFERABILITY MEASUREMENT WITH INTRA-CLASS FEATURE VARIANCE), a simple but effective method for transferability measurement. We deem transferability as the generalization of a pre-trained model on a target task. To quantify the degree of generalization, we measure the intra-class variance by the conditional entropy of target representation given a label. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. Compared to previous studies that estimate how discriminative the models are, intra-class variance is more accurate than those as it does not require an optimal feature extractor and classifier. Large intra-class variance implies the feature is efficient for learning tasks. TMI can be used for selecting not only the best supervised model but also the best self-supervised models and the best transferring layer."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.097,
+                0.59,
+                0.4,
+                0.605
+            ],
+            "angle": 0,
+            "content": "Our contributions are summarized as follows:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.097,
+                0.614,
+                0.47,
+                0.689
+            ],
+            "angle": 0,
+            "content": "- Algorithm. We propose TMI, a fast, accurate, and general method for measuring transferability. We view transferability as generalization ability of a pre-trained model on a target task. The model generalization is measured by intra-class feature variance."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.096,
+                0.69,
+                0.47,
+                0.825
+            ],
+            "angle": 0,
+            "content": "- Experiments. We conduct extensive experiments on seventeen datasets with fifty supervised pre-trained models and eleven self-supervised pre-trained models. TMI outperforms competitors for selecting the top-5 pre-trained architecture in 14 out of 17 cases and selecting the best source data in all cases. TMI also shows the best trade-off of the running time and transferability, outperforming competitors in 12 out of 17 cases (see Figure. 2)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.096,
+                0.826,
+                0.47,
+                0.902
+            ],
+            "angle": 0,
+            "content": "- Case study. We evaluate the transferability in more general cases, such as applying to self-supervised pretrained models and selecting the best transferring layer. Compared to competitors, TMI achieves the highest correlation coefficient in 13 out of 17 for selecting the"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.096,
+                0.614,
+                0.47,
+                0.902
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.531,
+                0.092,
+                0.892,
+                0.122
+            ],
+            "angle": 0,
+            "content": "best self-supervised pre-trained models, and in 16 out of 17 datasets for selecting the best transferring layer."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.133,
+                0.894,
+                0.208
+            ],
+            "angle": 0,
+            "content": "The rest of this paper is organized as follows. We introduce related works for transferability measurement in Section 2, and propose our method TMI in Section 3. After presenting experimental results in Section 4, we conclude in Section 5."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.5,
+                0.222,
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+            ],
+            "angle": 0,
+            "content": "2. Related Works"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.247,
+                0.895,
+                0.385
+            ],
+            "angle": 0,
+            "content": "We assume a pre-trained model \\(\\theta = (w, h)\\) in a source domain where \\(w\\) is a feature extractor, and \\(h\\) is a classifier (or head). We transfer the model \\(\\theta\\) to a target domain and add a new classifier \\(k\\) at the end of the feature extractor \\(w\\) before fine-tuning. Since the target task is different from the source task in most cases, the pre-trained classifier \\(h\\) is inappropriate to solve the target task. Thus, we fine-tune the model by minimizing the cross entropy loss on the target dataset \\(D = \\{(x_i, y_i)\\}_{i=1}^n\\):"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.608,
+                0.395,
+                0.892,
+                0.421
+            ],
+            "angle": 0,
+            "content": "\\[\nw ^ {*}, k ^ {*} = \\underset {w, k} {\\arg \\min } \\mathcal {L} (w, k) \\tag {1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.426,
+                0.894,
+                0.532
+            ],
+            "angle": 0,
+            "content": "where a feature extractor \\( w \\) maps an input \\( x_{i} \\) to a representation \\( r_i = w(x_i) \\), and a classifier \\( h \\) maps the representation \\( r_i \\) to a probability distribution \\( \\hat{y}_i = \\sigma(h(r_i)) \\) where \\( \\sigma(\\cdot) \\) is an activation function. Note that \\( \\hat{y}_i \\) is a vector of length \\( n_c \\), where \\( n_c \\) is the number of classes, and \\( \\mathcal{L}(\\cdot) \\) is the cross entropy loss, which is equivalent to the average negative log-likelihood:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.565,
+                0.542,
+                0.892,
+                0.582
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {L} (w, k) = - \\frac {1}{n} \\sum_ {i = 1} ^ {n} \\log P \\left(\\hat {y} _ {i, c} \\mid x _ {i}; w, k\\right) \\tag {2}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.587,
+                0.892,
+                0.616
+            ],
+            "angle": 0,
+            "content": "where \\(\\hat{y}_{i,c}\\) is the probability of \\(i\\)-th input instance belonging to its ground truth."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.616,
+                0.892,
+                0.647
+            ],
+            "angle": 0,
+            "content": "The transferability of a source task \\( S \\) to a target task \\( T \\) is defined by the log-likelihood [18]:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.555,
+                0.657,
+                0.892,
+                0.696
+            ],
+            "angle": 0,
+            "content": "\\[\nT r f _ {(S \\rightarrow T)} = \\frac {1}{n} \\sum_ {i = 1} ^ {n} \\log P \\left(\\hat {y} _ {i, c} \\mid x _ {i}; w ^ {*}, k ^ {*}\\right) \\tag {3}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.704,
+                0.895,
+                0.903
+            ],
+            "angle": 0,
+            "content": "Taskonomy [26] and Task2Vec [1] train the source model using target data to obtain the optimal feature extractor \\( w^{*} \\) and the optimal classifier \\( k^{*} \\). These methods are inconvenient to measure the transferability due to their training with data. There have been works focusing on estimating a bound on the log-likelihood to evaluate the transferability without training. NCE [18] proposes negative conditional entropy, which is drawn from the empirical joint distribution over ground truths and predicted labels from a pretrained model. Similarly, LEEP [13] replaces the ground truth with the average log-likelihood to evaluate the transferability. NCE and LEEP are fast, but are restricted to supervised pre-trained models because the algorithms rely"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.48,
+                0.945,
+                0.52,
+                0.958
+            ],
+            "angle": 0,
+            "content": "11475"
+        }
+    ],
+    [
+        {
+            "type": "image",
+            "bbox": [
+                0.1,
+                0.091,
+                0.222,
+                0.188
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.113,
+                0.19,
+                0.207,
+                0.202
+            ],
+            "angle": 0,
+            "content": "(a) Caltech-101"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.226,
+                0.092,
+                0.351,
+                0.188
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.24,
+                0.19,
+                0.335,
+                0.202
+            ],
+            "angle": 0,
+            "content": "(b) Caltech-256"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.353,
+                0.092,
+                0.475,
+                0.188
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.372,
+                0.19,
+                0.456,
+                0.202
+            ],
+            "angle": 0,
+            "content": "(c) CIFAR-10"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.483,
+                0.092,
+                0.607,
+                0.188
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
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+                0.19,
+                0.591,
+                0.202
+            ],
+            "angle": 0,
+            "content": "(d) CIFAR-100"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.616,
+                0.092,
+                0.738,
+                0.188
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.643,
+                0.19,
+                0.71,
+                0.202
+            ],
+            "angle": 0,
+            "content": "(e) MNIST"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.747,
+                0.092,
+                0.868,
+                0.188
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.751,
+                0.19,
+                0.863,
+                0.202
+            ],
+            "angle": 0,
+            "content": "(f) FashionMNIST"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.099,
+                0.209,
+                0.222,
+                0.302
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.13,
+                0.305,
+                0.192,
+                0.317
+            ],
+            "angle": 0,
+            "content": "(g) SVHN"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.227,
+                0.209,
+                0.35,
+                0.303
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.238,
+                0.305,
+                0.337,
+                0.316
+            ],
+            "angle": 0,
+            "content": "(h)FlowerPhotos"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.353,
+                0.208,
+                0.475,
+                0.303
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.377,
+                0.305,
+                0.45,
+                0.317
+            ],
+            "angle": 0,
+            "content": "(i) EuroSAT"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.48,
+                0.207,
+                0.61,
+                0.303
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.497,
+                0.305,
+                0.592,
+                0.318
+            ],
+            "angle": 0,
+            "content": "(j) Chest X-Ray"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.617,
+                0.208,
+                0.738,
+                0.303
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.646,
+                0.305,
+                0.707,
+                0.317
+            ],
+            "angle": 0,
+            "content": "(k) VisDA"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.746,
+                0.209,
+                0.868,
+                0.303
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.752,
+                0.305,
+                0.862,
+                0.317
+            ],
+            "angle": 0,
+            "content": "(1) FGVC-Aircraft"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.099,
+                0.322,
+                0.222,
+                0.417
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.119,
+                0.418,
+                0.203,
+                0.43
+            ],
+            "angle": 0,
+            "content": "(m) CUB-200"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.227,
+                0.322,
+                0.35,
+                0.417
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.263,
+                0.418,
+                0.313,
+                0.43
+            ],
+            "angle": 0,
+            "content": "(n) Cars"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.353,
+                0.322,
+                0.475,
+                0.417
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.387,
+                0.418,
+                0.44,
+                0.43
+            ],
+            "angle": 0,
+            "content": "(0) DTD"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.483,
+                0.322,
+                0.606,
+                0.417
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.505,
+                0.418,
+                0.583,
+                0.43
+            ],
+            "angle": 0,
+            "content": "(p) Food-101"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.616,
+                0.322,
+                0.739,
+                0.417
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.618,
+                0.418,
+                0.734,
+                0.43
+            ],
+            "angle": 0,
+            "content": "(q) Oxford-IIIT Pet"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.749,
+                0.33,
+                0.871,
+                0.417
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.076,
+                0.444,
+                0.892,
+                0.475
+            ],
+            "angle": 0,
+            "content": "Figure 2: Total running time (seconds) vs. Kendall's Correlation between transferability scores and test accuracies for selecting the best model among fifty pre-trained models. TMI provides the best trade-off between running time and correlation."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.501,
+                0.473,
+                0.804
+            ],
+            "angle": 0,
+            "content": "on the outputs of the pre-trained classifier. To apply measurements in more general cases, LogME [25] calculates the logarithm of maximum evidence given extracted features, and H-Score [2] solves an HGR maximum correlation problem. LogMe [25] and H-Score [2] still cannot be applied to select the best transferred layer since they extract features from only the ultimate layers of the pretrained feature extractors. TransRate [5] is the most recent algorithm and applicable to self-supervised models and transferring layer selection. However, it is time-consuming since it requires to calculate mutual information between extracted feature and ground truth. To simplify the problem, [18, 13, 25, 2, 5] assume that the optimal model can be obtained by re-training the classifier while freezing the feature extractor. They estimate the performance of re-training only the classifier, which is even worse than training a target model from scratch [24]. In this paper, we propose a fast, accurate, and general transferability measurement method that predicts fine-tuned performance of a pre-trained model to a new target task."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.078,
+                0.838,
+                0.25,
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+            ],
+            "angle": 0,
+            "content": "3. Proposed Method"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.077,
+                0.871,
+                0.469,
+                0.902
+            ],
+            "angle": 0,
+            "content": "We propose TMI to measure the transferability between a pre-trained model and a target task."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.5,
+                0.5,
+                0.61,
+                0.515
+            ],
+            "angle": 0,
+            "content": "3.1. Overview"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.527,
+                0.892,
+                0.573
+            ],
+            "angle": 0,
+            "content": "Our goal is to measure the transferability in a fast, accurate, and general manner. The following challenges need to be addressed for the goal."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.577,
+                0.892,
+                0.635
+            ],
+            "angle": 0,
+            "content": "C1 Speed. Training a pre-trained model on the target data requires a high computational cost. How can we measure the transferability without training on the target data?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.637,
+                0.892,
+                0.728
+            ],
+            "angle": 0,
+            "content": "C2 Accuracy. Existing methods evaluate the transferability that estimate the performance of the optimal target model obtained by re-training only the classifier, not the feature extractor, which is uncommon in transfer learning. How can we accurately measure the transferability for predicting the fine-tuning performance?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.728,
+                0.892,
+                0.804
+            ],
+            "angle": 0,
+            "content": "C3 Generality. How can we measure the transferability such that it is applicable to general settings, including selecting the best supervised pre-trained model, the best self-supervised pre-trained models, and the transferring layer?"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.505,
+                0.577,
+                0.892,
+                0.804
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.5,
+                0.807,
+                0.891,
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+            "angle": 0,
+            "content": "We address the aforementioned challenges with the following main ideas:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.51,
+                0.84,
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+            "angle": 0,
+            "content": "I1 We assess the transferability by evaluating the compactness and sparseness of target representations with only a one-way forward propagation."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.51,
+                0.886,
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+            ],
+            "angle": 0,
+            "content": "12 We relax the assumption proposed in existing methods"
+        },
+        {
+            "type": "list",
+            "bbox": [
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+                0.84,
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+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.481,
+                0.945,
+                0.519,
+                0.957
+            ],
+            "angle": 0,
+            "content": "11476"
+        }
+    ],
+    [
+        {
+            "type": "text",
+            "bbox": [
+                0.108,
+                0.092,
+                0.468,
+                0.182
+            ],
+            "angle": 0,
+            "content": "that the pre-trained feature extractor can extract all the information related to the target task. We view transferability as the generalization ability of a pre-trained model on a target task. We estimate the generalization ability using intra-class variance, which implies how efficient the feature is for learning tasks."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.088,
+                0.183,
+                0.468,
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+            ],
+            "angle": 0,
+            "content": "13 We measure the transferability using intra-class variance of latent representations, which is applicable to various settings including self-supervised pre-trained models and transferring layer selection."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.078,
+                0.25,
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+            ],
+            "angle": 0,
+            "content": "3.2.TMI"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.077,
+                0.273,
+                0.468,
+                0.454
+            ],
+            "angle": 0,
+            "content": "Given a pre-trained source model and a target dataset with labels, our goal is to measure the transferability between the source model and the target task to predict the fine-tuning performance. Existing methods measure the performance of re-trained classifier while freezing the feature extractor, which is time-consuming. To address the issues, our proposed TMI computes the transferability as the intra-class variance of a pre-trained model on a target task. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. It is unnecessary for TMI to obtain optimal classifier which is a time-consuming process."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.077,
+                0.455,
+                0.468,
+                0.515
+            ],
+            "angle": 0,
+            "content": "TMI The transferability of a pre-trained feature extractor \\(\\omega_{s}\\) from a source task \\(S\\) to a target task \\(T\\), denoted by \\(Trf_{(S\\rightarrow T)}\\), is measured by TMI with the intra-class feature variance obtained from the pre-trained model \\(\\omega_{s}\\)."
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.095,
+                0.532,
+                0.468,
+                0.573
+            ],
+            "angle": 0,
+            "content": "\\[\nT r f _ {(S \\rightarrow T)} := H \\left(\\omega_ {s} (X) | Y\\right) = \\sum_ {c = 1} ^ {C} \\frac {n _ {c}}{n} H \\left(\\omega_ {s} \\left(X _ {c}\\right)\\right) \\tag {4}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.077,
+                0.578,
+                0.468,
+                0.713
+            ],
+            "angle": 0,
+            "content": "where \\( X \\) and \\( Y \\) represent the target feature and the target label, respectively. \\( C \\) represents the number of classes and \\( X_{c} = \\{x|Y = c\\} \\) is the set of target features in the \\( c \\)-th class. \\( n \\) and \\( n_{c} \\) represent the number of total instances and number of instances in the \\( c \\)-th class, respectively. We measure the intra-class variance of target representations using conditional entropy. The intra-class variance encourages the pre-trained model to have a generalization ability for learning tasks."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.077,
+                0.713,
+                0.468,
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+            ],
+            "angle": 0,
+            "content": "How to utilize TMI? Given a set of pre-trained models and a target dataset with labels, we first obtain target representations of pre-trained models by one-way forward propagation. Next, we compute the transferability score using Equation (4). We select the pre-trained model with the highest transferability score and then fine-tune the model using the target data. We argue that the larger the TMI, the higher the transfer performance."
+        },
+        {
+            "type": "title",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "4. Experiments"
+        },
+        {
+            "type": "text",
+            "bbox": [
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+            ],
+            "angle": 0,
+            "content": "We present experimental results to answer the following questions about TMI:"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.092,
+                0.891,
+                0.167
+            ],
+            "angle": 0,
+            "content": "Q1 Evaluation of transferability estimation (Section 4.2). Does TMI accurately select the best pretrained model compared to baselines? Is the transferability score measured by TMI highly correlated to the transfer performance?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.167,
+                0.891,
+                0.211
+            ],
+            "angle": 0,
+            "content": "Q2 Comparison of ICV measures (Section 4.3). What is the best measurement to evaluate Intra-Class Variance (ICV)?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.213,
+                0.891,
+                0.257
+            ],
+            "angle": 0,
+            "content": "Q3 Evaluation of generalization ability (Section 4.4). Does intra-class variance estimate generalization ability of a pre-trained model on a target task?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.258,
+                0.891,
+                0.318
+            ],
+            "angle": 0,
+            "content": "Q4 General applications (Section 4.5). Does TMI accurately estimate the transferability in general cases, such as selecting the best self-supervised pre-trained model and the best transferring layer?"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.318,
+                0.89,
+                0.348
+            ],
+            "angle": 0,
+            "content": "Q5 Hyperparameter sensitivity (Section 4.6). How sensitive is TMI to hyperparameters?"
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.505,
+                0.092,
+                0.891,
+                0.348
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "title",
+            "bbox": [
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+                0.69,
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+            ],
+            "angle": 0,
+            "content": "4.1. Experimental Setup"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.5,
+                0.383,
+                0.891,
+                0.414
+            ],
+            "angle": 0,
+            "content": "We present pre-trained models, datasets, baselines, evaluation, estimator, and hyperparameters for our experiments."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.499,
+                0.435,
+                0.892,
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+            "angle": 0,
+            "content": "Pre-trained models. We use supervised model groups offered by Pytorch: ResNet, DenseNet, MnasNet, ShuffleNet V2, MobileNet V2, MobileNet V3, ResNeXt, Wide ResNet, EfficientNet, EfficientNet V2, and RegNet. We also utilize self-supervised model groups, which have only feature extractors without classifiers. We obtain Swin Transformer and Vision Transformer from Pytorch; we get MoCo v1, MoCo v2, Barlow Twins, SwAV, SimSiam, and Dino from Gitlab repositories. Note that we select fifty supervised pretrained models and eleven self-supervised pre-trained models from the aforementioned model groups."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.623,
+                0.891,
+                0.684
+            ],
+            "angle": 0,
+            "content": "Datasets. We use seventeen image datasets summarized in Table 1. We randomly split each dataset into a train set and test set by 8:2 ratio, except for datasets that already have been separated."
+        },
+        {
+            "type": "text",
+            "bbox": [
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+                0.892,
+                0.9
+            ],
+            "angle": 0,
+            "content": "Baselines. We compare the proposed method with the following baselines. NCE [18] defines the transferability as negative conditional entropy between ground truths and predicted labels from a pre-trained model. LEEP [13] measures negative conditional entropy between ground truths and predicted average log-likelihood. H-Score [2] estimates the transferability by solving an HGR maximum correlation problem. LogME [25] proposes the logarithm of maximum evidence given extracted features. TransRate [5] measures the transferability by the mutual information between target labels and target representations. We omit the comparison with Taskonomy [26] and Task2Vec [1] due to tremendous time costs."
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+        {
+            "type": "page_number",
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+            "content": "11477"
+        }
+    ],
+    [
+        {
+            "type": "table_caption",
+            "bbox": [
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+            "angle": 0,
+            "content": "Table 1: Summary of datasets."
+        },
+        {
+            "type": "table",
+            "bbox": [
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+            "content": "<table><tr><td>Dataset</td><td># of instances (train / test)</td><td># of classes</td><td>Category</td></tr><tr><td>Caltech-1011</td><td>7,315 / 1,829</td><td>101</td><td>Multiple domains</td></tr><tr><td>Caltech-2562</td><td>24,485 / 6,122</td><td>256</td><td>Multiple domains</td></tr><tr><td>CIFAR-103</td><td>50,000 / 10,000</td><td>10</td><td>Multiple domains</td></tr><tr><td>CIFAR-1003</td><td>50,000 / 10,000</td><td>100</td><td>Multiple domains</td></tr><tr><td>MNIST4</td><td>60,000 / 10,000</td><td>10</td><td>Digit</td></tr><tr><td>FashionMNIST5</td><td>60,000 / 10,000</td><td>10</td><td>Fashion</td></tr><tr><td>SVHN6</td><td>73,257 / 26,032</td><td>10</td><td>Digit</td></tr><tr><td>FlowerPhotos7</td><td>2,936 / 734</td><td>5</td><td>Flower</td></tr><tr><td>EuroSAT8</td><td>21,600 / 5,400</td><td>10</td><td>Landscape</td></tr><tr><td>Chest X-Ray9</td><td>5,216 / 623</td><td>2</td><td>Medicine</td></tr><tr><td>VisDA10</td><td>152,397 / 72,372</td><td>212</td><td>Multiple domains</td></tr><tr><td>FGVC-Aircraft11</td><td>3,334 / 3,333</td><td>100</td><td>Aircraft</td></tr><tr><td>CUB-20012</td><td>9,430 / 2,358</td><td>200</td><td>Animal</td></tr><tr><td>Cars13</td><td>8,144 / 8,041</td><td>196</td><td>Auto</td></tr><tr><td>DTD14</td><td>4,512 / 1,128</td><td>47</td><td>Texture</td></tr><tr><td>Food-10115</td><td>80,800 / 20,200</td><td>101</td><td>Food</td></tr><tr><td>Oxford-IIIT Pet16</td><td>3,680 / 3,669</td><td>37</td><td>Animal</td></tr></table>"
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+            "type": "list",
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+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.485,
+                0.47,
+                0.636
+            ],
+            "angle": 0,
+            "content": "Evaluation. We evaluate the performance of transferability measurement methods in two ways. First, do they accurately predict the best model? Second, are the transferability scores correlated to fine-tuned accuracy? We adopt Kendall correlation coefficient to measure the correlations between the transferability and fine-tuned accuracy. Kendall correlation coefficient tests the ordinal association between two measured quantities. Kendall correlation coefficient has values between -1 and 1, and correlations between two variables are high when the values are close to 1."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.67,
+                0.47,
+                0.777
+            ],
+            "angle": 0,
+            "content": "Estimator. We adopt \\( k \\)-nearest estimator [8] to evaluate the continuous entropy in Equation (4). Compared to kernel density estimators [12, 17], \\( k \\)-nearest estimator vastly reduces errors by sacrificing time, especially when data are too big. TMI greatly reduces the size of data to be calculated at once because it computes conditional entropy of representation given label."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.81,
+                0.468,
+                0.901
+            ],
+            "angle": 0,
+            "content": "Hyperparameters. We use greedy searching algorithm to find the best hyperparameters for fine-tuning. For transferability measurement, the only hyperparameter in our proposed TMI is the number \\( n_k \\) of neighbors, which is used in the entropy estimator [8]. We discuss the hyperparameter sensitivity to transferability in Section 4.6."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.089,
+                0.892,
+                0.164
+            ],
+            "angle": 0,
+            "content": "Table 2: Ratio of target tasks where their top-5 best source models include the model estimated by each transferability measurement method. The best result is in bold. Note that TMI gives the best ratio, accurately selecting the best pretrained models in 14 out of 17 cases."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.506,
+                0.177,
+                0.888,
+                0.214
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Methods</td><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Top-5</td><td>3 / 17</td><td>5 / 17</td><td>4 / 17</td><td>8 / 17</td><td>9 / 17</td><td>14 / 17</td></tr></table>"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.499,
+                0.236,
+                0.848,
+                0.252
+            ],
+            "angle": 0,
+            "content": "4.2. Evaluation of Transferability Estimation"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.259,
+                0.892,
+                0.349
+            ],
+            "angle": 0,
+            "content": "Remind that our goal is to quickly and accurately select the best pre-trained model among a pool of well-trained models. A pre-trained model is determined by two components, source data and model architecture. Thus, we conduct experiments to select the best model architecture and the best source data."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.367,
+                0.892,
+                0.502
+            ],
+            "angle": 0,
+            "content": "Selecting the best model architecture. We evaluate how the transferability measurement methods accurately select the best source model for a target model. Table 2 shows the ratio of targets tasks where their top-5 best source models include the model estimated by each transferability measurement method. Note that TMI gives the best ratio, accurately selecting the best pre-trained models in 14 out of 17 cases, which is much better than the ratio \\(9 / 17\\) of the second-best model TransRate."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.504,
+                0.892,
+                0.594
+            ],
+            "angle": 0,
+            "content": "We also compare the correlations between fine-tuned accuracies and transferability scores in Table 3. Note that our proposed TMI shows the best Kendall correlation in 13 out of 17 datasets, outperforming all the competitors. Some baselines even give negative coefficients, which means that their predictions are far from the correct ranking."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.612,
+                0.892,
+                0.763
+            ],
+            "angle": 0,
+            "content": "Selecting the best source data. We fine-tune eighteen pre-trained models with the same model architecture ResNet-50, but different source datasets, including ImageNet and seventeen datasets introduced in Section 4.1. We predict the best source data and compare correlations with the ground truths. Table 4 shows that our proposed TMI accurately predicts the best source data in all the 17 cases, outperforming all the competitors. Our proposed TMI gives the best Kendall correlation consistently in 12 out of 17 cases, as shown in Table 5."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.499,
+                0.772,
+                0.766,
+                0.788
+            ],
+            "angle": 0,
+            "content": "4.3. Comparison of ICV Measures"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.795,
+                0.892,
+                0.901
+            ],
+            "angle": 0,
+            "content": "Intra-Class Variance (ICV) is a general concept widely used in metric learning. In addition to the conditional entropy introduced in Section 3, we additionally choose four metric learning methods, Contrast [4], Center [20], SNCA [22], and MS [19]. We use the intra-class variance parts of their objective functions to measure the transferability. We apply the five aforementioned methods to measure"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.481,
+                0.945,
+                0.519,
+                0.957
+            ],
+            "angle": 0,
+            "content": "11478"
+        }
+    ],
+    [
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.076,
+                0.089,
+                0.895,
+                0.137
+            ],
+            "angle": 0,
+            "content": "Table 3: Correlations between fine-tuned accuracies and transferability scores when transferring supervised pre-trained models trained on ImageNet. The best result is in bold. Our proposed TMI gives the best Kendall correlation in 13 out of 17 cases, outperforming competitors."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.214,
+                0.147,
+                0.761,
+                0.451
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"6\">Kendall Correlation Coefficient</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>0.429</td><td>0.425</td><td>0.316</td><td>0.362</td><td>0.474</td><td>0.524</td></tr><tr><td>Caltech-256</td><td>0.686</td><td>0.455</td><td>0.492</td><td>0.503</td><td>0.509</td><td>0.481</td></tr><tr><td>CIFAR-10</td><td>-0.140</td><td>0.324</td><td>0.275</td><td>0.323</td><td>0.364</td><td>0.527</td></tr><tr><td>CIFAR-100</td><td>-0.202</td><td>0.340</td><td>0.351</td><td>0.398</td><td>0.432</td><td>0.497</td></tr><tr><td>MNIST</td><td>0.057</td><td>0.282</td><td>0.083</td><td>0.137</td><td>0.038</td><td>0.137</td></tr><tr><td>FashionMNIST</td><td>-0.328</td><td>0.127</td><td>-0.038</td><td>0.121</td><td>0.125</td><td>0.467</td></tr><tr><td>SVHN</td><td>-0.023</td><td>0.091</td><td>-0.002</td><td>0.092</td><td>0.206</td><td>0.394</td></tr><tr><td>FlowerPhotos</td><td>0.117</td><td>0.139</td><td>0.224</td><td>0.321</td><td>0.392</td><td>0.499</td></tr><tr><td>EuroSAT</td><td>0.269</td><td>0.082</td><td>-0.005</td><td>0.102</td><td>0.051</td><td>0.085</td></tr><tr><td>Chest X-Ray</td><td>0.071</td><td>0.152</td><td>0.171</td><td>0.333</td><td>0.285</td><td>0.395</td></tr><tr><td>VisDA</td><td>0.298</td><td>0.240</td><td>0.329</td><td>0.334</td><td>0.304</td><td>0.341</td></tr><tr><td>FGVC-Aircraft</td><td>0.366</td><td>-0.011</td><td>0.216</td><td>0.498</td><td>0.421</td><td>0.500</td></tr><tr><td>CUB-200</td><td>0.415</td><td>0.098</td><td>0.325</td><td>0.410</td><td>0.519</td><td>0.523</td></tr><tr><td>Cars</td><td>0.270</td><td>0.506</td><td>0.175</td><td>0.432</td><td>0.455</td><td>0.521</td></tr><tr><td>DTD</td><td>-0.074</td><td>0.501</td><td>0.479</td><td>0.366</td><td>0.499</td><td>0.401</td></tr><tr><td>Food-101</td><td>0.261</td><td>0.515</td><td>0.395</td><td>0.421</td><td>0.448</td><td>0.537</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.536</td><td>0.527</td><td>0.418</td><td>0.493</td><td>0.500</td><td>0.576</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.076,
+                0.473,
+                0.472,
+                0.536
+            ],
+            "angle": 0,
+            "content": "Table 4: Ratio of target tasks where the transferability measurement methods accurately find the best source data. The best method is in bold. TMI predicts the best source data in all cases, outperforming competitors."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.083,
+                0.545,
+                0.465,
+                0.582
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Methods</td><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Top-1</td><td>1 / 17</td><td>2 / 17</td><td>3 / 17</td><td>2 / 17</td><td>13 / 17</td><td>17 / 17</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.608,
+                0.47,
+                0.714
+            ],
+            "angle": 0,
+            "content": "the correlations between fine-tuned accuracies and transferability scores. Conditional Entropy (CE) shows the best correlations than the other four metric learning measurements in 15 out of 17 datasets, outperforming all the competitors, as shown in Table 6. M. Boudiaf et al. [3] demonstrate that the conditional entropy is a lower bound of intra-class variance of the four metric learning methods."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.077,
+                0.725,
+                0.394,
+                0.743
+            ],
+            "angle": 0,
+            "content": "4.4. Evaluation of Generalization Ability"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.75,
+                0.47,
+                0.869
+            ],
+            "angle": 0,
+            "content": "We perform experiments on two control groups to demonstrate that transferability is relevant to the generalization ability of a pre-trained model on a target task. In Group A, we use a pre-trained ResNet-50 model trained on ImageNet and fine-tune it using the dataset \\( D \\). In Group B, we train a ResNet-50 model trained from scratch on dataset \\( D \\) and fine-tune it on the same dataset. Dataset \\( D \\) is one of the datasets introduced in Section 4.1."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.077,
+                0.871,
+                0.47,
+                0.903
+            ],
+            "angle": 0,
+            "content": "Table 7 shows the difference of fine-tuning performance and that of transferability between two groups. “+” rep"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.475,
+                0.892,
+                0.552
+            ],
+            "angle": 0,
+            "content": "resents the difference is larger than 0 while “-” represents that is less than 0. Group A has higher transfer accuracies than Group B in most cases since models in Group A have learned a large amount of information from the ImageNet, which can potentially benefit downstream tasks."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.554,
+                0.892,
+                0.66
+            ],
+            "angle": 0,
+            "content": "Our proposed TMI and TransRate have the same symbols for transfer accuracy in 14 out of 17 cases, while the other four baselines do not provide similar symbols. TransRate also estimates well, but it is computationally expensive since it measures the mutual information between target representation and label. In contrast, TMI is more efficient as it considers only intra-class features."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.5,
+                0.678,
+                0.7,
+                0.695
+            ],
+            "angle": 0,
+            "content": "4.5. General Applications"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.704,
+                0.892,
+                0.901
+            ],
+            "angle": 0,
+            "content": "Applying to self-supervised pre-trained models. TMI not only measures the transferability between supervised pre-trained models and classification tasks but also can be extended to self-supervised pre-trained models. We prepare eleven self-supervised pre-trained models already trained on ImageNet provided by Github repositories introduced in Section 4.1. NCE [18] and LEEP [13] cannot measure the transferability between a self-supervised pre-trained model and a downstream task because they dig out the relationship between target ground truth and predicted label or probability. Thus, we compare the proposed TMI to LogME [25], H-Score [2], and TransRate [5] in Table 8; TMI outperforms competitors in 13 out of 17 classification datasets."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.48,
+                0.945,
+                0.521,
+                0.958
+            ],
+            "angle": 0,
+            "content": "11479"
+        }
+    ],
+    [
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.076,
+                0.089,
+                0.895,
+                0.137
+            ],
+            "angle": 0,
+            "content": "Table 5: Correlations between fine-tuned accuracies and transferability scores when transferring ResNet-50 models trained on different source data. The best result is in bold. TMI shows the best Kendall correlation in 12 out of 17 cases, outperforming all the competitors."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.212,
+                0.147,
+                0.761,
+                0.451
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"6\">Kendall Correlation Coefficient</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>-0.294</td><td>0.542</td><td>0.353</td><td>0.647</td><td>0.425</td><td>0.699</td></tr><tr><td>Caltech-256</td><td>-0.268</td><td>0.542</td><td>0.437</td><td>0.660</td><td>0.477</td><td>0.699</td></tr><tr><td>CIFAR-10</td><td>-0.216</td><td>0.373</td><td>0.383</td><td>0.490</td><td>0.399</td><td>0.630</td></tr><tr><td>CIFAR-100</td><td>-0.098</td><td>0.556</td><td>0.686</td><td>0.345</td><td>0.542</td><td>0.699</td></tr><tr><td>MNIST</td><td>-0.061</td><td>0.439</td><td>0.372</td><td>-0.007</td><td>0.115</td><td>0.162</td></tr><tr><td>FashionMNIST</td><td>-0.086</td><td>0.362</td><td>0.430</td><td>0.475</td><td>0.046</td><td>0.150</td></tr><tr><td>SVHN</td><td>0.039</td><td>0.285</td><td>-0.033</td><td>0.393</td><td>0.170</td><td>0.407</td></tr><tr><td>FlowerPhotos</td><td>-0.359</td><td>0.412</td><td>0.456</td><td>0.307</td><td>0.490</td><td>0.516</td></tr><tr><td>EuroSAT</td><td>-0.388</td><td>0.428</td><td>0.490</td><td>0.322</td><td>0.467</td><td>0.704</td></tr><tr><td>Chest X-Ray</td><td>-0.072</td><td>0.164</td><td>0.359</td><td>-0.111</td><td>0.425</td><td>-0.123</td></tr><tr><td>VisDA</td><td>-0.538</td><td>0.498</td><td>0.094</td><td>-0.039</td><td>0.498</td><td>0.546</td></tr><tr><td>FGVC-Aircraft</td><td>-0.033</td><td>0.373</td><td>0.339</td><td>0.556</td><td>0.399</td><td>0.688</td></tr><tr><td>CUB-200</td><td>-0.503</td><td>0.582</td><td>0.455</td><td>0.464</td><td>0.647</td><td>0.723</td></tr><tr><td>Cars</td><td>-0.294</td><td>0.529</td><td>0.403</td><td>0.752</td><td>0.542</td><td>0.804</td></tr><tr><td>DTD</td><td>-0.438</td><td>0.595</td><td>0.393</td><td>0.647</td><td>0.542</td><td>0.757</td></tr><tr><td>Food-101</td><td>-0.020</td><td>0.281</td><td>0.092</td><td>0.046</td><td>0.229</td><td>0.225</td></tr><tr><td>Oxford-IIIT Pet</td><td>-0.268</td><td>0.569</td><td>0.660</td><td>0.503</td><td>0.516</td><td>0.248</td></tr></table>"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.076,
+                0.473,
+                0.472,
+                0.521
+            ],
+            "angle": 0,
+            "content": "Table 6: Comparison of intra-class variance measurements. Conditional Entropy (CE) shows the best correlations consistently in 15 out of 17 cases. The best result is in bold."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.093,
+                0.53,
+                0.456,
+                0.774
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"5\">Kendall Correlation Coefficient</td></tr><tr><td>Contrast</td><td>Center</td><td>SNCA</td><td>MS</td><td>CE</td></tr><tr><td>Caltech-101</td><td>0.274</td><td>0.283</td><td>0.380</td><td>0.358</td><td>0.442</td></tr><tr><td>Caltech-256</td><td>0.295</td><td>0.347</td><td>0.261</td><td>0.347</td><td>0.354</td></tr><tr><td>CIFAR-10</td><td>0.328</td><td>0.322</td><td>0.275</td><td>0.299</td><td>0.369</td></tr><tr><td>CIFAR-100</td><td>0.374</td><td>0.282</td><td>0.207</td><td>0.200</td><td>0.376</td></tr><tr><td>MNIST</td><td>0.383</td><td>0.267</td><td>0.306</td><td>0.228</td><td>0.374</td></tr><tr><td>FashionMNIST</td><td>0.332</td><td>0.225</td><td>0.260</td><td>0.337</td><td>0.374</td></tr><tr><td>SVHN</td><td>0.303</td><td>0.359</td><td>0.369</td><td>0.360</td><td>0.383</td></tr><tr><td>FlowerPhotos</td><td>0.209</td><td>0.263</td><td>0.291</td><td>0.277</td><td>0.372</td></tr><tr><td>EuroSAT</td><td>0.292</td><td>0.237</td><td>0.367</td><td>0.268</td><td>0.400</td></tr><tr><td>Chest X-Ray</td><td>0.292</td><td>0.289</td><td>0.210</td><td>0.232</td><td>0.357</td></tr><tr><td>VisDA</td><td>0.259</td><td>0.352</td><td>0.200</td><td>0.329</td><td>0.352</td></tr><tr><td>FGVC-Aircraft</td><td>0.220</td><td>0.322</td><td>0.244</td><td>0.315</td><td>0.351</td></tr><tr><td>CUB-200</td><td>0.368</td><td>0.252</td><td>0.379</td><td>0.385</td><td>0.431</td></tr><tr><td>Cars</td><td>0.351</td><td>0.285</td><td>0.398</td><td>0.268</td><td>0.248</td></tr><tr><td>DTD</td><td>0.236</td><td>0.322</td><td>0.228</td><td>0.206</td><td>0.327</td></tr><tr><td>Food-101</td><td>0.322</td><td>0.267</td><td>0.246</td><td>0.269</td><td>0.484</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.301</td><td>0.202</td><td>0.363</td><td>0.274</td><td>0.449</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.81,
+                0.472,
+                0.903
+            ],
+            "angle": 0,
+            "content": "Selecting the best transferring layer. Given a ResNet-18 trained on ImageNet, we can easily determine the best transferring layer, which produces the best fine-tuning performance for a target task by measuring transferability. We compare the correlations of LogME [25], H-Score [2], and TransRate to select the best transferring layer for fine"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.498,
+                0.473,
+                0.895,
+                0.552
+            ],
+            "angle": 0,
+            "content": "Table 7: Comparison of the consistency of fine-tuning accuracy and transferability. “+” represents the difference is larger than 0 while “-” represents the difference is less than 0. TMI has consistent signs with fine-tuning accuracy in 14 out of 17 cases, outperforming its competitors."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.504,
+                0.56,
+                0.895,
+                0.791
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Dataset</td><td rowspan=\"2\">Accuracy (A-B)</td><td colspan=\"6\">Transferability (A-B)</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogME</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Caltech-256</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CIFAR-10</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CIFAR-100</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>MNIST</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>FashionMNIST</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>SVHN</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>FlowerPhotos</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>EuroSAT</td><td>+</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>Chest X-Ray</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>VisDA</td><td>+</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>FGVC-Aircraft</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CUB-200</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Cars</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>DTD</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Food-101</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Oxford-IIIT Pet</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.825,
+                0.895,
+                0.903
+            ],
+            "angle": 0,
+            "content": "tuning. Theoretically, LogME [25] and H-Score [2] cannot deal with layer selection because they rely on features from the ultimate layer of the feature extractor. However, we extract features from each layer and directly put them into LogME and H-Score algorithms to calculate the trans"
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.48,
+                0.945,
+                0.521,
+                0.958
+            ],
+            "angle": 0,
+            "content": "11480"
+        }
+    ],
+    [
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.076,
+                0.089,
+                0.472,
+                0.166
+            ],
+            "angle": 0,
+            "content": "Table 8: Correlations between fine-tuned accuracies and transferability scores when transferring self-supervised pretrained models to classification tasks. The best result is in bold. TMI shows the best Kendall correlation consistently in 13 out of 17 cases, outperforming all the competitors."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.103,
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+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"4\">Kendall Correlation Coefficient</td></tr><tr><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>-0.278</td><td>0.183</td><td>0.000</td><td>0.367</td></tr><tr><td>Caltech-256</td><td>-0.561</td><td>0.345</td><td>0.164</td><td>0.600</td></tr><tr><td>CIFAR-10</td><td>-0.337</td><td>0.018</td><td>0.164</td><td>0.624</td></tr><tr><td>CIFAR-100</td><td>-0.561</td><td>0.673</td><td>0.236</td><td>0.697</td></tr><tr><td>MNIST</td><td>-0.574</td><td>-0.262</td><td>-0.426</td><td>0.246</td></tr><tr><td>FashionMNIST</td><td>-0.330</td><td>-0.220</td><td>-0.330</td><td>0.094</td></tr><tr><td>SVHN</td><td>0.055</td><td>0.309</td><td>-0.345</td><td>0.075</td></tr><tr><td>FlowerPhotos</td><td>0.389</td><td>-0.241</td><td>0.130</td><td>0.075</td></tr><tr><td>EuroSAT</td><td>-0.073</td><td>-0.037</td><td>0.037</td><td>-0.208</td></tr><tr><td>Chest X-Ray</td><td>-0.037</td><td>-0.037</td><td>-0.257</td><td>0.175</td></tr><tr><td>VisDA</td><td>0.550</td><td>0.055</td><td>0.309</td><td>-0.597</td></tr><tr><td>FGVC-Aircraft</td><td>-0.345</td><td>0.091</td><td>-0.309</td><td>0.150</td></tr><tr><td>CUB-200</td><td>-0.112</td><td>-0.018</td><td>0.091</td><td>0.564</td></tr><tr><td>Cars</td><td>0.000</td><td>0.382</td><td>-0.200</td><td>0.262</td></tr><tr><td>DTD</td><td>0.382</td><td>0.018</td><td>0.224</td><td>0.455</td></tr><tr><td>Food-101</td><td>-0.411</td><td>-0.018</td><td>-0.018</td><td>0.204</td></tr><tr><td>Oxford-IIIT Pet</td><td>-0.585</td><td>0.294</td><td>0.073</td><td>0.807</td></tr></table>"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.142,
+                0.457,
+                0.408,
+                0.541
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.076,
+                0.558,
+                0.47,
+                0.587
+            ],
+            "angle": 0,
+            "content": "Figure 3: Sensitivity to the hyperparameter \\( n_k \\) in entropy estimator."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.622,
+                0.47,
+                0.789
+            ],
+            "angle": 0,
+            "content": "ferability. The size of extracted features is quite large in the early layers, resulting in much higher computational cost for transferability measurement. To reduce this cost, we add an avgpool2d layer to the end of transferring layer to reduce the dimension of extracted features. Table 9 shows that TMI achieves the highest correlation coefficients in 16 out of 17 datasets. Correlation coefficients of MNIST, SVHN, and Chest X-Ray are much lower than those of the other datasets because their ranges of minimum and maximum accuracies obtained by layer transfer are smaller than \\(0.5\\%\\), making it challenging to rank these dataset accurately."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.077,
+                0.801,
+                0.327,
+                0.818
+            ],
+            "angle": 0,
+            "content": "4.6. Hyperparameter Sensitivity"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.076,
+                0.825,
+                0.47,
+                0.903
+            ],
+            "angle": 0,
+            "content": "An advantage of TMI is that it does not train models on target datasets because it is time-consuming and sensitive to hyperparameters which may cause different results. The only hyperparameter in TMI is \\( n_k \\) used in the entropy estimator. In Figure. 3, the correlation between pre-trained"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.498,
+                0.089,
+                0.895,
+                0.151
+            ],
+            "angle": 0,
+            "content": "Table 9: Correlations between fine-tuned accuracies and transferability scores when transferring specific layers of pre-trained models. The best result is in bold. TMI shows the best correlations consistently in 16 out of 17 cases."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.527,
+                0.162,
+                0.866,
+                0.405
+            ],
+            "angle": 0,
+            "content": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"4\">Kendall Correlation Coefficient</td></tr><tr><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>0.346</td><td>0.935</td><td>0.752</td><td>0.941</td></tr><tr><td>Caltech-256</td><td>0.973</td><td>0.317</td><td>0.800</td><td>0.984</td></tr><tr><td>CIFAR-10</td><td>0.360</td><td>0.743</td><td>0.704</td><td>0.746</td></tr><tr><td>CIFAR-100</td><td>0.460</td><td>0.774</td><td>0.774</td><td>0.801</td></tr><tr><td>MNIST</td><td>0.180</td><td>0.055</td><td>0.179</td><td>0.184</td></tr><tr><td>FashionMNIST</td><td>0.294</td><td>0.241</td><td>0.298</td><td>0.308</td></tr><tr><td>SVHN</td><td>0.087</td><td>0.106</td><td>0.040</td><td>0.113</td></tr><tr><td>FlowerPhotos</td><td>0.779</td><td>0.198</td><td>0.687</td><td>0.792</td></tr><tr><td>EuroSAT</td><td>0.691</td><td>0.709</td><td>0.495</td><td>0.735</td></tr><tr><td>Chest X-Ray</td><td>0.113</td><td>0.154</td><td>0.100</td><td>0.054</td></tr><tr><td>VisDA</td><td>0.665</td><td>0.866</td><td>0.826</td><td>0.904</td></tr><tr><td>FGVC-Aircraft</td><td>0.791</td><td>0.302</td><td>0.595</td><td>0.817</td></tr><tr><td>CUB-200</td><td>0.317</td><td>0.970</td><td>0.787</td><td>0.954</td></tr><tr><td>Cars</td><td>0.330</td><td>0.869</td><td>0.752</td><td>0.882</td></tr><tr><td>DTD</td><td>0.924</td><td>0.401</td><td>0.761</td><td>0.931</td></tr><tr><td>Food-101</td><td>0.409</td><td>0.948</td><td>0.869</td><td>0.970</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.349</td><td>0.970</td><td>0.826</td><td>0.990</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.5,
+                0.431,
+                0.888,
+                0.446
+            ],
+            "angle": 0,
+            "content": "models and TMI shows consistency in most ranges for \\( n_k \\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.5,
+                0.459,
+                0.618,
+                0.475
+            ],
+            "angle": 0,
+            "content": "5. Conclusion"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.484,
+                0.892,
+                0.665
+            ],
+            "angle": 0,
+            "content": "We propose TMI, a fast, accurate, and general algorithm to measure the transferability. TMI quantifies the degree of generalization of a pre-trained model to a new task by estimating the intra-class feature variance. We conduct experiments on seventeen datasets with fifty supervised pretrained models and eleven self-supervised pre-trained models. TMI selects the top-5 pre-trained architecture in 14 out of 17 cases and selects the best source data in all cases, outperforming its competitors. We measure the correlation between the fine-tuning performance and transferability, and our proposed TMI outperforms competitors in 13 out of 17 cases."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.5,
+                0.68,
+                0.66,
+                0.696
+            ],
+            "angle": 0,
+            "content": "Acknowledgments"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.498,
+                0.704,
+                0.892,
+                0.901
+            ],
+            "angle": 0,
+            "content": "This work was supported by Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) [No.2020-0-00894, Flexible and Efficient Model Compression Method for Various Applications and Environments], [No.2021-0-01343, Artificial Intelligence Graduate School Program (Seoul National University)], and [NO.2021-0-02068, Artificial Intelligence Innovation Hub (Artificial Intelligence Institute, Seoul National University)]. The Institute of Engineering Research at Seoul National University provided research facilities for this work. The ICT at Seoul National University provides research facilities for this study. U Kang is the corresponding author."
+        },
+        {
+            "type": "page_number",
+            "bbox": [
+                0.481,
+                0.945,
+                0.518,
+                0.956
+            ],
+            "angle": 0,
+            "content": "11481"
+        }
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+            "angle": 0,
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2023/Fast and Accurate Transferability Measurement by Evaluating Intra-class Feature Variance/16d96c9b-c625-4fb0-aaab-b679d45a05de_origin.pdf

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2023/Fast and Accurate Transferability Measurement by Evaluating Intra-class Feature Variance/full.md

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+# Fast and Accurate Transferability Measurement by Evaluating Intra-class Feature Variance
+
+Huiwen Xu  
+Seoul National University  
+Seoul, Republic of Korea  
+xuhuiwen33@snu.ac.kr
+
+U Kang Seoul National University Seoul, Republic of Korea ukang@snu.ac.kr
+
+# Abstract
+
+Given a set of pre-trained models, how can we quickly and accurately find the most useful pre-trained model for a downstream task? Transferability measurement is to quantify how transferable is a pre-trained model learned on a source task to a target task. It is used for quickly ranking pre-trained models for a given task and thus becomes a crucial step for transfer learning. Existing methods measure transferability as the discrimination ability of a source model for a target data before transfer learning, which cannot accurately estimate the fine-tuning performance. Some of them restrict the application of transferability measurement in selecting the best supervised pre-trained models that have classifiers. It is important to have a general method for measuring transferability that can be applied in a variety of situations, such as selecting the best self-supervised pre-trained models that do not have classifiers, and selecting the best transferring layer for a target task.
+
+In this work, we propose TMI (TRANSFERABILITY MEASUREMENT WITH INTRA-CLASS FEATURE VARIANCE), a fast and accurate algorithm to measure transferability. We view transferability as the generalization of a pre-trained model on a target task by measuring intra-class feature variance. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. Compared to previous studies that estimate how discriminative the models are, intra-class variance is more accurate than those as it does not require an optimal feature extractor and classifier. Extensive experiments on real-world datasets show that TMI outperforms competitors for selecting the top-5 best models, and exhibits consistently better correlation in 13 out of 17 cases.
+
+# 1. Introduction
+
+Transfer learning is an important concept in the field of machine learning, revolutionizing the way models are
+
+![](images/f3be556d27f86e0b0aa81c21d32f1f474315371c44a0984b1ad15b1bc867c58b.jpg)  
+Figure 1: Illustration of selecting the best pre-trained model for a given target task.
+
+trained and applied to various domains. Transfer learning has proven to be particularly valuable when labeled data in the target domain are scarce or costly to obtain. Within the broader scope of transfer learning, several variants have emerged, such as source-free domain adaptation [10, 27, 23, 7], multi-source transfer learning [16, 14, 7, 9], heterogeneous transfer learning [11, 15], and open-set domain adaptation [6, 21].
+
+In transfer learning, fine-tuning a pre-trained model becomes a simple and effective way to improve the performance of a downstream task. A crucial step in transfer learning is to quickly and accurately select the most helpful pre-trained model in a set of given pre-trained models. Transferability measurement is to quantify how transferable is a pre-trained model learned on a source task to a target task. As shown in Figure. 1, transferability measurement is used for ranking pre-trained models for a given task, and is a crucial step for transfer learning.
+
+A desired method for transferability measurement should be faster than running a transfer learning algorithm, accurately select the best pre-trained model, and be general to be implemented in various common scenarios. Taskonomy [26] and Task2Vec [1] are computationally expensive due to their learning with data. NCE [18] and LEEP [13] are free of training but cannot be applied to general cases, such as selecting self-supervised pre-trained models and selecting the best transferring layer. LogMe [25] and H-Score [2] also cannot be applied theoretically to select the best transferring layer. Instead of measuring transferability, they measure how discriminative a pre-trained model is to a target task before implementing transfer learning, which cannot estimate fine-tuning performance. TransRate [5] is simple and general to be applied for transferability measurement, but too slow due to the computation of the entropy of whole hidden representations.
+
+In this paper, we propose TMI (TRANSFERABILITY MEASUREMENT WITH INTRA-CLASS FEATURE VARIANCE), a simple but effective method for transferability measurement. We deem transferability as the generalization of a pre-trained model on a target task. To quantify the degree of generalization, we measure the intra-class variance by the conditional entropy of target representation given a label. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. Compared to previous studies that estimate how discriminative the models are, intra-class variance is more accurate than those as it does not require an optimal feature extractor and classifier. Large intra-class variance implies the feature is efficient for learning tasks. TMI can be used for selecting not only the best supervised model but also the best self-supervised models and the best transferring layer.
+
+Our contributions are summarized as follows:
+
+- Algorithm. We propose TMI, a fast, accurate, and general method for measuring transferability. We view transferability as generalization ability of a pre-trained model on a target task. The model generalization is measured by intra-class feature variance.  
+- Experiments. We conduct extensive experiments on seventeen datasets with fifty supervised pre-trained models and eleven self-supervised pre-trained models. TMI outperforms competitors for selecting the top-5 pre-trained architecture in 14 out of 17 cases and selecting the best source data in all cases. TMI also shows the best trade-off of the running time and transferability, outperforming competitors in 12 out of 17 cases (see Figure. 2).  
+- Case study. We evaluate the transferability in more general cases, such as applying to self-supervised pretrained models and selecting the best transferring layer. Compared to competitors, TMI achieves the highest correlation coefficient in 13 out of 17 for selecting the
+
+best self-supervised pre-trained models, and in 16 out of 17 datasets for selecting the best transferring layer.
+
+The rest of this paper is organized as follows. We introduce related works for transferability measurement in Section 2, and propose our method TMI in Section 3. After presenting experimental results in Section 4, we conclude in Section 5.
+
+# 2. Related Works
+
+We assume a pre-trained model  $\theta = (w, h)$  in a source domain where  $w$  is a feature extractor, and  $h$  is a classifier (or head). We transfer the model  $\theta$  to a target domain and add a new classifier  $k$  at the end of the feature extractor  $w$  before fine-tuning. Since the target task is different from the source task in most cases, the pre-trained classifier  $h$  is inappropriate to solve the target task. Thus, we fine-tune the model by minimizing the cross entropy loss on the target dataset  $D = \{(x_i, y_i)\}_{i=1}^n$ :
+
+$$
+w ^ {*}, k ^ {*} = \underset {w, k} {\arg \min } \mathcal {L} (w, k) \tag {1}
+$$
+
+where a feature extractor  $w$  maps an input  $x_{i}$  to a representation  $r_i = w(x_i)$ , and a classifier  $h$  maps the representation  $r_i$  to a probability distribution  $\hat{y}_i = \sigma(h(r_i))$  where  $\sigma(\cdot)$  is an activation function. Note that  $\hat{y}_i$  is a vector of length  $n_c$ , where  $n_c$  is the number of classes, and  $\mathcal{L}(\cdot)$  is the cross entropy loss, which is equivalent to the average negative log-likelihood:
+
+$$
+\mathcal {L} (w, k) = - \frac {1}{n} \sum_ {i = 1} ^ {n} \log P \left(\hat {y} _ {i, c} \mid x _ {i}; w, k\right) \tag {2}
+$$
+
+where  $\hat{y}_{i,c}$  is the probability of  $i$ -th input instance belonging to its ground truth.
+
+The transferability of a source task  $S$  to a target task  $T$  is defined by the log-likelihood [18]:
+
+$$
+T r f _ {(S \rightarrow T)} = \frac {1}{n} \sum_ {i = 1} ^ {n} \log P \left(\hat {y} _ {i, c} \mid x _ {i}; w ^ {*}, k ^ {*}\right) \tag {3}
+$$
+
+Taskonomy [26] and Task2Vec [1] train the source model using target data to obtain the optimal feature extractor  $w^{*}$  and the optimal classifier  $k^{*}$ . These methods are inconvenient to measure the transferability due to their training with data. There have been works focusing on estimating a bound on the log-likelihood to evaluate the transferability without training. NCE [18] proposes negative conditional entropy, which is drawn from the empirical joint distribution over ground truths and predicted labels from a pretrained model. Similarly, LEEP [13] replaces the ground truth with the average log-likelihood to evaluate the transferability. NCE and LEEP are fast, but are restricted to supervised pre-trained models because the algorithms rely
+
+![](images/a9731248f7c35c157c6bdbcaeba07a3b58c6254bcfa6867e1c80149c32113399.jpg)  
+(a) Caltech-101
+
+![](images/f2afcec31899d62037b3846c2bce0ef393923ecb52c0bca1818c9bae03f88285.jpg)  
+(b) Caltech-256
+
+![](images/dae8c4d07881c56e467ea24debcf0df2b5294e9a7dcad15de1b74a7809b6da91.jpg)  
+(c) CIFAR-10
+
+![](images/62d8dba0ea2141c6f26444e17e624502d6a31a3907e837e37150af86bc68318f.jpg)  
+(d) CIFAR-100
+
+![](images/1ff58ff1b9aba23a0248e1fd808e99aff809542a7cf20344225a0fb800bc4de0.jpg)  
+(e) MNIST
+
+![](images/796bdb29190ae4c7ddbdfc8fea9dd77c86c1f589ef99bac0603dc9789815ea95.jpg)  
+(f) FashionMNIST
+
+![](images/47e5d3a24defa41fbb9996e6df75604c9c3fc6ac5129c586c2f6733d5c29a6a8.jpg)  
+(g) SVHN
+
+![](images/3e8269311f49908d6804ea194d7029f75205a9aa22efccf0d29a356ccc8522b2.jpg)  
+(h)FlowerPhotos
+
+![](images/563d544d5a3b8acfb5cf300e8b0e4890ddcfab206c133e280fac8885c3a4d437.jpg)  
+(i) EuroSAT
+
+![](images/231512f9b597aa897b309293edd853d04399ed061e886e74be47772a4f2774d6.jpg)  
+(j) Chest X-Ray
+
+![](images/a2d4e46bf4c2ba04433013f4b21e03d8178f515fe6ba4d679dc14d4cdf72fb27.jpg)  
+(k) VisDA
+
+![](images/1597aec12dfd9d10ffd9932fa5dadb1e6042b3553185e33d3edd482aa6d52216.jpg)  
+(1) FGVC-Aircraft
+
+![](images/490baa3b51fb19ee1879503f13f2cab59a100f99020df1bcbf1a6b1d206ddaca.jpg)  
+(m) CUB-200
+
+![](images/6442787cf94ddc90ad887a2d8906d67e4abfa52fd238b4d731296a587c0375bc.jpg)  
+(n) Cars
+
+![](images/60f66bbe2c739c9e84b585b92a6da91dcee89670844ec0f0da5951447e45cfff.jpg)  
+(0) DTD
+
+![](images/6ab8e30a03644ed04531acad2e869b8f6d6e25941bb4e8c96b7c64038f320935.jpg)  
+(p) Food-101  
+Figure 2: Total running time (seconds) vs. Kendall's Correlation between transferability scores and test accuracies for selecting the best model among fifty pre-trained models. TMI provides the best trade-off between running time and correlation.
+
+![](images/a23dacbc109931c9493e82f68bfee0bfbe78b2a186e7887b2355b77982df373a.jpg)  
+(q) Oxford-IIIT Pet
+
+![](images/6f9ba93d8c11955ece7591be98cff4c64123f005d356866d04f60ab5c3d85e8e.jpg)
+
+on the outputs of the pre-trained classifier. To apply measurements in more general cases, LogME [25] calculates the logarithm of maximum evidence given extracted features, and H-Score [2] solves an HGR maximum correlation problem. LogMe [25] and H-Score [2] still cannot be applied to select the best transferred layer since they extract features from only the ultimate layers of the pretrained feature extractors. TransRate [5] is the most recent algorithm and applicable to self-supervised models and transferring layer selection. However, it is time-consuming since it requires to calculate mutual information between extracted feature and ground truth. To simplify the problem, [18, 13, 25, 2, 5] assume that the optimal model can be obtained by re-training the classifier while freezing the feature extractor. They estimate the performance of re-training only the classifier, which is even worse than training a target model from scratch [24]. In this paper, we propose a fast, accurate, and general transferability measurement method that predicts fine-tuned performance of a pre-trained model to a new target task.
+
+# 3. Proposed Method
+
+We propose TMI to measure the transferability between a pre-trained model and a target task.
+
+# 3.1. Overview
+
+Our goal is to measure the transferability in a fast, accurate, and general manner. The following challenges need to be addressed for the goal.
+
+C1 Speed. Training a pre-trained model on the target data requires a high computational cost. How can we measure the transferability without training on the target data?  
+C2 Accuracy. Existing methods evaluate the transferability that estimate the performance of the optimal target model obtained by re-training only the classifier, not the feature extractor, which is uncommon in transfer learning. How can we accurately measure the transferability for predicting the fine-tuning performance?  
+C3 Generality. How can we measure the transferability such that it is applicable to general settings, including selecting the best supervised pre-trained model, the best self-supervised pre-trained models, and the transferring layer?
+
+We address the aforementioned challenges with the following main ideas:
+
+I1 We assess the transferability by evaluating the compactness and sparseness of target representations with only a one-way forward propagation.  
+12 We relax the assumption proposed in existing methods
+
+that the pre-trained feature extractor can extract all the information related to the target task. We view transferability as the generalization ability of a pre-trained model on a target task. We estimate the generalization ability using intra-class variance, which implies how efficient the feature is for learning tasks.
+
+13 We measure the transferability using intra-class variance of latent representations, which is applicable to various settings including self-supervised pre-trained models and transferring layer selection.
+
+# 3.2.TMI
+
+Given a pre-trained source model and a target dataset with labels, our goal is to measure the transferability between the source model and the target task to predict the fine-tuning performance. Existing methods measure the performance of re-trained classifier while freezing the feature extractor, which is time-consuming. To address the issues, our proposed TMI computes the transferability as the intra-class variance of a pre-trained model on a target task. Intra-class variance evaluates the adaptability of the model to a new task, which measures how transferable the model is. It is unnecessary for TMI to obtain optimal classifier which is a time-consuming process.
+
+TMI The transferability of a pre-trained feature extractor  $\omega_{s}$  from a source task  $S$  to a target task  $T$ , denoted by  $Trf_{(S\rightarrow T)}$ , is measured by TMI with the intra-class feature variance obtained from the pre-trained model  $\omega_{s}$ .
+
+$$
+T r f _ {(S \rightarrow T)} := H \left(\omega_ {s} (X) | Y\right) = \sum_ {c = 1} ^ {C} \frac {n _ {c}}{n} H \left(\omega_ {s} \left(X _ {c}\right)\right) \tag {4}
+$$
+
+where  $X$  and  $Y$  represent the target feature and the target label, respectively.  $C$  represents the number of classes and  $X_{c} = \{x|Y = c\}$  is the set of target features in the  $c$ -th class.  $n$  and  $n_{c}$  represent the number of total instances and number of instances in the  $c$ -th class, respectively. We measure the intra-class variance of target representations using conditional entropy. The intra-class variance encourages the pre-trained model to have a generalization ability for learning tasks.
+
+How to utilize TMI? Given a set of pre-trained models and a target dataset with labels, we first obtain target representations of pre-trained models by one-way forward propagation. Next, we compute the transferability score using Equation (4). We select the pre-trained model with the highest transferability score and then fine-tune the model using the target data. We argue that the larger the TMI, the higher the transfer performance.
+
+# 4. Experiments
+
+We present experimental results to answer the following questions about TMI:
+
+Q1 Evaluation of transferability estimation (Section 4.2). Does TMI accurately select the best pretrained model compared to baselines? Is the transferability score measured by TMI highly correlated to the transfer performance?  
+Q2 Comparison of ICV measures (Section 4.3). What is the best measurement to evaluate Intra-Class Variance (ICV)?  
+Q3 Evaluation of generalization ability (Section 4.4). Does intra-class variance estimate generalization ability of a pre-trained model on a target task?  
+Q4 General applications (Section 4.5). Does TMI accurately estimate the transferability in general cases, such as selecting the best self-supervised pre-trained model and the best transferring layer?  
+Q5 Hyperparameter sensitivity (Section 4.6). How sensitive is TMI to hyperparameters?
+
+# 4.1. Experimental Setup
+
+We present pre-trained models, datasets, baselines, evaluation, estimator, and hyperparameters for our experiments.
+
+Pre-trained models. We use supervised model groups offered by Pytorch: ResNet, DenseNet, MnasNet, ShuffleNet V2, MobileNet V2, MobileNet V3, ResNeXt, Wide ResNet, EfficientNet, EfficientNet V2, and RegNet. We also utilize self-supervised model groups, which have only feature extractors without classifiers. We obtain Swin Transformer and Vision Transformer from Pytorch; we get MoCo v1, MoCo v2, Barlow Twins, SwAV, SimSiam, and Dino from Gitlab repositories. Note that we select fifty supervised pretrained models and eleven self-supervised pre-trained models from the aforementioned model groups.
+
+Datasets. We use seventeen image datasets summarized in Table 1. We randomly split each dataset into a train set and test set by 8:2 ratio, except for datasets that already have been separated.
+
+Baselines. We compare the proposed method with the following baselines. NCE [18] defines the transferability as negative conditional entropy between ground truths and predicted labels from a pre-trained model. LEEP [13] measures negative conditional entropy between ground truths and predicted average log-likelihood. H-Score [2] estimates the transferability by solving an HGR maximum correlation problem. LogME [25] proposes the logarithm of maximum evidence given extracted features. TransRate [5] measures the transferability by the mutual information between target labels and target representations. We omit the comparison with Taskonomy [26] and Task2Vec [1] due to tremendous time costs.
+
+Table 1: Summary of datasets.  
+
+<table><tr><td>Dataset</td><td># of instances (train / test)</td><td># of classes</td><td>Category</td></tr><tr><td>Caltech-1011</td><td>7,315 / 1,829</td><td>101</td><td>Multiple domains</td></tr><tr><td>Caltech-2562</td><td>24,485 / 6,122</td><td>256</td><td>Multiple domains</td></tr><tr><td>CIFAR-103</td><td>50,000 / 10,000</td><td>10</td><td>Multiple domains</td></tr><tr><td>CIFAR-1003</td><td>50,000 / 10,000</td><td>100</td><td>Multiple domains</td></tr><tr><td>MNIST4</td><td>60,000 / 10,000</td><td>10</td><td>Digit</td></tr><tr><td>FashionMNIST5</td><td>60,000 / 10,000</td><td>10</td><td>Fashion</td></tr><tr><td>SVHN6</td><td>73,257 / 26,032</td><td>10</td><td>Digit</td></tr><tr><td>FlowerPhotos7</td><td>2,936 / 734</td><td>5</td><td>Flower</td></tr><tr><td>EuroSAT8</td><td>21,600 / 5,400</td><td>10</td><td>Landscape</td></tr><tr><td>Chest X-Ray9</td><td>5,216 / 623</td><td>2</td><td>Medicine</td></tr><tr><td>VisDA10</td><td>152,397 / 72,372</td><td>212</td><td>Multiple domains</td></tr><tr><td>FGVC-Aircraft11</td><td>3,334 / 3,333</td><td>100</td><td>Aircraft</td></tr><tr><td>CUB-20012</td><td>9,430 / 2,358</td><td>200</td><td>Animal</td></tr><tr><td>Cars13</td><td>8,144 / 8,041</td><td>196</td><td>Auto</td></tr><tr><td>DTD14</td><td>4,512 / 1,128</td><td>47</td><td>Texture</td></tr><tr><td>Food-10115</td><td>80,800 / 20,200</td><td>101</td><td>Food</td></tr><tr><td>Oxford-IIIT Pet16</td><td>3,680 / 3,669</td><td>37</td><td>Animal</td></tr></table>
+
+<sup>1</sup>https://data.caltech.edu/records/mzrjq-6wc02  
+2 https://data.caltech.edu/records/nyy15-4j048  
+<sup>3</sup> https://www.cs.toronto.edu/~kriz/cifar.html  
+4 http://yann.learun.com/exdb/mnist/  
+5 https://github.com/zalandoresearch/fashion-mnist  
+$^{6}$  http://ufldl.stanford.edu/housenumbers/  
+$^7$  https://www.tensorflow.org/datasets/catalog/tf_flowers  
+8 https://github.com/phelber/eurosat  
+9 https://data.mendeley.com/datasets/rscbjbr9sj/2  
+10 http://ai.bu.edu/visda-2017/#browse  
+11 https://www Robots.ox.ac.uk/~vgg/data/fgvc-aircraft/  
+12 http://www.vision.caltech.edu/datasets/cub_200_2011/  
+13 https://ai.stanford.edu/~jkrause/cars/car_dataset.html  
+14 https://www.robots.ox.ac.uk/~vgg/data/dtd/  
+15 https://data.vision.ee.ethz.ch/cvl/datasetsextra/food-101/  
+16 https://www Robots.ox.ac.uk/~vgg/data/pets/
+
+Evaluation. We evaluate the performance of transferability measurement methods in two ways. First, do they accurately predict the best model? Second, are the transferability scores correlated to fine-tuned accuracy? We adopt Kendall correlation coefficient to measure the correlations between the transferability and fine-tuned accuracy. Kendall correlation coefficient tests the ordinal association between two measured quantities. Kendall correlation coefficient has values between -1 and 1, and correlations between two variables are high when the values are close to 1.
+
+Estimator. We adopt  $k$ -nearest estimator [8] to evaluate the continuous entropy in Equation (4). Compared to kernel density estimators [12, 17],  $k$ -nearest estimator vastly reduces errors by sacrificing time, especially when data are too big. TMI greatly reduces the size of data to be calculated at once because it computes conditional entropy of representation given label.
+
+Hyperparameters. We use greedy searching algorithm to find the best hyperparameters for fine-tuning. For transferability measurement, the only hyperparameter in our proposed TMI is the number  $n_k$  of neighbors, which is used in the entropy estimator [8]. We discuss the hyperparameter sensitivity to transferability in Section 4.6.
+
+Table 2: Ratio of target tasks where their top-5 best source models include the model estimated by each transferability measurement method. The best result is in bold. Note that TMI gives the best ratio, accurately selecting the best pretrained models in 14 out of 17 cases.
+
+<table><tr><td>Methods</td><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Top-5</td><td>3 / 17</td><td>5 / 17</td><td>4 / 17</td><td>8 / 17</td><td>9 / 17</td><td>14 / 17</td></tr></table>
+
+# 4.2. Evaluation of Transferability Estimation
+
+Remind that our goal is to quickly and accurately select the best pre-trained model among a pool of well-trained models. A pre-trained model is determined by two components, source data and model architecture. Thus, we conduct experiments to select the best model architecture and the best source data.
+
+Selecting the best model architecture. We evaluate how the transferability measurement methods accurately select the best source model for a target model. Table 2 shows the ratio of targets tasks where their top-5 best source models include the model estimated by each transferability measurement method. Note that TMI gives the best ratio, accurately selecting the best pre-trained models in 14 out of 17 cases, which is much better than the ratio  $9 / 17$  of the second-best model TransRate.
+
+We also compare the correlations between fine-tuned accuracies and transferability scores in Table 3. Note that our proposed TMI shows the best Kendall correlation in 13 out of 17 datasets, outperforming all the competitors. Some baselines even give negative coefficients, which means that their predictions are far from the correct ranking.
+
+Selecting the best source data. We fine-tune eighteen pre-trained models with the same model architecture ResNet-50, but different source datasets, including ImageNet and seventeen datasets introduced in Section 4.1. We predict the best source data and compare correlations with the ground truths. Table 4 shows that our proposed TMI accurately predicts the best source data in all the 17 cases, outperforming all the competitors. Our proposed TMI gives the best Kendall correlation consistently in 12 out of 17 cases, as shown in Table 5.
+
+# 4.3. Comparison of ICV Measures
+
+Intra-Class Variance (ICV) is a general concept widely used in metric learning. In addition to the conditional entropy introduced in Section 3, we additionally choose four metric learning methods, Contrast [4], Center [20], SNCA [22], and MS [19]. We use the intra-class variance parts of their objective functions to measure the transferability. We apply the five aforementioned methods to measure
+
+Table 3: Correlations between fine-tuned accuracies and transferability scores when transferring supervised pre-trained models trained on ImageNet. The best result is in bold. Our proposed TMI gives the best Kendall correlation in 13 out of 17 cases, outperforming competitors.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="6">Kendall Correlation Coefficient</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>0.429</td><td>0.425</td><td>0.316</td><td>0.362</td><td>0.474</td><td>0.524</td></tr><tr><td>Caltech-256</td><td>0.686</td><td>0.455</td><td>0.492</td><td>0.503</td><td>0.509</td><td>0.481</td></tr><tr><td>CIFAR-10</td><td>-0.140</td><td>0.324</td><td>0.275</td><td>0.323</td><td>0.364</td><td>0.527</td></tr><tr><td>CIFAR-100</td><td>-0.202</td><td>0.340</td><td>0.351</td><td>0.398</td><td>0.432</td><td>0.497</td></tr><tr><td>MNIST</td><td>0.057</td><td>0.282</td><td>0.083</td><td>0.137</td><td>0.038</td><td>0.137</td></tr><tr><td>FashionMNIST</td><td>-0.328</td><td>0.127</td><td>-0.038</td><td>0.121</td><td>0.125</td><td>0.467</td></tr><tr><td>SVHN</td><td>-0.023</td><td>0.091</td><td>-0.002</td><td>0.092</td><td>0.206</td><td>0.394</td></tr><tr><td>FlowerPhotos</td><td>0.117</td><td>0.139</td><td>0.224</td><td>0.321</td><td>0.392</td><td>0.499</td></tr><tr><td>EuroSAT</td><td>0.269</td><td>0.082</td><td>-0.005</td><td>0.102</td><td>0.051</td><td>0.085</td></tr><tr><td>Chest X-Ray</td><td>0.071</td><td>0.152</td><td>0.171</td><td>0.333</td><td>0.285</td><td>0.395</td></tr><tr><td>VisDA</td><td>0.298</td><td>0.240</td><td>0.329</td><td>0.334</td><td>0.304</td><td>0.341</td></tr><tr><td>FGVC-Aircraft</td><td>0.366</td><td>-0.011</td><td>0.216</td><td>0.498</td><td>0.421</td><td>0.500</td></tr><tr><td>CUB-200</td><td>0.415</td><td>0.098</td><td>0.325</td><td>0.410</td><td>0.519</td><td>0.523</td></tr><tr><td>Cars</td><td>0.270</td><td>0.506</td><td>0.175</td><td>0.432</td><td>0.455</td><td>0.521</td></tr><tr><td>DTD</td><td>-0.074</td><td>0.501</td><td>0.479</td><td>0.366</td><td>0.499</td><td>0.401</td></tr><tr><td>Food-101</td><td>0.261</td><td>0.515</td><td>0.395</td><td>0.421</td><td>0.448</td><td>0.537</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.536</td><td>0.527</td><td>0.418</td><td>0.493</td><td>0.500</td><td>0.576</td></tr></table>
+
+Table 4: Ratio of target tasks where the transferability measurement methods accurately find the best source data. The best method is in bold. TMI predicts the best source data in all cases, outperforming competitors.  
+
+<table><tr><td>Methods</td><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Top-1</td><td>1 / 17</td><td>2 / 17</td><td>3 / 17</td><td>2 / 17</td><td>13 / 17</td><td>17 / 17</td></tr></table>
+
+the correlations between fine-tuned accuracies and transferability scores. Conditional Entropy (CE) shows the best correlations than the other four metric learning measurements in 15 out of 17 datasets, outperforming all the competitors, as shown in Table 6. M. Boudiaf et al. [3] demonstrate that the conditional entropy is a lower bound of intra-class variance of the four metric learning methods.
+
+# 4.4. Evaluation of Generalization Ability
+
+We perform experiments on two control groups to demonstrate that transferability is relevant to the generalization ability of a pre-trained model on a target task. In Group A, we use a pre-trained ResNet-50 model trained on ImageNet and fine-tune it using the dataset  $D$ . In Group B, we train a ResNet-50 model trained from scratch on dataset  $D$  and fine-tune it on the same dataset. Dataset  $D$  is one of the datasets introduced in Section 4.1.
+
+Table 7 shows the difference of fine-tuning performance and that of transferability between two groups. “+” rep
+
+resents the difference is larger than 0 while “-” represents that is less than 0. Group A has higher transfer accuracies than Group B in most cases since models in Group A have learned a large amount of information from the ImageNet, which can potentially benefit downstream tasks.
+
+Our proposed TMI and TransRate have the same symbols for transfer accuracy in 14 out of 17 cases, while the other four baselines do not provide similar symbols. TransRate also estimates well, but it is computationally expensive since it measures the mutual information between target representation and label. In contrast, TMI is more efficient as it considers only intra-class features.
+
+# 4.5. General Applications
+
+Applying to self-supervised pre-trained models. TMI not only measures the transferability between supervised pre-trained models and classification tasks but also can be extended to self-supervised pre-trained models. We prepare eleven self-supervised pre-trained models already trained on ImageNet provided by Github repositories introduced in Section 4.1. NCE [18] and LEEP [13] cannot measure the transferability between a self-supervised pre-trained model and a downstream task because they dig out the relationship between target ground truth and predicted label or probability. Thus, we compare the proposed TMI to LogME [25], H-Score [2], and TransRate [5] in Table 8; TMI outperforms competitors in 13 out of 17 classification datasets.
+
+Table 5: Correlations between fine-tuned accuracies and transferability scores when transferring ResNet-50 models trained on different source data. The best result is in bold. TMI shows the best Kendall correlation in 12 out of 17 cases, outperforming all the competitors.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="6">Kendall Correlation Coefficient</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>-0.294</td><td>0.542</td><td>0.353</td><td>0.647</td><td>0.425</td><td>0.699</td></tr><tr><td>Caltech-256</td><td>-0.268</td><td>0.542</td><td>0.437</td><td>0.660</td><td>0.477</td><td>0.699</td></tr><tr><td>CIFAR-10</td><td>-0.216</td><td>0.373</td><td>0.383</td><td>0.490</td><td>0.399</td><td>0.630</td></tr><tr><td>CIFAR-100</td><td>-0.098</td><td>0.556</td><td>0.686</td><td>0.345</td><td>0.542</td><td>0.699</td></tr><tr><td>MNIST</td><td>-0.061</td><td>0.439</td><td>0.372</td><td>-0.007</td><td>0.115</td><td>0.162</td></tr><tr><td>FashionMNIST</td><td>-0.086</td><td>0.362</td><td>0.430</td><td>0.475</td><td>0.046</td><td>0.150</td></tr><tr><td>SVHN</td><td>0.039</td><td>0.285</td><td>-0.033</td><td>0.393</td><td>0.170</td><td>0.407</td></tr><tr><td>FlowerPhotos</td><td>-0.359</td><td>0.412</td><td>0.456</td><td>0.307</td><td>0.490</td><td>0.516</td></tr><tr><td>EuroSAT</td><td>-0.388</td><td>0.428</td><td>0.490</td><td>0.322</td><td>0.467</td><td>0.704</td></tr><tr><td>Chest X-Ray</td><td>-0.072</td><td>0.164</td><td>0.359</td><td>-0.111</td><td>0.425</td><td>-0.123</td></tr><tr><td>VisDA</td><td>-0.538</td><td>0.498</td><td>0.094</td><td>-0.039</td><td>0.498</td><td>0.546</td></tr><tr><td>FGVC-Aircraft</td><td>-0.033</td><td>0.373</td><td>0.339</td><td>0.556</td><td>0.399</td><td>0.688</td></tr><tr><td>CUB-200</td><td>-0.503</td><td>0.582</td><td>0.455</td><td>0.464</td><td>0.647</td><td>0.723</td></tr><tr><td>Cars</td><td>-0.294</td><td>0.529</td><td>0.403</td><td>0.752</td><td>0.542</td><td>0.804</td></tr><tr><td>DTD</td><td>-0.438</td><td>0.595</td><td>0.393</td><td>0.647</td><td>0.542</td><td>0.757</td></tr><tr><td>Food-101</td><td>-0.020</td><td>0.281</td><td>0.092</td><td>0.046</td><td>0.229</td><td>0.225</td></tr><tr><td>Oxford-IIIT Pet</td><td>-0.268</td><td>0.569</td><td>0.660</td><td>0.503</td><td>0.516</td><td>0.248</td></tr></table>
+
+Table 6: Comparison of intra-class variance measurements. Conditional Entropy (CE) shows the best correlations consistently in 15 out of 17 cases. The best result is in bold.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="5">Kendall Correlation Coefficient</td></tr><tr><td>Contrast</td><td>Center</td><td>SNCA</td><td>MS</td><td>CE</td></tr><tr><td>Caltech-101</td><td>0.274</td><td>0.283</td><td>0.380</td><td>0.358</td><td>0.442</td></tr><tr><td>Caltech-256</td><td>0.295</td><td>0.347</td><td>0.261</td><td>0.347</td><td>0.354</td></tr><tr><td>CIFAR-10</td><td>0.328</td><td>0.322</td><td>0.275</td><td>0.299</td><td>0.369</td></tr><tr><td>CIFAR-100</td><td>0.374</td><td>0.282</td><td>0.207</td><td>0.200</td><td>0.376</td></tr><tr><td>MNIST</td><td>0.383</td><td>0.267</td><td>0.306</td><td>0.228</td><td>0.374</td></tr><tr><td>FashionMNIST</td><td>0.332</td><td>0.225</td><td>0.260</td><td>0.337</td><td>0.374</td></tr><tr><td>SVHN</td><td>0.303</td><td>0.359</td><td>0.369</td><td>0.360</td><td>0.383</td></tr><tr><td>FlowerPhotos</td><td>0.209</td><td>0.263</td><td>0.291</td><td>0.277</td><td>0.372</td></tr><tr><td>EuroSAT</td><td>0.292</td><td>0.237</td><td>0.367</td><td>0.268</td><td>0.400</td></tr><tr><td>Chest X-Ray</td><td>0.292</td><td>0.289</td><td>0.210</td><td>0.232</td><td>0.357</td></tr><tr><td>VisDA</td><td>0.259</td><td>0.352</td><td>0.200</td><td>0.329</td><td>0.352</td></tr><tr><td>FGVC-Aircraft</td><td>0.220</td><td>0.322</td><td>0.244</td><td>0.315</td><td>0.351</td></tr><tr><td>CUB-200</td><td>0.368</td><td>0.252</td><td>0.379</td><td>0.385</td><td>0.431</td></tr><tr><td>Cars</td><td>0.351</td><td>0.285</td><td>0.398</td><td>0.268</td><td>0.248</td></tr><tr><td>DTD</td><td>0.236</td><td>0.322</td><td>0.228</td><td>0.206</td><td>0.327</td></tr><tr><td>Food-101</td><td>0.322</td><td>0.267</td><td>0.246</td><td>0.269</td><td>0.484</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.301</td><td>0.202</td><td>0.363</td><td>0.274</td><td>0.449</td></tr></table>
+
+Selecting the best transferring layer. Given a ResNet-18 trained on ImageNet, we can easily determine the best transferring layer, which produces the best fine-tuning performance for a target task by measuring transferability. We compare the correlations of LogME [25], H-Score [2], and TransRate to select the best transferring layer for fine
+
+Table 7: Comparison of the consistency of fine-tuning accuracy and transferability. “+” represents the difference is larger than 0 while “-” represents the difference is less than 0. TMI has consistent signs with fine-tuning accuracy in 14 out of 17 cases, outperforming its competitors.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Accuracy (A-B)</td><td colspan="6">Transferability (A-B)</td></tr><tr><td>NCE</td><td>LEEP</td><td>LogME</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Caltech-256</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CIFAR-10</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CIFAR-100</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>MNIST</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>FashionMNIST</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>SVHN</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>FlowerPhotos</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>EuroSAT</td><td>+</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>Chest X-Ray</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>VisDA</td><td>+</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>FGVC-Aircraft</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>CUB-200</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Cars</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>DTD</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Food-101</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Oxford-IIIT Pet</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr></table>
+
+tuning. Theoretically, LogME [25] and H-Score [2] cannot deal with layer selection because they rely on features from the ultimate layer of the feature extractor. However, we extract features from each layer and directly put them into LogME and H-Score algorithms to calculate the trans
+
+Table 8: Correlations between fine-tuned accuracies and transferability scores when transferring self-supervised pretrained models to classification tasks. The best result is in bold. TMI shows the best Kendall correlation consistently in 13 out of 17 cases, outperforming all the competitors.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="4">Kendall Correlation Coefficient</td></tr><tr><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>-0.278</td><td>0.183</td><td>0.000</td><td>0.367</td></tr><tr><td>Caltech-256</td><td>-0.561</td><td>0.345</td><td>0.164</td><td>0.600</td></tr><tr><td>CIFAR-10</td><td>-0.337</td><td>0.018</td><td>0.164</td><td>0.624</td></tr><tr><td>CIFAR-100</td><td>-0.561</td><td>0.673</td><td>0.236</td><td>0.697</td></tr><tr><td>MNIST</td><td>-0.574</td><td>-0.262</td><td>-0.426</td><td>0.246</td></tr><tr><td>FashionMNIST</td><td>-0.330</td><td>-0.220</td><td>-0.330</td><td>0.094</td></tr><tr><td>SVHN</td><td>0.055</td><td>0.309</td><td>-0.345</td><td>0.075</td></tr><tr><td>FlowerPhotos</td><td>0.389</td><td>-0.241</td><td>0.130</td><td>0.075</td></tr><tr><td>EuroSAT</td><td>-0.073</td><td>-0.037</td><td>0.037</td><td>-0.208</td></tr><tr><td>Chest X-Ray</td><td>-0.037</td><td>-0.037</td><td>-0.257</td><td>0.175</td></tr><tr><td>VisDA</td><td>0.550</td><td>0.055</td><td>0.309</td><td>-0.597</td></tr><tr><td>FGVC-Aircraft</td><td>-0.345</td><td>0.091</td><td>-0.309</td><td>0.150</td></tr><tr><td>CUB-200</td><td>-0.112</td><td>-0.018</td><td>0.091</td><td>0.564</td></tr><tr><td>Cars</td><td>0.000</td><td>0.382</td><td>-0.200</td><td>0.262</td></tr><tr><td>DTD</td><td>0.382</td><td>0.018</td><td>0.224</td><td>0.455</td></tr><tr><td>Food-101</td><td>-0.411</td><td>-0.018</td><td>-0.018</td><td>0.204</td></tr><tr><td>Oxford-IIIT Pet</td><td>-0.585</td><td>0.294</td><td>0.073</td><td>0.807</td></tr></table>
+
+![](images/c98a0030dc307929921556676a29d661d875515f1787899a5231af92281004ef.jpg)  
+Figure 3: Sensitivity to the hyperparameter  $n_k$  in entropy estimator.
+
+ferability. The size of extracted features is quite large in the early layers, resulting in much higher computational cost for transferability measurement. To reduce this cost, we add an avgpool2d layer to the end of transferring layer to reduce the dimension of extracted features. Table 9 shows that TMI achieves the highest correlation coefficients in 16 out of 17 datasets. Correlation coefficients of MNIST, SVHN, and Chest X-Ray are much lower than those of the other datasets because their ranges of minimum and maximum accuracies obtained by layer transfer are smaller than  $0.5\%$ , making it challenging to rank these dataset accurately.
+
+# 4.6. Hyperparameter Sensitivity
+
+An advantage of TMI is that it does not train models on target datasets because it is time-consuming and sensitive to hyperparameters which may cause different results. The only hyperparameter in TMI is  $n_k$  used in the entropy estimator. In Figure. 3, the correlation between pre-trained
+
+Table 9: Correlations between fine-tuned accuracies and transferability scores when transferring specific layers of pre-trained models. The best result is in bold. TMI shows the best correlations consistently in 16 out of 17 cases.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="4">Kendall Correlation Coefficient</td></tr><tr><td>LogMe</td><td>H-Score</td><td>TransRate</td><td>TMI</td></tr><tr><td>Caltech-101</td><td>0.346</td><td>0.935</td><td>0.752</td><td>0.941</td></tr><tr><td>Caltech-256</td><td>0.973</td><td>0.317</td><td>0.800</td><td>0.984</td></tr><tr><td>CIFAR-10</td><td>0.360</td><td>0.743</td><td>0.704</td><td>0.746</td></tr><tr><td>CIFAR-100</td><td>0.460</td><td>0.774</td><td>0.774</td><td>0.801</td></tr><tr><td>MNIST</td><td>0.180</td><td>0.055</td><td>0.179</td><td>0.184</td></tr><tr><td>FashionMNIST</td><td>0.294</td><td>0.241</td><td>0.298</td><td>0.308</td></tr><tr><td>SVHN</td><td>0.087</td><td>0.106</td><td>0.040</td><td>0.113</td></tr><tr><td>FlowerPhotos</td><td>0.779</td><td>0.198</td><td>0.687</td><td>0.792</td></tr><tr><td>EuroSAT</td><td>0.691</td><td>0.709</td><td>0.495</td><td>0.735</td></tr><tr><td>Chest X-Ray</td><td>0.113</td><td>0.154</td><td>0.100</td><td>0.054</td></tr><tr><td>VisDA</td><td>0.665</td><td>0.866</td><td>0.826</td><td>0.904</td></tr><tr><td>FGVC-Aircraft</td><td>0.791</td><td>0.302</td><td>0.595</td><td>0.817</td></tr><tr><td>CUB-200</td><td>0.317</td><td>0.970</td><td>0.787</td><td>0.954</td></tr><tr><td>Cars</td><td>0.330</td><td>0.869</td><td>0.752</td><td>0.882</td></tr><tr><td>DTD</td><td>0.924</td><td>0.401</td><td>0.761</td><td>0.931</td></tr><tr><td>Food-101</td><td>0.409</td><td>0.948</td><td>0.869</td><td>0.970</td></tr><tr><td>Oxford-IIIT Pet</td><td>0.349</td><td>0.970</td><td>0.826</td><td>0.990</td></tr></table>
+
+models and TMI shows consistency in most ranges for  $n_k$ .
+
+# 5. Conclusion
+
+We propose TMI, a fast, accurate, and general algorithm to measure the transferability. TMI quantifies the degree of generalization of a pre-trained model to a new task by estimating the intra-class feature variance. We conduct experiments on seventeen datasets with fifty supervised pretrained models and eleven self-supervised pre-trained models. TMI selects the top-5 pre-trained architecture in 14 out of 17 cases and selects the best source data in all cases, outperforming its competitors. We measure the correlation between the fine-tuning performance and transferability, and our proposed TMI outperforms competitors in 13 out of 17 cases.
+
+# Acknowledgments
+
+This work was supported by Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) [No.2020-0-00894, Flexible and Efficient Model Compression Method for Various Applications and Environments], [No.2021-0-01343, Artificial Intelligence Graduate School Program (Seoul National University)], and [NO.2021-0-02068, Artificial Intelligence Innovation Hub (Artificial Intelligence Institute, Seoul National University)]. The Institute of Engineering Research at Seoul National University provided research facilities for this work. The ICT at Seoul National University provides research facilities for this study. U Kang is the corresponding author.
+
+# References
+
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+[2] Yajie Bao, Yang Li, Shao-Lun Huang, Lin Zhang, Lizhong Zheng, Amir Roshan Zamir, and Leonidas J. Guibas. An information-theoretic approach to transferability in task transfer learning. In ICIP, 2019.  
+[3] Malik Boudiaf, Jérôme Rony, Imtiaz Masud Ziko, Eric Granger, Marco Pedersoli, Pablo Piantanida, and Ismail Ben Ayed. A unifying mutual information view of metric learning: Cross-entropy vs. pairwise losses. In Computer Vision - ECCV 2020 - 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part VI, volume 12351 of Lecture Notes in Computer Science, pages 548-564. Springer, 2020.  
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+        "text": "FastRecon: Few-shot Industrial Anomaly Detection via Fast Feature Reconstruction",
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+        "text": "Zheng Fang $^{1,2\\dagger}$  Xiaoyang Wang $^{1,2\\dagger}$  Haocheng Li $^{3}$  Jiejie Liu $^{1,3}$  Qiugui Hu $^{3}$  Jimin Xiao $^{1*}$ $^{1}$ XJTLU  $^{2}$ Metavisioncn  $^{3}$ Dinnar Automation Technology  \n{fz.jun26th, haocheng.li789}@gmail.com,  \n{xiaoyang.wang20, jiejie.liu22}@student.xjtlu.edu.cn, jimin.xiao@xjtlu.edu.cn",
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+        "text": "In industrial anomaly detection, data efficiency and the ability for fast migration across products become the main concerns when developing detection algorithms. Existing methods tend to be data-hungry and work in the one-model-one-category way, which hinders their effectiveness in real-world industrial scenarios. In this paper, we propose a few-shot anomaly detection strategy that works in a low-data regime and can generalize across products at no cost. Given a defective query sample, we propose to utilize a few normal samples as a reference to reconstruct its normal version, where the final anomaly detection can be achieved by sample alignment. Specifically, we introduce a novel regression with distribution regularization to obtain the optimal transformation from support to query features, which guarantees the reconstruction result shares visual similarity with the query sample and meanwhile maintains the property of normal samples. Experimental results show that our method significantly outperforms previous state-of-the-art at both image and pixel-level AUROC performances from 2 to 8-shot scenarios. Besides, with only a limited number of training samples (less than 8 samples), our method reaches competitive performance with vanilla AD methods which are trained with extensive normal samples. The code is available at https://github.com/FzJun26th/ FastRecon.",
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+        "text": "1. Introduction",
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+        "text": "Anomaly detection (AD) in computer vision aims to identify rare observations that deviate significantly from the majority of the data and do not conform to a well-defined notion of normal behavior [4]. In industrial settings, AD is widely used to detect the anomalies on the surface of the product, which has been attracting a lot of attention re",
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+        "image_caption": [
+            "Figure 1: Comparison of our FastRecon with (a) Vanilla Reconstruction-based method and (b) Vanilla Feature-Matching-based method. Compared with (a), our method needs no training session and requires less normal samples for feature reconstruction. Compared with (b), our method applies direct pixel-to-pixel alignment between the test sample and reconstruction result for anomaly detection without an exhaustive feature searching process, thus being more efficient in inference."
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+    {
+        "type": "text",
+        "text": "cently [31, 16, 9]. The difficulty of industrial AD lies in the diversity of the defects, which can take different forms, from subtle flaws like slight bruises, to obvious ones such as missing components [3]. This makes it difficult to detect and classify anomalies. Primal researches focus on utilizing supervised learning to tackle industrial AD [5, 17]. However, such a learning paradigm often requires an exhaustive set of anomalous samples for each category, which can be",
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+    {
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+        "text": "This ICCV paper is the Open Access version, provided by the Computer Vision Foundation. Except for this watermark, it is identical to the accepted version; the final published version of the proceedings is available on IEEE Xplore.",
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+        "text": "$\\dagger$  Equal contribution.",
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+    {
+        "type": "page_footnote",
+        "text": "* Corresponding author.",
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+    },
+    {
+        "type": "page_number",
+        "text": "17481",
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+    {
+        "type": "text",
+        "text": "naturally rare to collect, making it inefficient for real-world industrial scenarios.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recent works pay attention to learning with limited or even no supervision to overcome the problem caused by the lack of anomalous samples. As shown in Fig. 1 (a), reconstruction-based industrial AD [34, 10, 26, 14] utilizes generative models such as autoencoder to learn the distribution of normal samples through the cumbersome training process. As in Fig. 1 (b), recent researches favor feature-matching strategy [28, 7] that conducts feature comparison between the test sample and memory bank of normal features, where the feature searching causes high computational complexity. Both lines of work require massive normal samples either for model training or memory construction, only to achieve detection for a single product category, and fail to quickly generalize across products.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we state that well-generalized industrial AD can be achieved in an extremely data-efficient way. We propose a training-free, few-shot AD method named FastRecon, to meet the demand for industrial scenarios, which works under the situation that only a limited number of normal samples are available. Given a query sample to detect, we propose to construct its normal version at the feature level through a linear transformation based on features extracted from limited normal samples. This constructed sample shares low-frequency features with the original query image but differs in the defect regions, making the anomaly apparent when directly compared to the original. The comparisons on the methodology of our method and previous mainstream works are shown in Fig. 1. The key lies in finding the optimal transformation. On the one hand, the transformation should bring the constructed sample somewhere close to the query sample so they share visual features on the normal parts. On the other hand, the constructed sample should maintain the property of a normal sample so the difference between the two samples can fully reveal the anomaly regions. With this goal, we proposed a distributional regularization term upon the classical ridge regression algorithm. We first utilize normal features to fit a Gaussian distribution. When finding the transformation from normal features to the query sample, we simultaneously push the reconstructed sample toward the center of the built distribution to enhance its normality. Note that our proposed optimization process admits a closed-form solution just as ridge regression, which allows fast computation with low cost, ensuring the adaptation and migration of our method from old products to novel ones.",
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+        "text": "Experiments on the benchmark datasets for industrial defection, MVTec [3] and MDPP [15], as well as comparisons with various existing state-of-the-art FSAD methods [19, 13], indicate the effectiveness of our model. Compared with previous state-of-the-art [13], our method achieves improvements of  $5.27\\%$ ,  $6.04\\%$ ,  $3.99\\%$  on MVTec, and",
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+        "text": "10.25%, 11.55%, 10.60% on MPDD, at image-level AUROC under 2-shot, 4-shot and 8-shot scenarios, respectively. The contributions of this work are summarized below:",
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+        "list_items": [
+            "1. We propose a few-shot and category-agnostic industrial AD strategy which adopts sample alignment between the original test sample and its reconstructed normal version.",
+            "2. We propose a regression algorithm with distribution regularization which has a closed-form solution for the transformation estimation, keeping the reconstructed result close to the query image while maintaining the property of normal samples.",
+            "3. Our proposed method achieves state-of-the-art few-shot AD performance under various datasets and settings."
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+        "text": "2. Related Work",
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+        "text": "2.1. Anomaly Detection",
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+    {
+        "type": "text",
+        "text": "Industrial visual anomaly detection (AD) is a rising topic in computer vision. The mainstream AD methods can be categorized as reconstruction-based model and feature-matching-based models. Previous reconstruction-based models classified anomalous samples either though reconstruction loss [2, 33, 30] or via comparison between reconstructed normal samples and test sample [21, 22, 29] with generative models. Another line of works, feature-matching-based models, take full advantage of pre-trained features. For example, SPADE [6] firstly put forward memory banks comprising various feature hierarchies for fine-grained, KNN-based [8] anomaly segmentation and image-level anomaly detection. Similarly, PaDiM [7] and PatchCore [28] investigated AD performance on MVTec in unsupervised learning settings. However, both two lines of methods show greatly high data dependence which would hinder their industrial application.",
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+        "type": "text",
+        "text": "Few-shot anomaly detection (FSAD) has developed to meet the demands of fast manufacturing changeover. TDG [23] proposes a hierarchical generative model that captures the multi-scale patch distribution of each support image. DiffNet [19] leverages the descriptiveness of features extracted by convolutional neural networks to estimate their density using a normalizing flow, which is a tool well-suited to estimate distributions from a few support samples. RegAD [13] explored a new paradigm for FSAD, by learning a common model shared among multiple categories in a meta-learning style. Despite of their decent inference performances under few-shot settings, the training process be",
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+        "image_caption": [
+            "Figure 2: Overview of our method. Feature maps of each query sample and support samples are exacted by a pre-trained encoder. Features from support images are aggregated into a support feature pool. This pool is down-sampled through greedy coreset selection as  $S$  to reduce data redundancy and improve inference speed. The coreset  $S$  and the original query feature map  $Q$  are then input to our proposed regression with distribution regularization as shown in the grey region. An optimal transformation  $\\bar{W}$  between  $S$  and  $Q$  is obtained by the regression to make sure the reconstructed sample  $\\bar{W} S$ , denoted as  $\\bar{Q}$ , to share similarity with  $Q$  but keeps all the property of normal samples. Finally, we align  $\\bar{Q}$  and  $Q$  for direct comparison to obtaining the anomaly estimation."
+        ],
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+    {
+        "type": "text",
+        "text": "comes the main obstacle for fast model migration in real industrial scenarios.",
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+    {
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+        "text": "3. Methodology",
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+        "text": "In this section, we introduce the details of FastRecon. Firstly, we define the one-class classification problem in industrial anomaly detection. FastRecon contains 3 parts: feature extraction and coreset selection, regression with distribution regularization, and anomaly detection with reconstruction. The overview of our method is shown in Fig. 2.",
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+        "text": "3.1. Problem Definition",
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+        "text": "We formally define the problem for one-class classification of industrial anomaly detection. Following the few-shot learning setting, in an  $n$ -way and  $k$ -shot episode, a support set consisting of normal samples from  $n$  categories is given, i.e.,  $S = \\bigcup_{i=1}^{n} T_i$ , where the subset  $T_i$  consists of  $k$  normal samples from category  $c_i$ . During test time, a query sample, whether normal or anomalous, is presented. The model predicts whether or not the sample is anomalous at the pixel and image level.",
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+        "text": "Existing studies [7, 28] have shown that extracted features from pre-trained CNNs perform well in the anomaly",
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+        "text": "detection task. We also utilize a pre-trained ResNet-50 to output a patch-level feature map of an image.  $\\phi_{i,j}$  is used to denote hierarchy-level  $j$  of the pre-trained network  $\\phi$  for the image  $x_{i} \\in X$  (dataset  $X$ ), where  $j \\in \\{1,2,3,4\\}$ . We adopt the intermediate features of the hierarchy-level, i.e.,  $j \\in \\{2,3\\}$ . The resulting feature map has a dimension of  $H \\times W \\times D$ , where  $H$ ,  $W$  are the height and width of the largest activation map used to generate the embedding, and  $D$  is the total number of channels of both activation maps used to generate embedding.",
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+        "text": "To facilitate calculations, the feature map is reshaped from 3-dimension  $H \\times W \\times D$  to 2-dimension  $R \\times D$ , where  $R = H \\times W$ . Features in the reshaped map represent embedding vectors, i.e., the local aware patch features. In a  $k$ -shot setting, our ultimate support feature pool is obtained by concatenating features of  $k$  support images, with a size of  $k \\times R \\times D$ . Besides, following [18], we take a coreset sampling operation to get a coreset  $S \\in R^{c \\times d}$ , where  $c$  is the number of clustering centers computed with K-greedy algorithm. The coverage of  $S$  is comparable to the original support feature pools but with a smaller scale so that the computational complexity and the performance of detection could be balanced. For a query image, with the same pretrained network, it is also extracted to patch-level feature map  $Q$ . The process of feature extraction and coreset is",
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+        "image_caption": [
+            "(d)",
+            "Figure 3: Visual comparison of decoding results on reconstructed features. We train a decoder on features of the whole dataset for image recovery, which is used to probe the transformed features from the encoder. (a) The original query image with defects region in red boxes. (b) Decoding results on original query features. (c) Decoding results on transformed features by ridge regression. The defects regions are reconstructed as on original query features. (d) Decoding results on transformed features by the proposed regression, which shares visual similarity with the original query image but with the property of normal samples."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            348,
+            88,
+            429,
+            212
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "shown in the brown region in Fig. 2.",
+        "bbox": [
+            76,
+            429,
+            318,
+            445
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3. Regression with Distribution Regularization",
+        "text_level": 1,
+        "bbox": [
+            76,
+            452,
+            450,
+            469
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Why Ridge Regression fails. The goal is to reconstruct query feature map  $Q$  as a linear transformation of S-coreset  $S$  by finding an optimal transformation weight  $W$  such that  $WS \\approx Q$  and meanwhile  $WS$  has most properties of normal samples. Searching for optimal weight  $W$  amounts to solving the linear least-squares problem [25]. With original ridge regression, the optimization takes the following form:",
+        "bbox": [
+            76,
+            476,
+            468,
+            583
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {\\boldsymbol {W}} = \\underset {\\boldsymbol {W}} {\\arg \\min } \\| \\boldsymbol {Q} - \\boldsymbol {W} \\boldsymbol {S} \\| ^ {2} + \\lambda \\| \\boldsymbol {W} \\| ^ {2}, \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            137,
+            594,
+            468,
+            619
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\lambda$  is the weight of the penalty term and  $\\| \\cdot \\|$  representing the Frobenius norm. The penalty term in ridge regression, i.e.,  $\\| W\\|^2$ , ensures tractability when the linear system is over- or under-constrained. However, the transformation obtained from ridge regression can cause the reconstructed sample to deviate significantly from the distribution of normal samples, i.e., after the transformation, the result  $WS$  can be too close to  $Q$  so that features of defect regions are reconstructed as in  $Q$ . As shown in Fig. 3 column (c), the decoding results on reconstructed features share excessively high similarity with the original query image with defects regions being reconstructed.",
+        "bbox": [
+            75,
+            626,
+            468,
+            806
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The distribution regularization. In our work, we propose a distributional regularization term to replace the original penalty term in ridge regression, which effectively enhances the normality of the reconstruction result  $W S$ . We first model the distribution of normal feature maps by an isotropic Gaussian density function denoted as  $f(\\boldsymbol{X}|\\boldsymbol{\\mu},\\boldsymbol{\\Sigma})$",
+        "bbox": [
+            75,
+            810,
+            468,
+            901
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\pmb{\\mu},\\pmb{\\Sigma}$  is the mean and co-variance estimated by the support set feature maps. We design the regularization term as  $|f(WS) - f(\\pmb{\\mu})|$  and the proposed regression has the form as:",
+        "bbox": [
+            498,
+            90,
+            890,
+            150
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {\\boldsymbol {W}} = \\underset {\\boldsymbol {W}} {\\arg \\min } \\| \\boldsymbol {Q} - \\boldsymbol {W} \\boldsymbol {S} \\| ^ {2} + \\lambda | f (\\boldsymbol {W} \\boldsymbol {S}) - f (\\boldsymbol {\\mu}) |, \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            509,
+            160,
+            890,
+            186
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\lambda$  denotes the coefficient that controls the contribution of the regularization. When searching the optimal transformation  $W$ , the optimization of the first term  $\\|Q - WS\\|^2$  naturally pushes  $WS$  towards original query feature maps  $Q$ , while with the proposed regularization  $|f(WS) - f(\\pmb{\\mu})|$ , we force the constructed  $WS$  to lie in the high probability density region in the distribution, that is, somewhere close to the distribution center. The visualization of distribution regularization is shown in the grey region of Fig. 2.",
+        "bbox": [
+            498,
+            196,
+            890,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The approximation for closed-form solution. The density function inside the regularization term in Eq. (2) prevents us from having a closed-form solution for the regression, which hinders fast calculation of the transformation. Thus, we propose an approximated version of Eq. (2) with the help of Taylor's theorem for multivariable functions on the distribution density function  $f$ . Taylor's expansion for  $f(x)$  at  $a$  is shown as follows:",
+        "bbox": [
+            498,
+            351,
+            890,
+            472
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nf (\\boldsymbol {x}) \\approx f (\\boldsymbol {a}) + \\nabla f (\\boldsymbol {a}) (\\mathbf {x} - \\boldsymbol {a}) + \\frac {1}{2} (\\mathbf {x} - \\boldsymbol {a}) ^ {T} H f (\\boldsymbol {a}) (\\mathbf {x} - \\boldsymbol {a}), \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            500,
+            482,
+            890,
+            523
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $Hf(\\pmb{a})$  denotes Hessian matrix of  $f(\\pmb{x})$  at  $\\pmb{a}$ . And we approximate the density function with the first and second-order term of Taylor's expansion, where the first-order term is 0 at  $\\pmb{\\mu}$ . Following Eq. (3), we expand the probability density function  $f(WS)$  at  $\\pmb{\\mu}$  as:",
+        "bbox": [
+            498,
+            523,
+            890,
+            599
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} f (\\boldsymbol {W} \\boldsymbol {S}) \\approx f (\\boldsymbol {\\mu}) + \\nabla f (\\boldsymbol {\\mu}) (\\boldsymbol {W} \\boldsymbol {S} - \\boldsymbol {\\mu}) \\\\ + \\frac {1}{2} (\\boldsymbol {W} \\boldsymbol {S} - \\boldsymbol {\\mu}) ^ {T} H f (\\boldsymbol {\\mu}) (\\boldsymbol {W} \\boldsymbol {S} - \\boldsymbol {\\mu}). \\tag {4} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            504,
+            609,
+            890,
+            669
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since our assumed data distribution is isotropic Gaussian, its covariance matrix can be represented as  $\\pmb{\\Sigma} = \\sigma^2\\pmb{I}$ , where  $\\sigma^2$  is the variance for all dimensions. Then, we have the term  $\\nabla f(\\pmb{\\mu})(\\pmb{W}\\pmb{S} - \\pmb{\\mu})$  as 0, and  $Hf(\\pmb{\\mu})$  equals the  $\\pmb{\\Sigma}^{-1}$  which is the inverse of the covariance matrix. Then, by placing  $f(\\pmb{\\mu})$  to the left, we have",
+        "bbox": [
+            498,
+            670,
+            890,
+            762
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} f (\\boldsymbol {W} \\boldsymbol {S}) - f (\\boldsymbol {\\mu}) \\approx \\frac {1}{2} \\left(\\boldsymbol {W} \\boldsymbol {S} - \\boldsymbol {\\mu}\\right) ^ {T} \\boldsymbol {\\Sigma} ^ {- 1} (\\boldsymbol {W} \\boldsymbol {S} - \\boldsymbol {\\mu}) \\\\ = \\frac {1}{2 \\sigma^ {2}} \\left(\\boldsymbol {W} \\boldsymbol {S} - \\boldsymbol {\\mu}\\right) ^ {T} \\boldsymbol {I} \\left(\\boldsymbol {W} \\boldsymbol {S} - \\boldsymbol {\\mu}\\right) \\tag {5} \\\\ = \\frac {1}{2 \\sigma^ {2}} \\| \\boldsymbol {W} \\boldsymbol {S} - \\boldsymbol {\\mu} \\| ^ {2}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            506,
+            771,
+            890,
+            861
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $I$  is an identity matrix. Then we merge the coefficient  $\\frac{1}{2\\sigma^2}$  into  $\\lambda$  and the approximated version of the optimization",
+        "bbox": [
+            500,
+            869,
+            890,
+            902
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "17484",
+        "bbox": [
+            480,
+            944,
+            517,
+            955
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "target in Eq. (2) is:",
+        "bbox": [
+            76,
+            90,
+            204,
+            104
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {W} = \\underset {W} {\\arg \\min } \\| Q - W S \\| ^ {2} + \\lambda \\| W S - \\mu \\| ^ {2}. \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            102,
+            112,
+            468,
+            138
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finding the optimal transformation. Knowing that the following optimization target of least square error (LSE) of",
+        "bbox": [
+            76,
+            146,
+            468,
+            176
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {W} ^ {\\prime} = \\underset {\\boldsymbol {W}} {\\arg \\min } \\| \\boldsymbol {Q} - \\boldsymbol {W} \\boldsymbol {S} \\| ^ {2}, \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            168,
+            188,
+            468,
+            214
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "has the closed-form solution as",
+        "bbox": [
+            76,
+            219,
+            285,
+            232
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {W} ^ {\\prime} = \\boldsymbol {Q} \\boldsymbol {S} ^ {\\top} \\left(\\boldsymbol {S} ^ {\\top} \\boldsymbol {S}\\right) ^ {- 1}, \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            238,
+            468,
+            267
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "to utilize the conclusion from Eq. (8), we transform Eq. (6) by variable substitution to fit the form of Eq. (8). Let  $Q_*$  and  $S_*$  be the column-wise augmented version of  $Q$  and  $S$  respectively:",
+        "bbox": [
+            76,
+            273,
+            468,
+            334
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {S} _ {*} = \\left[ \\boldsymbol {S} \\sqrt {\\lambda} \\boldsymbol {S} \\right], \\quad \\boldsymbol {Q} _ {*} = \\left[ \\boldsymbol {Q} \\sqrt {\\lambda} \\boldsymbol {\\mu} \\right], \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            143,
+            340,
+            468,
+            359
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\lambda$  is the coefficient of the regularization term which is always set as a positive number. Hence, equality holds for",
+        "bbox": [
+            76,
+            367,
+            468,
+            411
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\| Q _ {*} - W S _ {*} \\right\\| ^ {2} = \\left\\| Q - W S \\right\\| ^ {2} + \\lambda \\left\\| \\boldsymbol {\\mu} - W S \\right\\| ^ {2}, \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            93,
+            419,
+            468,
+            450
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and the form in Eq. (6) can be written as",
+        "bbox": [
+            76,
+            452,
+            346,
+            465
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {W} = \\underset {W} {\\arg \\min } \\| Q _ {*} - W S _ {*} \\| ^ {2}. \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            163,
+            473,
+            468,
+            500
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Based on the conclusion in Eq. (8), the solution for our optimization process is",
+        "bbox": [
+            76,
+            507,
+            468,
+            537
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\bar {W} = Q _ {*} S _ {*} ^ {\\top} \\left(S _ {*} ^ {\\top} S _ {*}\\right) ^ {- 1} \\tag {12} \\\\ = \\left(\\boldsymbol {Q} \\boldsymbol {S} ^ {\\top} + \\lambda \\cdot \\boldsymbol {\\mu} \\boldsymbol {S} ^ {\\top}\\right) \\left(\\boldsymbol {S} ^ {\\top} \\boldsymbol {S} + \\lambda \\cdot \\boldsymbol {S} \\boldsymbol {S} ^ {\\top}\\right) ^ {- 1}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            89,
+            544,
+            468,
+            603
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4. Anomaly Detection with Reconstruction",
+        "text_level": 1,
+        "bbox": [
+            76,
+            609,
+            419,
+            625
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "After getting the reconstruction query feature maps  $\\bar{Q}$ , we align it with those from the original query sample  $\\mathbf{Q}$  to estimate the pixel-level anomaly segmentation map and image-level anomaly score. For pixel  $i$  in the query image, the anomaly score  $m_{i}$  is calculated as:",
+        "bbox": [
+            76,
+            632,
+            468,
+            708
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nm _ {i} = \\left\\| Q _ {i} - \\bar {Q} _ {i} \\right\\| ^ {2}, i \\in \\mathcal {N}, \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            176,
+            715,
+            468,
+            736
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathcal{N}$  denotes the collection of all pixel indices in  $Q$ .  $Q_{i}$  denotes the relevant feature vector of pixel  $i$ . For image-level classification, we use the maximum distance score  $s^{*}$  among all the pixels to represent the image-level score as",
+        "bbox": [
+            76,
+            743,
+            467,
+            804
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns ^ {*} = \\max  \\left\\{\\left\\| Q _ {i} - \\bar {Q} _ {i} \\right\\| ^ {2} \\mid i \\in \\mathcal {N} \\right\\}. \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            153,
+            811,
+            468,
+            833
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Besides, similar to [28, 7], to match the original input resolution, the segmentation map should be up-scaled with bilinear interpolation. Finally, we smoothed the segmentation map with a Gaussian of kernel width of 4.",
+        "bbox": [
+            76,
+            839,
+            468,
+            900
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4. Experiment",
+        "text_level": 1,
+        "bbox": [
+            500,
+            90,
+            624,
+            106
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1. Experimental Setups",
+        "text_level": 1,
+        "bbox": [
+            500,
+            114,
+            696,
+            131
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Dataset. We evaluated the performance of our model by using the MVTec AD dataset [3] and MPDD [15], which are the real-world benchmarks for AD. MVTec dataset includes 5354 images in 15 categories. 3629 of these images are defect-free and the remaining 1725 have defects. Each category has an average of five different types of defects. The image resolution ranges from  $700 \\times 700$  to  $1024 \\times 1024$ . MPDD dataset contains 6 classes of metal productions. Images taken from different distances and spatial directions, as well as the non-uniform backgrounds, make the AD task on this dataset extremely difficult. The MPDD dataset includes 888 normal images in the training set and 176 normal images and 282 abnormal images in the test set with the resolution of  $1024 \\times 1024$ . Additionally, both datasets provide ground truth of defective regions on pixel level.",
+        "bbox": [
+            496,
+            138,
+            890,
+            366
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Competing Methods. We compare our method with previous state-of-the-art FSAD and also full-data AD methods. The methods for comparison include GANomaly [1], ARNet [27], MKD [20], CutPaste [24], FYD [32], PaDiM [7], PatchCore [18], CflowAD [11], TDG [23], DiffNet [19] and RegAD [13].",
+        "bbox": [
+            496,
+            372,
+            890,
+            463
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Protocols. We measure the model performance with the area under the Receiver Operating Characteristic curve (AUROC) on classification results. We use image-level AUROC to evaluate the sample detection results and pixel-level AUROC to measure the defects localization performance.",
+        "bbox": [
+            496,
+            470,
+            890,
+            560
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details. In our experiment, we used a ResNet-50 [12] model pre-trained on ImageNet for feature extraction. We extract features from the 2 middle-level layers of the model. For both MVtec and MPDD datasets, all the support and query images are reshaped to  $368 \\times 368$  for training and inference. For all the experiments, the coefficient  $\\lambda$  for our regularization term is set as 0.3, and the corset selection sampling rate  $s$  is set as 0.15. All the experiments are conducted on one NVIDIA GTX 3090 GPU.",
+        "bbox": [
+            496,
+            568,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2. Experiment Results",
+        "text_level": 1,
+        "bbox": [
+            500,
+            715,
+            687,
+            731
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 Comparison with State-of-the-art Methods",
+        "text_level": 1,
+        "bbox": [
+            500,
+            739,
+            849,
+            753
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Few-shot AD averaged performance. Tab. 1 shows FSAD averaged performance on both MVTec and MPDD datasets. Our method (FastRecon) obtains the best performances under all experimental settings. Specifically, FastRecon improves image-level AUROC by  $5.27\\%$ ,  $6.04\\%$ ,  $3.99\\%$  on MVTec and  $10.25\\%$ ,  $11.55\\%$ $10.60\\%$  on MPDD over the current state-of-the-art FSAD method RegAD for 2-shot, 4-shot, and 8-shot scenarios respectively. For pixel-level AUROC, FastRecon also outperforms RegAD",
+        "bbox": [
+            496,
+            763,
+            890,
+            900
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "17485",
+        "bbox": [
+            480,
+            944,
+            517,
+            955
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/67397eb50dafb5311e2bfdc10313eb969415c67e8c2422813060e7e6d7bb936d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"6\">AUROC at Image-level</td><td colspan=\"4\">AUROC at Pixel-level</td></tr><tr><td>Dataset</td><td>k</td><td>TDG [23]</td><td>DiffNet [19]</td><td>RegAD [13]</td><td>PaDiM [7]</td><td>PatchCore [18]</td><td>FastRcon (ours)</td><td>RegAD [13]</td><td>PaDiM [7]</td><td>PatchCore [18]</td><td>FastRcon (ours)</td></tr><tr><td rowspan=\"3\">MVTec</td><td>2</td><td>71.20</td><td>80.60</td><td>85.70</td><td>78.90</td><td>87.81</td><td>90.97</td><td>94.60</td><td>90.50</td><td>94.75</td><td>95.86</td></tr><tr><td>4</td><td>72.70</td><td>81.30</td><td>88.20</td><td>71.60</td><td>89.49</td><td>94.24</td><td>95.80</td><td>80.20</td><td>94.99</td><td>96.98</td></tr><tr><td>8</td><td>75.20</td><td>83.20</td><td>91.20</td><td>75.30</td><td>94.31</td><td>95.19</td><td>96.70</td><td>80.50</td><td>95.60</td><td>97.27</td></tr><tr><td rowspan=\"3\">MPDD</td><td>2</td><td>60.30</td><td>60.20</td><td>63.40</td><td>-</td><td>59.55</td><td>73.65</td><td>93.20</td><td>-</td><td>79.15</td><td>97.03</td></tr><tr><td>4</td><td>63.50</td><td>63.30</td><td>68.30</td><td>-</td><td>59.78</td><td>79.85</td><td>93.90</td><td>-</td><td>79.82</td><td>97.60</td></tr><tr><td>8</td><td>68.20</td><td>68.50</td><td>71.90</td><td>-</td><td>59.95</td><td>82.50</td><td>95.10</td><td>-</td><td>80.30</td><td>97.92</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/e8dbc819ef7009f2d24cac97c5af964650068c0bfd71eb3b802bda6e6a68870f.jpg",
+        "table_caption": [
+            "Table 1: FSAD performance comparisons on MVtec and MPDD dataset. The results are averaged over all categories. Both image-level and pixel-level performances are reported in AUROC (\\%)  $\\uparrow$ .  $k$  denotes the number of shots in our few-shot settings. The best results for different settings are in bold."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">RegAD [13]</td><td colspan=\"2\">PatchCore [18]</td><td colspan=\"2\">FastRecon</td></tr><tr><td>Category</td><td>Image</td><td>Pixel</td><td>Image</td><td>Pixel</td><td>Image</td><td>Pixel</td></tr><tr><td>Bottle</td><td>99.40</td><td>98.40</td><td>99.60</td><td>98.60</td><td>99.44</td><td>98.50</td></tr><tr><td>Cable</td><td>76.10</td><td>92.70</td><td>97.40</td><td>97.90</td><td>93.79</td><td>96.12</td></tr><tr><td>Capsule</td><td>72.40</td><td>97.60</td><td>66.30</td><td>97.70</td><td>90.07</td><td>98.96</td></tr><tr><td>Carpet</td><td>97.90</td><td>98.90</td><td>99.00</td><td>99.00</td><td>99.90</td><td>99.15</td></tr><tr><td>Grid</td><td>91.20</td><td>85.70</td><td>63.00</td><td>70.60</td><td>88.81</td><td>86.32</td></tr><tr><td>Hazelnut</td><td>95.80</td><td>98.00</td><td>92.80</td><td>97.00</td><td>99.32</td><td>98.59</td></tr><tr><td>Leather</td><td>100.00</td><td>99.10</td><td>100.00</td><td>96.90</td><td>100.00</td><td>99.20</td></tr><tr><td>Metal_Nut</td><td>94.60</td><td>97.80</td><td>94.70</td><td>97.00</td><td>99.12</td><td>98.72</td></tr><tr><td>Pill</td><td>80.80</td><td>97.40</td><td>89.00</td><td>96.90</td><td>93.48</td><td>98.32</td></tr><tr><td>Screw</td><td>56.60</td><td>95.00</td><td>54.10</td><td>92.10</td><td>62.46</td><td>97.10</td></tr><tr><td>Tile</td><td>95.50</td><td>94.90</td><td>100.00</td><td>96.00</td><td>100.00</td><td>96.73</td></tr><tr><td>Toothbrush</td><td>90.90</td><td>98.50</td><td>95.20</td><td>98.80</td><td>93.61</td><td>99.04</td></tr><tr><td>Transistor</td><td>85.20</td><td>93.80</td><td>98.40</td><td>95.00</td><td>97.29</td><td>94.18</td></tr><tr><td>Wood</td><td>98.60</td><td>94.70</td><td>97.40</td><td>93.10</td><td>99.29</td><td>94.94</td></tr><tr><td>Zipper</td><td>88.50</td><td>94.00</td><td>95.50</td><td>98.30</td><td>96.95</td><td>98.89</td></tr><tr><td>Average</td><td>88.20</td><td>95.80</td><td>89.49</td><td>94.99</td><td>94.24</td><td>96.98</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "by  $1.26\\%$ ,  $1.18\\%$ ,  $0.57\\%$  on MVTev and  $3.83\\%$ ,  $3.70\\%$ ,  $2.82\\%$  on MPDD respectively.",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Few-shot AD categorical performances. Tab. 2 and Tab. 3 show the FSAD categorical performance. As shown in Tab. 2, FastRecon outperforms all the baselines in 10 out of 15 categories at the image level and in 12 out of 15 categories at the pixel level. And as shown in Tab. 3, FastRecon outperforms all the baselines in 4 out of 6 categories at the image level and in all categories at the pixel level.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Comparison with full-data AD. Tab. 4 shows the comparison of our method with AD methods which utilize the whole normal samples. Results show that our FastRecon has competitive performance compared with full-data methods, indicating the high data efficiency of our model.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/3a50dbce58c7fc218e572f8e1bc126b4818ffcc3dd12f7efde7d9a76d3eee651.jpg",
+        "table_caption": [
+            "Table 2: FSAD categorical performance comparisons in AUROC (\\%)  $\\uparrow$  on MVTec dataset. Results are from the setting of shot  $k = 4$ . The best results are in bold. Our method achieves the best performance in most categories."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">RegAD [13]</td><td colspan=\"2\">PatchCore [18]</td><td colspan=\"2\">FastRecon</td></tr><tr><td>Category</td><td>Image</td><td>Pixel</td><td>Image</td><td>Pixel</td><td>Image</td><td>Pixel</td></tr><tr><td>Bracket/black</td><td>63.80</td><td>-</td><td>58.90</td><td>79.10</td><td>71.81</td><td>95.54</td></tr><tr><td>Bracket_brown</td><td>66.10</td><td>-</td><td>70.80</td><td>77.30</td><td>63.34</td><td>95.93</td></tr><tr><td>Bracket_white</td><td>59.30</td><td>-</td><td>70.70</td><td>69.30</td><td>69.44</td><td>98.89</td></tr><tr><td>Connector</td><td>77.20</td><td>-</td><td>59.40</td><td>86.40</td><td>97.62</td><td>98.04</td></tr><tr><td>Metal_plate</td><td>78.60</td><td>-</td><td>64.40</td><td>86.70</td><td>100.00</td><td>99.29</td></tr><tr><td>Tubes</td><td>67.50</td><td>-</td><td>34.50</td><td>80.10</td><td>76.90</td><td>97.89</td></tr><tr><td>Average</td><td>68.30</td><td>93.90</td><td>59.78</td><td>79.82</td><td>79.85</td><td>97.60</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/78140d570edcf7928347bb6461ba46bdf8bb9fa1b11d87734be76999e08da429.jpg",
+        "table_caption": [
+            "Table 3: FSAD categorical performance comparisons in AUROC (\\%)  $\\uparrow$  on MPDD dataset. Results are from the setting of shot  $k = 4$ . The best results are in bold. Our method achieves the best performance in most categories."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\"></td><td colspan=\"2\">MVTec AD</td><td colspan=\"2\">MPDD</td></tr><tr><td>Method</td><td>Data</td><td>Backbone</td><td>Image</td><td>Pixel</td><td>Image</td><td>Pixel</td></tr><tr><td>FastRecon</td><td>2</td><td>WRN50</td><td>90.97</td><td>95.86</td><td>73.65</td><td>97.03</td></tr><tr><td>FastRecon</td><td>4</td><td>WRN50</td><td>94.24</td><td>96.98</td><td>79.85</td><td>97.60</td></tr><tr><td>FastRecon</td><td>8</td><td>WRN50</td><td>95.19</td><td>97.27</td><td>82.50</td><td>97.92</td></tr><tr><td>GANomaly [1]</td><td>Full</td><td>Unet</td><td>80.50</td><td>-</td><td>64.80</td><td>-</td></tr><tr><td>ARNet [27]</td><td>Full</td><td>Unet</td><td>83.90</td><td>-</td><td>69.70</td><td>-</td></tr><tr><td>MKD [20]</td><td>Full</td><td>Res18</td><td>87.70</td><td>90.70</td><td>-</td><td>-</td></tr><tr><td>FYD [32]</td><td>Full</td><td>Res18</td><td>97.30</td><td>97.40</td><td>-</td><td>-</td></tr><tr><td>PaDiM [7]</td><td>Full</td><td>WRN50</td><td>97.90</td><td>97.50</td><td>74.80</td><td>96.70</td></tr><tr><td>PatchCore [18]</td><td>Full</td><td>WRN50</td><td>99.10</td><td>98.10</td><td>82.10</td><td>95.70</td></tr><tr><td>CflowAD [11]</td><td>Full</td><td>WRN50</td><td>98.60</td><td>98.60</td><td>86.10</td><td>97.70</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 4: AD performance comparisons in AUROC  $(\\%)$  ↑ on MVTec and MPDD dataset with full-data methods which utilize the whole dataset in training. Both image-level and pixel-level performances are averaged over all categories. The Data column denotes different usage of the dataset.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Speed performance of FastRecon. Speed comparisons on both training and inference are shown in the following Tab. 5. For FastRecon, training stage consists of feature extraction, feature subsampling and calculation on transfor",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "17486",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/d776065b7544db0615f0589064a6374347446c6200f668fc5ea0292674a077b7.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative results of anomaly localization for both MVTec and MPDD datasets. The first row in the red box presents the support sample for each category while the second row indicates the query samples. The results show that our method can provide accurate localization of defect regions even for more complicated patterns in MPDD."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            99,
+            88,
+            869,
+            329
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/c0cd66bc11cfb004d56653a5475a5a52a02710423a7fe22f3d1a845ef5523617.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of decoding results on reconstructed query features. With a pretrained image decoder, we investigate the transformed query features in the MVtec dataset. The first row shows the original query image, and the middle row shows the images decoded from the original query features. The last row represents the image decoded from the reconstructed query feature map. Our method can reveal the normal side of the defective query samples."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/6d8c5a4ab2a871f8420b283964a81fe28fc54fa83741b4d8628b4a5fcf41713a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Aggregated Training</td><td>Training Time</td><td>Inference Time</td></tr><tr><td>PatchCore</td><td>No</td><td>80.37</td><td>0.60</td></tr><tr><td>RegAD (k=2)</td><td>Yes</td><td>660.80</td><td>0.05</td></tr><tr><td>FastRecon (k=8)</td><td>No</td><td>1.95</td><td>0.04</td></tr><tr><td>FastRecon (k=2)</td><td>No</td><td>0.52</td><td>0.04</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Training and inference speed comparisons in seconds. RegAD requires additional training on aggregated features. The inference speed is evaluated as averaged inference time per query sample. FastRecon achieves overall best speed performance with 2 and 8 shots.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "mation matrix. FastRecon achieves the best performance.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3. Visualization",
+        "text_level": 1,
+        "bbox": [
+            500,
+            676,
+            635,
+            690
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Localization of anomaly detection Fig. 4 demonstrates visual examples of anomaly detection on MVTec and MPDD. Most defects can be accurately identified.",
+        "bbox": [
+            498,
+            704,
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+            750
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Decoding results on reconstructed query feature. In order to better reflect the reconstruction ability of FastRecon, a feature decoder is built to visualize the query sample from reconstructed feature maps. The decoder is trained with full MVTec dataset to learn to decode image sample from feature maps. Fig. 5 shows that, images decoded from reconstructed feature maps maintain the normality of the normal samples, which illustrates the effectiveness of our approach.",
+        "bbox": [
+            498,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "17487",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/b04dbb283130a3c3550b987e19a2e32336bb190697feeef75804ea3f163e4a16.jpg",
+        "image_caption": [
+            "(a) Sampling Rate  $s$"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            84,
+            88,
+            349,
+            247
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/9bf3c50330417414c6c713a50630eee1a728d5061ed36a96435c04aacd3a323e.jpg",
+        "image_caption": [
+            "(b) Coefficient  $\\lambda$  of Regularization Term"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            352,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/f5b51b470c35bdefd27e1e6afcba4f56ecfa70483c4117db9eecfe649271df19.jpg",
+        "image_caption": [
+            "(c) Resolution  $r$"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            619,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/fc77ce6651bcabc7e720c752faaf4c21744a254ecb3fec5714b576c3795631ee.jpg",
+        "image_caption": [
+            "Figure 6: Effect of three hyper-parameters comparisons in AUROC (\\%) on MVTec dataset. (a) AD performance with different sampling rate  $s$ . (b) AD performance with different coefficient  $\\lambda$ . (c) AD performance with different resolution  $r$ ."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            88,
+            337,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/92a55d12dd9fb1a76aaf04e9858e180b5aeda32c34fe0c914e37d594757be312.jpg",
+        "image_caption": [
+            "Figure 7: AD performance comparisons in AUROC (\\%) on MVTec dataset between ridge regression and proposed regularization. Our results, displayed in blue, are better than ridge regression at both image and pixel levels."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            86,
+            476,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4. Ablation Studies",
+        "text_level": 1,
+        "bbox": [
+            76,
+            717,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we ablate the core parts of our proposed method to investigate its impact on the final performance. We also discuss the effect of hyper-parameters, which are the coefficient of regularization term  $\\lambda$ , the resolution of input data, and the coreset selection sampling ratio  $s$ .",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Effect of sampling Rate  $s$ . Fig. 6 (a) illustrates the relationship between the Sampling Rate  $s$  and AUROC performance. The sampling rate  $s$  decides the number of feature points in the support feature pool. As shown in Fig. 6 (a), the optimal sampling rate is around 0.45.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Impact of the distribution regularization term and coefficient  $\\lambda$ . In Fig. 7, FastRecon obtains better AUROC performance at both image and pixel levels. It reflects the significance of our proposed distribution regularization. Fig. 6 (b) shows the relationship between coefficient  $\\lambda$  of regularization term and AUROC performance for 4-shot and 8-shot scenarios. When  $\\lambda$  equals 0.3, the AUROC performances are best at both image and pixel levels.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Effect of image resolution  $r$ . In our experiment, images, no matter how much the original resolutions are, should be reshaped to new resolution  $r \\times r$ . Fig. 6 (c) reflects the relationship between resolution  $r$  and AUROC performance. The resolution will affect the feature points. As shown in Fig. 6 (c), when the resolution equals 368, FastRecon performs well at both image and pixel levels.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5. Conclusion",
+        "text_level": 1,
+        "bbox": [
+            500,
+            601,
+            617,
+            616
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, a novel method FastRecon is proposed for few-shot anomaly detection based on feature reconstruction. For each query sample, we construct their normal version, referenced by a limited number of normal features. To ensure the reconstructed result shares high visual similarity with the query sample while maintaining the property of the normal sample, a regression equation with proposed distribution regularization is proposed, to obtain the optimal transformation from support to query features. With this fast and accurate reconstruction, the final anomaly detection could be accomplished by a simple alignment. Experimental results show the state-of-the-art performance of our method for 2-shot, 4-shot, and 8-shot scenarios.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements. This work was supported by National Key R&D Program of China (No.2022YFE0200300), National Natural Science Foundation of China (No.61972323).",
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "17488",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            78,
+            89,
+            174,
+            106
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
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