| Level | ABB. | Description |
| Environment | DS | Dark Scene. The light is too low to distinguish the target. |
| OE | Overexposure. The illumination is too high to distinguish the target. |
| IC | Illumination Change. There are illumination changes during one video sequence. |
| Target | OV | Out of View. Object partially or fully leave the view. |
| FM | Fast Motion. The average per-frame object motion is larger than 20 pixels. |
| TR | Target Rotation. Target rotates in plane or out of plane. |
| SV | Scale Variation. Ratio change of target size between minimum and maximum is more than 50%. |
| DF | Deformation. The object is deformable. |
| CO | Composite Objects. The target object is an ensemble of multiple objects (e.g. man with a basketball). |
| Platform | LR | Low Resolution. The ratio of the object area to the image size is lower than 5%. |
| CM | Camera Motion. The camera moves/shakes. |
| MB | Motion Blur. The target is blurred due to the motion of itself or the camera. |
| VC | Viewpoint Change. The viewpoint is not fixed because the capturing angle changes. |
| SF | Sensor Failure. At least one camera cannot provide useful information. |
| Interaction | PO | Partial Occlusion. The object is partially occluded. |
| ST | Similar Targets. There are similar objects. |
| FO | Full Occlusion. The object is fully occluded. |
| BC | Background Clutter. There are distractors around the target object. |
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+ "text": "dation sets by themselves.",
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+ "text": "Objects. Unlike previous drone-based tracking datasets which only contain limited object categories, our $\\mathrm{D}^2\\mathrm{Cube}$ covers generic objects. Fig. 2 shows the object classes in our test set, in which 34 classes are included. Specifically, we include some classes rarely appeared in the semantic area, e.g., part of an entire object (upper part of a body) or composite object (man holding a basketball). The whole $\\mathrm{D}^2\\mathrm{Cube}$ includes more than 100 categories and covers diverse objects, it thus is representative for daily scenarios.",
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+ "text": "Scenarios. In this work, we mainly focus on daily life scenarios. We involve many applicable scenarios for RGBD aerial tracking, e.g., sports, work, service and entertainment, in which aerial robots and depth cameras can both work well. In detail, our recording scenarios cover daily life scenes, including office, bedroom, meeting room, gym, stadium, kitchen, and so on. Both short-term and long-term",
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+ "type": "text",
+ "text": "Annotations. As shown in Fig. 1, we provide tight axis-aligned bounding box annotations for the target objects at the frame level. A professional team annotates $\\mathrm{D}^2\\mathrm{Cube}$ rigorously. The annotation process follows the following rules: (i) if the target appears in the frame, we annotate the visible part of the target by the tightest bounding box. (ii) if the target does not appear in the frame, we will mark this frame with an \"target loss\" tag.",
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+ "text": "Attributes. We define 18 tracking challenges in RGBD aerial tracking and classify the attributes in a hierarchical manner. All the attributes are defined in four levels: environment, target, platform and interaction. At different levels, correspondingly there are different challenges. Details of each attribute are given in Table 2. With such a hierarchical classification of different challenges, we can justify what challenges RGBD trackers are indeed suffering from. We also give an annotated example in Fig. 3. The distribution of each attribute in our test set is given in Fig. 4.",
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+ "text": "4. Efficient Multimodal Tracker",
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+ "text": "To achieve RGBD tracking on UAV platforms, trackers' ability to run on edge platforms with limited resources is of importance. However, the vast majority of RGBD trackers focus on architectural design with heavy backbones and additional modules. Such complex frameworks cannot satisfy the real-time requirements of aerial tracking. In this section, we propose Efficient Multimodal Tracker (EMT) for RGBD aerial tracking, which discards the heavy backbones and additional modules.",
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+ "text": "The proposed EMT contains four main steps, i.e., efficient modality-aware fusion, feature extraction, efficient attention-based feature matching, and target prediction. The overall architecture is shown in Fig. 5.",
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+ "text": "Efficient modality-aware fusion. Firstly, to speed up the fusion, we design a novel Efficient Modality-Aware Fusion (EMAF) module that can first fuse the raw data from",
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+ "img_path": "images/33d5d6c1bbf0318d7a27bf61ffaba04f0f8046bcdd9295163207af9ebfeb3e15.jpg",
+ "image_caption": [
+ "Figure 5. Overview of our proposed Efficient Multimodal Tracker (EMT). Left: Pipeline for EMT. Right: Architecture of Efficient Modality-Aware Fusion (EMAF) module."
+ ],
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+ "text": "multiple modalities at a very early stage. The key to speeding up is that once the high-dimensional features are well integrated, the amount of computation that follows is greatly reduced. In addition, in order to effectively adapt to the scene, we also employ a fusion strategy that can dynamically estimate the importance of features from two modalities. The details of the fusion module to obtain fused image patches $\\mathbf{T}_{\\mathrm{fused}}$ and $\\mathbf{X}_{\\mathrm{fused}}$ are referred to 4.2.",
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+ "text": "Feature extraction. Secondly, we assume that the fused template patch is $\\mathbf{T}_{\\mathrm{fused}} \\in \\mathbb{R}_{3 \\times \\mathbf{H}_{\\mathrm{t0}} \\times \\mathbf{W}_{\\mathrm{t0}}}$ and the fused search patch is $\\mathbf{X}_{\\mathrm{fused}} \\in \\mathbb{R}_{3 \\times \\mathbf{H}_{\\mathrm{x0}} \\times \\mathbf{W}_{\\mathrm{x0}}}$ . Then, we treat them as the input of the parameter-sharing backbone network for feature extraction. A modified ResNet-18 Network [6] is used as backbone network to obtain features maps for templates $f_{t} \\in \\mathbb{C} \\times \\mathbb{H}_{t} \\times \\mathbb{W}_{t}$ and search regions $f_{x} \\in \\mathbb{C} \\times \\mathbb{H}_{x} \\times \\mathbb{W}_{x}$ . Here $(H_{t}, W_{t}) = (\\frac{H_{t0}}{16}, \\frac{W_{t0}}{16})$ , $(H_{x}, W_{x}) = (\\frac{H_{x0}}{16}, \\frac{W_{x0}}{16})$ and $C = 256$ .",
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+ "text": "Efficient attention-based feature matching. Thirdly, as we obtain the multimodal feature maps for the template and the search region, the next step is to match the corresponding features. To further speed up this procedure, we first use a trainable embedding to reduce the dimension of template features and then design a one-way attention-based fusion module to efficiently fuse the template features and search area features. We give a detailed description of template-to-search matching in Sec. 4.3.",
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+ "text": "Target prediction. With the template-to-search map, we obtain target predictions by using regression head and classification head. The regression head is to regress the overlap between groundtruth and bounding box candidates. The classification head is to classify the objects and background.",
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+ "text": "4.2. Efficient Modality-Aware Fusion",
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+ "text": "Our EMT takes the dual-modal image patches as input, and performs a weighted fusion of modalities online. Unlike the modal independent backbone network design of traditional RGBD trackers, our proposed EMT reduces the model's size through a very early learnable fusion.",
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+ "text": "Raw patch preparation. Specifically, four image patches will be treated as the input, including the color and depth template patches, and the color and depth patches for search regions. On the one hand, the template image patches $\\mathbf{T}_{\\mathrm{rgb}}$ and $\\mathbf{T}_{\\mathrm{depth}}$ are obtained by expanding the target bounding box of the first frame twice in the video. To effectively enhance discrimination, these patches should include the local surrounding information. And, the perturbation is also added to the target to avoid learning location bias. On the other hand, the patches for search regions $\\mathbf{X}_{\\mathrm{rgb}}$ and $\\mathbf{X}_{\\mathrm{depth}}$ are obtained by expanding the target box in the previous frame by four times instead of the whole original image, which utilizes the temporal context in the video sequence and reduces the computational cost.",
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+ "text": "Dynamic cross-modal fusion. The aim of this step is to fuse the data and reduce the dimension at a very early stage. The intrinsic reason we can fuse these two types of data at such an early stage is that they are pixel-level aligned in image space. Moreover, it requires to dynamically judge the environment of the tracking target by extracting the global information of the image patches. To this end, we calculate the importance of the two modalities in the current frame based on the context of the two modalities. Based on the importance ratio, the RGB image patch and depth image patch can be fused by the weights for early fusion.",
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+ "text": "Taking the search branch as an example, the RGB and",
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+ "text": "depth image patches go through $3 \\times 3$ convolution layer (Conv) and a batch norm layer (BN) to extract discriminative features. Then, a global pooling layer (GAP) is used to extract the global context of the two modalities. Next, the features are concatenated and as input of the fully connected layers (FC) to output the importance $(\\alpha$ and $\\beta)$ of the two modalities in the current frame. Finally, RGB image patches and depth image patches can be fused by importance weights. The process can be formulated as follows:",
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+ "type": "equation",
+ "text": "\n$$\n\\mathbf {X} _ {\\mathbf {r g b}} = \\left(\\operatorname {G A P} (\\operatorname {B N} (\\operatorname {C o n v} (\\mathbf {X} _ {\\mathbf {r g b}})) \\right.,\n$$\n",
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+ "text": "\n$$\n\\mathbf {X} _ {\\text {d e p t h}} = \\operatorname {G A P} (\\operatorname {B N} (\\operatorname {C o n v} (\\mathbf {X} _ {\\text {d e p t h}}))), \\tag {1}\n$$\n",
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+ "text": "\n$$\n\\alpha , \\beta = \\operatorname {F C} \\left(\\operatorname {C a t} \\left(\\mathbf {X} _ {\\mathbf {r g b}}, \\mathbf {X} _ {\\mathbf {d e p t h}}\\right)\\right),\n$$\n",
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+ "text": "\n$$\n\\mathbf {X} _ {\\text {f u s e d}} = \\alpha * \\mathbf {X} _ {\\text {r g b}} + \\beta * \\mathbf {X} _ {\\text {d e p t h}},\n$$\n",
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+ "type": "text",
+ "text": "where $\\mathbf{X}_{\\mathrm{fused}}$ has the same size of original images.",
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+ "text": "The immediate benefit of this early fusion is that we avoid extracting features from two modalities by using two separate backbones. Therefore, both the size of memory and the amount of computation have been greatly reduced.",
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+ "text": "4.3. Efficient Attention-based Feature Matching",
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+ "type": "text",
+ "text": "For a given template feature $f_{t}$ , we will use an attention-based module to find corresponding features in $f_{x}$ for a search region. Naturally, similar to [7], both cross-attention between features and self-attention within features can be directly used. However, to speed computation up, we design a more streamlined network to achieve feature matching.",
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+ "text": "Compact template representation. To make the template compact, we use a learnable embedding to reduce the dimensions of template vectors $f_{t}$ . As shown in Fig. 5, through the cross-attention with dimension reduction embedding, we obtain a compact template representation $f_{t}^{c}$ .",
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+ "type": "text",
+ "text": "One-Way Attention (OWA) module. With such a compact template representation $f_{t}^{c}$ , we then utilize an efficient matching from template to search area. Here, we merely deploy the cross-attention based Cross-Feature Augment (CFA) module [7] for fusion, in which multi-head cross-attention is used in a residual form. However, CFA is utilized in a one-way manner, due to the fact that we only need the template project on search areas to make a prediction in the search area [5]. OWA can be repeated by several times. With such a one-way cross-attention module, we can achieve feature matching more efficiently.",
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+ "type": "text",
+ "text": "4.4. Training and Inference",
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+ "text": "Loss Function. The losses are computed between the outputs of target prediction and groundtruth:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathcal {L} _ {\\text {t o t a l}} = \\lambda_ {1} \\mathcal {L} _ {\\text {c l s}} + \\lambda_ {2} \\mathcal {L} _ {\\text {b b o x}} + \\lambda_ {3} \\mathcal {L} _ {\\text {G I o u}}. \\tag {2}\n$$\n",
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+ "type": "text",
+ "text": "Here, classification loss $\\mathcal{L}_{cls}$ is to dicriminate the object from background. Regression loss consists of $\\mathcal{L}_{bbox}$ -",
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+ {
+ "type": "text",
+ "text": "Mean Squared Error (MSE) between the predicted bounding boxes and the groundtruth bounding boxes, and $\\mathcal{L}_{GIou}$ - generalized IoU loss [29]. We use $\\lambda_{1} = 0.8344$ , $\\lambda_{2} = 5$ , $\\lambda_{3} = 2$ in our experiments following [7].",
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+ "type": "text",
+ "text": "Training phase. The training process follows the standard training recipe of current trackers [13,28,34]. We use ResNet-18 [16] pretrained on the ImageNet as our backbone and fine-tune the whole tracking framework with the training set of our proposed $\\mathrm{D}^2\\mathrm{Cube}$ and the training data recipe of [34]. We randomly sample two frames within an interval of 30 frames in sequences as template and search region. Subsequently, templates and search regions are obtained with some data augmentation such as jitter, brightness change, and then resized to $128\\times 128$ and $256\\times 256$ pixels, respectively. The template embedding size is 16. The model is trained with AdamW [25] optimizer, and the learning rate for the backbone and EMAF module are $1e - 5$ and $1e - 4$ , respectively. Weight decay is $1e - 4$ . The learning rate decays 10 times every 50 epochs. We sample 128,000 pairs in each epoch and the whole tracker is trained for 100 epochs on a single 32GB Tesla V100 GPU with a batch size of 128.",
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+ "type": "text",
+ "text": "Tracking phase. During inference, the template and search image are resized to fixed size. Dynamic cross-modal fusion module performs early fusion of the two modalities. After feature extraction and correlation, the prediction head outputs 256 bounding boxes and classification scores. A penalty window is employed to filter distractors.",
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+ "type": "text",
+ "text": "5. Experiments",
+ "text_level": 1,
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+ {
+ "type": "text",
+ "text": "5.1. Experimental Settings",
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+ "type": "text",
+ "text": "Hardware. All comparison experiments, except for the onboard tests, are executed on a single 32GB Tesla V100 GPU. A widely-used UAV onboard processor NVidia Jetson NX Xavier is used for onboard tests.",
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+ "type": "text",
+ "text": "Evaluation protocols. We follow the evaluation principles in long-term RGBD tracking from VOT challenge [19]. One-Pass Evaluation (OPE) is used to test trackers' performance on our proposed $\\mathrm{D}^2\\mathrm{Cube}$ . At frame $t$ , $\\theta_t$ is a prediction confidence score and $\\tau_{\\theta}$ is a classification threshold. If the predicted $\\theta_t$ is not below $\\tau_{\\theta}$ , $A_t(\\tau_{\\theta})$ is used to denote the corresponding prediction. Otherwise, we set $A_t(\\tau_{\\theta}) = \\emptyset$ . Thus, $\\Omega(A_t(\\tau_{\\theta}), G_t)$ indicates the intersection-over-union (IoU) between the prediction result $A_t(\\tau_{\\theta})$ and the groundtruth $G_t$ . We here calculate the precision-recall over the whole test set as follows:",
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+ "type": "equation",
+ "text": "\n$$\nP r (\\tau_ {\\theta}) = \\frac {1}{M} \\sum_ {i = 1} ^ {M} \\frac {1}{N _ {p}} \\sum_ {A _ {t} (\\tau_ {\\theta}) \\neq \\varnothing} \\Omega \\left(A _ {t} (\\tau_ {\\theta}), G _ {t}\\right), \\tag {3}\n$$\n",
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+ "text": "\n$$\nR e (\\tau_ {\\theta}) = \\frac {1}{M} \\sum_ {i = 1} ^ {M} \\frac {1}{N _ {g}} \\sum_ {G _ {t} \\neq \\emptyset} \\Omega (A _ {t} (\\tau_ {\\theta}), G _ {t}),\n$$\n",
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+ "type": "page_number",
+ "text": "13379",
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+ "type": "table",
+ "img_path": "images/9c471c06335e3fa834263e6a12901e99045b1651f1d7e9320bf3da34dda800cc.jpg",
+ "table_caption": [
+ "Table 3. Performance comparison of state-of-the-art RGB aerial trackers on $\\mathrm{D}^2$ Cube dataset. The top 3 results are shown in red, green, and blue. Speed in FPS (frames per second)."
+ ],
+ "table_footnote": [],
+ "table_body": "| Level | ABB. | Description |
| Environment | DS | Dark Scene. The light is too low to distinguish the target. |
| OE | Overexposure. The illumination is too high to distinguish the target. |
| IC | Illumination Change. There are illumination changes during one video sequence. |
| Target | OV | Out of View. Object partially or fully leave the view. |
| FM | Fast Motion. The average per-frame object motion is larger than 20 pixels. |
| TR | Target Rotation. Target rotates in plane or out of plane. |
| SV | Scale Variation. Ratio change of target size between minimum and maximum is more than 50%. |
| DF | Deformation. The object is deformable. |
| CO | Composite Objects. The target object is an ensemble of multiple objects (e.g. man with a basketball). |
| Platform | LR | Low Resolution. The ratio of the object area to the image size is lower than 5%. |
| CM | Camera Motion. The camera moves/shakes. |
| MB | Motion Blur. The target is blurred due to the motion of itself or the camera. |
| VC | Viewpoint Change. The viewpoint is not fixed because the capturing angle changes. |
| SF | Sensor Failure. At least one camera cannot provide useful information. |
| Interaction | PO | Partial Occlusion. The object is partially occluded. |
| ST | Similar Targets. There are similar objects. |
| FO | Full Occlusion. The object is fully occluded. |
| BC | Background Clutter. There are distractors around the target object. |
"
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+ "content": "dation sets by themselves."
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+ "content": "Objects. Unlike previous drone-based tracking datasets which only contain limited object categories, our \\(\\mathrm{D}^2\\mathrm{Cube}\\) covers generic objects. Fig. 2 shows the object classes in our test set, in which 34 classes are included. Specifically, we include some classes rarely appeared in the semantic area, e.g., part of an entire object (upper part of a body) or composite object (man holding a basketball). The whole \\(\\mathrm{D}^2\\mathrm{Cube}\\) includes more than 100 categories and covers diverse objects, it thus is representative for daily scenarios."
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+ "content": "Scenarios. In this work, we mainly focus on daily life scenarios. We involve many applicable scenarios for RGBD aerial tracking, e.g., sports, work, service and entertainment, in which aerial robots and depth cameras can both work well. In detail, our recording scenarios cover daily life scenes, including office, bedroom, meeting room, gym, stadium, kitchen, and so on. Both short-term and long-term"
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+ "angle": 0,
+ "content": "tracking scenarios are included."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.31,
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+ 0.431
+ ],
+ "angle": 0,
+ "content": "Annotations. As shown in Fig. 1, we provide tight axis-aligned bounding box annotations for the target objects at the frame level. A professional team annotates \\(\\mathrm{D}^2\\mathrm{Cube}\\) rigorously. The annotation process follows the following rules: (i) if the target appears in the frame, we annotate the visible part of the target by the tightest bounding box. (ii) if the target does not appear in the frame, we will mark this frame with an \"target loss\" tag."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.432,
+ 0.892,
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+ ],
+ "angle": 0,
+ "content": "Attributes. We define 18 tracking challenges in RGBD aerial tracking and classify the attributes in a hierarchical manner. All the attributes are defined in four levels: environment, target, platform and interaction. At different levels, correspondingly there are different challenges. Details of each attribute are given in Table 2. With such a hierarchical classification of different challenges, we can justify what challenges RGBD trackers are indeed suffering from. We also give an annotated example in Fig. 3. The distribution of each attribute in our test set is given in Fig. 4."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "4. Efficient Multimodal Tracker"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.623,
+ 0.892,
+ 0.759
+ ],
+ "angle": 0,
+ "content": "To achieve RGBD tracking on UAV platforms, trackers' ability to run on edge platforms with limited resources is of importance. However, the vast majority of RGBD trackers focus on architectural design with heavy backbones and additional modules. Such complex frameworks cannot satisfy the real-time requirements of aerial tracking. In this section, we propose Efficient Multimodal Tracker (EMT) for RGBD aerial tracking, which discards the heavy backbones and additional modules."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.499,
+ 0.771,
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+ ],
+ "angle": 0,
+ "content": "4.1. Multimodal Fusion and Matching Architecture"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.795,
+ 0.892,
+ 0.855
+ ],
+ "angle": 0,
+ "content": "The proposed EMT contains four main steps, i.e., efficient modality-aware fusion, feature extraction, efficient attention-based feature matching, and target prediction. The overall architecture is shown in Fig. 5."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.856,
+ 0.892,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "Efficient modality-aware fusion. Firstly, to speed up the fusion, we design a novel Efficient Modality-Aware Fusion (EMAF) module that can first fuse the raw data from"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 5. Overview of our proposed Efficient Multimodal Tracker (EMT). Left: Pipeline for EMT. Right: Architecture of Efficient Modality-Aware Fusion (EMAF) module."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.396,
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+ ],
+ "angle": 0,
+ "content": "multiple modalities at a very early stage. The key to speeding up is that once the high-dimensional features are well integrated, the amount of computation that follows is greatly reduced. In addition, in order to effectively adapt to the scene, we also employ a fusion strategy that can dynamically estimate the importance of features from two modalities. The details of the fusion module to obtain fused image patches \\(\\mathbf{T}_{\\mathrm{fused}}\\) and \\(\\mathbf{X}_{\\mathrm{fused}}\\) are referred to 4.2."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Feature extraction. Secondly, we assume that the fused template patch is \\(\\mathbf{T}_{\\mathrm{fused}} \\in \\mathbb{R}_{3 \\times \\mathbf{H}_{\\mathrm{t0}} \\times \\mathbf{W}_{\\mathrm{t0}}}\\) and the fused search patch is \\(\\mathbf{X}_{\\mathrm{fused}} \\in \\mathbb{R}_{3 \\times \\mathbf{H}_{\\mathrm{x0}} \\times \\mathbf{W}_{\\mathrm{x0}}}\\). Then, we treat them as the input of the parameter-sharing backbone network for feature extraction. A modified ResNet-18 Network [6] is used as backbone network to obtain features maps for templates \\(f_{t} \\in \\mathbb{C} \\times \\mathbb{H}_{t} \\times \\mathbb{W}_{t}\\) and search regions \\(f_{x} \\in \\mathbb{C} \\times \\mathbb{H}_{x} \\times \\mathbb{W}_{x}\\). Here \\((H_{t}, W_{t}) = (\\frac{H_{t0}}{16}, \\frac{W_{t0}}{16})\\), \\((H_{x}, W_{x}) = (\\frac{H_{x0}}{16}, \\frac{W_{x0}}{16})\\) and \\(C = 256\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.678,
+ 0.47,
+ 0.814
+ ],
+ "angle": 0,
+ "content": "Efficient attention-based feature matching. Thirdly, as we obtain the multimodal feature maps for the template and the search region, the next step is to match the corresponding features. To further speed up this procedure, we first use a trainable embedding to reduce the dimension of template features and then design a one-way attention-based fusion module to efficiently fuse the template features and search area features. We give a detailed description of template-to-search matching in Sec. 4.3."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Target prediction. With the template-to-search map, we obtain target predictions by using regression head and classification head. The regression head is to regress the overlap between groundtruth and bounding box candidates. The classification head is to classify the objects and background."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "4.2. Efficient Modality-Aware Fusion"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.422,
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+ ],
+ "angle": 0,
+ "content": "Our EMT takes the dual-modal image patches as input, and performs a weighted fusion of modalities online. Unlike the modal independent backbone network design of traditional RGBD trackers, our proposed EMT reduces the model's size through a very early learnable fusion."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.502,
+ 0.893,
+ 0.713
+ ],
+ "angle": 0,
+ "content": "Raw patch preparation. Specifically, four image patches will be treated as the input, including the color and depth template patches, and the color and depth patches for search regions. On the one hand, the template image patches \\(\\mathbf{T}_{\\mathrm{rgb}}\\) and \\(\\mathbf{T}_{\\mathrm{depth}}\\) are obtained by expanding the target bounding box of the first frame twice in the video. To effectively enhance discrimination, these patches should include the local surrounding information. And, the perturbation is also added to the target to avoid learning location bias. On the other hand, the patches for search regions \\(\\mathbf{X}_{\\mathrm{rgb}}\\) and \\(\\mathbf{X}_{\\mathrm{depth}}\\) are obtained by expanding the target box in the previous frame by four times instead of the whole original image, which utilizes the temporal context in the video sequence and reduces the computational cost."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.717,
+ 0.893,
+ 0.883
+ ],
+ "angle": 0,
+ "content": "Dynamic cross-modal fusion. The aim of this step is to fuse the data and reduce the dimension at a very early stage. The intrinsic reason we can fuse these two types of data at such an early stage is that they are pixel-level aligned in image space. Moreover, it requires to dynamically judge the environment of the tracking target by extracting the global information of the image patches. To this end, we calculate the importance of the two modalities in the current frame based on the context of the two modalities. Based on the importance ratio, the RGB image patch and depth image patch can be fused by the weights for early fusion."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.52,
+ 0.886,
+ 0.892,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "Taking the search branch as an example, the RGB and"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.481,
+ 0.945,
+ 0.519,
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+ "angle": 0,
+ "content": "13378"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.092,
+ 0.473,
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+ ],
+ "angle": 0,
+ "content": "depth image patches go through \\(3 \\times 3\\) convolution layer (Conv) and a batch norm layer (BN) to extract discriminative features. Then, a global pooling layer (GAP) is used to extract the global context of the two modalities. Next, the features are concatenated and as input of the fully connected layers (FC) to output the importance \\((\\alpha\\) and \\(\\beta)\\) of the two modalities in the current frame. Finally, RGB image patches and depth image patches can be fused by importance weights. The process can be formulated as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.152,
+ 0.25,
+ 0.395,
+ 0.268
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {X} _ {\\mathbf {r g b}} = \\left(\\operatorname {G A P} (\\operatorname {B N} (\\operatorname {C o n v} (\\mathbf {X} _ {\\mathbf {r g b}})) \\right.,\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.139,
+ 0.269,
+ 0.469,
+ 0.293
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {X} _ {\\text {d e p t h}} = \\operatorname {G A P} (\\operatorname {B N} (\\operatorname {C o n v} (\\mathbf {X} _ {\\text {d e p t h}}))), \\tag {1}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.165,
+ 0.288,
+ 0.39,
+ 0.305
+ ],
+ "angle": 0,
+ "content": "\\[\n\\alpha , \\beta = \\operatorname {F C} \\left(\\operatorname {C a t} \\left(\\mathbf {X} _ {\\mathbf {r g b}}, \\mathbf {X} _ {\\mathbf {d e p t h}}\\right)\\right),\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.143,
+ 0.307,
+ 0.382,
+ 0.323
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {X} _ {\\text {f u s e d}} = \\alpha * \\mathbf {X} _ {\\text {r g b}} + \\beta * \\mathbf {X} _ {\\text {d e p t h}},\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.332,
+ 0.418,
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+ ],
+ "angle": 0,
+ "content": "where \\(\\mathbf{X}_{\\mathrm{fused}}\\) has the same size of original images."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "The immediate benefit of this early fusion is that we avoid extracting features from two modalities by using two separate backbones. Therefore, both the size of memory and the amount of computation have been greatly reduced."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.416,
+ 0.449,
+ 0.433
+ ],
+ "angle": 0,
+ "content": "4.3. Efficient Attention-based Feature Matching"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.439,
+ 0.469,
+ 0.53
+ ],
+ "angle": 0,
+ "content": "For a given template feature \\( f_{t} \\), we will use an attention-based module to find corresponding features in \\( f_{x} \\) for a search region. Naturally, similar to [7], both cross-attention between features and self-attention within features can be directly used. However, to speed computation up, we design a more streamlined network to achieve feature matching."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.53,
+ 0.469,
+ 0.606
+ ],
+ "angle": 0,
+ "content": "Compact template representation. To make the template compact, we use a learnable embedding to reduce the dimensions of template vectors \\( f_{t} \\). As shown in Fig. 5, through the cross-attention with dimension reduction embedding, we obtain a compact template representation \\( f_{t}^{c} \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.606,
+ 0.47,
+ 0.773
+ ],
+ "angle": 0,
+ "content": "One-Way Attention (OWA) module. With such a compact template representation \\( f_{t}^{c} \\), we then utilize an efficient matching from template to search area. Here, we merely deploy the cross-attention based Cross-Feature Augment (CFA) module [7] for fusion, in which multi-head cross-attention is used in a residual form. However, CFA is utilized in a one-way manner, due to the fact that we only need the template project on search areas to make a prediction in the search area [5]. OWA can be repeated by several times. With such a one-way cross-attention module, we can achieve feature matching more efficiently."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.78,
+ 0.292,
+ 0.796
+ ],
+ "angle": 0,
+ "content": "4.4. Training and Inference"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.804,
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+ ],
+ "angle": 0,
+ "content": "Loss Function. The losses are computed between the outputs of target prediction and groundtruth:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.138,
+ 0.845,
+ 0.469,
+ 0.862
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} _ {\\text {t o t a l}} = \\lambda_ {1} \\mathcal {L} _ {\\text {c l s}} + \\lambda_ {2} \\mathcal {L} _ {\\text {b b o x}} + \\lambda_ {3} \\mathcal {L} _ {\\text {G I o u}}. \\tag {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.871,
+ 0.47,
+ 0.902
+ ],
+ "angle": 0,
+ "content": "Here, classification loss \\(\\mathcal{L}_{cls}\\) is to dicriminate the object from background. Regression loss consists of \\(\\mathcal{L}_{bbox}\\) -"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.092,
+ 0.892,
+ 0.152
+ ],
+ "angle": 0,
+ "content": "Mean Squared Error (MSE) between the predicted bounding boxes and the groundtruth bounding boxes, and \\(\\mathcal{L}_{GIou}\\) - generalized IoU loss [29]. We use \\(\\lambda_{1} = 0.8344\\), \\(\\lambda_{2} = 5\\), \\(\\lambda_{3} = 2\\) in our experiments following [7]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.153,
+ 0.893,
+ 0.424
+ ],
+ "angle": 0,
+ "content": "Training phase. The training process follows the standard training recipe of current trackers [13,28,34]. We use ResNet-18 [16] pretrained on the ImageNet as our backbone and fine-tune the whole tracking framework with the training set of our proposed \\(\\mathrm{D}^2\\mathrm{Cube}\\) and the training data recipe of [34]. We randomly sample two frames within an interval of 30 frames in sequences as template and search region. Subsequently, templates and search regions are obtained with some data augmentation such as jitter, brightness change, and then resized to \\(128\\times 128\\) and \\(256\\times 256\\) pixels, respectively. The template embedding size is 16. The model is trained with AdamW [25] optimizer, and the learning rate for the backbone and EMAF module are \\(1e - 5\\) and \\(1e - 4\\), respectively. Weight decay is \\(1e - 4\\). The learning rate decays 10 times every 50 epochs. We sample 128,000 pairs in each epoch and the whole tracker is trained for 100 epochs on a single 32GB Tesla V100 GPU with a batch size of 128."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.424,
+ 0.892,
+ 0.515
+ ],
+ "angle": 0,
+ "content": "Tracking phase. During inference, the template and search image are resized to fixed size. Dynamic cross-modal fusion module performs early fusion of the two modalities. After feature extraction and correlation, the prediction head outputs 256 bounding boxes and classification scores. A penalty window is employed to filter distractors."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.53,
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+ ],
+ "angle": 0,
+ "content": "5. Experiments"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.554,
+ 0.707,
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+ ],
+ "angle": 0,
+ "content": "5.1. Experimental Settings"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.578,
+ 0.892,
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+ ],
+ "angle": 0,
+ "content": "Hardware. All comparison experiments, except for the onboard tests, are executed on a single 32GB Tesla V100 GPU. A widely-used UAV onboard processor NVidia Jetson NX Xavier is used for onboard tests."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.639,
+ 0.893,
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+ ],
+ "angle": 0,
+ "content": "Evaluation protocols. We follow the evaluation principles in long-term RGBD tracking from VOT challenge [19]. One-Pass Evaluation (OPE) is used to test trackers' performance on our proposed \\(\\mathrm{D}^2\\mathrm{Cube}\\). At frame \\(t\\), \\(\\theta_t\\) is a prediction confidence score and \\(\\tau_{\\theta}\\) is a classification threshold. If the predicted \\(\\theta_t\\) is not below \\(\\tau_{\\theta}\\), \\(A_t(\\tau_{\\theta})\\) is used to denote the corresponding prediction. Otherwise, we set \\(A_t(\\tau_{\\theta}) = \\emptyset\\). Thus, \\(\\Omega(A_t(\\tau_{\\theta}), G_t)\\) indicates the intersection-over-union (IoU) between the prediction result \\(A_t(\\tau_{\\theta})\\) and the groundtruth \\(G_t\\). We here calculate the precision-recall over the whole test set as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.54,
+ 0.814,
+ 0.892,
+ 0.865
+ ],
+ "angle": 0,
+ "content": "\\[\nP r (\\tau_ {\\theta}) = \\frac {1}{M} \\sum_ {i = 1} ^ {M} \\frac {1}{N _ {p}} \\sum_ {A _ {t} (\\tau_ {\\theta}) \\neq \\varnothing} \\Omega \\left(A _ {t} (\\tau_ {\\theta}), G _ {t}\\right), \\tag {3}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.54,
+ 0.862,
+ 0.83,
+ 0.904
+ ],
+ "angle": 0,
+ "content": "\\[\nR e (\\tau_ {\\theta}) = \\frac {1}{M} \\sum_ {i = 1} ^ {M} \\frac {1}{N _ {g}} \\sum_ {G _ {t} \\neq \\emptyset} \\Omega (A _ {t} (\\tau_ {\\theta}), G _ {t}),\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.481,
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+ ],
+ "angle": 0,
+ "content": "13379"
+ }
+ ],
+ [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.077,
+ 0.089,
+ 0.473,
+ 0.133
+ ],
+ "angle": 0,
+ "content": "Table 3. Performance comparison of state-of-the-art RGB aerial trackers on \\(\\mathrm{D}^2\\)Cube dataset. The top 3 results are shown in red, green, and blue. Speed in FPS (frames per second)."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.099,
+ 0.142,
+ 0.451,
+ 0.314
+ ],
+ "angle": 0,
+ "content": "| Level | ABB. | Description |
| Environment | DS | Dark Scene. The light is too low to distinguish the target. |
| OE | Overexposure. The illumination is too high to distinguish the target. |
| IC | Illumination Change. There are illumination changes during one video sequence. |
| Target | OV | Out of View. Object partially or fully leave the view. |
| FM | Fast Motion. The average per-frame object motion is larger than 20 pixels. |
| TR | Target Rotation. Target rotates in plane or out of plane. |
| SV | Scale Variation. Ratio change of target size between minimum and maximum is more than 50%. |
| DF | Deformation. The object is deformable. |
| CO | Composite Objects. The target object is an ensemble of multiple objects (e.g. man with a basketball). |
| Platform | LR | Low Resolution. The ratio of the object area to the image size is lower than 5%. |
| CM | Camera Motion. The camera moves/shakes. |
| MB | Motion Blur. The target is blurred due to the motion of itself or the camera. |
| VC | Viewpoint Change. The viewpoint is not fixed because the capturing angle changes. |
| SF | Sensor Failure. At least one camera cannot provide useful information. |
| Interaction | PO | Partial Occlusion. The object is partially occluded. |
| ST | Similar Targets. There are similar objects. |
| FO | Full Occlusion. The object is fully occluded. |
| BC | Background Clutter. There are distractors around the target object. |
+
+dation sets by themselves.
+
+Objects. Unlike previous drone-based tracking datasets which only contain limited object categories, our $\mathrm{D}^2\mathrm{Cube}$ covers generic objects. Fig. 2 shows the object classes in our test set, in which 34 classes are included. Specifically, we include some classes rarely appeared in the semantic area, e.g., part of an entire object (upper part of a body) or composite object (man holding a basketball). The whole $\mathrm{D}^2\mathrm{Cube}$ includes more than 100 categories and covers diverse objects, it thus is representative for daily scenarios.
+
+Scenarios. In this work, we mainly focus on daily life scenarios. We involve many applicable scenarios for RGBD aerial tracking, e.g., sports, work, service and entertainment, in which aerial robots and depth cameras can both work well. In detail, our recording scenarios cover daily life scenes, including office, bedroom, meeting room, gym, stadium, kitchen, and so on. Both short-term and long-term
+
+tracking scenarios are included.
+
+Annotations. As shown in Fig. 1, we provide tight axis-aligned bounding box annotations for the target objects at the frame level. A professional team annotates $\mathrm{D}^2\mathrm{Cube}$ rigorously. The annotation process follows the following rules: (i) if the target appears in the frame, we annotate the visible part of the target by the tightest bounding box. (ii) if the target does not appear in the frame, we will mark this frame with an "target loss" tag.
+
+Attributes. We define 18 tracking challenges in RGBD aerial tracking and classify the attributes in a hierarchical manner. All the attributes are defined in four levels: environment, target, platform and interaction. At different levels, correspondingly there are different challenges. Details of each attribute are given in Table 2. With such a hierarchical classification of different challenges, we can justify what challenges RGBD trackers are indeed suffering from. We also give an annotated example in Fig. 3. The distribution of each attribute in our test set is given in Fig. 4.
+
+# 4. Efficient Multimodal Tracker
+
+To achieve RGBD tracking on UAV platforms, trackers' ability to run on edge platforms with limited resources is of importance. However, the vast majority of RGBD trackers focus on architectural design with heavy backbones and additional modules. Such complex frameworks cannot satisfy the real-time requirements of aerial tracking. In this section, we propose Efficient Multimodal Tracker (EMT) for RGBD aerial tracking, which discards the heavy backbones and additional modules.
+
+# 4.1. Multimodal Fusion and Matching Architecture
+
+The proposed EMT contains four main steps, i.e., efficient modality-aware fusion, feature extraction, efficient attention-based feature matching, and target prediction. The overall architecture is shown in Fig. 5.
+
+Efficient modality-aware fusion. Firstly, to speed up the fusion, we design a novel Efficient Modality-Aware Fusion (EMAF) module that can first fuse the raw data from
+
+
+Figure 5. Overview of our proposed Efficient Multimodal Tracker (EMT). Left: Pipeline for EMT. Right: Architecture of Efficient Modality-Aware Fusion (EMAF) module.
+
+multiple modalities at a very early stage. The key to speeding up is that once the high-dimensional features are well integrated, the amount of computation that follows is greatly reduced. In addition, in order to effectively adapt to the scene, we also employ a fusion strategy that can dynamically estimate the importance of features from two modalities. The details of the fusion module to obtain fused image patches $\mathbf{T}_{\mathrm{fused}}$ and $\mathbf{X}_{\mathrm{fused}}$ are referred to 4.2.
+
+Feature extraction. Secondly, we assume that the fused template patch is $\mathbf{T}_{\mathrm{fused}} \in \mathbb{R}_{3 \times \mathbf{H}_{\mathrm{t0}} \times \mathbf{W}_{\mathrm{t0}}}$ and the fused search patch is $\mathbf{X}_{\mathrm{fused}} \in \mathbb{R}_{3 \times \mathbf{H}_{\mathrm{x0}} \times \mathbf{W}_{\mathrm{x0}}}$ . Then, we treat them as the input of the parameter-sharing backbone network for feature extraction. A modified ResNet-18 Network [6] is used as backbone network to obtain features maps for templates $f_{t} \in \mathbb{C} \times \mathbb{H}_{t} \times \mathbb{W}_{t}$ and search regions $f_{x} \in \mathbb{C} \times \mathbb{H}_{x} \times \mathbb{W}_{x}$ . Here $(H_{t}, W_{t}) = (\frac{H_{t0}}{16}, \frac{W_{t0}}{16})$ , $(H_{x}, W_{x}) = (\frac{H_{x0}}{16}, \frac{W_{x0}}{16})$ and $C = 256$ .
+
+Efficient attention-based feature matching. Thirdly, as we obtain the multimodal feature maps for the template and the search region, the next step is to match the corresponding features. To further speed up this procedure, we first use a trainable embedding to reduce the dimension of template features and then design a one-way attention-based fusion module to efficiently fuse the template features and search area features. We give a detailed description of template-to-search matching in Sec. 4.3.
+
+Target prediction. With the template-to-search map, we obtain target predictions by using regression head and classification head. The regression head is to regress the overlap between groundtruth and bounding box candidates. The classification head is to classify the objects and background.
+
+# 4.2. Efficient Modality-Aware Fusion
+
+Our EMT takes the dual-modal image patches as input, and performs a weighted fusion of modalities online. Unlike the modal independent backbone network design of traditional RGBD trackers, our proposed EMT reduces the model's size through a very early learnable fusion.
+
+Raw patch preparation. Specifically, four image patches will be treated as the input, including the color and depth template patches, and the color and depth patches for search regions. On the one hand, the template image patches $\mathbf{T}_{\mathrm{rgb}}$ and $\mathbf{T}_{\mathrm{depth}}$ are obtained by expanding the target bounding box of the first frame twice in the video. To effectively enhance discrimination, these patches should include the local surrounding information. And, the perturbation is also added to the target to avoid learning location bias. On the other hand, the patches for search regions $\mathbf{X}_{\mathrm{rgb}}$ and $\mathbf{X}_{\mathrm{depth}}$ are obtained by expanding the target box in the previous frame by four times instead of the whole original image, which utilizes the temporal context in the video sequence and reduces the computational cost.
+
+Dynamic cross-modal fusion. The aim of this step is to fuse the data and reduce the dimension at a very early stage. The intrinsic reason we can fuse these two types of data at such an early stage is that they are pixel-level aligned in image space. Moreover, it requires to dynamically judge the environment of the tracking target by extracting the global information of the image patches. To this end, we calculate the importance of the two modalities in the current frame based on the context of the two modalities. Based on the importance ratio, the RGB image patch and depth image patch can be fused by the weights for early fusion.
+
+Taking the search branch as an example, the RGB and
+
+depth image patches go through $3 \times 3$ convolution layer (Conv) and a batch norm layer (BN) to extract discriminative features. Then, a global pooling layer (GAP) is used to extract the global context of the two modalities. Next, the features are concatenated and as input of the fully connected layers (FC) to output the importance $(\alpha$ and $\beta)$ of the two modalities in the current frame. Finally, RGB image patches and depth image patches can be fused by importance weights. The process can be formulated as follows:
+
+$$
+\mathbf {X} _ {\mathbf {r g b}} = \left(\operatorname {G A P} (\operatorname {B N} (\operatorname {C o n v} (\mathbf {X} _ {\mathbf {r g b}})) \right.,
+$$
+
+$$
+\mathbf {X} _ {\text {d e p t h}} = \operatorname {G A P} (\operatorname {B N} (\operatorname {C o n v} (\mathbf {X} _ {\text {d e p t h}}))), \tag {1}
+$$
+
+$$
+\alpha , \beta = \operatorname {F C} \left(\operatorname {C a t} \left(\mathbf {X} _ {\mathbf {r g b}}, \mathbf {X} _ {\mathbf {d e p t h}}\right)\right),
+$$
+
+$$
+\mathbf {X} _ {\text {f u s e d}} = \alpha * \mathbf {X} _ {\text {r g b}} + \beta * \mathbf {X} _ {\text {d e p t h}},
+$$
+
+where $\mathbf{X}_{\mathrm{fused}}$ has the same size of original images.
+
+The immediate benefit of this early fusion is that we avoid extracting features from two modalities by using two separate backbones. Therefore, both the size of memory and the amount of computation have been greatly reduced.
+
+# 4.3. Efficient Attention-based Feature Matching
+
+For a given template feature $f_{t}$ , we will use an attention-based module to find corresponding features in $f_{x}$ for a search region. Naturally, similar to [7], both cross-attention between features and self-attention within features can be directly used. However, to speed computation up, we design a more streamlined network to achieve feature matching.
+
+Compact template representation. To make the template compact, we use a learnable embedding to reduce the dimensions of template vectors $f_{t}$ . As shown in Fig. 5, through the cross-attention with dimension reduction embedding, we obtain a compact template representation $f_{t}^{c}$ .
+
+One-Way Attention (OWA) module. With such a compact template representation $f_{t}^{c}$ , we then utilize an efficient matching from template to search area. Here, we merely deploy the cross-attention based Cross-Feature Augment (CFA) module [7] for fusion, in which multi-head cross-attention is used in a residual form. However, CFA is utilized in a one-way manner, due to the fact that we only need the template project on search areas to make a prediction in the search area [5]. OWA can be repeated by several times. With such a one-way cross-attention module, we can achieve feature matching more efficiently.
+
+# 4.4. Training and Inference
+
+Loss Function. The losses are computed between the outputs of target prediction and groundtruth:
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \lambda_ {1} \mathcal {L} _ {\text {c l s}} + \lambda_ {2} \mathcal {L} _ {\text {b b o x}} + \lambda_ {3} \mathcal {L} _ {\text {G I o u}}. \tag {2}
+$$
+
+Here, classification loss $\mathcal{L}_{cls}$ is to dicriminate the object from background. Regression loss consists of $\mathcal{L}_{bbox}$ -
+
+Mean Squared Error (MSE) between the predicted bounding boxes and the groundtruth bounding boxes, and $\mathcal{L}_{GIou}$ - generalized IoU loss [29]. We use $\lambda_{1} = 0.8344$ , $\lambda_{2} = 5$ , $\lambda_{3} = 2$ in our experiments following [7].
+
+Training phase. The training process follows the standard training recipe of current trackers [13,28,34]. We use ResNet-18 [16] pretrained on the ImageNet as our backbone and fine-tune the whole tracking framework with the training set of our proposed $\mathrm{D}^2\mathrm{Cube}$ and the training data recipe of [34]. We randomly sample two frames within an interval of 30 frames in sequences as template and search region. Subsequently, templates and search regions are obtained with some data augmentation such as jitter, brightness change, and then resized to $128\times 128$ and $256\times 256$ pixels, respectively. The template embedding size is 16. The model is trained with AdamW [25] optimizer, and the learning rate for the backbone and EMAF module are $1e - 5$ and $1e - 4$ , respectively. Weight decay is $1e - 4$ . The learning rate decays 10 times every 50 epochs. We sample 128,000 pairs in each epoch and the whole tracker is trained for 100 epochs on a single 32GB Tesla V100 GPU with a batch size of 128.
+
+Tracking phase. During inference, the template and search image are resized to fixed size. Dynamic cross-modal fusion module performs early fusion of the two modalities. After feature extraction and correlation, the prediction head outputs 256 bounding boxes and classification scores. A penalty window is employed to filter distractors.
+
+# 5. Experiments
+
+# 5.1. Experimental Settings
+
+Hardware. All comparison experiments, except for the onboard tests, are executed on a single 32GB Tesla V100 GPU. A widely-used UAV onboard processor NVidia Jetson NX Xavier is used for onboard tests.
+
+Evaluation protocols. We follow the evaluation principles in long-term RGBD tracking from VOT challenge [19]. One-Pass Evaluation (OPE) is used to test trackers' performance on our proposed $\mathrm{D}^2\mathrm{Cube}$ . At frame $t$ , $\theta_t$ is a prediction confidence score and $\tau_{\theta}$ is a classification threshold. If the predicted $\theta_t$ is not below $\tau_{\theta}$ , $A_t(\tau_{\theta})$ is used to denote the corresponding prediction. Otherwise, we set $A_t(\tau_{\theta}) = \emptyset$ . Thus, $\Omega(A_t(\tau_{\theta}), G_t)$ indicates the intersection-over-union (IoU) between the prediction result $A_t(\tau_{\theta})$ and the groundtruth $G_t$ . We here calculate the precision-recall over the whole test set as follows:
+
+$$
+P r (\tau_ {\theta}) = \frac {1}{M} \sum_ {i = 1} ^ {M} \frac {1}{N _ {p}} \sum_ {A _ {t} (\tau_ {\theta}) \neq \varnothing} \Omega \left(A _ {t} (\tau_ {\theta}), G _ {t}\right), \tag {3}
+$$
+
+$$
+R e (\tau_ {\theta}) = \frac {1}{M} \sum_ {i = 1} ^ {M} \frac {1}{N _ {g}} \sum_ {G _ {t} \neq \emptyset} \Omega (A _ {t} (\tau_ {\theta}), G _ {t}),
+$$
+
+Table 3. Performance comparison of state-of-the-art RGB aerial trackers on $\mathrm{D}^2$ Cube dataset. The top 3 results are shown in red, green, and blue. Speed in FPS (frames per second).
+
+| Level | ABB. | Description |
| Environment | DS | Dark Scene. The light is too low to distinguish the target. |
| OE | Overexposure. The illumination is too high to distinguish the target. |
| IC | Illumination Change. There are illumination changes during one video sequence. |
| Target | OV | Out of View. Object partially or fully leave the view. |
| FM | Fast Motion. The average per-frame object motion is larger than 20 pixels. |
| TR | Target Rotation. Target rotates in plane or out of plane. |
| SV | Scale Variation. Ratio change of target size between minimum and maximum is more than 50%. |
| DF | Deformation. The object is deformable. |
| CO | Composite Objects. The target object is an ensemble of multiple objects (e.g. man with a basketball). |
| Platform | LR | Low Resolution. The ratio of the object area to the image size is lower than 5%. |
| CM | Camera Motion. The camera moves/shakes. |
| MB | Motion Blur. The target is blurred due to the motion of itself or the camera. |
| VC | Viewpoint Change. The viewpoint is not fixed because the capturing angle changes. |
| SF | Sensor Failure. At least one camera cannot provide useful information. |
| Interaction | PO | Partial Occlusion. The object is partially occluded. |
| ST | Similar Targets. There are similar objects. |
| FO | Full Occlusion. The object is fully occluded. |
| BC | Background Clutter. There are distractors around the target object. |
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+ "content": ". The learning rate decays 10 times every 50 epochs. We sample 128,000 pairs in each epoch and the whole tracker is trained for 100 epochs on a single 32GB Tesla V100 GPU with a batch size of 128."
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+ "type": "text",
+ "content": ". We here calculate the precision-recall over the whole test set as follows:"
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+ "content": "P r (\\tau_ {\\theta}) = \\frac {1}{M} \\sum_ {i = 1} ^ {M} \\frac {1}{N _ {p}} \\sum_ {A _ {t} (\\tau_ {\\theta}) \\neq \\varnothing} \\Omega \\left(A _ {t} (\\tau_ {\\theta}), G _ {t}\\right), \\tag {3}",
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+ "content": "R e (\\tau_ {\\theta}) = \\frac {1}{M} \\sum_ {i = 1} ^ {M} \\frac {1}{N _ {g}} \\sum_ {G _ {t} \\neq \\emptyset} \\Omega (A _ {t} (\\tau_ {\\theta}), G _ {t}),",
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+ "content": "Cube dataset. The top 3 results are shown in red, green, and blue. Speed in FPS (frames per second)."
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+ "html": " | Loss | Medium | Hard |
| \\( \\mathcal{L}_{cls} \\) | \\( \\mathcal{L}_{con} \\) | \\( \\mathcal{R}\\mathrm{Oxf} \\) | +1M | \\( \\mathcal{R}\\mathrm{{Par}} \\) | +1M | \\( \\mathcal{R}\\mathrm{Oxf} \\) | +1M | \\( \\mathcal{R}\\mathrm{{Par}} \\) | +1M |
| (a) Local feature aggregation |
| R101-HOW-VLAD (GLDv2-clean) [13, 38, 43] | ✓ | | 73.5 | 60.4 | 82.3 | 62.6 | 51.9 | 33.2 | 67.0 | 41.8 |
| R101-HOW-ASMK (GLDv2-clean) [37, 38, 43] | ✓ | | 80.4 | 70.2 | 85.4 | 68.8 | 62.5 | 45.4 | 70.8 | 45.4 |
| R50-FIRE-ASMK (SfM-120k) [41] | | ✓ | 81.8 | 66.5 | 85.3 | 67.6 | 61.2 | 40.1 | 70.0 | 42.9 |
| R50-MDA-ASMK (SfM-120k) [44] | | ✓ | 81.8 | 68.7 | 83.3 | 64.7 | 62.2 | 45.3 | 66.2 | 38.9 |
| R50-Token (GLDv2-clean) [43] | ✓ | | 80.5 | 68.3 | 87.6 | 73.9 | 62.1 | 43.4 | 73.8 | 53.3 |
| R101-Token (GLDv2-clean) [43] | ✓ | | 82.3 | 70.5 | 89.3 | 76.7 | 66.6 | 47.3 | 78.6 | 55.9 |
| (b) Global single-pass |
| R101-GeM (SfM-120k) [29] | | ✓ | 65.3 | 46.1 | 77.3 | 52.6 | 39.6 | 22.2 | 56.6 | 24.8 |
| R101-GeM-AP* (GLDv1) [28, 32] | | ✓ | 66.3 | - | 80.2 | - | 42.5 | - | 60.8 | - |
| R101-GeM-ArcFace (GLDv2-clean) [42] | ✓ | | 74.2 | - | 84.9 | - | 51.6 | - | 70.3 | - |
| R101-SOLAR (GLDv1) [23] | | ✓ | 69.9 | 53.5 | 81.6 | 59.2 | 47.9 | 29.9 | 65.5 | 33.4 |
| R50-DELG (GLDv2-clean) [3] | ✓ | | 73.6 | 60.6 | 85.7 | 68.6 | 51.0 | 32.7 | 71.5 | 44.4 |
| R101-DELG (GLDv2-clean) [3] | ✓ | | 76.3 | 63.7 | 86.6 | 70.6 | 55.6 | 37.5 | 72.4 | 46.9 |
| \\( R50-DOLG^‡ \\) (GLDv2-clean) [46] | ✓ | | 78.6 | 68.9 | 87.5 | 76.7 | 58.2 | 44.1 | 73.7 | 56.2 |
| \\( R101-DOLG^‡ \\) (GLDv2-clean) [46] | ✓ | | 79.5 | 72.1 | 89.7 | 80.3 | 59.5 | 47.8 | 78.1 | 61.5 |
| R101-GLAM (GLDv2-clean) [34] | ✓ | | 78.6 | 68.0 | 88.5 | 73.5 | 60.2 | 43.5 | 76.8 | 53.1 |
| R50-CVNet-Global (GLDv2-clean) [18] | ✓ | ✓ | 81.0 | 72.6 | 88.8 | 79.0 | 62.1 | 50.2 | 76.5 | 60.2 |
| R101-CVNet-Global (GLDv2-clean) [18] | ✓ | ✓ | 80.2 | 74.0 | 90.3 | 80.6 | 63.1 | 53.7 | 79.1 | 62.2 |
| (c) Ours |
| R50-SENet (GLDv2-clean) | ✓ | | 81.4 | 72.9 | 90.5 | 79.0 | 62.3 | 48.7 | 80.3 | 59.9 |
| R50-SENet (GLDv2-clean) | ✓ | ✓ | 81.9 | 74.2 | 90.0 | 79.1 | 63.0 | 52.0 | 78.1 | 59.9 |
| R101-SENet (GLDv2-clean) | ✓ | | 80.0 | 72.5 | 91.6 | 82.1 | 61.7 | 49.2 | 82.2 | 64.6 |
| R101-SENet (GLDv2-clean) | ✓ | ✓ | 82.8 | 76.1 | 91.7 | 83.6 | 66.0 | 55.7 | 82.8 | 67.8 |
",
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+ "text": "Table 1. Comparison with the state-of-the-art models. evaluated performances on Revisited Oxford (ROxf) and Revisited Paris (RPar) with adding 1M distractors experiments (+1M). The best scores for each group and backbone are presented as boldfaced text. $\\star$ denotes using AP-Loss. $\\ddagger$ denotes reproduced model with the official model and weights.",
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+ "text": "with their 1 million distractors. In this table, we divided the previous global embedding models into two groups: (a) Local feature aggregation and (b) Global single-pass. (a)",
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+ "text": "Local feature aggregation. Our proposed network shows better overall performance than solutions using classical aggregators such as VLAD [12] or ASMK [37]. However, they consume more extract and matching time due to the aggregator. Recently, a learnable aggregator named Token [43] has been proposed and sets a new state-of-the-art performance with reasonable extraction latency and matching time. However, due to the limitation that structural characteristics cannot be considered, it shows a substantial performance decrease when a 1M distractor is added. Our proposed method leads the performance by up to $8.7\\%$ (RPar-Hard+1M) when adding 1M distractor experiments in the setting using the same loss function (R101-SENet- $\\mathcal{L}_{cls}$ ). (b) Global single-pass. From some point of view, our proposed networks belong to this group. In this group, our proposed networks outperform existing solutions in most experiments regardless of the setting. Among the solutions using classification loss only, the state-of-the-art solution is DOLG | Loss | Medium | Hard |
| \\( \\mathcal{L}_{cls} \\) | \\( \\mathcal{L}_{con} \\) | \\( \\mathcal{R}\\mathrm{Oxf} \\) | +1M | \\( \\mathcal{R}\\mathrm{{Par}} \\) | +1M | \\( \\mathcal{R}\\mathrm{Oxf} \\) | +1M | \\( \\mathcal{R}\\mathrm{{Par}} \\) | +1M |
| (a) Local feature aggregation |
| R101-HOW-VLAD (GLDv2-clean) [13, 38, 43] | ✓ | | 73.5 | 60.4 | 82.3 | 62.6 | 51.9 | 33.2 | 67.0 | 41.8 |
| R101-HOW-ASMK (GLDv2-clean) [37, 38, 43] | ✓ | | 80.4 | 70.2 | 85.4 | 68.8 | 62.5 | 45.4 | 70.8 | 45.4 |
| R50-FIRE-ASMK (SfM-120k) [41] | | ✓ | 81.8 | 66.5 | 85.3 | 67.6 | 61.2 | 40.1 | 70.0 | 42.9 |
| R50-MDA-ASMK (SfM-120k) [44] | | ✓ | 81.8 | 68.7 | 83.3 | 64.7 | 62.2 | 45.3 | 66.2 | 38.9 |
| R50-Token (GLDv2-clean) [43] | ✓ | | 80.5 | 68.3 | 87.6 | 73.9 | 62.1 | 43.4 | 73.8 | 53.3 |
| R101-Token (GLDv2-clean) [43] | ✓ | | 82.3 | 70.5 | 89.3 | 76.7 | 66.6 | 47.3 | 78.6 | 55.9 |
| (b) Global single-pass |
| R101-GeM (SfM-120k) [29] | | ✓ | 65.3 | 46.1 | 77.3 | 52.6 | 39.6 | 22.2 | 56.6 | 24.8 |
| R101-GeM-AP* (GLDv1) [28, 32] | | ✓ | 66.3 | - | 80.2 | - | 42.5 | - | 60.8 | - |
| R101-GeM-ArcFace (GLDv2-clean) [42] | ✓ | | 74.2 | - | 84.9 | - | 51.6 | - | 70.3 | - |
| R101-SOLAR (GLDv1) [23] | | ✓ | 69.9 | 53.5 | 81.6 | 59.2 | 47.9 | 29.9 | 65.5 | 33.4 |
| R50-DELG (GLDv2-clean) [3] | ✓ | | 73.6 | 60.6 | 85.7 | 68.6 | 51.0 | 32.7 | 71.5 | 44.4 |
| R101-DELG (GLDv2-clean) [3] | ✓ | | 76.3 | 63.7 | 86.6 | 70.6 | 55.6 | 37.5 | 72.4 | 46.9 |
| \\( R50-DOLG^‡ \\) (GLDv2-clean) [46] | ✓ | | 78.6 | 68.9 | 87.5 | 76.7 | 58.2 | 44.1 | 73.7 | 56.2 |
| \\( R101-DOLG^‡ \\) (GLDv2-clean) [46] | ✓ | | 79.5 | 72.1 | 89.7 | 80.3 | 59.5 | 47.8 | 78.1 | 61.5 |
| R101-GLAM (GLDv2-clean) [34] | ✓ | | 78.6 | 68.0 | 88.5 | 73.5 | 60.2 | 43.5 | 76.8 | 53.1 |
| R50-CVNet-Global (GLDv2-clean) [18] | ✓ | ✓ | 81.0 | 72.6 | 88.8 | 79.0 | 62.1 | 50.2 | 76.5 | 60.2 |
| R101-CVNet-Global (GLDv2-clean) [18] | ✓ | ✓ | 80.2 | 74.0 | 90.3 | 80.6 | 63.1 | 53.7 | 79.1 | 62.2 |
| (c) Ours |
| R50-SENet (GLDv2-clean) | ✓ | | 81.4 | 72.9 | 90.5 | 79.0 | 62.3 | 48.7 | 80.3 | 59.9 |
| R50-SENet (GLDv2-clean) | ✓ | ✓ | 81.9 | 74.2 | 90.0 | 79.1 | 63.0 | 52.0 | 78.1 | 59.9 |
| R101-SENet (GLDv2-clean) | ✓ | | 80.0 | 72.5 | 91.6 | 82.1 | 61.7 | 49.2 | 82.2 | 64.6 |
| R101-SENet (GLDv2-clean) | ✓ | ✓ | 82.8 | 76.1 | 91.7 | 83.6 | 66.0 | 55.7 | 82.8 | 67.8 |
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+ "content": "with their 1 million distractors. In this table, we divided the previous global embedding models into two groups: (a) Local feature aggregation and (b) Global single-pass. (a)"
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+ "content": "Local feature aggregation. Our proposed network shows better overall performance than solutions using classical aggregators such as VLAD [12] or ASMK [37]. However, they consume more extract and matching time due to the aggregator. Recently, a learnable aggregator named Token [43] has been proposed and sets a new state-of-the-art performance with reasonable extraction latency and matching time. However, due to the limitation that structural characteristics cannot be considered, it shows a substantial performance decrease when a 1M distractor is added. Our proposed method leads the performance by up to \\(8.7\\%\\) (RPar-Hard+1M) when adding 1M distractor experiments in the setting using the same loss function (R101-SENet-\\(\\mathcal{L}_{cls}\\)). (b) Global single-pass. From some point of view, our proposed networks belong to this group. In this group, our proposed networks outperform existing solutions in most experiments regardless of the setting. Among the solutions using classification loss only, the state-of-the-art solution is DOLG | Loss | Medium | Hard |
| \( \mathcal{L}_{cls} \) | \( \mathcal{L}_{con} \) | \( \mathcal{R}\mathrm{Oxf} \) | +1M | \( \mathcal{R}\mathrm{{Par}} \) | +1M | \( \mathcal{R}\mathrm{Oxf} \) | +1M | \( \mathcal{R}\mathrm{{Par}} \) | +1M |
| (a) Local feature aggregation |
| R101-HOW-VLAD (GLDv2-clean) [13, 38, 43] | ✓ | | 73.5 | 60.4 | 82.3 | 62.6 | 51.9 | 33.2 | 67.0 | 41.8 |
| R101-HOW-ASMK (GLDv2-clean) [37, 38, 43] | ✓ | | 80.4 | 70.2 | 85.4 | 68.8 | 62.5 | 45.4 | 70.8 | 45.4 |
| R50-FIRE-ASMK (SfM-120k) [41] | | ✓ | 81.8 | 66.5 | 85.3 | 67.6 | 61.2 | 40.1 | 70.0 | 42.9 |
| R50-MDA-ASMK (SfM-120k) [44] | | ✓ | 81.8 | 68.7 | 83.3 | 64.7 | 62.2 | 45.3 | 66.2 | 38.9 |
| R50-Token (GLDv2-clean) [43] | ✓ | | 80.5 | 68.3 | 87.6 | 73.9 | 62.1 | 43.4 | 73.8 | 53.3 |
| R101-Token (GLDv2-clean) [43] | ✓ | | 82.3 | 70.5 | 89.3 | 76.7 | 66.6 | 47.3 | 78.6 | 55.9 |
| (b) Global single-pass |
| R101-GeM (SfM-120k) [29] | | ✓ | 65.3 | 46.1 | 77.3 | 52.6 | 39.6 | 22.2 | 56.6 | 24.8 |
| R101-GeM-AP* (GLDv1) [28, 32] | | ✓ | 66.3 | - | 80.2 | - | 42.5 | - | 60.8 | - |
| R101-GeM-ArcFace (GLDv2-clean) [42] | ✓ | | 74.2 | - | 84.9 | - | 51.6 | - | 70.3 | - |
| R101-SOLAR (GLDv1) [23] | | ✓ | 69.9 | 53.5 | 81.6 | 59.2 | 47.9 | 29.9 | 65.5 | 33.4 |
| R50-DELG (GLDv2-clean) [3] | ✓ | | 73.6 | 60.6 | 85.7 | 68.6 | 51.0 | 32.7 | 71.5 | 44.4 |
| R101-DELG (GLDv2-clean) [3] | ✓ | | 76.3 | 63.7 | 86.6 | 70.6 | 55.6 | 37.5 | 72.4 | 46.9 |
| \( R50-DOLG^‡ \) (GLDv2-clean) [46] | ✓ | | 78.6 | 68.9 | 87.5 | 76.7 | 58.2 | 44.1 | 73.7 | 56.2 |
| \( R101-DOLG^‡ \) (GLDv2-clean) [46] | ✓ | | 79.5 | 72.1 | 89.7 | 80.3 | 59.5 | 47.8 | 78.1 | 61.5 |
| R101-GLAM (GLDv2-clean) [34] | ✓ | | 78.6 | 68.0 | 88.5 | 73.5 | 60.2 | 43.5 | 76.8 | 53.1 |
| R50-CVNet-Global (GLDv2-clean) [18] | ✓ | ✓ | 81.0 | 72.6 | 88.8 | 79.0 | 62.1 | 50.2 | 76.5 | 60.2 |
| R101-CVNet-Global (GLDv2-clean) [18] | ✓ | ✓ | 80.2 | 74.0 | 90.3 | 80.6 | 63.1 | 53.7 | 79.1 | 62.2 |
| (c) Ours |
| R50-SENet (GLDv2-clean) | ✓ | | 81.4 | 72.9 | 90.5 | 79.0 | 62.3 | 48.7 | 80.3 | 59.9 |
| R50-SENet (GLDv2-clean) | ✓ | ✓ | 81.9 | 74.2 | 90.0 | 79.1 | 63.0 | 52.0 | 78.1 | 59.9 |
| R101-SENet (GLDv2-clean) | ✓ | | 80.0 | 72.5 | 91.6 | 82.1 | 61.7 | 49.2 | 82.2 | 64.6 |
| R101-SENet (GLDv2-clean) | ✓ | ✓ | 82.8 | 76.1 | 91.7 | 83.6 | 66.0 | 55.7 | 82.8 | 67.8 |
+
+Table 1. Comparison with the state-of-the-art models. evaluated performances on Revisited Oxford (ROxf) and Revisited Paris (RPar) with adding 1M distractors experiments (+1M). The best scores for each group and backbone are presented as boldfaced text. $\star$ denotes using AP-Loss. $\ddagger$ denotes reproduced model with the official model and weights.
+
+with their 1 million distractors. In this table, we divided the previous global embedding models into two groups: (a) Local feature aggregation and (b) Global single-pass. (a)
+
+Local feature aggregation. Our proposed network shows better overall performance than solutions using classical aggregators such as VLAD [12] or ASMK [37]. However, they consume more extract and matching time due to the aggregator. Recently, a learnable aggregator named Token [43] has been proposed and sets a new state-of-the-art performance with reasonable extraction latency and matching time. However, due to the limitation that structural characteristics cannot be considered, it shows a substantial performance decrease when a 1M distractor is added. Our proposed method leads the performance by up to $8.7\%$ (RPar-Hard+1M) when adding 1M distractor experiments in the setting using the same loss function (R101-SENet- $\mathcal{L}_{cls}$ ). (b) Global single-pass. From some point of view, our proposed networks belong to this group. In this group, our proposed networks outperform existing solutions in most experiments regardless of the setting. Among the solutions using classification loss only, the state-of-the-art solution is DOLG | Loss | Medium | Hard |
| \\( \\mathcal{L}_{cls} \\) | \\( \\mathcal{L}_{con} \\) | \\( \\mathcal{R}\\mathrm{Oxf} \\) | +1M | \\( \\mathcal{R}\\mathrm{{Par}} \\) | +1M | \\( \\mathcal{R}\\mathrm{Oxf} \\) | +1M | \\( \\mathcal{R}\\mathrm{{Par}} \\) | +1M |
| (a) Local feature aggregation |
| R101-HOW-VLAD (GLDv2-clean) [13, 38, 43] | ✓ | | 73.5 | 60.4 | 82.3 | 62.6 | 51.9 | 33.2 | 67.0 | 41.8 |
| R101-HOW-ASMK (GLDv2-clean) [37, 38, 43] | ✓ | | 80.4 | 70.2 | 85.4 | 68.8 | 62.5 | 45.4 | 70.8 | 45.4 |
| R50-FIRE-ASMK (SfM-120k) [41] | | ✓ | 81.8 | 66.5 | 85.3 | 67.6 | 61.2 | 40.1 | 70.0 | 42.9 |
| R50-MDA-ASMK (SfM-120k) [44] | | ✓ | 81.8 | 68.7 | 83.3 | 64.7 | 62.2 | 45.3 | 66.2 | 38.9 |
| R50-Token (GLDv2-clean) [43] | ✓ | | 80.5 | 68.3 | 87.6 | 73.9 | 62.1 | 43.4 | 73.8 | 53.3 |
| R101-Token (GLDv2-clean) [43] | ✓ | | 82.3 | 70.5 | 89.3 | 76.7 | 66.6 | 47.3 | 78.6 | 55.9 |
| (b) Global single-pass |
| R101-GeM (SfM-120k) [29] | | ✓ | 65.3 | 46.1 | 77.3 | 52.6 | 39.6 | 22.2 | 56.6 | 24.8 |
| R101-GeM-AP* (GLDv1) [28, 32] | | ✓ | 66.3 | - | 80.2 | - | 42.5 | - | 60.8 | - |
| R101-GeM-ArcFace (GLDv2-clean) [42] | ✓ | | 74.2 | - | 84.9 | - | 51.6 | - | 70.3 | - |
| R101-SOLAR (GLDv1) [23] | | ✓ | 69.9 | 53.5 | 81.6 | 59.2 | 47.9 | 29.9 | 65.5 | 33.4 |
| R50-DELG (GLDv2-clean) [3] | ✓ | | 73.6 | 60.6 | 85.7 | 68.6 | 51.0 | 32.7 | 71.5 | 44.4 |
| R101-DELG (GLDv2-clean) [3] | ✓ | | 76.3 | 63.7 | 86.6 | 70.6 | 55.6 | 37.5 | 72.4 | 46.9 |
| \\( R50-DOLG^‡ \\) (GLDv2-clean) [46] | ✓ | | 78.6 | 68.9 | 87.5 | 76.7 | 58.2 | 44.1 | 73.7 | 56.2 |
| \\( R101-DOLG^‡ \\) (GLDv2-clean) [46] | ✓ | | 79.5 | 72.1 | 89.7 | 80.3 | 59.5 | 47.8 | 78.1 | 61.5 |
| R101-GLAM (GLDv2-clean) [34] | ✓ | | 78.6 | 68.0 | 88.5 | 73.5 | 60.2 | 43.5 | 76.8 | 53.1 |
| R50-CVNet-Global (GLDv2-clean) [18] | ✓ | ✓ | 81.0 | 72.6 | 88.8 | 79.0 | 62.1 | 50.2 | 76.5 | 60.2 |
| R101-CVNet-Global (GLDv2-clean) [18] | ✓ | ✓ | 80.2 | 74.0 | 90.3 | 80.6 | 63.1 | 53.7 | 79.1 | 62.2 |
| (c) Ours |
| R50-SENet (GLDv2-clean) | ✓ | | 81.4 | 72.9 | 90.5 | 79.0 | 62.3 | 48.7 | 80.3 | 59.9 |
| R50-SENet (GLDv2-clean) | ✓ | ✓ | 81.9 | 74.2 | 90.0 | 79.1 | 63.0 | 52.0 | 78.1 | 59.9 |
| R101-SENet (GLDv2-clean) | ✓ | | 80.0 | 72.5 | 91.6 | 82.1 | 61.7 | 49.2 | 82.2 | 64.6 |
| R101-SENet (GLDv2-clean) | ✓ | ✓ | 82.8 | 76.1 | 91.7 | 83.6 | 66.0 | 55.7 | 82.8 | 67.8 |
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+ "content": "Local feature aggregation. Our proposed network shows better overall performance than solutions using classical aggregators such as VLAD [12] or ASMK [37]. However, they consume more extract and matching time due to the aggregator. Recently, a learnable aggregator named Token [43] has been proposed and sets a new state-of-the-art performance with reasonable extraction latency and matching time. However, due to the limitation that structural characteristics cannot be considered, it shows a substantial performance decrease when a 1M distractor is added. Our proposed method leads the performance by up to "
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+ "content": " (RPar-Hard+1M) when adding 1M distractor experiments in the setting using the same loss function (R101-SENet-"
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+ "content": "\\mathcal{L}_{cls}"
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+ ],
+ "type": "text",
+ "content": "). (b) Global single-pass. From some point of view, our proposed networks belong to this group. In this group, our proposed networks outperform existing solutions in most experiments regardless of the setting. Among the solutions using classification loss only, the state-of-the-art solution is DOLG | Known camera poses | Unknown camera poses |
| Static scene | NeRF [44], SVS [59], NeRF++ [82], Mip-NeRF [4], Mip-NeRF 360 [5], DirectXVoxGO [68], Plenoxels [23], Instant-ngp [45], TenorRF [12] | NeRF - - [73], BARF [40], SC-NeRF [31], NeRF-SLAM [60] |
| Dynamic scene | NV [43], D-NeRF [56], NR-NeRF [71], NSFF [39], DynamicNeRF [24], Nerfies [52], HyperNeRF [53], TiNeuVox [20], T-NeRF [25] | Ours |
",
+ "bbox": [
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+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "many recent dynamic view synthesis methods for unstructured videos [24,25,39,52,53,56,71,76] and new methods based on deformable fields [20]. However, these techniques require precise camera poses typically estimated via SfM systems such as COLMAP [62] (bottom left of Table 1).",
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+ {
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+ "text": "However, SfM systems are not robust to many issues, such as noisy images from low-light conditions, motion blur caused by users, or dynamic objects in the scene, such as people, cars, and animals. The robustness problem of the SfM systems causes the existing dynamic view synthesis methods to be fragile and impractical for many challenging videos. Recently, several NeRF-based methods [31, 40, 60, 73] have proposed jointly optimizing the camera poses with the scene geometry. Nevertheless, these methods can only handle strictly static scenes (top right of Table 1).",
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+ {
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+ "text": "We introduce RoDynRF, an algorithm for reconstructing dynamic radiance fields from casual videos. Unlike existing approaches, we do not require accurate camera poses as input. Our method optimizes camera poses and two radiance fields, modeling static and dynamic elements. Our approach includes a coarse-to-fine strategy and epipolar geometry to exclude moving pixels, deformation fields, time-dependent appearance models, and regularization losses for improved consistency. We evaluate the algorithm on multiple datasets, including Sintel [9], Dynamic View Synthesis [79], iPhone [25], and DAVIS [55], and show visual comparisons with existing methods.",
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+ "- We present a space-time synthesis algorithm from a dynamic monocular video that does not require known camera poses and camera intrinsics as input.",
+ "- Our proposed careful architecture designs and auxiliary losses improve the robustness of camera pose estimation and dynamic radiance field reconstruction.",
+ "- Quantitative and qualitative evaluations demonstrate the robustness of our method over other state-of-the-art methods on several challenging datasets that typical SfM systems fail to estimate camera poses."
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+ {
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+ "text": "Static view synthesis. Many view synthesis techniques construct specific scene geometry from images captured at various positions [8] and use local warps [11] to synthesize high-quality novel views of a scene. Approaches to",
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+ "text": "light field rendering use implicit scene geometry to create photorealistic novel views, but they require densely captured images [27,37]. By using soft 3D reconstruction [54], learning-based dense depth maps [22], multiplane images (MPIs) [14,21,67], additional learned deep features [30,58], or voxel-based implicit scene representations [66], several earlier work attempt to use proxy scene geometry to enhance rendering quality.",
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+ "text": "Recent methods implicitly model the scene as a continuous neural radiance field (NeRF) [4, 44, 82] with multi-layer perceptrons (MLPs). However, NeRF requires days of training time to represent a scene. Therefore, recent methods [12, 23, 45, 68] replace the implicit MLPs with explicit voxels and significantly improve the training speed.",
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+ "text": "Several approaches synthesize novel views from a single RGB input image. These methods often fill up holes in the disoccluded regions and predict depth [38, 49], additionally learned features [75], multiplane images [72], and layered depth images [34, 64]. Although these techniques have produced excellent view synthesis results, they can only handle static scenes. Our approach performs view synthesis of dynamic scenes from a single monocular video, in contrast to existing view synthesis techniques focusing on static scenes.",
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+ "text": "Dynamic view synthesis. By focusing on human bodies [74], using RGBD data [16], reconstructing sparse geometry [51], or producing minimal stereoscopic disparity transitions between input views [1], many techniques reconstruct and synthesize novel views from non-rigid dynamic scenes. Other techniques break down dynamic scenes into piece-wise rigid parts using hand-crafted priors [36, 61]. Many systems cannot handle scenes with complicated geometry and instead require multi-view and time-synchronized videos as input to provide interactive view manipulation [3, 7, 41, 85]. Yoon et al. [79] used depth from single-view and multi-view stereo to synthesize novel views of dynamic scenes from a single video using explicit depth-based 3D warping.",
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+ "text": "A recent line of work extends NeRF to handle dynamic scenes [20, 24, 39, 52, 53, 56, 71, 76]. Although these space-time synthesis results are impressive, these techniques rely on precise camera pose input. Consequently, these techniques are not applicable to challenging scenes where COLMAP [62] or current SfM systems fail. Our approach, in contrast, can handle complex dynamic scenarios without known camera poses.",
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+ "text": "Visual odometry and camera pose estimation. From a collection of images, visual odometry estimates the 3D camera poses [18, 19, 46-48]. These techniques mainly fall into two categories: direct methods that maximize photometric consistency [78, 84] and feature-based methods that rely on manually created or learned features [46, 47, 63]. Self-supervised image reconstruction losses have recently",
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+ "Figure 2. Overall framework. We model the dynamic scene with static and dynamic radiance fields. The static radiance fields take both the sampled coordinates $(x, y, z)$ and the viewing direction $\\mathbf{d}$ as input and predict the density $\\sigma^s$ and color $\\mathbf{c}^s$ . Note that the density of the static part is invariant to time and viewing direction, therefore, we use summation of the queried features as the density (instead of using an MLP). We only compute the losses over the static regions. The computed gradients backpropagate not only to the static voxel field and MLPs but also to the camera parameters. The dynamic radiance fields take the sampled coordinates and the time $t$ to obtain the deformed coordinates $(x', y', z')$ in the canonical space. Then we query the features using these deformed coordinates from the dynamic voxel fields and pass the features along with the time index to a time-dependent shallow MLPs to get the color $\\mathbf{c}^d$ , density $\\sigma^d$ , and nonrigidity $m^d$ of the dynamic part. Finally, after the volume rendering, we can obtain the RGB images $\\mathbf{C}^{\\{s,d\\}}$ and the depth maps $\\mathbf{D}^{\\{s,d\\}}$ from the static and dynamic parts along with a nonrigidity mask $\\mathbf{M}^d$ . Finally, we calculate the per-frame reconstruction loss. Note that here we only include per-frame losses."
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+ "text": "been used in learning-based systems to tackle visual odometry [2, 6, 26, 35, 70, 77, 80, 81, 83, 84]. Estimating camera poses from casually captured videos remains challenging. NeRF-based techniques have been proposed to combine neural 3D representation and camera poses for optimization [31, 40, 60, 73], although they are limited to static sequences. In contrast to the visual odometry techniques outlined above, our system simultaneously optimizes camera poses and models dynamic objects models.",
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+ "text": "In this section, we first briefly introduce the background of neural radiance fields and their extension of camera pose estimation and dynamic scene representation in Section 3.1. We then describe the overview of our method in Section 3.2. Next, we discuss the details of camera pose estimation with the static radiance field reconstruction in Section 3.3. After that, we show how to model the dynamic scene in Section 3.4. Finally, we outline the implementation details in Section 3.5.",
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+ "text": "NeRF. Neural radiance fields (NeRF) [44] represent a static 3D scene with implicit MLPs parameterized by $\\Theta$ and map",
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+ "text": "the 3D position $(x,y,z)$ and viewing direction $(\\theta ,\\phi)$ to its corresponding color $\\mathbf{c}$ and density $\\sigma$ ..",
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+ "text": "\n$$\n(\\mathbf {c}, \\sigma) = \\operatorname {M L P} _ {\\Theta} (x, y, z, \\theta , \\phi). \\tag {1}\n$$\n",
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+ "text": "We can compute the pixel color by applying volume rendering [17, 32] along the ray $\\mathbf{r}$ emitted from the camera origin:",
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+ "text": "\n$$\n\\hat {\\mathbf {C}} (\\mathbf {r}) = \\sum_ {i = 1} ^ {N} T (i) \\left(1 - \\exp (- \\sigma (i) \\delta (i))\\right) \\mathbf {c} (i), \\tag {2}\n$$\n",
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+ "text": "\n$$\nT (i) = \\exp (- \\sum_ {i - 1} ^ {j} \\sigma (j) \\delta (j)),\n$$\n",
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+ "text": "where $\\delta (i)$ represents the distance between two consecutive sample points along the ray, $N$ is the number of samples along each ray, and $T(i)$ indicates the accumulated transparency. As the volume rendering procedure is differentiable, we can optimize the radiance fields by minimizing the reconstruction error between the rendered color $\\hat{\\mathbf{C}}$ and the ground truth color $\\mathbf{C}$ :",
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+ "text": "\n$$\n\\mathcal {L} = \\left\\| \\hat {\\mathbf {C}} (\\mathbf {r}) - \\mathbf {C} (\\mathbf {r}) \\right\\| _ {2} ^ {2}. \\tag {3}\n$$\n",
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+ "text": "Explicit neural voxel radiance fields. Although with compelling rendering quality, NeRF-based methods model",
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+ "(a) Static radiance field reconstruction and pose estimation"
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+ "(b) Dynamic radiance field reconstruction",
+ "Figure 3. Training losses. For both the (a) static and (b) dynamic parts, we introduce three auxiliary losses to encourage the consistency of the modeling: reprojection loss, disparity loss, and monocular depth loss. The reprojection loss encourages the projection of the 3D volume rendered points onto neighbor frames to be similar to the pre-calculated flow. The disparity loss forces the volume rendered 3D points from two corresponding points of neighbor frames to have similar z values. Finally, the monocular depth loss calculates the scale-and shift-invariant loss between the volume rendered depth and the pre-calculated MiDaS depth. (a) We use the motion mask to exclude the dynamic regions from the loss calculation. (b) We use a scene flow MLP to model the 3D movement of the volume rendered 3D points."
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+ {
+ "type": "text",
+ "text": "the scene with implicit representations such as MLPs for high storage efficiency. These methods, however, are very slow to train. To overcome this drawback, recent methods [12, 23, 45, 68] propose to model the radiance fields with explicit voxels. Specifically, these methods replace the mapping function with voxel grids and directly optimize the features sampled from the voxels. They usually apply shallow MLPs to handle the view-dependent effects. By eliminating the heavy usage of the MLPs, the training time of these methods reduces from days to hours. We also leverage explicit representation in this work.",
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+ "text": "We show our proposed framework in Figure 2. Given an input video sequence with $N$ frames, our method jointly optimizes the camera poses, focal length, and static and dynamic radiance fields. We represent both the static and dynamic parts with explicit neural voxels $\\mathbf{V}^s$ and $\\mathbf{V}^d$ , respectively. The static radiance fields are responsible for reconstructing the static scene and estimating the camera poses and focal length. At the same time, the goal of dynamic radiance fields is to model the scene dynamics in the video (usually caused by moving objects).",
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+ "text": "3.3. Camera Pose Estimation",
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+ "text": "Motion mask generation. Excluding dynamic regions in the video helps improve the robustness of camera pose estimation. Existing methods [39] often leverage off-the-shelf instance segmentation methods such as Mask R-CNN [29] to mask out the common moving objects. However, many moving objects are hard to detect/segment in the input video, such as drifting water or swaying trees. Therefore, in",
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+ "image_caption": [
+ "(a) w/o coarse-to-fine",
+ "(c) w/o late viewing direction conditioning",
+ "Figure 4. The impact of design choices on camera pose estimation. (a) No coarse-to-fine strategy leads to sub-optimal solutions. (b) No single-image depth prior results in poor scene geometry for challenging camera trajectories. (c) The absence of late viewing direction conditioning leads to wrong geometry and poses due to minimizing photometric loss instead of consistent voxel space using MLP. (d) Our proposed method incorporates all components and yields reasonable scene geometry and camera trajectory."
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+ "(b) w/o monocular depth prior",
+ "(d) Full model"
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+ "text": "addition to the masks from Mask R-CNN, we also estimate the fundamental matrix using the optical flow from consecutive frames. We then calculate and threshold the Sampson distance (the distance of each pixel to the estimated epipolar line) to obtain a binary motion mask. Finally, we combine the results from Mask R-CNN and epipolar distance thresholding to obtain our final motion masks.",
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+ "text": "Coarse-to-fine static scene reconstruction. The first part of our method is reconstructing the static radiance fields along with the camera poses. We jointly optimize the 6D camera poses $[R|t]_i, i \\in [1..N]$ and the focal length $f$ shared by all input frames simultaneously. Similar to ex",
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+ "type": "page_number",
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+ {
+ "type": "text",
+ "text": "isting pose estimation methods [40], we optimize the static scene representation in a coarse-to-fine manner. Specifically, we start with a smaller static voxel resolution and progressively increase the voxel resolution during the training. This coarse-to-fine strategy is essential to the camera pose estimation as the energy surface will become smoother. Thus, the optimizer will have less chance of getting stuck in sub-optimal solutions (Figure 4(a) vs. Figure 4(d)).",
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+ "text": "Late viewing direction conditioning. As our primary supervision is the photometric consistency loss, the optimization could bypass the neural voxel and directly learn a mapping function from the viewing direction to the output sample color. Therefore, we choose to fuse the viewing direction only in the last layer of the color MLP as shown in Figure 2. This design choice is critical because we are reconstructing not only the scene geometry but also the camera poses. Figure 4(c) shows that without the late viewing direction conditioning, the optimization could minimize the photometric loss by optimizing the MLP and lead to erroneous camera poses and geometry estimation.",
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+ "text": "Losses. We minimize the photometric loss between the prediction $\\hat{\\mathbf{C}}^s (\\mathbf{r})$ and the captured images in the static regions:",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {L} _ {c} ^ {s} = \\left\\| (\\hat {\\mathbf {C}} ^ {s} (\\mathbf {r}) - \\mathbf {C} (\\mathbf {r})) \\cdot (1 - \\mathbf {M} (\\mathbf {r})) \\right\\| _ {2} ^ {2}, \\tag {4}\n$$\n",
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+ "type": "text",
+ "text": "where $\\mathbf{M}$ denotes the motion mask.",
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+ {
+ "type": "text",
+ "text": "To handle casually-captured but challenging camera trajectories such as fast-moving or pure rotating, we introduce auxiliary losses to regularize the training, similar to [24,39].",
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+ "(1) Reprojection loss $\\mathcal{L}_{\\mathrm{reproj}}^s$ : We use 2D optical flow estimated by RAFT [69] to guide the training. First, we volume render all the sampled 3D points along a ray to generate a surface point. We then reproject this point onto its neighbor frame and calculate the reprojection error with the correspondence estimated from RAFT.",
+ "(2) Disparity loss $\\mathcal{L}_{\\mathrm{disp}}^s$ : Similar to the reprojection loss above, we also regularize the error in the $z$ -direction (in the camera coordinate). We volume render the two corresponding points into 3D space and calculate the error of the $z$ component. As we care more about the near than the far, we compute this loss in the inverse-depth domain.",
+ "(3) Monocular depth loss $\\mathcal{L}_{\\mathrm{monodepth}}^s$ : The two losses above cannot handle pure rotating cameras and often lead to the incorrect camera poses and geometry (Figure 4(b)). We enforce the depth order from multiple pixels of the same frame to match the order of a monocular depth map. We pre-calculate the depth map using MiDaSv2.1 [57]. The depth prediction from MiDaS is up to an unknown scale and shift. Therefore, we use the same scale- and shift-invariant loss in MiDaS to constrain our rendered depth values."
+ ],
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+ "type": "text",
+ "text": "We illustrate these auxiliary losses in Figure 3(a). Since the optical flow and depth map may not be accurate, we",
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+ {
+ "type": "table",
+ "img_path": "images/088ab38e572f20ded45d9e7e2352af36896eda7e7e60044121970b9fdce4bba7.jpg",
+ "table_caption": [
+ "Table 2. Quantitative evaluation of camera poses estimation on the MPI Sintel dataset. The methods of the top block discard the dynamic components and do not reconstruct the radiance fields; thus they cannot render novel views. We exclude the COLMAP results since it fails to produce poses in 5 out of 14 sequences."
+ ],
+ "table_footnote": [],
+ "table_body": " | Known camera poses | Unknown camera poses |
| Static scene | NeRF [44], SVS [59], NeRF++ [82], Mip-NeRF [4], Mip-NeRF 360 [5], DirectXVoxGO [68], Plenoxels [23], Instant-ngp [45], TenorRF [12] | NeRF - - [73], BARF [40], SC-NeRF [31], NeRF-SLAM [60] |
| Dynamic scene | NV [43], D-NeRF [56], NR-NeRF [71], NSFF [39], DynamicNeRF [24], Nerfies [52], HyperNeRF [53], TiNeuVox [20], T-NeRF [25] | Ours |
"
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "many recent dynamic view synthesis methods for unstructured videos [24,25,39,52,53,56,71,76] and new methods based on deformable fields [20]. However, these techniques require precise camera poses typically estimated via SfM systems such as COLMAP [62] (bottom left of Table 1)."
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+ "angle": 0,
+ "content": "However, SfM systems are not robust to many issues, such as noisy images from low-light conditions, motion blur caused by users, or dynamic objects in the scene, such as people, cars, and animals. The robustness problem of the SfM systems causes the existing dynamic view synthesis methods to be fragile and impractical for many challenging videos. Recently, several NeRF-based methods [31, 40, 60, 73] have proposed jointly optimizing the camera poses with the scene geometry. Nevertheless, these methods can only handle strictly static scenes (top right of Table 1)."
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "We introduce RoDynRF, an algorithm for reconstructing dynamic radiance fields from casual videos. Unlike existing approaches, we do not require accurate camera poses as input. Our method optimizes camera poses and two radiance fields, modeling static and dynamic elements. Our approach includes a coarse-to-fine strategy and epipolar geometry to exclude moving pixels, deformation fields, time-dependent appearance models, and regularization losses for improved consistency. We evaluate the algorithm on multiple datasets, including Sintel [9], Dynamic View Synthesis [79], iPhone [25], and DAVIS [55], and show visual comparisons with existing methods."
+ },
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+ "content": "We summarize our core contributions as follows:"
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+ "content": "- We present a space-time synthesis algorithm from a dynamic monocular video that does not require known camera poses and camera intrinsics as input."
+ },
+ {
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+ "angle": 0,
+ "content": "- Our proposed careful architecture designs and auxiliary losses improve the robustness of camera pose estimation and dynamic radiance field reconstruction."
+ },
+ {
+ "type": "text",
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+ "content": "- Quantitative and qualitative evaluations demonstrate the robustness of our method over other state-of-the-art methods on several challenging datasets that typical SfM systems fail to estimate camera poses."
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+ "content": "2. Related Work"
+ },
+ {
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+ "angle": 0,
+ "content": "Static view synthesis. Many view synthesis techniques construct specific scene geometry from images captured at various positions [8] and use local warps [11] to synthesize high-quality novel views of a scene. Approaches to"
+ },
+ {
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+ "angle": 0,
+ "content": "light field rendering use implicit scene geometry to create photorealistic novel views, but they require densely captured images [27,37]. By using soft 3D reconstruction [54], learning-based dense depth maps [22], multiplane images (MPIs) [14,21,67], additional learned deep features [30,58], or voxel-based implicit scene representations [66], several earlier work attempt to use proxy scene geometry to enhance rendering quality."
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+ {
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+ "angle": 0,
+ "content": "Recent methods implicitly model the scene as a continuous neural radiance field (NeRF) [4, 44, 82] with multi-layer perceptrons (MLPs). However, NeRF requires days of training time to represent a scene. Therefore, recent methods [12, 23, 45, 68] replace the implicit MLPs with explicit voxels and significantly improve the training speed."
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+ "content": "Several approaches synthesize novel views from a single RGB input image. These methods often fill up holes in the disoccluded regions and predict depth [38, 49], additionally learned features [75], multiplane images [72], and layered depth images [34, 64]. Although these techniques have produced excellent view synthesis results, they can only handle static scenes. Our approach performs view synthesis of dynamic scenes from a single monocular video, in contrast to existing view synthesis techniques focusing on static scenes."
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+ "content": "Dynamic view synthesis. By focusing on human bodies [74], using RGBD data [16], reconstructing sparse geometry [51], or producing minimal stereoscopic disparity transitions between input views [1], many techniques reconstruct and synthesize novel views from non-rigid dynamic scenes. Other techniques break down dynamic scenes into piece-wise rigid parts using hand-crafted priors [36, 61]. Many systems cannot handle scenes with complicated geometry and instead require multi-view and time-synchronized videos as input to provide interactive view manipulation [3, 7, 41, 85]. Yoon et al. [79] used depth from single-view and multi-view stereo to synthesize novel views of dynamic scenes from a single video using explicit depth-based 3D warping."
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+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "A recent line of work extends NeRF to handle dynamic scenes [20, 24, 39, 52, 53, 56, 71, 76]. Although these space-time synthesis results are impressive, these techniques rely on precise camera pose input. Consequently, these techniques are not applicable to challenging scenes where COLMAP [62] or current SfM systems fail. Our approach, in contrast, can handle complex dynamic scenarios without known camera poses."
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+ "content": "Visual odometry and camera pose estimation. From a collection of images, visual odometry estimates the 3D camera poses [18, 19, 46-48]. These techniques mainly fall into two categories: direct methods that maximize photometric consistency [78, 84] and feature-based methods that rely on manually created or learned features [46, 47, 63]. Self-supervised image reconstruction losses have recently"
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+ 0.086,
+ 0.898,
+ 0.355
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.076,
+ 0.367,
+ 0.895,
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+ ],
+ "angle": 0,
+ "content": "Figure 2. Overall framework. We model the dynamic scene with static and dynamic radiance fields. The static radiance fields take both the sampled coordinates \\((x, y, z)\\) and the viewing direction \\(\\mathbf{d}\\) as input and predict the density \\(\\sigma^s\\) and color \\(\\mathbf{c}^s\\). Note that the density of the static part is invariant to time and viewing direction, therefore, we use summation of the queried features as the density (instead of using an MLP). We only compute the losses over the static regions. The computed gradients backpropagate not only to the static voxel field and MLPs but also to the camera parameters. The dynamic radiance fields take the sampled coordinates and the time \\(t\\) to obtain the deformed coordinates \\((x', y', z')\\) in the canonical space. Then we query the features using these deformed coordinates from the dynamic voxel fields and pass the features along with the time index to a time-dependent shallow MLPs to get the color \\(\\mathbf{c}^d\\), density \\(\\sigma^d\\), and nonrigidity \\(m^d\\) of the dynamic part. Finally, after the volume rendering, we can obtain the RGB images \\(\\mathbf{C}^{\\{s,d\\}}\\) and the depth maps \\(\\mathbf{D}^{\\{s,d\\}}\\) from the static and dynamic parts along with a nonrigidity mask \\(\\mathbf{M}^d\\). Finally, we calculate the per-frame reconstruction loss. Note that here we only include per-frame losses."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.531,
+ 0.473,
+ 0.668
+ ],
+ "angle": 0,
+ "content": "been used in learning-based systems to tackle visual odometry [2, 6, 26, 35, 70, 77, 80, 81, 83, 84]. Estimating camera poses from casually captured videos remains challenging. NeRF-based techniques have been proposed to combine neural 3D representation and camera poses for optimization [31, 40, 60, 73], although they are limited to static sequences. In contrast to the visual odometry techniques outlined above, our system simultaneously optimizes camera poses and models dynamic objects models."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.677,
+ 0.17,
+ 0.693
+ ],
+ "angle": 0,
+ "content": "3. Method"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.702,
+ 0.47,
+ 0.837
+ ],
+ "angle": 0,
+ "content": "In this section, we first briefly introduce the background of neural radiance fields and their extension of camera pose estimation and dynamic scene representation in Section 3.1. We then describe the overview of our method in Section 3.2. Next, we discuss the details of camera pose estimation with the static radiance field reconstruction in Section 3.3. After that, we show how to model the dynamic scene in Section 3.4. Finally, we outline the implementation details in Section 3.5."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.845,
+ 0.219,
+ 0.859
+ ],
+ "angle": 0,
+ "content": "3.1. Preliminaries"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.871,
+ 0.47,
+ 0.903
+ ],
+ "angle": 0,
+ "content": "NeRF. Neural radiance fields (NeRF) [44] represent a static 3D scene with implicit MLPs parameterized by \\(\\Theta\\) and map"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.531,
+ 0.892,
+ 0.562
+ ],
+ "angle": 0,
+ "content": "the 3D position \\((x,y,z)\\) and viewing direction \\((\\theta ,\\phi)\\) to its corresponding color \\(\\mathbf{c}\\) and density \\(\\sigma\\) .."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.597,
+ 0.568,
+ 0.892,
+ 0.586
+ ],
+ "angle": 0,
+ "content": "\\[\n(\\mathbf {c}, \\sigma) = \\operatorname {M L P} _ {\\Theta} (x, y, z, \\theta , \\phi). \\tag {1}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.592,
+ 0.892,
+ 0.623
+ ],
+ "angle": 0,
+ "content": "We can compute the pixel color by applying volume rendering [17, 32] along the ray \\(\\mathbf{r}\\) emitted from the camera origin:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.549,
+ 0.632,
+ 0.892,
+ 0.682
+ ],
+ "angle": 0,
+ "content": "\\[\n\\hat {\\mathbf {C}} (\\mathbf {r}) = \\sum_ {i = 1} ^ {N} T (i) \\left(1 - \\exp (- \\sigma (i) \\delta (i))\\right) \\mathbf {c} (i), \\tag {2}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.598,
+ 0.679,
+ 0.796,
+ 0.719
+ ],
+ "angle": 0,
+ "content": "\\[\nT (i) = \\exp (- \\sum_ {i - 1} ^ {j} \\sigma (j) \\delta (j)),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.723,
+ 0.894,
+ 0.828
+ ],
+ "angle": 0,
+ "content": "where \\(\\delta (i)\\) represents the distance between two consecutive sample points along the ray, \\(N\\) is the number of samples along each ray, and \\(T(i)\\) indicates the accumulated transparency. As the volume rendering procedure is differentiable, we can optimize the radiance fields by minimizing the reconstruction error between the rendered color \\(\\hat{\\mathbf{C}}\\) and the ground truth color \\(\\mathbf{C}\\):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.616,
+ 0.833,
+ 0.892,
+ 0.863
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} = \\left\\| \\hat {\\mathbf {C}} (\\mathbf {r}) - \\mathbf {C} (\\mathbf {r}) \\right\\| _ {2} ^ {2}. \\tag {3}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.499,
+ 0.871,
+ 0.892,
+ 0.902
+ ],
+ "angle": 0,
+ "content": "Explicit neural voxel radiance fields. Although with compelling rendering quality, NeRF-based methods model"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.492,
+ 0.946,
+ 0.507,
+ 0.956
+ ],
+ "angle": 0,
+ "content": "15"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.109,
+ 0.087,
+ 0.466,
+ 0.278
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.093,
+ 0.281,
+ 0.476,
+ 0.295
+ ],
+ "angle": 0,
+ "content": "(a) Static radiance field reconstruction and pose estimation"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.491,
+ 0.087,
+ 0.882,
+ 0.278
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.547,
+ 0.28,
+ 0.822,
+ 0.295
+ ],
+ "angle": 0,
+ "content": "(b) Dynamic radiance field reconstruction"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.076,
+ 0.302,
+ 0.894,
+ 0.385
+ ],
+ "angle": 0,
+ "content": "Figure 3. Training losses. For both the (a) static and (b) dynamic parts, we introduce three auxiliary losses to encourage the consistency of the modeling: reprojection loss, disparity loss, and monocular depth loss. The reprojection loss encourages the projection of the 3D volume rendered points onto neighbor frames to be similar to the pre-calculated flow. The disparity loss forces the volume rendered 3D points from two corresponding points of neighbor frames to have similar z values. Finally, the monocular depth loss calculates the scale-and shift-invariant loss between the volume rendered depth and the pre-calculated MiDaS depth. (a) We use the motion mask to exclude the dynamic regions from the loss calculation. (b) We use a scene flow MLP to model the 3D movement of the volume rendered 3D points."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.411,
+ 0.473,
+ 0.576
+ ],
+ "angle": 0,
+ "content": "the scene with implicit representations such as MLPs for high storage efficiency. These methods, however, are very slow to train. To overcome this drawback, recent methods [12, 23, 45, 68] propose to model the radiance fields with explicit voxels. Specifically, these methods replace the mapping function with voxel grids and directly optimize the features sampled from the voxels. They usually apply shallow MLPs to handle the view-dependent effects. By eliminating the heavy usage of the MLPs, the training time of these methods reduces from days to hours. We also leverage explicit representation in this work."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.586,
+ 0.253,
+ 0.601
+ ],
+ "angle": 0,
+ "content": "3.2. Method Overview"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.609,
+ 0.47,
+ 0.76
+ ],
+ "angle": 0,
+ "content": "We show our proposed framework in Figure 2. Given an input video sequence with \\(N\\) frames, our method jointly optimizes the camera poses, focal length, and static and dynamic radiance fields. We represent both the static and dynamic parts with explicit neural voxels \\(\\mathbf{V}^s\\) and \\(\\mathbf{V}^d\\), respectively. The static radiance fields are responsible for reconstructing the static scene and estimating the camera poses and focal length. At the same time, the goal of dynamic radiance fields is to model the scene dynamics in the video (usually caused by moving objects)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.769,
+ 0.304,
+ 0.784
+ ],
+ "angle": 0,
+ "content": "3.3. Camera Pose Estimation"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.796,
+ 0.471,
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+ ],
+ "angle": 0,
+ "content": "Motion mask generation. Excluding dynamic regions in the video helps improve the robustness of camera pose estimation. Existing methods [39] often leverage off-the-shelf instance segmentation methods such as Mask R-CNN [29] to mask out the common moving objects. However, many moving objects are hard to detect/segment in the input video, such as drifting water or swaying trees. Therefore, in"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.561,
+ 0.408,
+ 0.672,
+ 0.477
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.564,
+ 0.479,
+ 0.671,
+ 0.489
+ ],
+ "angle": 0,
+ "content": "(a) w/o coarse-to-fine"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.57,
+ 0.489,
+ 0.664,
+ 0.557
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.512,
+ 0.559,
+ 0.722,
+ 0.571
+ ],
+ "angle": 0,
+ "content": "(c) w/o late viewing direction conditioning"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.745,
+ 0.409,
+ 0.861,
+ 0.477
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.733,
+ 0.479,
+ 0.882,
+ 0.49
+ ],
+ "angle": 0,
+ "content": "(b) w/o monocular depth prior"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.741,
+ 0.491,
+ 0.872,
+ 0.558
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.771,
+ 0.559,
+ 0.843,
+ 0.569
+ ],
+ "angle": 0,
+ "content": "(d) Full model"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.498,
+ 0.576,
+ 0.892,
+ 0.687
+ ],
+ "angle": 0,
+ "content": "Figure 4. The impact of design choices on camera pose estimation. (a) No coarse-to-fine strategy leads to sub-optimal solutions. (b) No single-image depth prior results in poor scene geometry for challenging camera trajectories. (c) The absence of late viewing direction conditioning leads to wrong geometry and poses due to minimizing photometric loss instead of consistent voxel space using MLP. (d) Our proposed method incorporates all components and yields reasonable scene geometry and camera trajectory."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.716,
+ 0.892,
+ 0.821
+ ],
+ "angle": 0,
+ "content": "addition to the masks from Mask R-CNN, we also estimate the fundamental matrix using the optical flow from consecutive frames. We then calculate and threshold the Sampson distance (the distance of each pixel to the estimated epipolar line) to obtain a binary motion mask. Finally, we combine the results from Mask R-CNN and epipolar distance thresholding to obtain our final motion masks."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.826,
+ 0.894,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "Coarse-to-fine static scene reconstruction. The first part of our method is reconstructing the static radiance fields along with the camera poses. We jointly optimize the 6D camera poses \\([R|t]_i, i \\in [1..N]\\) and the focal length \\(f\\) shared by all input frames simultaneously. Similar to ex"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.491,
+ 0.946,
+ 0.508,
+ 0.956
+ ],
+ "angle": 0,
+ "content": "16"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.092,
+ 0.473,
+ 0.212
+ ],
+ "angle": 0,
+ "content": "isting pose estimation methods [40], we optimize the static scene representation in a coarse-to-fine manner. Specifically, we start with a smaller static voxel resolution and progressively increase the voxel resolution during the training. This coarse-to-fine strategy is essential to the camera pose estimation as the energy surface will become smoother. Thus, the optimizer will have less chance of getting stuck in sub-optimal solutions (Figure 4(a) vs. Figure 4(d))."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.216,
+ 0.473,
+ 0.395
+ ],
+ "angle": 0,
+ "content": "Late viewing direction conditioning. As our primary supervision is the photometric consistency loss, the optimization could bypass the neural voxel and directly learn a mapping function from the viewing direction to the output sample color. Therefore, we choose to fuse the viewing direction only in the last layer of the color MLP as shown in Figure 2. This design choice is critical because we are reconstructing not only the scene geometry but also the camera poses. Figure 4(c) shows that without the late viewing direction conditioning, the optimization could minimize the photometric loss by optimizing the MLP and lead to erroneous camera poses and geometry estimation."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.399,
+ 0.473,
+ 0.445
+ ],
+ "angle": 0,
+ "content": "Losses. We minimize the photometric loss between the prediction \\(\\hat{\\mathbf{C}}^s (\\mathbf{r})\\) and the captured images in the static regions:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.135,
+ 0.455,
+ 0.47,
+ 0.485
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} _ {c} ^ {s} = \\left\\| (\\hat {\\mathbf {C}} ^ {s} (\\mathbf {r}) - \\mathbf {C} (\\mathbf {r})) \\cdot (1 - \\mathbf {M} (\\mathbf {r})) \\right\\| _ {2} ^ {2}, \\tag {4}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.495,
+ 0.315,
+ 0.509
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathbf{M}\\) denotes the motion mask."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.511,
+ 0.47,
+ 0.556
+ ],
+ "angle": 0,
+ "content": "To handle casually-captured but challenging camera trajectories such as fast-moving or pure rotating, we introduce auxiliary losses to regularize the training, similar to [24,39]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.556,
+ 0.47,
+ 0.645
+ ],
+ "angle": 0,
+ "content": "(1) Reprojection loss \\(\\mathcal{L}_{\\mathrm{reproj}}^s\\): We use 2D optical flow estimated by RAFT [69] to guide the training. First, we volume render all the sampled 3D points along a ray to generate a surface point. We then reproject this point onto its neighbor frame and calculate the reprojection error with the correspondence estimated from RAFT."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.646,
+ 0.47,
+ 0.735
+ ],
+ "angle": 0,
+ "content": "(2) Disparity loss \\(\\mathcal{L}_{\\mathrm{disp}}^s\\): Similar to the reprojection loss above, we also regularize the error in the \\(z\\)-direction (in the camera coordinate). We volume render the two corresponding points into 3D space and calculate the error of the \\(z\\) component. As we care more about the near than the far, we compute this loss in the inverse-depth domain."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.736,
+ 0.47,
+ 0.87
+ ],
+ "angle": 0,
+ "content": "(3) Monocular depth loss \\(\\mathcal{L}_{\\mathrm{monodepth}}^s\\): The two losses above cannot handle pure rotating cameras and often lead to the incorrect camera poses and geometry (Figure 4(b)). We enforce the depth order from multiple pixels of the same frame to match the order of a monocular depth map. We pre-calculate the depth map using MiDaSv2.1 [57]. The depth prediction from MiDaS is up to an unknown scale and shift. Therefore, we use the same scale- and shift-invariant loss in MiDaS to constrain our rendered depth values."
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.076,
+ 0.556,
+ 0.47,
+ 0.87
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.871,
+ 0.47,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "We illustrate these auxiliary losses in Figure 3(a). Since the optical flow and depth map may not be accurate, we"
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.499,
+ 0.09,
+ 0.894,
+ 0.16
+ ],
+ "angle": 0,
+ "content": "Table 2. Quantitative evaluation of camera poses estimation on the MPI Sintel dataset. The methods of the top block discard the dynamic components and do not reconstruct the radiance fields; thus they cannot render novel views. We exclude the COLMAP results since it fails to produce poses in 5 out of 14 sequences."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.502,
+ 0.163,
+ 0.894,
+ 0.273
+ ],
+ "angle": 0,
+ "content": " | Known camera poses | Unknown camera poses |
| Static scene | NeRF [44], SVS [59], NeRF++ [82], Mip-NeRF [4], Mip-NeRF 360 [5], DirectXVoxGO [68], Plenoxels [23], Instant-ngp [45], TenorRF [12] | NeRF - - [73], BARF [40], SC-NeRF [31], NeRF-SLAM [60] |
| Dynamic scene | NV [43], D-NeRF [56], NR-NeRF [71], NSFF [39], DynamicNeRF [24], Nerfies [52], HyperNeRF [53], TiNeuVox [20], T-NeRF [25] | Ours |
+
+many recent dynamic view synthesis methods for unstructured videos [24,25,39,52,53,56,71,76] and new methods based on deformable fields [20]. However, these techniques require precise camera poses typically estimated via SfM systems such as COLMAP [62] (bottom left of Table 1).
+
+However, SfM systems are not robust to many issues, such as noisy images from low-light conditions, motion blur caused by users, or dynamic objects in the scene, such as people, cars, and animals. The robustness problem of the SfM systems causes the existing dynamic view synthesis methods to be fragile and impractical for many challenging videos. Recently, several NeRF-based methods [31, 40, 60, 73] have proposed jointly optimizing the camera poses with the scene geometry. Nevertheless, these methods can only handle strictly static scenes (top right of Table 1).
+
+We introduce RoDynRF, an algorithm for reconstructing dynamic radiance fields from casual videos. Unlike existing approaches, we do not require accurate camera poses as input. Our method optimizes camera poses and two radiance fields, modeling static and dynamic elements. Our approach includes a coarse-to-fine strategy and epipolar geometry to exclude moving pixels, deformation fields, time-dependent appearance models, and regularization losses for improved consistency. We evaluate the algorithm on multiple datasets, including Sintel [9], Dynamic View Synthesis [79], iPhone [25], and DAVIS [55], and show visual comparisons with existing methods.
+
+We summarize our core contributions as follows:
+
+- We present a space-time synthesis algorithm from a dynamic monocular video that does not require known camera poses and camera intrinsics as input.
+- Our proposed careful architecture designs and auxiliary losses improve the robustness of camera pose estimation and dynamic radiance field reconstruction.
+- Quantitative and qualitative evaluations demonstrate the robustness of our method over other state-of-the-art methods on several challenging datasets that typical SfM systems fail to estimate camera poses.
+
+# 2. Related Work
+
+Static view synthesis. Many view synthesis techniques construct specific scene geometry from images captured at various positions [8] and use local warps [11] to synthesize high-quality novel views of a scene. Approaches to
+
+light field rendering use implicit scene geometry to create photorealistic novel views, but they require densely captured images [27,37]. By using soft 3D reconstruction [54], learning-based dense depth maps [22], multiplane images (MPIs) [14,21,67], additional learned deep features [30,58], or voxel-based implicit scene representations [66], several earlier work attempt to use proxy scene geometry to enhance rendering quality.
+
+Recent methods implicitly model the scene as a continuous neural radiance field (NeRF) [4, 44, 82] with multi-layer perceptrons (MLPs). However, NeRF requires days of training time to represent a scene. Therefore, recent methods [12, 23, 45, 68] replace the implicit MLPs with explicit voxels and significantly improve the training speed.
+
+Several approaches synthesize novel views from a single RGB input image. These methods often fill up holes in the disoccluded regions and predict depth [38, 49], additionally learned features [75], multiplane images [72], and layered depth images [34, 64]. Although these techniques have produced excellent view synthesis results, they can only handle static scenes. Our approach performs view synthesis of dynamic scenes from a single monocular video, in contrast to existing view synthesis techniques focusing on static scenes.
+
+Dynamic view synthesis. By focusing on human bodies [74], using RGBD data [16], reconstructing sparse geometry [51], or producing minimal stereoscopic disparity transitions between input views [1], many techniques reconstruct and synthesize novel views from non-rigid dynamic scenes. Other techniques break down dynamic scenes into piece-wise rigid parts using hand-crafted priors [36, 61]. Many systems cannot handle scenes with complicated geometry and instead require multi-view and time-synchronized videos as input to provide interactive view manipulation [3, 7, 41, 85]. Yoon et al. [79] used depth from single-view and multi-view stereo to synthesize novel views of dynamic scenes from a single video using explicit depth-based 3D warping.
+
+A recent line of work extends NeRF to handle dynamic scenes [20, 24, 39, 52, 53, 56, 71, 76]. Although these space-time synthesis results are impressive, these techniques rely on precise camera pose input. Consequently, these techniques are not applicable to challenging scenes where COLMAP [62] or current SfM systems fail. Our approach, in contrast, can handle complex dynamic scenarios without known camera poses.
+
+Visual odometry and camera pose estimation. From a collection of images, visual odometry estimates the 3D camera poses [18, 19, 46-48]. These techniques mainly fall into two categories: direct methods that maximize photometric consistency [78, 84] and feature-based methods that rely on manually created or learned features [46, 47, 63]. Self-supervised image reconstruction losses have recently
+
+
+Figure 2. Overall framework. We model the dynamic scene with static and dynamic radiance fields. The static radiance fields take both the sampled coordinates $(x, y, z)$ and the viewing direction $\mathbf{d}$ as input and predict the density $\sigma^s$ and color $\mathbf{c}^s$ . Note that the density of the static part is invariant to time and viewing direction, therefore, we use summation of the queried features as the density (instead of using an MLP). We only compute the losses over the static regions. The computed gradients backpropagate not only to the static voxel field and MLPs but also to the camera parameters. The dynamic radiance fields take the sampled coordinates and the time $t$ to obtain the deformed coordinates $(x', y', z')$ in the canonical space. Then we query the features using these deformed coordinates from the dynamic voxel fields and pass the features along with the time index to a time-dependent shallow MLPs to get the color $\mathbf{c}^d$ , density $\sigma^d$ , and nonrigidity $m^d$ of the dynamic part. Finally, after the volume rendering, we can obtain the RGB images $\mathbf{C}^{\{s,d\}}$ and the depth maps $\mathbf{D}^{\{s,d\}}$ from the static and dynamic parts along with a nonrigidity mask $\mathbf{M}^d$ . Finally, we calculate the per-frame reconstruction loss. Note that here we only include per-frame losses.
+
+been used in learning-based systems to tackle visual odometry [2, 6, 26, 35, 70, 77, 80, 81, 83, 84]. Estimating camera poses from casually captured videos remains challenging. NeRF-based techniques have been proposed to combine neural 3D representation and camera poses for optimization [31, 40, 60, 73], although they are limited to static sequences. In contrast to the visual odometry techniques outlined above, our system simultaneously optimizes camera poses and models dynamic objects models.
+
+# 3. Method
+
+In this section, we first briefly introduce the background of neural radiance fields and their extension of camera pose estimation and dynamic scene representation in Section 3.1. We then describe the overview of our method in Section 3.2. Next, we discuss the details of camera pose estimation with the static radiance field reconstruction in Section 3.3. After that, we show how to model the dynamic scene in Section 3.4. Finally, we outline the implementation details in Section 3.5.
+
+# 3.1. Preliminaries
+
+NeRF. Neural radiance fields (NeRF) [44] represent a static 3D scene with implicit MLPs parameterized by $\Theta$ and map
+
+the 3D position $(x,y,z)$ and viewing direction $(\theta ,\phi)$ to its corresponding color $\mathbf{c}$ and density $\sigma$ ..
+
+$$
+(\mathbf {c}, \sigma) = \operatorname {M L P} _ {\Theta} (x, y, z, \theta , \phi). \tag {1}
+$$
+
+We can compute the pixel color by applying volume rendering [17, 32] along the ray $\mathbf{r}$ emitted from the camera origin:
+
+$$
+\hat {\mathbf {C}} (\mathbf {r}) = \sum_ {i = 1} ^ {N} T (i) \left(1 - \exp (- \sigma (i) \delta (i))\right) \mathbf {c} (i), \tag {2}
+$$
+
+$$
+T (i) = \exp (- \sum_ {i - 1} ^ {j} \sigma (j) \delta (j)),
+$$
+
+where $\delta (i)$ represents the distance between two consecutive sample points along the ray, $N$ is the number of samples along each ray, and $T(i)$ indicates the accumulated transparency. As the volume rendering procedure is differentiable, we can optimize the radiance fields by minimizing the reconstruction error between the rendered color $\hat{\mathbf{C}}$ and the ground truth color $\mathbf{C}$ :
+
+$$
+\mathcal {L} = \left\| \hat {\mathbf {C}} (\mathbf {r}) - \mathbf {C} (\mathbf {r}) \right\| _ {2} ^ {2}. \tag {3}
+$$
+
+Explicit neural voxel radiance fields. Although with compelling rendering quality, NeRF-based methods model
+
+
+(a) Static radiance field reconstruction and pose estimation
+
+
+(b) Dynamic radiance field reconstruction
+Figure 3. Training losses. For both the (a) static and (b) dynamic parts, we introduce three auxiliary losses to encourage the consistency of the modeling: reprojection loss, disparity loss, and monocular depth loss. The reprojection loss encourages the projection of the 3D volume rendered points onto neighbor frames to be similar to the pre-calculated flow. The disparity loss forces the volume rendered 3D points from two corresponding points of neighbor frames to have similar z values. Finally, the monocular depth loss calculates the scale-and shift-invariant loss between the volume rendered depth and the pre-calculated MiDaS depth. (a) We use the motion mask to exclude the dynamic regions from the loss calculation. (b) We use a scene flow MLP to model the 3D movement of the volume rendered 3D points.
+
+the scene with implicit representations such as MLPs for high storage efficiency. These methods, however, are very slow to train. To overcome this drawback, recent methods [12, 23, 45, 68] propose to model the radiance fields with explicit voxels. Specifically, these methods replace the mapping function with voxel grids and directly optimize the features sampled from the voxels. They usually apply shallow MLPs to handle the view-dependent effects. By eliminating the heavy usage of the MLPs, the training time of these methods reduces from days to hours. We also leverage explicit representation in this work.
+
+# 3.2. Method Overview
+
+We show our proposed framework in Figure 2. Given an input video sequence with $N$ frames, our method jointly optimizes the camera poses, focal length, and static and dynamic radiance fields. We represent both the static and dynamic parts with explicit neural voxels $\mathbf{V}^s$ and $\mathbf{V}^d$ , respectively. The static radiance fields are responsible for reconstructing the static scene and estimating the camera poses and focal length. At the same time, the goal of dynamic radiance fields is to model the scene dynamics in the video (usually caused by moving objects).
+
+# 3.3. Camera Pose Estimation
+
+Motion mask generation. Excluding dynamic regions in the video helps improve the robustness of camera pose estimation. Existing methods [39] often leverage off-the-shelf instance segmentation methods such as Mask R-CNN [29] to mask out the common moving objects. However, many moving objects are hard to detect/segment in the input video, such as drifting water or swaying trees. Therefore, in
+
+
+
+
+(a) w/o coarse-to-fine
+(c) w/o late viewing direction conditioning
+Figure 4. The impact of design choices on camera pose estimation. (a) No coarse-to-fine strategy leads to sub-optimal solutions. (b) No single-image depth prior results in poor scene geometry for challenging camera trajectories. (c) The absence of late viewing direction conditioning leads to wrong geometry and poses due to minimizing photometric loss instead of consistent voxel space using MLP. (d) Our proposed method incorporates all components and yields reasonable scene geometry and camera trajectory.
+
+
+
+
+(b) w/o monocular depth prior
+(d) Full model
+
+addition to the masks from Mask R-CNN, we also estimate the fundamental matrix using the optical flow from consecutive frames. We then calculate and threshold the Sampson distance (the distance of each pixel to the estimated epipolar line) to obtain a binary motion mask. Finally, we combine the results from Mask R-CNN and epipolar distance thresholding to obtain our final motion masks.
+
+Coarse-to-fine static scene reconstruction. The first part of our method is reconstructing the static radiance fields along with the camera poses. We jointly optimize the 6D camera poses $[R|t]_i, i \in [1..N]$ and the focal length $f$ shared by all input frames simultaneously. Similar to ex
+
+isting pose estimation methods [40], we optimize the static scene representation in a coarse-to-fine manner. Specifically, we start with a smaller static voxel resolution and progressively increase the voxel resolution during the training. This coarse-to-fine strategy is essential to the camera pose estimation as the energy surface will become smoother. Thus, the optimizer will have less chance of getting stuck in sub-optimal solutions (Figure 4(a) vs. Figure 4(d)).
+
+Late viewing direction conditioning. As our primary supervision is the photometric consistency loss, the optimization could bypass the neural voxel and directly learn a mapping function from the viewing direction to the output sample color. Therefore, we choose to fuse the viewing direction only in the last layer of the color MLP as shown in Figure 2. This design choice is critical because we are reconstructing not only the scene geometry but also the camera poses. Figure 4(c) shows that without the late viewing direction conditioning, the optimization could minimize the photometric loss by optimizing the MLP and lead to erroneous camera poses and geometry estimation.
+
+Losses. We minimize the photometric loss between the prediction $\hat{\mathbf{C}}^s (\mathbf{r})$ and the captured images in the static regions:
+
+$$
+\mathcal {L} _ {c} ^ {s} = \left\| (\hat {\mathbf {C}} ^ {s} (\mathbf {r}) - \mathbf {C} (\mathbf {r})) \cdot (1 - \mathbf {M} (\mathbf {r})) \right\| _ {2} ^ {2}, \tag {4}
+$$
+
+where $\mathbf{M}$ denotes the motion mask.
+
+To handle casually-captured but challenging camera trajectories such as fast-moving or pure rotating, we introduce auxiliary losses to regularize the training, similar to [24,39].
+
+(1) Reprojection loss $\mathcal{L}_{\mathrm{reproj}}^s$ : We use 2D optical flow estimated by RAFT [69] to guide the training. First, we volume render all the sampled 3D points along a ray to generate a surface point. We then reproject this point onto its neighbor frame and calculate the reprojection error with the correspondence estimated from RAFT.
+(2) Disparity loss $\mathcal{L}_{\mathrm{disp}}^s$ : Similar to the reprojection loss above, we also regularize the error in the $z$ -direction (in the camera coordinate). We volume render the two corresponding points into 3D space and calculate the error of the $z$ component. As we care more about the near than the far, we compute this loss in the inverse-depth domain.
+(3) Monocular depth loss $\mathcal{L}_{\mathrm{monodepth}}^s$ : The two losses above cannot handle pure rotating cameras and often lead to the incorrect camera poses and geometry (Figure 4(b)). We enforce the depth order from multiple pixels of the same frame to match the order of a monocular depth map. We pre-calculate the depth map using MiDaSv2.1 [57]. The depth prediction from MiDaS is up to an unknown scale and shift. Therefore, we use the same scale- and shift-invariant loss in MiDaS to constrain our rendered depth values.
+
+We illustrate these auxiliary losses in Figure 3(a). Since the optical flow and depth map may not be accurate, we
+
+Table 2. Quantitative evaluation of camera poses estimation on the MPI Sintel dataset. The methods of the top block discard the dynamic components and do not reconstruct the radiance fields; thus they cannot render novel views. We exclude the COLMAP results since it fails to produce poses in 5 out of 14 sequences.
+
+ | Known camera poses | Unknown camera poses |
| Static scene | NeRF [44], SVS [59], NeRF++ [82], Mip-NeRF [4], Mip-NeRF 360 [5], DirectXVoxGO [68], Plenoxels [23], Instant-ngp [45], TenorRF [12] | NeRF - - [73], BARF [40], SC-NeRF [31], NeRF-SLAM [60] |
| Dynamic scene | NV [43], D-NeRF [56], NR-NeRF [71], NSFF [39], DynamicNeRF [24], Nerfies [52], HyperNeRF [53], TiNeuVox [20], T-NeRF [25] | Ours |
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+ "content": "We introduce RoDynRF, an algorithm for reconstructing dynamic radiance fields from casual videos. Unlike existing approaches, we do not require accurate camera poses as input. Our method optimizes camera poses and two radiance fields, modeling static and dynamic elements. Our approach includes a coarse-to-fine strategy and epipolar geometry to exclude moving pixels, deformation fields, time-dependent appearance models, and regularization losses for improved consistency. We evaluate the algorithm on multiple datasets, including Sintel [9], Dynamic View Synthesis [79], iPhone [25], and DAVIS [55], and show visual comparisons with existing methods."
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+ "html": "| Model | 3DMatch | 3DLoMatch (FCGF) | 3DLoMatch (Predator) | Sec. |
| FCGF | FPFH | 5000 | 2500 | 1000 | 500 | 250 | 5000 | 2500 | 1000 | 500 | 250 |
| PointDSC w/ SCNonlocalxyz | 92.42 | 77.51 | 55.8 | 52.6 | 46.8 | 37.7 | 26.7 | 68.1 | 67.3 | 66.5 | 63.4 | 60.5 | 0.11 |
| PointDSC w/ VBNonlocalxyz | 93.04 | 80.16 | 57.7 | 55.6 | 50.2 | 39.9 | 26.1 | 69.7 | 69.6 | 67.9 | 64.9 | 61.9 | 0.17 |
| PointDSC w/ SCNonlocalfeat | 92.36 | 77.76 | 54.6 | 50.6 | 44.9 | 36.8 | 25.4 | 69.2 | 68.6 | 67.9 | 63.5 | 59.9 | 0.13 |
| PointDSC w/ VBNonlocalfeat | 93.04 | 80.53 | 56.9 | 56.5 | 50.9 | 42.2 | 28.9 | 69.2 | 68.7 | 68.0 | 64.6 | 60.6 | 0.18 |
| PointDSC w/ SCNonlocalcls | 92.98 | 78.99 | 54.1 | 52.2 | 46.0 | 38.7 | 27.7 | 67.6 | 66.9 | 67.2 | 63.7 | 60.2 | 0.11 |
| PointDSC w/ VBNonlocalfeat+Vote | 93.41 | 81.21 | 58.3 | 56.5 | 51.9 | 44.7 | 31.1 | 69.3 | 69.5 | 68.2 | 65.3 | 61.2 | 0.20 |
| PointDSC w/ VBNonlocalfeat+Vote+CS | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 68.7 | 66.4 | 63.0 | 0.22 |
| Iteration times L = 6 | 93.41 | 82.32 | 58.1 | 57.1 | 52.9 | 48.3 | 34.8 | 69.7 | 69.7 | 68.7 | 66.3 | 63.8 | 0.19 |
| Iteration times L = 9 | 93.41 | 81.58 | 58.2 | 57.0 | 53.2 | 47.6 | 32.5 | 70.0 | 69.4 | 68.5 | 66.9 | 63.2 | 0.20 |
| Iteration times L = 12* | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 68.7 | 66.4 | 63.0 | 0.22 |
| Random feat. dim. d̅ = 32 | 93.41 | 82.38 | 58.0 | 57.5 | 53.5 | 48.1 | 34.8 | 69.7 | 70.0 | 68.6 | 66.3 | 63.3 | 0.20 |
| Random feat. dim. d̅ = 64 | 93.41 | 81.45 | 57.9 | 56.9 | 52.8 | 48.6 | 35.0 | 69.6 | 69.7 | 68.4 | 66.5 | 62.3 | 0.21 |
| Random feat. dim. d̅ = 128* | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 69.3 | 66.4 | 63.3 | 0.22 |
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+ "text": "eling can effectively learn the discriminative inlier/outlier features for reliable inlier search. (2): Then, to further highlight the superiority of our variational inference-guided feature learning, we also try to add classification loss on the features produced by each iteration in SCNonlocal to guide their learning (denoted as $SCNonlocal^{cls}$ ). As shown in the fifth row in Table 3, such loss-based label-propagation way just can achieve very limited performance gain and even sometimes degrades score. It demonstrates the superiority of our label-dependent posterior guidance for (prior) feature learning. Also, owing to such posterior guidance, the training curves in Fig. 4 (d) show that our method can achieve significantly faster convergence speed than SCNonlocal.",
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+ "text": "Discriminative Feature Learning? In order to verify whether VBNonlocal can learn more discriminative features than SCNonlocal, we visualize the distribution of the feature similarity of inliers on 3DMatch (Fig. 4 (a)) and 3DLoMatch (Fig. 4 (b)). As we can see, our learned inlier features own much higher similarities (approximate to 1) than SCNonlocal on both datasets, which demonstrates that our proposed Bayesian-inspired non-local network truly can promote more discriminative correspondence-feature learning.",
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+ "text": "Variational Non-local Setting. We further test the performance changes under different VBNonlocal settings. (1) We first test the model robustness under different numbers of non-local iterations. The second block in Table 3 verifies that our method is robust to the iteration time and tends to consistently achieve outstanding $RR$ score. (2) Then, the bottom block in Table 3 further shows our model stability to different dimensions of random features.",
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+ "text": "Voting-based Inlier Searching. Furthermore, we evaluate the performance contribution of the proposed voting-based inlier searching strategy (Vote). As we can see in the sixth row of Table 3, voting strategy can consistently achieve performance improvement regardless in 3DMatch or in more",
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+ "text": "challenging 3DLoMatch. It mainly benefits from the high-quality hypothetical inliers sampled by our voting policy.",
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+ "text": "Conservative Seed Selection. Finally, we test the effectiveness of the proposed conservative seed selection strategy (CS) motivated by Theorem 1. As we can see in the seventh row of Table 3, CS can achieve consistent performance gain in each setting. Especially, in the cases of fewer points (e.g., FCGF setting with 250 and 500 points), the improvement is much more significant $(+3.4\\%, +2.5\\%)$ . As the analysis before, in sparse point clouds, the original inlier selection strategy like in [2] is aggressive and prone to miss too many inlier seeds. Instead, our conservative selection strategy can effectively mitigate it as well as keep registration efficiency.",
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+ "text": "In this paper, we adapted the variational Bayesian inference into the non-local network and developed the effective Bayesian-guided long-term dependencies for discriminative correspondence-feature learning. To achieve effective variational inference, a probabilistic graphical model was customized over our non-local network, and the variational low bound was derived as the optimization objective for model training. In addition, we proposed a Wilson score-based voting mechanism for high-quality inlier sampling and theoretically verified its superiority over RANSAC. Extensive experiments on indoor/outdoor datasets demonstrated its promising performance.",
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+ "content": "This CVPR 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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+ "content": "\\(^{1}\\)PCA Lab, Nanjing University of Science and Technology, China \\(^{2}\\)CVLab, EPFL, Switzerland \\(\\{\\text{jiang.hao.bo, csjxie, csjyang}\\} \\text{@njust.edu.cn}\\) \\(\\{\\text{zheng.dang, zhen.wei, mathieu.salzmann}\\} \\text{@epfl.ch}\\)"
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+ "content": "Learning-based outlier (mismatched correspondence) rejection for robust 3D registration generally formulates the outlier removal as an inlier/outlier classification problem. The core for this to be successful is to learn the discriminative inlier/outlier feature representations. In this paper, we develop a novel variational non-local network-based outlier rejection framework for robust alignment. By reformulating the non-local feature learning with variational Bayesian inference, the Bayesian-driven long-range dependencies can be modeled to aggregate discriminative geometric context information for inlier/outlier distinction. Specifically, to achieve such Bayesian-driven contextual dependencies, each query/value component in our nonlocal network predicts a prior feature distribution and a posterior one. Embedded with the inlier/outlier label, the posterior feature distribution is label-dependent and discriminative. Thus, pushing the prior to be close to the discriminative posterior in the training step enables the features sampled from this prior at test time to model high-quality long-range dependencies. Notably, to achieve effective posterior feature guidance, a specific probabilistic graphical model is designed over our non-local model, which lets us derive a variational low bound as our optimization objective for model training. Finally, we propose a voting-based inlier searching strategy to cluster the high-quality hypothetical inliers for transformation estimation. Extensive experiments on 3DMatch, 3DLoMatch, and KITTI datasets verify the effectiveness of our method. Code is available at https://github.com/Jiang-HB/VBReg."
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+ "content": "Point cloud registration is a fundamental but challenging 3D computer vision task, with many potential applications such as 3D scene reconstruction [1,39], object pose estimation [13,44], and Lidar SLAM [15, 51]. It aims to align two partially overlapping point clouds by estimating their relative rigid transformation, i.e., 3D rotation and 3D translation. A popular approach to address the large-scale scene registration problem consists of extracting point descriptors [11, 14, 17, 37, 38, 50] and establishing correspondences between the two point clouds, from which the transformation can be obtained geometrically. In this context, much effort has been dedicated to designing traditional and deep learning-based descriptors [3, 11, 21, 41, 50]. However, the resulting correspondences inevitably still suffer from outliers (wrong matchings), particularly in challenging cases, such as low-overlap, repetitive structures, or noisy point sets, leading to registration failure."
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+ "content": "To address this, many outlier filtering strategies have been developed to robustify the registration process. These include traditional rejection methods using random sample consensus [16], point-wise descriptor similarity [7, 32] or group-wise spatial consistency [46]. Deep learning methods have also been proposed, focusing on learning correspondence features used to estimate inlier confidence values [2, 10, 33]. In particular, the current state-of-the-art method, PointDSC [2], relies on a spatial consistency-driven non-local network to capture long-range context in its learned correspondence features. While effective, PointDSC still yields limited registration robustness, particularly for scenes with a high outlier ratio, where the spatial consistency constraints may become ambiguous [36], thereby degrading the correspondence features' quality."
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+ "content": "Our experimental results on extensive benchmark datasets demonstrate that our framework outperforms the state-of-the-art registration methods."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.078,
+ 0.785,
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+ "angle": 0,
+ "content": "2. Related Work"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.81,
+ 0.473,
+ 0.903
+ ],
+ "angle": 0,
+ "content": "End-to-end Registration Methods. With the advances of deep learning in the 3D vision field [34], the learning-based end-to-end registration model has achieved increasing research attention. DCP [42] uses the feature similarity to establish pseudo correspondences for SVD-based transformation estimation. RPM-Net [47] exploits the Sinkhorn"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.092,
+ 0.895,
+ 0.259
+ ],
+ "angle": 0,
+ "content": "layer and annealing for discriminative matching map generation. [22, 23] integrate the cross-entropy method into the deep model for robust registration. RIENet [40] uses the structure difference between the source neighborhood and the pseudo-target one for inlier confidence evaluation. With the powerful feature representation of Transformer, RegTR [48] effectively aligns large-scale indoor scenes in an end-to-end manner. [13] propose a match-normalization layer for robust registration in the real-world 6D object pose estimation task. More end-to-end models such as [10, 18, 28–30, 33, 53] also present impressive precisions."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.262,
+ 0.895,
+ 0.565
+ ],
+ "angle": 0,
+ "content": "Learning-based Feature Descriptors. To align the complex scenes, a popular pipeline is to exploit feature descriptors for 3D matching. Compared to hand-crafted descriptors such as [17, 37, 38], the deep feature descriptor presents superior registration precision and has achieved much more attention in recent years. The pioneering 3DMatch [50] exploits the Siamese 3D CNN to learn the local geometric feature via contrastive loss. FCGF [11] exploits a fully convolutional network for dense feature extraction in a one-shot fashion. Furthermore, D3feat [3] jointly learns the dense feature descriptor and the detection score for each point. By integrating the overlap-attention module into D3feat, Predictor [21] largely improves the registration reliability in low-overlapping point clouds. YOHO [41] utilizes the group equivariant feature learning to achieve the rotation invariance and shows great robustness to the point density and the noise interference. [35] develops a geometric transformer to learn the geometric context for robust super-point matching. Lepard [31] embeds the relative 3D positional encoding into the transformer for discriminative descriptor learning."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.569,
+ 0.895,
+ 0.903
+ ],
+ "angle": 0,
+ "content": "Outlier Rejection Methods. Despite significant progress in learning-based feature descriptor, generating mismatched correspondences (outliers) in some challenging scenes remains unavoidable. Traditional outlier filtering methods, such as RANSAC [16] and its variants [4, 24, 27], use repeated sampling and verification for outlier rejection. However, these methods tend to have a high time cost, particularly in scenes with a high outlier ratio. Instead, FGR [52] and TEASER [45] integrate the robust loss function into the optimization objective to weaken the interference from outliers. Recently, Chen et al. [9] developed second-order spatial compatibility for robust consensus sampling. With the rise of deep 3D vision, most learnable outlier rejection models [10, 33] formulate outlier rejection as a binary classification task and reject correspondences with low confidence. Yi et al. [49] proposed a context normalization-embedded deep network for inlier evaluation, while Brachmann et al. [6] enhanced classical RANSAC with neural-guided prior confidence. As our baseline, PointDSC [2] proposes exploiting a spatial consistency-guided non-local inlier classifier for inlier evaluation, followed by neural spectral matching for robust registration. However, under high outlier"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.483,
+ 0.945,
+ 0.517,
+ 0.957
+ ],
+ "angle": 0,
+ "content": "1149"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.092,
+ 0.47,
+ 0.152
+ ],
+ "angle": 0,
+ "content": "ratios, spatial consistency can be ambiguous (as shown in Fig. 1), misleading non-local feature aggregation. Instead, we propose exploiting Bayesian-driven long-range dependencies for discriminative non-local feature learning."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.168,
+ 0.186,
+ 0.185
+ ],
+ "angle": 0,
+ "content": "3. Approach"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.078,
+ 0.193,
+ 0.208,
+ 0.209
+ ],
+ "angle": 0,
+ "content": "3.1. Background"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.216,
+ 0.469,
+ 0.412
+ ],
+ "angle": 0,
+ "content": "Problem Setting. In the pairwise 3D registration task, given a source point cloud \\(\\mathbf{X} = \\{\\mathbf{x}_i\\in \\mathbb{R}^3\\mid i = 1,\\dots,|\\mathbf{X}|\\}\\) and a target point cloud \\(\\mathbf{Y} = \\{\\mathbf{y}_j\\in \\mathbb{R}^3\\mid j = 1,\\dots,|\\mathbf{Y}|\\}\\), we aim to find their optimal rigid transformation consisting of a rotation matrix \\(\\mathbf{R}^{*}\\in SO(3)\\) and a translation vector \\(\\mathbf{t}^{*}\\in \\mathbb{R}^{3}\\) to align their overlapping region precisely. In this work, we focus on the descriptor-based pipeline for large-scale scene registration. Based on the feature-level nearest neighbor, we construct a set of putative correspondences \\(\\mathcal{C} = \\left\\{\\mathbf{c}_i = (\\mathbf{x}_i,\\mathbf{y}_i)\\in \\mathbb{R}^6\\mid i = 1,\\dots,|\\mathcal{C}|\\right\\}\\). The inlier (correctly matched correspondence) is defined as the correspondence satisfying \\(\\| \\mathbf{R}^{*}\\mathbf{x}_{i} + \\mathbf{t}^{*} - \\mathbf{y}_{i}\\| < \\varepsilon\\), where \\(\\varepsilon\\) indicates the inlier threshold."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.413,
+ 0.469,
+ 0.533
+ ],
+ "angle": 0,
+ "content": "Vanilla Non-local Feature Embedding. Given the putative correspondence set \\(\\mathcal{C}\\), [2] leverages the spatial consistency-guided non-local network (SCNonlocal) for their feature embedding. The injected geometric compatibility matrix can effectively regularize the long-range dependencies for discriminative inlier/outlier feature learning. In detail, it contains \\(L\\) iterations and the feature aggregation in \\(l\\)-th iteration can be formulated as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.126,
+ 0.534,
+ 0.469,
+ 0.572
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {F} _ {i} ^ {l + 1} = \\mathbf {F} _ {i} ^ {l} + \\operatorname {M L P} \\left(\\sum_ {j = 1} ^ {| \\mathcal {C} |} \\operatorname {s o f t m a x} _ {j} \\left(\\boldsymbol {\\alpha} ^ {l} \\boldsymbol {\\beta}\\right) \\mathbf {V} _ {j} ^ {l}\\right), \\tag {1}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.574,
+ 0.469,
+ 0.716
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathbf{F}_i^l\\in \\mathbb{R}^d\\) indicates the feature embedding of correspondence \\(\\mathbf{c}_i\\) in \\(l\\)-th iteration (the initial feature \\(\\mathbf{F}_i^0\\) is obtained via linear projection on \\(\\mathbf{c}_i\\)) and \\(\\mathbf{V}_i^l = f_v^l (\\mathbf{F}_i^l)\\in \\mathbb{R}^d\\) is the projected value feature. \\(\\alpha^l\\in \\mathbb{R}^{|\\mathcal{C}|\\times |\\mathcal{C}|}\\) is the non-local attention map whose entry \\(\\alpha_{i,j}^{l}\\) reflects the feature similarity between the projected query feature \\(\\mathbf{Q}_i^l = f_q^l (\\mathbf{F}_i^l)\\in \\mathbb{R}^d\\) and the key feature \\(\\mathbf{K}_i^l = f_k^l (\\mathbf{F}_j^l)\\in \\mathbb{R}^d\\). \\(\\beta \\in \\mathbb{R}^{|\\mathcal{C}|\\times |\\mathcal{C}|}\\) represents the geometric compatibility matrix of correspondences, where the compatibility between \\(\\mathbf{c}_i\\) and \\(\\mathbf{c}_j\\) is:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.085,
+ 0.717,
+ 0.469,
+ 0.758
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\beta} _ {i, j} = \\max \\left(0, 1 - \\frac {d _ {i j} ^ {2}}{\\varepsilon^ {2}}\\right), d _ {i j} = \\left\\| \\mathbf {x} _ {i} - \\mathbf {x} _ {j} \\right\\| - \\left\\| \\mathbf {y} _ {i} - \\mathbf {y} _ {j} \\right\\| |. \\tag {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.76,
+ 0.469,
+ 0.837
+ ],
+ "angle": 0,
+ "content": "Based on the fact that the geometric distance \\(d_{i,j}\\) of inliers \\(\\mathbf{c}_i\\) and \\(\\mathbf{c}_j\\) tend to be minor, Eq. 2 will assign a high compatibility value on the inlier pair, thereby promoting the nonlocal network to effectively cluster the inlier features for discriminative inlier/outlier feature learning."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.847,
+ 0.435,
+ 0.864
+ ],
+ "angle": 0,
+ "content": "3.2. Variational Non-local Feature Embedding"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.871,
+ 0.469,
+ 0.902
+ ],
+ "angle": 0,
+ "content": "While effective, SCNonlocal still suffers from ambiguous long-range dependencies, especially in some challeng-"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.518,
+ 0.088,
+ 0.872,
+ 0.213
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.5,
+ 0.212,
+ 0.892,
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+ ],
+ "angle": 0,
+ "content": "Figure 1. The ratio of inlier-outlier pairs with positive compatibilities in 3DLoMatch [21] using FCGF and Predator descriptors."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.258,
+ 0.892,
+ 0.41
+ ],
+ "angle": 0,
+ "content": "ing scenes (e.g., the low-overlapping case). Two essential reasons are: (i) Wrong geometric compatibility. As shown in Fig. 1, for 3DLoMatch dataset with Predator and FCGF descriptors, almost \\(30\\%\\) and \\(17\\%\\) of inlier-outlier pairs own the positive compatibility values, respectively, which potentially misleads the attention weight for wrong feature clustering. (ii) Lack of uncertainty modeling. In symmetric or repetitive scenes, the inlier/outlier prediction contains significant uncertainty. Therefore, it's necessary to design a robust feature representation to capture such uncertainty."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.412,
+ 0.893,
+ 0.717
+ ],
+ "angle": 0,
+ "content": "To overcome them, we develop a variational non-local network, a probabilistic feature learning framework, for discriminative correspondence embedding. Our core idea is to inject random features into our model to capture the ambiguous nature of long-range dependencies, and then leverage the variational Bayesian inference to model the Bayesian-driven long-range dependencies for discriminative feature aggregation. Specifically, we first introduce the random feature variables \\( z_{k,i}^{l}, z_{q,i}^{l} \\) and \\( z_{v,i}^{l} \\) into the key \\( \\mathbf{K}_i^l \\), query \\( \\mathbf{Q}_i^l \\) and value \\( \\mathbf{V}_i^l \\) components in our non-local module to capture their potential uncertainty in the long-range dependency modeling. Then, the prior/posterior encoders are constructed to predict their prior feature distribution and the posterior one, respectively. Embedded with the inlier/outlier label, the posterior feature distribution is label-dependent and discriminative. Thus, by pushing the prior close to the discriminative posterior in the training phase, this prior at test time also tends to sample discriminative query, key, and value features for high-quality long-range dependency modeling."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.72,
+ 0.893,
+ 0.903
+ ],
+ "angle": 0,
+ "content": "Probabilistic Graphical Model over Variational Nonlocal Network. To achieve effective variational Bayesian inference, we need to first characterize the conditional dependencies of the injected random features so that the variational lower bound can be derived as the optimization objective for model training. As shown in Fig. 2, we customize the probabilistic graphical model over our non-local network to clarify the dependencies of random features (the circles). The solid line denotes the label prediction process, while the dashed line represents our label-dependent posterior encoder (i.e., inference model). Notably, the deterministic hidden query/key/value features \\(\\mathbf{h}_{k,i}^{l}\\in \\mathbb{R}^{d^{\\prime}}\\)"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.483,
+ 0.945,
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+ ],
+ "angle": 0,
+ "content": "1150"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.09,
+ 0.47,
+ 0.136
+ ],
+ "angle": 0,
+ "content": "\\(\\mathbf{h}_{q,i}^{l}\\in \\mathbb{R}^{d^{\\prime}}\\) , and \\(\\mathbf{h}_{v,i}^{l}\\in \\mathbb{R}^{d^{\\prime}}\\) are also introduced to summarize the historical information for better feature updating in each iteration."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.14,
+ 0.47,
+ 0.32
+ ],
+ "angle": 0,
+ "content": "Inlier/outlier Prediction Process. Based on the defined conditional dependencies in Fig. 2, the prediction process of correspondence labels \\(\\mathbf{b} = \\{b_1, b_2, \\dots, b_{|\\mathcal{C}|} \\mid b_i \\in \\{0,1\\}\\}\\) (1 indicates inlier and 0 outlier) is formulated as follows. Beginning with the initial linear projection \\(\\tilde{\\mathbf{F}}^0 \\in \\mathbb{R}^{|\\mathcal{C}| \\times d}\\) of correspondences \\(\\mathcal{C} = \\{\\mathbf{c}_i\\}\\), we iteratively perform the probabilistic non-local aggregation for feature updating. In the \\(l\\)-th iteration, we first employ a Gated Recurrent Unit (GRU) [12] to predict the hidden query/key/value features which summarize the historical query/key/value features (sampled from the prior distributions) and the correspondence features in previous iterations:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.156,
+ 0.323,
+ 0.39,
+ 0.343
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {h} _ {q, i} ^ {l} = \\operatorname {G R U} _ {q} \\left(\\mathbf {h} _ {q, i} ^ {l - 1}, \\left[ \\mathbf {z} _ {q, i} ^ {l - 1}, \\tilde {\\mathbf {F}} _ {i} ^ {l - 1} \\right]\\right),\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.156,
+ 0.345,
+ 0.469,
+ 0.365
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {h} _ {k, i} ^ {l} = \\operatorname {G R U} _ {k} \\left(\\mathbf {h} _ {k, i} ^ {l - 1}, \\left[ \\mathbf {z} _ {k, i} ^ {l - 1}, \\tilde {\\mathbf {F}} _ {i} ^ {l - 1} \\right]\\right), \\tag {3}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.157,
+ 0.367,
+ 0.39,
+ 0.387
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {h} _ {v, i} ^ {l} = \\operatorname {G R U} _ {v} \\left(\\mathbf {h} _ {v, i} ^ {l - 1}, \\left[ \\mathbf {z} _ {v, i} ^ {l - 1}, \\tilde {\\mathbf {F}} _ {i} ^ {l - 1} \\right]\\right),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.389,
+ 0.47,
+ 0.547
+ ],
+ "angle": 0,
+ "content": "where \\([\\cdot ,\\cdot ]\\) denotes the feature concatenation and \\(\\tilde{\\mathbf{F}}_i^{l - 1}\\) is the learned correspondence features of \\(\\mathbf{c}_i\\) in iteration \\(l - 1\\) Then, with as input the predicted hidden features, the prior encoder \\(p_{\\theta}(\\cdot)\\) is utilized to predict the prior feature distribution for query/key/value, respectively. Furthermore, we sample features \\(\\mathbf{z}_{q,i}^{l}\\in \\mathbb{R}^{\\bar{d}}\\) \\(\\mathbf{z}_{k,i}^{l}\\in \\mathbb{R}^{\\bar{d}}\\) and \\(\\mathbf{z}_{v,i}^{l}\\in \\mathbb{R}^{\\bar{d}}\\) from the predicted prior query/key/value distribution and combine them with the hidden features to predict the corresponding query \\(\\tilde{\\mathbf{Q}}_i^l\\in \\mathbb{R}^d\\) , key \\(\\tilde{\\mathbf{K}}_i^l\\in \\mathbb{R}^d\\) and value \\(\\tilde{\\mathbf{V}}_i^l\\in \\mathbb{R}^d\\) through a neural network \\(f_{\\theta}^{q,k,v}:\\mathbb{R}^{d' + \\tilde{d}}\\to \\mathbb{R}^d\\) .."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.117,
+ 0.563,
+ 0.424,
+ 0.584
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {z} _ {q, i} ^ {l} \\sim p _ {\\theta} (z _ {q, i} ^ {l} \\mid \\mathbf {h} _ {q, i} ^ {l}), \\tilde {\\mathbf {Q}} _ {i} ^ {l} = f _ {\\theta} ^ {q} ([ \\mathbf {z} _ {q, i} ^ {l}, \\mathbf {h} _ {q, i} ^ {l} ]),\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.119,
+ 0.585,
+ 0.469,
+ 0.605
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {z} _ {k, i} ^ {l} \\sim p _ {\\theta} \\left(z _ {k, i} ^ {l} \\mid \\mathbf {h} _ {k, i} ^ {l}\\right), \\quad \\tilde {\\mathbf {K}} _ {i} ^ {l} = f _ {\\theta} ^ {k} \\left(\\left[ \\mathbf {z} _ {k, i} ^ {l}, \\mathbf {h} _ {k, i} ^ {l} \\right]\\right), \\tag {4}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.12,
+ 0.607,
+ 0.426,
+ 0.627
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {z} _ {v, i} ^ {l} \\sim p _ {\\theta} (z _ {v, i} ^ {l} \\mid \\mathbf {h} _ {v, i} ^ {l}), \\tilde {\\mathbf {V}} _ {i} ^ {l} = f _ {\\theta} ^ {v} ([ \\mathbf {z} _ {v, i} ^ {l}, \\mathbf {h} _ {v, i} ^ {l} ]),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.642,
+ 0.469,
+ 0.807
+ ],
+ "angle": 0,
+ "content": "where the prior feature distribution is the Gaussian distribution with the mean and the standard deviation parameterized by a neural network. Finally, with the learned \\(\\tilde{\\mathbf{Q}}_i^l\\), \\(\\tilde{\\mathbf{K}}_i^l\\) and \\(\\tilde{\\mathbf{V}}_i^l\\), the correspondence feature \\(\\tilde{\\mathbf{F}}_i^l\\) in \\(l\\)-th iteration can be aggregated with the same non-local operation in Eq. 1. After \\(L\\) feature iterations, we feed the correspondence feature \\(\\tilde{\\mathbf{F}}_i^L\\) in the last iteration into a label prediction model \\(y_{\\theta}\\) to predict the inlier/outlier labels \\(b_i \\sim y_{\\theta}(b_i \\mid \\tilde{\\mathbf{F}}_i^L)\\), where the label prediction model outputs a scalar Gaussian distribution with the mean parameterized by the neural network and the unit variance."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.81,
+ 0.469,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "Variational Posterior Encoder. Due to the nonlinearity of our variational non-local model, we cannot directly derive the precise posterior distribution for random query/key/value features using the standard Bayes' theorem. Taking inspiration from the Variational Bayesian inference, we construct a label-dependent posterior encoder"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.501,
+ 0.089,
+ 0.892,
+ 0.198
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.498,
+ 0.207,
+ 0.892,
+ 0.306
+ ],
+ "angle": 0,
+ "content": "Figure 2. Probabilistic graphical model for our variational nonlocal network. For simplicity, we just demonstrate two iterations. The white circles indicate the random features and the white squares denote the deterministic hidden features. The solid line represents the inlier/outlier prediction process and the dashed line denotes the label-dependent variational posterior encoder. We just show the variational posterior for \\( z_{k}^{1} \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.5,
+ 0.317,
+ 0.797,
+ 0.333
+ ],
+ "angle": 0,
+ "content": "\\(q_{\\phi}(\\cdot)\\) to approximate the feature posterior:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.594,
+ 0.341,
+ 0.793,
+ 0.361
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {z} _ {q, i} ^ {l} \\sim q _ {\\phi} \\left(z _ {q, i} ^ {l} \\mid [ \\mathbf {h} _ {q, i} ^ {l}, [ b _ {i} ] _ {\\times k} ]\\right)\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.595,
+ 0.364,
+ 0.892,
+ 0.382
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {z} _ {k, i} ^ {l} \\sim q _ {\\phi} \\left(z _ {k, i} ^ {l} \\mid \\left[ \\mathbf {h} _ {k, i} ^ {l}, [ b _ {i} ] _ {\\times k} \\right]\\right) \\tag {5}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.595,
+ 0.384,
+ 0.797,
+ 0.403
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {z} _ {v, i} ^ {l} \\sim q _ {\\phi} \\left(z _ {v, i} ^ {l} \\mid [ \\mathbf {h} _ {v, i} ^ {l}, [ b _ {i} ] _ {\\times k} ]\\right),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.412,
+ 0.892,
+ 0.471
+ ],
+ "angle": 0,
+ "content": "where \\([b_i]_{\\times k}\\) indicates a label vector generated by tiling the scalar label \\(k\\) times. The output of each posterior encoder is a diagonal Gaussian distribution with parameterized mean and standard deviation."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.473,
+ 0.892,
+ 0.548
+ ],
+ "angle": 0,
+ "content": "Variational Lower Bound. Finally, we derive the optimization objective \\(\\mathrm{ELBO}(\\theta ,\\phi)\\), the variational (evidence) lower bound of log-likelihood correspondence labels \\(\\ln y_{\\theta}(\\mathbf{b}\\mid \\mathcal{C})\\), to train our variational non-local network (Please refer to Appendix A for the detailed derivation):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.512,
+ 0.556,
+ 0.88,
+ 0.581
+ ],
+ "angle": 0,
+ "content": "\\[\n\\operatorname {E L B O} (\\theta , \\phi) = \\mathbb {E} _ {\\prod_ {l = 0} ^ {L - 1} q _ {\\phi} (z _ {q, k, v} ^ {l} | \\mathbf {h} _ {q, k, v} ^ {l}, \\mathbf {b})} \\left[ \\ln y _ {\\theta} (\\mathbf {b} \\mid \\tilde {\\mathbf {F}} ^ {L}) \\right] -\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.515,
+ 0.583,
+ 0.892,
+ 0.634
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {l = 0} ^ {L - 1} \\mathbb {E} _ {q _ {\\phi}} \\left[ \\mathrm {D} _ {\\mathrm {K L}} \\left(q _ {\\phi} \\left(z _ {q, k, v} ^ {l} \\mid \\mathbf {h} _ {q, k, v} ^ {l}, \\mathbf {b}\\right) \\mid p _ {\\theta} \\left(z _ {q, k, v} ^ {l} \\mid \\mathbf {h} _ {q, k, v} ^ {l}\\right)\\right) \\right] \\tag {6}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.643,
+ 0.892,
+ 0.823
+ ],
+ "angle": 0,
+ "content": "where for clarity, we utilize the subscript \\(q,k,v\\) to denote the same operator performed on query/key/value. \\(\\mathrm{D}_{\\mathrm{KL}}(\\cdot ||\\cdot)\\) denotes the Kullback-Leibler (KL) divergence between two distributions. By maximizing the variational lower bound above, we can optimize the network parameters to indirectly maximize the log-likelihood value of correspondence labels. Eq. 6 indicates that the discriminative, label-dependent feature posterior explicitly constrains the prior by reducing their KL divergence in the training phase, which promotes the query, key, and value features sampled from the prior to model the high-quality long-term dependencies at test time."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.833,
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+ ],
+ "angle": 0,
+ "content": "3.3. Voting-based Inlier Searching"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.856,
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+ ],
+ "angle": 0,
+ "content": "With the learned correspondence features above, we then propose a voting-based sampling strategy to search the desired inlier subset from the entire putative correspondences"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.484,
+ 0.945,
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+ ],
+ "angle": 0,
+ "content": "1151"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.091,
+ 0.473,
+ 0.318
+ ],
+ "angle": 0,
+ "content": "for optimal transformation estimation. Our sampling mechanism is deterministic and efficient. We first select a set of high-confidence seeds \\(\\mathcal{C}_{seed}\\) from \\(\\mathcal{C}\\) based on their inlier confidence (predicted in § 3.2) and the Non-Maximum Suppression (as performed in [2]), where the number of seeds \\(|\\mathcal{C}_{seed}| = \\lfloor |C|*v\\rfloor\\) (\\(v\\) is a fixed seed ratio). Then, for each seed, we cluster its most compatible correspondences into it to form the hypothetical inliers. Ideally, if the seed is an inlier and the compatible value is correct (without ambiguity problem as in Fig. 2), its most compatible correspondences are also inliers, thus we can cluster the desired inlier subset successfully. To cluster high-confidence hypothetical inliers, we perform a coarse-to-fine clustering strategy, including the voting-based coarse clustering and the Wilson score-based fine clustering."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.32,
+ 0.473,
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+ ],
+ "angle": 0,
+ "content": "Voting-based Coarse Clustering. We view the correspondence features \\(\\{\\tilde{\\mathbf{F}}^1,\\dots,\\tilde{\\mathbf{F}}^L\\}\\) learned from all non-local iterations as \\(L\\) voters. For \\(l\\)-th voter, it deterministically clusters \\(\\kappa - 1\\) most compatible correspondences \\((\\kappa - 1)\\mathcal{A}_{\\mathbf{c}_i}^l\\) for each seed \\(\\mathbf{c}_i \\in \\mathcal{C}_{\\text{seed}}\\) to from the hypothetical inliers \\((\\kappa)\\mathcal{M}_{\\mathbf{c}_i}^l = \\{\\mathbf{c}_i\\} \\cup (\\kappa - 1)\\mathcal{A}_{\\mathbf{c}_i}^l\\). To this end, we first compute the compatibility matrix \\(\\mathbf{S}^l \\in \\mathbb{R}^{|\\mathcal{C}| \\times |\\mathcal{C}|}\\) for \\(l\\)-th voter, where each entry \\(\\mathbf{S}_{i,j}^l = \\mathrm{clip}(1 - (1 - \\cos(\\tilde{\\mathbf{F}}_i^l, \\tilde{\\mathbf{F}}_j^l) / \\sigma^2), 0, 1) \\cdot \\beta_{i,j}\\), where the parameter \\(\\sigma\\) is to control the feature compatibility and geometric compatibility \\(\\beta_{i,j}\\) is defined in Eq. 2. Thus, \\((\\kappa - 1)\\mathcal{A}_{\\mathbf{c}_i}^l\\) can be determined as \\(\\{\\mathbf{c}_j | \\mathbf{S}_{i,j}^l \\geq \\gamma_{\\kappa - 1}\\}\\), where threshold \\(\\gamma_{\\kappa - 1}\\) is the \\((\\kappa - 1)\\)-th largest compatibility. Benefiting from our discriminative feature representations, \\(\\mathbf{S}^l\\) can effectively handle the ambiguity problem in Fig. 2 and promote robust inlier clustering. After the voting process above, each seed \\(\\mathbf{c}_i\\) can achieve \\(L\\) candidate hypothetical inliers \\(\\left\\{\\left(\\kappa\\right)\\mathcal{M}_{\\mathbf{c}_i}^1, \\dots, \\left(\\kappa\\right)\\mathcal{M}_{\\mathbf{c}_i}^L\\right\\}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.583,
+ 0.472,
+ 0.693
+ ],
+ "angle": 0,
+ "content": "Wilson score-based Fine Clustering. Based on the voted hypothetical inliers above, we then fuse these voted results with Wilson score [43] to form the high-quality hypothetical inliers \\((^{\\kappa})\\tilde{\\mathcal{M}}_{\\mathbf{c}_i}\\). In detail, we denote \\(\\mathcal{T}_{i,j}^{l} = \\mathbb{I}\\{\\mathbf{c}_{j}\\in\\) \\((^{\\kappa})\\mathcal{M}_{\\mathbf{c}_{i}}^{l}\\}\\) to indicate whether \\((^{\\kappa})\\mathcal{M}_{\\mathbf{c}_{i}}^{l}\\) contains \\(\\mathbf{c}_j\\). The Wilson score \\(\\mathbf{W}_{i,j}^{n}\\) of accepting \\(\\mathbf{c}_j\\) into \\((^{\\kappa})\\tilde{\\mathcal{M}}_{\\mathbf{c}_i}\\) can be computed as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.088,
+ 0.695,
+ 0.47,
+ 0.75
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {W} _ {i, j} ^ {n} = \\frac {1}{1 + \\frac {z ^ {2}}{n}} \\left[ \\hat {p} _ {i, j} ^ {(n)} + \\frac {z ^ {2}}{2 n} - z \\sqrt {\\frac {\\hat {p} _ {i , j} ^ {(n)} \\left(1 - \\hat {p} _ {i , j} ^ {(n)}\\right)}{n} + \\frac {z ^ {2}}{4 n ^ {2}}} \\right], \\tag {7}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.753,
+ 0.473,
+ 0.902
+ ],
+ "angle": 0,
+ "content": "where \\(\\hat{p}_{i,j}^{(n)} = \\frac{1}{n}\\sum_{\\tau = 1}^{n}\\mathcal{I}_{i,j}^{\\tau}\\) is the average acceptance ratio of top \\(n\\) voters and \\(z = 1.96\\) (i.e., z-score at \\(95\\%\\) confidence level). Eq. 7 indicates that the Wilson score not only considers the sample mean but also the confidence (positively related to sample number \\(n\\)). Finally, among the set of Wilson scores under different sample numbers \\(\\{\\mathbf{W}_{i,j}^{1},\\dots,\\mathbf{W}_{i,j}^{L}\\}\\), we choose the largest one as the final Wilson score \\(\\bar{\\mathbf{W}}_{i,j} = \\max_{n\\in \\{1,\\dots,L\\}}\\mathbf{W}_{i,j}^{n}\\) for \\(\\mathbf{c}_j\\). Thus, the final hypothetical inliers set of \\(\\mathbf{c}_i\\) can be determined as"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.089,
+ 0.892,
+ 0.121
+ ],
+ "angle": 0,
+ "content": "\\((^{\\kappa})\\tilde{\\mathcal{M}}_{\\mathbf{c}_i} = \\{\\mathbf{c}_i\\} \\cup \\{\\mathbf{c}_j\\mid \\tilde{\\mathbf{W}}_{i,j}\\geq \\gamma_{\\kappa -1}^\\prime \\}\\) , where \\(\\gamma_{\\kappa -1}^{\\prime}\\) is the \\((\\kappa -1)\\) -th largest Wilson score."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.121,
+ 0.894,
+ 0.304
+ ],
+ "angle": 0,
+ "content": "Theoretical Analysis and Conservative Seed Selection. Finally, we try to theoretically analyze our deterministic inlier searching mechanism compared to RANSAC [16] and further propose a simple but effective conservative seed selection strategy for more robust 3D registration in sparse point clouds. We let \\(\\{^{(\\kappa)}\\mathcal{M}_i^{sac}\\}_{i=1}^J\\) be the randomly sampled hypothetical inlier subset in RANSAC, and let \\(C_{in}, C_{out}\\) and \\(p_{in}\\) be the inlier subset, outlier subset and the inlier ratio, respectively, \\(\\mathcal{C} = C_{in} \\cup C_{out}\\), \\(p_{in} = |C_{in}| / |\\mathcal{C}|\\). We also denote the inliers in seeds as \\(\\tilde{C}_{in} = C_{in} \\cap C_{seed}\\). Then, we can derive the following theorem (Please refer to Appendix B for the derivation process)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.309,
+ 0.895,
+ 0.392
+ ],
+ "angle": 0,
+ "content": "Theorem 1 Assume the number of outliers in \\({}^{(\\kappa)}\\tilde{\\mathcal{M}}_{\\mathbf{c}_i}\\) (\\(\\mathbf{c}_i \\in \\tilde{\\mathcal{C}}_{in}\\)) follows a Poisson distribution \\(Pois(\\alpha \\cdot \\kappa)\\). Then, if \\(\\alpha < -\\frac{1}{\\kappa} \\cdot \\log \\left[1 - (1 - p_{in}^{\\kappa})^{J / |\\tilde{\\mathcal{C}}_{in}|}\\right] \\triangleq \\mathcal{U}\\), the probability of our method achieving the inlier subset is greater than or equal to that of RANSAC."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.575,
+ 0.394,
+ 0.892,
+ 0.429
+ ],
+ "angle": 0,
+ "content": "\\[\nP \\left(\\max _ {\\mathbf {c} _ {i} \\in \\mathcal {C} _ {s e e d}} | ^ {(\\kappa)} \\tilde {\\mathcal {M}} _ {\\mathbf {c} _ {i}} \\cap \\mathcal {C} _ {i n} | = \\kappa\\right) \\geq \\tag {8}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.577,
+ 0.424,
+ 0.8,
+ 0.451
+ ],
+ "angle": 0,
+ "content": "\\[\nP \\Big (\\max _ {1 \\leq i \\leq J} | ^ {(\\kappa)} \\mathcal {M} _ {i} ^ {s a c} \\cap \\mathcal {C} _ {i n} | = \\kappa \\Big).\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.457,
+ 0.895,
+ 0.744
+ ],
+ "angle": 0,
+ "content": "Theorem 1 shows that as the inlier ratio \\( p_{in} \\) degrades, the upper bound \\( \\mathcal{U} \\) will increase, indicating our method tends to be more likely to achieve the inlier subset than RANSAC in high outlier cases. Therefore, our method tends to be more robust to outliers than RANSAC. In addition, Theorem 1 also shows that the more inliers \\( |\\tilde{\\mathcal{C}}_{in}| \\) in \\( \\mathcal{C}_{\\text{seed}} \\), the better for our method. It means selecting all correspondences in \\( \\mathcal{C} \\) as seeds theoretically seems the best choice since it can avoid missing any inlier. However, in real implementation, we have to perform inlier selection to accelerate the registration speed. Nevertheless, in sparse point clouds, the number of correspondences is small. If we still perform the original inlier selection, fewer seeds even may not contain any inlier, leading to registration failure. To overcome it, we develop a conservative seed selection strategy, which changes the seed number to \\( \\max \\{\\lfloor v \\cdot |\\mathcal{C}| \\rfloor, n\\} (n < |\\mathcal{C}|) \\), where the lower bar \\( n \\) of the seed number can effectively avoid selecting too fewer inliers in sparse cases and in dense cases, it would degrade to the original selection strategy."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.5,
+ 0.752,
+ 0.794,
+ 0.768
+ ],
+ "angle": 0,
+ "content": "3.4. Rigid Transformation Estimation"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.774,
+ 0.892,
+ 0.836
+ ],
+ "angle": 0,
+ "content": "With the voted hypothetical inliers \\(\\{\\mathbf{\\Psi}^{(\\kappa)}\\tilde{\\mathcal{M}}_{\\mathbf{c}_i}\\mid \\mathbf{c}_i\\in\\) \\(\\mathcal{C}_{seed}\\}\\) above, we estimate the optimal transformation parameter based on the Procrustes method [20] to minimize the least-squares errors for each hypothetical inlier group:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.522,
+ 0.837,
+ 0.892,
+ 0.87
+ ],
+ "angle": 0,
+ "content": "\\[\n\\hat {\\mathbf {R}} _ {i}, \\hat {\\mathbf {t}} _ {i} = \\arg \\min _ {\\mathbf {R}, \\mathbf {t}} \\sum_ {\\mathbf {c} _ {j} \\in {} ^ {(\\kappa)} \\tilde {\\mathcal {M}} _ {\\mathbf {c} _ {i}}} \\omega_ {j} \\cdot \\left\\| \\mathbf {R} ^ {\\top} \\mathbf {x} _ {j} + \\mathbf {t} - \\mathbf {y} _ {j} \\right\\| _ {2}, \\tag {9}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.871,
+ 0.892,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "where \\(\\omega_{j}\\) is the error weight computed by the neural spectral matching as in [2]. Then, we select the transformation"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.483,
+ 0.945,
+ 0.516,
+ 0.957
+ ],
+ "angle": 0,
+ "content": "1152"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.092,
+ 0.468,
+ 0.121
+ ],
+ "angle": 0,
+ "content": "parameter that maximizes the number of overlapped correspondences as the final optimal transformation estimation:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.088,
+ 0.124,
+ 0.468,
+ 0.169
+ ],
+ "angle": 0,
+ "content": "\\[\n\\hat {\\mathbf {R}} ^ {*}, \\hat {\\mathbf {t}} ^ {*} = \\underset {\\{\\hat {\\mathbf {R}} _ {i}, \\hat {\\mathbf {t}} _ {i} \\} _ {i = 1} ^ {| C |}} {\\arg \\max } \\sum_ {\\mathbf {c} _ {j} \\in \\mathcal {C}} v _ {i} \\cdot \\mathbb {I} \\left\\{\\| \\hat {\\mathbf {R}} _ {i} ^ {\\top} \\mathbf {x} _ {j} + \\hat {\\mathbf {t}} _ {i} - \\mathbf {y} _ {j} \\| _ {2} < \\varepsilon \\right\\}, \\tag {10}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.17,
+ 0.469,
+ 0.233
+ ],
+ "angle": 0,
+ "content": "where \\( v_{i} = 1 - \\|\\hat{\\mathbf{R}}_{i}^{\\top}\\mathbf{x}_{j} + \\hat{\\mathbf{t}}_{i} - \\mathbf{y}_{j}\\|_{2}^{2} / \\varepsilon^{2} \\) is used to re-weight the inlier count as performed in [23]. Finally, we refine it using all recovered inliers in a least-squares optimization as a common practice in [2,4]."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.249,
+ 0.21,
+ 0.266
+ ],
+ "angle": 0,
+ "content": "4. Experiments"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.274,
+ 0.284,
+ 0.291
+ ],
+ "angle": 0,
+ "content": "4.1. Experimental Settings"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.298,
+ 0.469,
+ 0.494
+ ],
+ "angle": 0,
+ "content": "Implementation Details. For our variational non-local module, the number of iterations \\( L \\) is 12, and the dimensions of the correspondence feature, random feature, and hidden feature are set to 128, 128, and 256, respectively. For our voting-based inlier sampling module, the size of hypothetical inliers \\( \\kappa \\) is 40. For seed selection, seed ratio \\( v \\) is 0.1 and the lower bar of seed number \\( n \\) is 1000. Our model is trained with 50 epochs using Adam optimizer with learning rate \\( 10^{-4} \\) and weight decay \\( 10^{-6} \\). We utilize PyTorch to implement our project and perform all experiments on the server equipped with an Intel i5 2.2 GHz CPU and one Tesla V100 GPU. For simplicity, we name our Variational Bayesian-based Registration framework as VBReg."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.496,
+ 0.469,
+ 0.571
+ ],
+ "angle": 0,
+ "content": "Evaluation Metric. We use three metrics to evaluate our method, including (1) Registration Recall \\((RR)\\), the percent of the successful registration satisfying the error thresholds of rotation and translation at the same time, (2) average Rotation Error \\((RE)\\) and (3) average Translation Error \\((TE)\\):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.093,
+ 0.572,
+ 0.469,
+ 0.619
+ ],
+ "angle": 0,
+ "content": "\\[\n\\operatorname {R E} (\\hat {\\mathbf {R}}) = \\arccos \\frac {\\operatorname {T r} \\left(\\hat {\\mathbf {R}} ^ {\\top} \\mathbf {R} ^ {*}\\right) - 1}{2}, \\operatorname {T E} (\\hat {\\mathbf {t}}) = \\left\\| \\hat {\\mathbf {t}} - \\mathbf {t} ^ {*} \\right\\| _ {2} ^ {2}, \\tag {11}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.623,
+ 0.469,
+ 0.684
+ ],
+ "angle": 0,
+ "content": "where \\(\\hat{\\mathbf{R}}\\) and \\(\\hat{\\mathbf{t}}\\) are the predicted rotation matrix and rotation vector, respectively, while \\(\\mathbf{R}^*\\) and \\(\\mathbf{t}^*\\) are the corresponding ground truth. The average \\(RE\\) and \\(TE\\) are computed only on successful aligned point cloud pairs."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.077,
+ 0.696,
+ 0.385,
+ 0.712
+ ],
+ "angle": 0,
+ "content": "4.2. Comparison with Existing Methods"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.076,
+ 0.72,
+ 0.469,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "Evaluation on 3DMatch. We first evaluate our method on 3DMatch benchmark [50], which contains 46 training scenes, 8 validation scenes, and 8 test scenes. We first voxelize and down-sample the point cloud with \\(5\\mathrm{cm}\\) voxel size and then leverage FCGF [11] and FPFH [37] descriptors to construct the putative correspondences based on the feature nearest neighbor. We compare our method with eight state-of-the-art (SOTA) correspondence-based methods, where FGR [52], SM [26], RANSAC (50k) [16], TEASER++ [45], and SC2_PCR [9] are representative traditional methods, while DGR [10], DHVR [25], and PointDSC [2] are advanced deep learning-based methods. As shown in Table 1,"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.503,
+ 0.089,
+ 0.892,
+ 0.187
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.526,
+ 0.199,
+ 0.866,
+ 0.213
+ ],
+ "angle": 0,
+ "content": "Figure 3. Registration visualization on 3DLoMatch [19]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.22,
+ 0.892,
+ 0.431
+ ],
+ "angle": 0,
+ "content": "in the FCGF setting, our method achieves the best performance in \\(RR\\) and \\(RE\\) criteria while the same \\(TE\\) with PointDSC. We need to highlight that \\(RR\\) is a more important criterion than \\(RE\\) and \\(TE\\) since the rotation and translation errors are just calculated in a successful registration. It means that the higher \\(RR\\) may include more challenging but successful registration cases, potentially increasing their errors. In the FPFH setting, it can be observed that our method can still achieve the best \\(RR\\) score among all deep methods, but perform slightly worse than SC2_PCR. Notably, compared to PointDSC (our baseline), the precision gain on \\(RR\\) is impressive \\((5.24\\% \\uparrow)\\), which benefits from the effectiveness of our variational non-local feature learning and the voting-based inlier searching."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.433,
+ 0.893,
+ 0.809
+ ],
+ "angle": 0,
+ "content": "Evaluation on 3DLoMatch. We further test our method on 3DLoMatch benchmark [21]. Compared to 3DMatch sharing more than \\(30\\%\\) overlap, the overlaps of point cloud pairs in 3DLoMatch just lie in \\(10\\% \\sim 30\\%\\), thus presenting much more challenges. We leverage FCGF [11] and recently popular Predator [21] as the feature descriptors for putative correspondence generation. We choose six traditional methods: FGR [52], SM [26], RANSAC (50k) [16], TEASER++ [45], SC2_PCR [9], and TR_DE [8], and two deep methods: DHVR [25], and PointDSC [2] for comparison. The registration recalls \\((RR)\\) under different numbers of correspondences are listed in Table 2. It can be observed that regardless of FCGF or Predator descriptor, our method almost achieves the best performance on all settings, except for FCGF setting with 250 points. Notably, in more challenging cases, the performance advantage over PointDSC is further expanded \\((+9.5\\%)\\) and \\(+7.8\\%\\) in the cases of FCGF with 500 and 250 points. These experimental results further demonstrate the outstanding robustness of our method when encountering those extremely low-overlapping cases (high outlier ratios). Some visualization results are listed in Fig. 3 and the \\(RR\\) changes under different inlier ratios are presented in Fig. 4 (c), which suggests that our performance gains are mainly brought by model robustness in extremely high outlier situations."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.498,
+ 0.811,
+ 0.894,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "Evaluation on KITTI. Finally, we evaluate our method on the outdoor LIDAR-scanned driving scenarios from KITTI dataset [19]. In line with [11], we utilize sequences 0-5, 6-7, and 8-10 as the training set, validation set, and test set, respectively. Also, as the setting in [3, 11], we further refine the ground-truth rigid transformations using ICP [5] and"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "content": "1153"
+ }
+ ],
+ [
+ {
+ "type": "table",
+ "bbox": [
+ 0.082,
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+ "angle": 0,
+ "content": "| Model | 3DMatch | 3DLoMatch (FCGF) | 3DLoMatch (Predator) | Sec. |
| FCGF | FPFH | 5000 | 2500 | 1000 | 500 | 250 | 5000 | 2500 | 1000 | 500 | 250 |
| PointDSC w/ SCNonlocalxyz | 92.42 | 77.51 | 55.8 | 52.6 | 46.8 | 37.7 | 26.7 | 68.1 | 67.3 | 66.5 | 63.4 | 60.5 | 0.11 |
| PointDSC w/ VBNonlocalxyz | 93.04 | 80.16 | 57.7 | 55.6 | 50.2 | 39.9 | 26.1 | 69.7 | 69.6 | 67.9 | 64.9 | 61.9 | 0.17 |
| PointDSC w/ SCNonlocalfeat | 92.36 | 77.76 | 54.6 | 50.6 | 44.9 | 36.8 | 25.4 | 69.2 | 68.6 | 67.9 | 63.5 | 59.9 | 0.13 |
| PointDSC w/ VBNonlocalfeat | 93.04 | 80.53 | 56.9 | 56.5 | 50.9 | 42.2 | 28.9 | 69.2 | 68.7 | 68.0 | 64.6 | 60.6 | 0.18 |
| PointDSC w/ SCNonlocalcls | 92.98 | 78.99 | 54.1 | 52.2 | 46.0 | 38.7 | 27.7 | 67.6 | 66.9 | 67.2 | 63.7 | 60.2 | 0.11 |
| PointDSC w/ VBNonlocalfeat+Vote | 93.41 | 81.21 | 58.3 | 56.5 | 51.9 | 44.7 | 31.1 | 69.3 | 69.5 | 68.2 | 65.3 | 61.2 | 0.20 |
| PointDSC w/ VBNonlocalfeat+Vote+CS | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 68.7 | 66.4 | 63.0 | 0.22 |
| Iteration times L = 6 | 93.41 | 82.32 | 58.1 | 57.1 | 52.9 | 48.3 | 34.8 | 69.7 | 69.7 | 68.7 | 66.3 | 63.8 | 0.19 |
| Iteration times L = 9 | 93.41 | 81.58 | 58.2 | 57.0 | 53.2 | 47.6 | 32.5 | 70.0 | 69.4 | 68.5 | 66.9 | 63.2 | 0.20 |
| Iteration times L = 12* | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 68.7 | 66.4 | 63.0 | 0.22 |
| Random feat. dim. d̅ = 32 | 93.41 | 82.38 | 58.0 | 57.5 | 53.5 | 48.1 | 34.8 | 69.7 | 70.0 | 68.6 | 66.3 | 63.3 | 0.20 |
| Random feat. dim. d̅ = 64 | 93.41 | 81.45 | 57.9 | 56.9 | 52.8 | 48.6 | 35.0 | 69.6 | 69.7 | 68.4 | 66.5 | 62.3 | 0.21 |
| Random feat. dim. d̅ = 128* | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 69.3 | 66.4 | 63.3 | 0.22 |
"
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.077,
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+ 0.342
+ ],
+ "angle": 0,
+ "content": "Table 3. Ablation studies on 3DMatch [50] and 3DLoMatch [21] datasets. SCNonlocal: Spatial consistency-guided non-local network; VBNonlocal: Variational Bayesian-based non-local network; Vote: Voting-based inlier searching; CS: Conservative seed selection."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "eling can effectively learn the discriminative inlier/outlier features for reliable inlier search. (2): Then, to further highlight the superiority of our variational inference-guided feature learning, we also try to add classification loss on the features produced by each iteration in SCNonlocal to guide their learning (denoted as \\(SCNonlocal^{cls}\\)). As shown in the fifth row in Table 3, such loss-based label-propagation way just can achieve very limited performance gain and even sometimes degrades score. It demonstrates the superiority of our label-dependent posterior guidance for (prior) feature learning. Also, owing to such posterior guidance, the training curves in Fig. 4 (d) show that our method can achieve significantly faster convergence speed than SCNonlocal."
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+ "content": "Discriminative Feature Learning? In order to verify whether VBNonlocal can learn more discriminative features than SCNonlocal, we visualize the distribution of the feature similarity of inliers on 3DMatch (Fig. 4 (a)) and 3DLoMatch (Fig. 4 (b)). As we can see, our learned inlier features own much higher similarities (approximate to 1) than SCNonlocal on both datasets, which demonstrates that our proposed Bayesian-inspired non-local network truly can promote more discriminative correspondence-feature learning."
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+ "content": "Variational Non-local Setting. We further test the performance changes under different VBNonlocal settings. (1) We first test the model robustness under different numbers of non-local iterations. The second block in Table 3 verifies that our method is robust to the iteration time and tends to consistently achieve outstanding \\(RR\\) score. (2) Then, the bottom block in Table 3 further shows our model stability to different dimensions of random features."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Voting-based Inlier Searching. Furthermore, we evaluate the performance contribution of the proposed voting-based inlier searching strategy (Vote). As we can see in the sixth row of Table 3, voting strategy can consistently achieve performance improvement regardless in 3DMatch or in more"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "challenging 3DLoMatch. It mainly benefits from the high-quality hypothetical inliers sampled by our voting policy."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Conservative Seed Selection. Finally, we test the effectiveness of the proposed conservative seed selection strategy (CS) motivated by Theorem 1. As we can see in the seventh row of Table 3, CS can achieve consistent performance gain in each setting. Especially, in the cases of fewer points (e.g., FCGF setting with 250 and 500 points), the improvement is much more significant \\((+3.4\\%, +2.5\\%)\\). As the analysis before, in sparse point clouds, the original inlier selection strategy like in [2] is aggressive and prone to miss too many inlier seeds. Instead, our conservative selection strategy can effectively mitigate it as well as keep registration efficiency."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5. Conclusion"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "In this paper, we adapted the variational Bayesian inference into the non-local network and developed the effective Bayesian-guided long-term dependencies for discriminative correspondence-feature learning. To achieve effective variational inference, a probabilistic graphical model was customized over our non-local network, and the variational low bound was derived as the optimization objective for model training. In addition, we proposed a Wilson score-based voting mechanism for high-quality inlier sampling and theoretically verified its superiority over RANSAC. Extensive experiments on indoor/outdoor datasets demonstrated its promising performance."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "6. Acknowledgments"
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+ "content": "This work was supported by the National Science Fund of China (Grant Nos. U1713208, U62276144)."
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+ 0.26
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+ "angle": 0,
+ "content": "[51] Ji Zhang and Sanjiv Singh. Loam: Lidar odometry and mapping in real-time. In Robotics: Science and Systems, 2014. 1"
+ },
+ {
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+ "bbox": [
+ 0.503,
+ 0.262,
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+ "content": "[52] Qian-Yi Zhou, Jaesik Park, and Vladlen Koltun. Fast global registration. In ECCV, 2016. 2, 6, 7"
+ },
+ {
+ "type": "ref_text",
+ "bbox": [
+ 0.503,
+ 0.292,
+ 0.892,
+ 0.332
+ ],
+ "angle": 0,
+ "content": "[53] Jing Zhu and Yi Fang. Reference grid-assisted network for 3D point signature learning from point clouds. In WACV (2020). 2"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.503,
+ 0.093,
+ 0.892,
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+# Robust Outlier Rejection for 3D Registration with Variational Bayes
+
+Haobo Jiang $^{1}$ , Zheng Dang $^{2}$ , Zhen Wei $^{2}$ , Jin Xie $^{*1}$ , Jian Yang $^{*1}$ , and Mathieu Salzmann $^{*2}$
+
+$^{1}$ PCA Lab, Nanjing University of Science and Technology, China $^{2}$ CVLab, EPFL, Switzerland $\{\text{jiang.hao.bo, csjxie, csjyang}\} \text{@njust.edu.cn}$ $\{\text{zheng.dang, zhen.wei, mathieu.salzmann}\} \text{@epfl.ch}$
+
+# Abstract
+
+Learning-based outlier (mismatched correspondence) rejection for robust 3D registration generally formulates the outlier removal as an inlier/outlier classification problem. The core for this to be successful is to learn the discriminative inlier/outlier feature representations. In this paper, we develop a novel variational non-local network-based outlier rejection framework for robust alignment. By reformulating the non-local feature learning with variational Bayesian inference, the Bayesian-driven long-range dependencies can be modeled to aggregate discriminative geometric context information for inlier/outlier distinction. Specifically, to achieve such Bayesian-driven contextual dependencies, each query/value component in our nonlocal network predicts a prior feature distribution and a posterior one. Embedded with the inlier/outlier label, the posterior feature distribution is label-dependent and discriminative. Thus, pushing the prior to be close to the discriminative posterior in the training step enables the features sampled from this prior at test time to model high-quality long-range dependencies. Notably, to achieve effective posterior feature guidance, a specific probabilistic graphical model is designed over our non-local model, which lets us derive a variational low bound as our optimization objective for model training. Finally, we propose a voting-based inlier searching strategy to cluster the high-quality hypothetical inliers for transformation estimation. Extensive experiments on 3DMatch, 3DLoMatch, and KITTI datasets verify the effectiveness of our method. Code is available at https://github.com/Jiang-HB/VBReg.
+
+# 1. Introduction
+
+Point cloud registration is a fundamental but challenging 3D computer vision task, with many potential applications such as 3D scene reconstruction [1,39], object pose estimation [13,44], and Lidar SLAM [15, 51]. It aims to align two partially overlapping point clouds by estimating their relative rigid transformation, i.e., 3D rotation and 3D translation. A popular approach to address the large-scale scene registration problem consists of extracting point descriptors [11, 14, 17, 37, 38, 50] and establishing correspondences between the two point clouds, from which the transformation can be obtained geometrically. In this context, much effort has been dedicated to designing traditional and deep learning-based descriptors [3, 11, 21, 41, 50]. However, the resulting correspondences inevitably still suffer from outliers (wrong matchings), particularly in challenging cases, such as low-overlap, repetitive structures, or noisy point sets, leading to registration failure.
+
+To address this, many outlier filtering strategies have been developed to robustify the registration process. These include traditional rejection methods using random sample consensus [16], point-wise descriptor similarity [7, 32] or group-wise spatial consistency [46]. Deep learning methods have also been proposed, focusing on learning correspondence features used to estimate inlier confidence values [2, 10, 33]. In particular, the current state-of-the-art method, PointDSC [2], relies on a spatial consistency-driven non-local network to capture long-range context in its learned correspondence features. While effective, PointDSC still yields limited registration robustness, particularly for scenes with a high outlier ratio, where the spatial consistency constraints may become ambiguous [36], thereby degrading the correspondence features' quality.
+
+In this paper, we propose to explicitly account for the ambiguities arising from high outlier ratios by developing a probabilistic feature learning framework. To this end, we introduce a variational non-local network based on an at
+
+tention mechanism to learn discriminative inlier/outlier feature representations for robust outlier rejection. Specifically, to capture the ambiguous nature of long-range contextual dependencies, we inject a random feature in each query, key, and value component in our non-local network. The prior/posterior distributions of such random features are predicted by prior/posterior encoders. To encourage the resulting features to be discriminative, we make the posterior feature distribution label-dependent. During training, we then push the prior distribution close to the label-dependent posterior, thus allowing the prior encoder to also learn discriminative query, key, and value features. This enables the features sampled from this prior at test time to model high-quality long-range dependencies.
+
+To achieve effective variational inference, we customize a probabilistic graphical model over our variational non-local network to characterize the conditional dependencies of the random features. This lets us derive a variational lower bound as the optimization objective for network training. Finally, we propose a voting-based deterministic inlier searching mechanism for transformation estimation, where the correspondence features learned from all non-local iterations jointly vote for high-confidence hypothetical inliers for SVD-based transformation estimation. We theoretically analyze the robustness of our deterministic inlier searching strategy compared to RANSAC, which also motivates us to design a conservative seed selection mechanism to improve robustness in sparse point clouds.
+
+To summarize, our contributions are as follows:
+
+- We propose a novel variational non-local network for outlier rejection, learning discriminative correspondence features with Bayesian-driven long-range contextual dependencies.
+- We customize the probabilistic graphical model over our variational non-local network and derive the variational low bound for effective model optimization.
+- We introduce a Wilson score-based voting mechanism to search high-quality hypothetical inliers, and theoretically demonstrate its superiority over RANSAC.
+
+Our experimental results on extensive benchmark datasets demonstrate that our framework outperforms the state-of-the-art registration methods.
+
+# 2. Related Work
+
+End-to-end Registration Methods. With the advances of deep learning in the 3D vision field [34], the learning-based end-to-end registration model has achieved increasing research attention. DCP [42] uses the feature similarity to establish pseudo correspondences for SVD-based transformation estimation. RPM-Net [47] exploits the Sinkhorn
+
+layer and annealing for discriminative matching map generation. [22, 23] integrate the cross-entropy method into the deep model for robust registration. RIENet [40] uses the structure difference between the source neighborhood and the pseudo-target one for inlier confidence evaluation. With the powerful feature representation of Transformer, RegTR [48] effectively aligns large-scale indoor scenes in an end-to-end manner. [13] propose a match-normalization layer for robust registration in the real-world 6D object pose estimation task. More end-to-end models such as [10, 18, 28–30, 33, 53] also present impressive precisions.
+
+Learning-based Feature Descriptors. To align the complex scenes, a popular pipeline is to exploit feature descriptors for 3D matching. Compared to hand-crafted descriptors such as [17, 37, 38], the deep feature descriptor presents superior registration precision and has achieved much more attention in recent years. The pioneering 3DMatch [50] exploits the Siamese 3D CNN to learn the local geometric feature via contrastive loss. FCGF [11] exploits a fully convolutional network for dense feature extraction in a one-shot fashion. Furthermore, D3feat [3] jointly learns the dense feature descriptor and the detection score for each point. By integrating the overlap-attention module into D3feat, Predictor [21] largely improves the registration reliability in low-overlapping point clouds. YOHO [41] utilizes the group equivariant feature learning to achieve the rotation invariance and shows great robustness to the point density and the noise interference. [35] develops a geometric transformer to learn the geometric context for robust super-point matching. Lepard [31] embeds the relative 3D positional encoding into the transformer for discriminative descriptor learning.
+
+Outlier Rejection Methods. Despite significant progress in learning-based feature descriptor, generating mismatched correspondences (outliers) in some challenging scenes remains unavoidable. Traditional outlier filtering methods, such as RANSAC [16] and its variants [4, 24, 27], use repeated sampling and verification for outlier rejection. However, these methods tend to have a high time cost, particularly in scenes with a high outlier ratio. Instead, FGR [52] and TEASER [45] integrate the robust loss function into the optimization objective to weaken the interference from outliers. Recently, Chen et al. [9] developed second-order spatial compatibility for robust consensus sampling. With the rise of deep 3D vision, most learnable outlier rejection models [10, 33] formulate outlier rejection as a binary classification task and reject correspondences with low confidence. Yi et al. [49] proposed a context normalization-embedded deep network for inlier evaluation, while Brachmann et al. [6] enhanced classical RANSAC with neural-guided prior confidence. As our baseline, PointDSC [2] proposes exploiting a spatial consistency-guided non-local inlier classifier for inlier evaluation, followed by neural spectral matching for robust registration. However, under high outlier
+
+ratios, spatial consistency can be ambiguous (as shown in Fig. 1), misleading non-local feature aggregation. Instead, we propose exploiting Bayesian-driven long-range dependencies for discriminative non-local feature learning.
+
+# 3. Approach
+
+# 3.1. Background
+
+Problem Setting. In the pairwise 3D registration task, given a source point cloud $\mathbf{X} = \{\mathbf{x}_i\in \mathbb{R}^3\mid i = 1,\dots,|\mathbf{X}|\}$ and a target point cloud $\mathbf{Y} = \{\mathbf{y}_j\in \mathbb{R}^3\mid j = 1,\dots,|\mathbf{Y}|\}$ , we aim to find their optimal rigid transformation consisting of a rotation matrix $\mathbf{R}^{*}\in SO(3)$ and a translation vector $\mathbf{t}^{*}\in \mathbb{R}^{3}$ to align their overlapping region precisely. In this work, we focus on the descriptor-based pipeline for large-scale scene registration. Based on the feature-level nearest neighbor, we construct a set of putative correspondences $\mathcal{C} = \left\{\mathbf{c}_i = (\mathbf{x}_i,\mathbf{y}_i)\in \mathbb{R}^6\mid i = 1,\dots,|\mathcal{C}|\right\}$ . The inlier (correctly matched correspondence) is defined as the correspondence satisfying $\| \mathbf{R}^{*}\mathbf{x}_{i} + \mathbf{t}^{*} - \mathbf{y}_{i}\| < \varepsilon$ , where $\varepsilon$ indicates the inlier threshold.
+
+Vanilla Non-local Feature Embedding. Given the putative correspondence set $\mathcal{C}$ , [2] leverages the spatial consistency-guided non-local network (SCNonlocal) for their feature embedding. The injected geometric compatibility matrix can effectively regularize the long-range dependencies for discriminative inlier/outlier feature learning. In detail, it contains $L$ iterations and the feature aggregation in $l$ -th iteration can be formulated as:
+
+$$
+\mathbf {F} _ {i} ^ {l + 1} = \mathbf {F} _ {i} ^ {l} + \operatorname {M L P} \left(\sum_ {j = 1} ^ {| \mathcal {C} |} \operatorname {s o f t m a x} _ {j} \left(\boldsymbol {\alpha} ^ {l} \boldsymbol {\beta}\right) \mathbf {V} _ {j} ^ {l}\right), \tag {1}
+$$
+
+where $\mathbf{F}_i^l\in \mathbb{R}^d$ indicates the feature embedding of correspondence $\mathbf{c}_i$ in $l$ -th iteration (the initial feature $\mathbf{F}_i^0$ is obtained via linear projection on $\mathbf{c}_i$ ) and $\mathbf{V}_i^l = f_v^l (\mathbf{F}_i^l)\in \mathbb{R}^d$ is the projected value feature. $\alpha^l\in \mathbb{R}^{|\mathcal{C}|\times |\mathcal{C}|}$ is the non-local attention map whose entry $\alpha_{i,j}^{l}$ reflects the feature similarity between the projected query feature $\mathbf{Q}_i^l = f_q^l (\mathbf{F}_i^l)\in \mathbb{R}^d$ and the key feature $\mathbf{K}_i^l = f_k^l (\mathbf{F}_j^l)\in \mathbb{R}^d$ . $\beta \in \mathbb{R}^{|\mathcal{C}|\times |\mathcal{C}|}$ represents the geometric compatibility matrix of correspondences, where the compatibility between $\mathbf{c}_i$ and $\mathbf{c}_j$ is:
+
+$$
+\boldsymbol {\beta} _ {i, j} = \max \left(0, 1 - \frac {d _ {i j} ^ {2}}{\varepsilon^ {2}}\right), d _ {i j} = \left\| \mathbf {x} _ {i} - \mathbf {x} _ {j} \right\| - \left\| \mathbf {y} _ {i} - \mathbf {y} _ {j} \right\| |. \tag {2}
+$$
+
+Based on the fact that the geometric distance $d_{i,j}$ of inliers $\mathbf{c}_i$ and $\mathbf{c}_j$ tend to be minor, Eq. 2 will assign a high compatibility value on the inlier pair, thereby promoting the nonlocal network to effectively cluster the inlier features for discriminative inlier/outlier feature learning.
+
+# 3.2. Variational Non-local Feature Embedding
+
+While effective, SCNonlocal still suffers from ambiguous long-range dependencies, especially in some challeng-
+
+
+Figure 1. The ratio of inlier-outlier pairs with positive compatibilities in 3DLoMatch [21] using FCGF and Predator descriptors.
+
+ing scenes (e.g., the low-overlapping case). Two essential reasons are: (i) Wrong geometric compatibility. As shown in Fig. 1, for 3DLoMatch dataset with Predator and FCGF descriptors, almost $30\%$ and $17\%$ of inlier-outlier pairs own the positive compatibility values, respectively, which potentially misleads the attention weight for wrong feature clustering. (ii) Lack of uncertainty modeling. In symmetric or repetitive scenes, the inlier/outlier prediction contains significant uncertainty. Therefore, it's necessary to design a robust feature representation to capture such uncertainty.
+
+To overcome them, we develop a variational non-local network, a probabilistic feature learning framework, for discriminative correspondence embedding. Our core idea is to inject random features into our model to capture the ambiguous nature of long-range dependencies, and then leverage the variational Bayesian inference to model the Bayesian-driven long-range dependencies for discriminative feature aggregation. Specifically, we first introduce the random feature variables $z_{k,i}^{l}, z_{q,i}^{l}$ and $z_{v,i}^{l}$ into the key $\mathbf{K}_i^l$ , query $\mathbf{Q}_i^l$ and value $\mathbf{V}_i^l$ components in our non-local module to capture their potential uncertainty in the long-range dependency modeling. Then, the prior/posterior encoders are constructed to predict their prior feature distribution and the posterior one, respectively. Embedded with the inlier/outlier label, the posterior feature distribution is label-dependent and discriminative. Thus, by pushing the prior close to the discriminative posterior in the training phase, this prior at test time also tends to sample discriminative query, key, and value features for high-quality long-range dependency modeling.
+
+Probabilistic Graphical Model over Variational Nonlocal Network. To achieve effective variational Bayesian inference, we need to first characterize the conditional dependencies of the injected random features so that the variational lower bound can be derived as the optimization objective for model training. As shown in Fig. 2, we customize the probabilistic graphical model over our non-local network to clarify the dependencies of random features (the circles). The solid line denotes the label prediction process, while the dashed line represents our label-dependent posterior encoder (i.e., inference model). Notably, the deterministic hidden query/key/value features $\mathbf{h}_{k,i}^{l}\in \mathbb{R}^{d^{\prime}}$
+
+$\mathbf{h}_{q,i}^{l}\in \mathbb{R}^{d^{\prime}}$ , and $\mathbf{h}_{v,i}^{l}\in \mathbb{R}^{d^{\prime}}$ are also introduced to summarize the historical information for better feature updating in each iteration.
+
+Inlier/outlier Prediction Process. Based on the defined conditional dependencies in Fig. 2, the prediction process of correspondence labels $\mathbf{b} = \{b_1, b_2, \dots, b_{|\mathcal{C}|} \mid b_i \in \{0,1\}\}$ (1 indicates inlier and 0 outlier) is formulated as follows. Beginning with the initial linear projection $\tilde{\mathbf{F}}^0 \in \mathbb{R}^{|\mathcal{C}| \times d}$ of correspondences $\mathcal{C} = \{\mathbf{c}_i\}$ , we iteratively perform the probabilistic non-local aggregation for feature updating. In the $l$ -th iteration, we first employ a Gated Recurrent Unit (GRU) [12] to predict the hidden query/key/value features which summarize the historical query/key/value features (sampled from the prior distributions) and the correspondence features in previous iterations:
+
+$$
+\mathbf {h} _ {q, i} ^ {l} = \operatorname {G R U} _ {q} \left(\mathbf {h} _ {q, i} ^ {l - 1}, \left[ \mathbf {z} _ {q, i} ^ {l - 1}, \tilde {\mathbf {F}} _ {i} ^ {l - 1} \right]\right),
+$$
+
+$$
+\mathbf {h} _ {k, i} ^ {l} = \operatorname {G R U} _ {k} \left(\mathbf {h} _ {k, i} ^ {l - 1}, \left[ \mathbf {z} _ {k, i} ^ {l - 1}, \tilde {\mathbf {F}} _ {i} ^ {l - 1} \right]\right), \tag {3}
+$$
+
+$$
+\mathbf {h} _ {v, i} ^ {l} = \operatorname {G R U} _ {v} \left(\mathbf {h} _ {v, i} ^ {l - 1}, \left[ \mathbf {z} _ {v, i} ^ {l - 1}, \tilde {\mathbf {F}} _ {i} ^ {l - 1} \right]\right),
+$$
+
+where $[\cdot ,\cdot ]$ denotes the feature concatenation and $\tilde{\mathbf{F}}_i^{l - 1}$ is the learned correspondence features of $\mathbf{c}_i$ in iteration $l - 1$ Then, with as input the predicted hidden features, the prior encoder $p_{\theta}(\cdot)$ is utilized to predict the prior feature distribution for query/key/value, respectively. Furthermore, we sample features $\mathbf{z}_{q,i}^{l}\in \mathbb{R}^{\bar{d}}$ $\mathbf{z}_{k,i}^{l}\in \mathbb{R}^{\bar{d}}$ and $\mathbf{z}_{v,i}^{l}\in \mathbb{R}^{\bar{d}}$ from the predicted prior query/key/value distribution and combine them with the hidden features to predict the corresponding query $\tilde{\mathbf{Q}}_i^l\in \mathbb{R}^d$ , key $\tilde{\mathbf{K}}_i^l\in \mathbb{R}^d$ and value $\tilde{\mathbf{V}}_i^l\in \mathbb{R}^d$ through a neural network $f_{\theta}^{q,k,v}:\mathbb{R}^{d' + \tilde{d}}\to \mathbb{R}^d$ ..
+
+$$
+\mathbf {z} _ {q, i} ^ {l} \sim p _ {\theta} (z _ {q, i} ^ {l} \mid \mathbf {h} _ {q, i} ^ {l}), \tilde {\mathbf {Q}} _ {i} ^ {l} = f _ {\theta} ^ {q} ([ \mathbf {z} _ {q, i} ^ {l}, \mathbf {h} _ {q, i} ^ {l} ]),
+$$
+
+$$
+\mathbf {z} _ {k, i} ^ {l} \sim p _ {\theta} \left(z _ {k, i} ^ {l} \mid \mathbf {h} _ {k, i} ^ {l}\right), \quad \tilde {\mathbf {K}} _ {i} ^ {l} = f _ {\theta} ^ {k} \left(\left[ \mathbf {z} _ {k, i} ^ {l}, \mathbf {h} _ {k, i} ^ {l} \right]\right), \tag {4}
+$$
+
+$$
+\mathbf {z} _ {v, i} ^ {l} \sim p _ {\theta} (z _ {v, i} ^ {l} \mid \mathbf {h} _ {v, i} ^ {l}), \tilde {\mathbf {V}} _ {i} ^ {l} = f _ {\theta} ^ {v} ([ \mathbf {z} _ {v, i} ^ {l}, \mathbf {h} _ {v, i} ^ {l} ]),
+$$
+
+where the prior feature distribution is the Gaussian distribution with the mean and the standard deviation parameterized by a neural network. Finally, with the learned $\tilde{\mathbf{Q}}_i^l$ , $\tilde{\mathbf{K}}_i^l$ and $\tilde{\mathbf{V}}_i^l$ , the correspondence feature $\tilde{\mathbf{F}}_i^l$ in $l$ -th iteration can be aggregated with the same non-local operation in Eq. 1. After $L$ feature iterations, we feed the correspondence feature $\tilde{\mathbf{F}}_i^L$ in the last iteration into a label prediction model $y_{\theta}$ to predict the inlier/outlier labels $b_i \sim y_{\theta}(b_i \mid \tilde{\mathbf{F}}_i^L)$ , where the label prediction model outputs a scalar Gaussian distribution with the mean parameterized by the neural network and the unit variance.
+
+Variational Posterior Encoder. Due to the nonlinearity of our variational non-local model, we cannot directly derive the precise posterior distribution for random query/key/value features using the standard Bayes' theorem. Taking inspiration from the Variational Bayesian inference, we construct a label-dependent posterior encoder
+
+
+Figure 2. Probabilistic graphical model for our variational nonlocal network. For simplicity, we just demonstrate two iterations. The white circles indicate the random features and the white squares denote the deterministic hidden features. The solid line represents the inlier/outlier prediction process and the dashed line denotes the label-dependent variational posterior encoder. We just show the variational posterior for $z_{k}^{1}$ .
+
+$q_{\phi}(\cdot)$ to approximate the feature posterior:
+
+$$
+\mathbf {z} _ {q, i} ^ {l} \sim q _ {\phi} \left(z _ {q, i} ^ {l} \mid [ \mathbf {h} _ {q, i} ^ {l}, [ b _ {i} ] _ {\times k} ]\right)
+$$
+
+$$
+\mathbf {z} _ {k, i} ^ {l} \sim q _ {\phi} \left(z _ {k, i} ^ {l} \mid \left[ \mathbf {h} _ {k, i} ^ {l}, [ b _ {i} ] _ {\times k} \right]\right) \tag {5}
+$$
+
+$$
+\mathbf {z} _ {v, i} ^ {l} \sim q _ {\phi} \left(z _ {v, i} ^ {l} \mid [ \mathbf {h} _ {v, i} ^ {l}, [ b _ {i} ] _ {\times k} ]\right),
+$$
+
+where $[b_i]_{\times k}$ indicates a label vector generated by tiling the scalar label $k$ times. The output of each posterior encoder is a diagonal Gaussian distribution with parameterized mean and standard deviation.
+
+Variational Lower Bound. Finally, we derive the optimization objective $\mathrm{ELBO}(\theta ,\phi)$ , the variational (evidence) lower bound of log-likelihood correspondence labels $\ln y_{\theta}(\mathbf{b}\mid \mathcal{C})$ , to train our variational non-local network (Please refer to Appendix A for the detailed derivation):
+
+$$
+\operatorname {E L B O} (\theta , \phi) = \mathbb {E} _ {\prod_ {l = 0} ^ {L - 1} q _ {\phi} (z _ {q, k, v} ^ {l} | \mathbf {h} _ {q, k, v} ^ {l}, \mathbf {b})} \left[ \ln y _ {\theta} (\mathbf {b} \mid \tilde {\mathbf {F}} ^ {L}) \right] -
+$$
+
+$$
+\sum_ {l = 0} ^ {L - 1} \mathbb {E} _ {q _ {\phi}} \left[ \mathrm {D} _ {\mathrm {K L}} \left(q _ {\phi} \left(z _ {q, k, v} ^ {l} \mid \mathbf {h} _ {q, k, v} ^ {l}, \mathbf {b}\right) \mid p _ {\theta} \left(z _ {q, k, v} ^ {l} \mid \mathbf {h} _ {q, k, v} ^ {l}\right)\right) \right] \tag {6}
+$$
+
+where for clarity, we utilize the subscript $q,k,v$ to denote the same operator performed on query/key/value. $\mathrm{D}_{\mathrm{KL}}(\cdot ||\cdot)$ denotes the Kullback-Leibler (KL) divergence between two distributions. By maximizing the variational lower bound above, we can optimize the network parameters to indirectly maximize the log-likelihood value of correspondence labels. Eq. 6 indicates that the discriminative, label-dependent feature posterior explicitly constrains the prior by reducing their KL divergence in the training phase, which promotes the query, key, and value features sampled from the prior to model the high-quality long-term dependencies at test time.
+
+# 3.3. Voting-based Inlier Searching
+
+With the learned correspondence features above, we then propose a voting-based sampling strategy to search the desired inlier subset from the entire putative correspondences
+
+for optimal transformation estimation. Our sampling mechanism is deterministic and efficient. We first select a set of high-confidence seeds $\mathcal{C}_{seed}$ from $\mathcal{C}$ based on their inlier confidence (predicted in § 3.2) and the Non-Maximum Suppression (as performed in [2]), where the number of seeds $|\mathcal{C}_{seed}| = \lfloor |C|*v\rfloor$ ( $v$ is a fixed seed ratio). Then, for each seed, we cluster its most compatible correspondences into it to form the hypothetical inliers. Ideally, if the seed is an inlier and the compatible value is correct (without ambiguity problem as in Fig. 2), its most compatible correspondences are also inliers, thus we can cluster the desired inlier subset successfully. To cluster high-confidence hypothetical inliers, we perform a coarse-to-fine clustering strategy, including the voting-based coarse clustering and the Wilson score-based fine clustering.
+
+Voting-based Coarse Clustering. We view the correspondence features $\{\tilde{\mathbf{F}}^1,\dots,\tilde{\mathbf{F}}^L\}$ learned from all non-local iterations as $L$ voters. For $l$ -th voter, it deterministically clusters $\kappa - 1$ most compatible correspondences $(\kappa - 1)\mathcal{A}_{\mathbf{c}_i}^l$ for each seed $\mathbf{c}_i \in \mathcal{C}_{\text{seed}}$ to from the hypothetical inliers $(\kappa)\mathcal{M}_{\mathbf{c}_i}^l = \{\mathbf{c}_i\} \cup (\kappa - 1)\mathcal{A}_{\mathbf{c}_i}^l$ . To this end, we first compute the compatibility matrix $\mathbf{S}^l \in \mathbb{R}^{|\mathcal{C}| \times |\mathcal{C}|}$ for $l$ -th voter, where each entry $\mathbf{S}_{i,j}^l = \mathrm{clip}(1 - (1 - \cos(\tilde{\mathbf{F}}_i^l, \tilde{\mathbf{F}}_j^l) / \sigma^2), 0, 1) \cdot \beta_{i,j}$ , where the parameter $\sigma$ is to control the feature compatibility and geometric compatibility $\beta_{i,j}$ is defined in Eq. 2. Thus, $(\kappa - 1)\mathcal{A}_{\mathbf{c}_i}^l$ can be determined as $\{\mathbf{c}_j | \mathbf{S}_{i,j}^l \geq \gamma_{\kappa - 1}\}$ , where threshold $\gamma_{\kappa - 1}$ is the $(\kappa - 1)$ -th largest compatibility. Benefiting from our discriminative feature representations, $\mathbf{S}^l$ can effectively handle the ambiguity problem in Fig. 2 and promote robust inlier clustering. After the voting process above, each seed $\mathbf{c}_i$ can achieve $L$ candidate hypothetical inliers $\left\{\left(\kappa\right)\mathcal{M}_{\mathbf{c}_i}^1, \dots, \left(\kappa\right)\mathcal{M}_{\mathbf{c}_i}^L\right\}$ .
+
+Wilson score-based Fine Clustering. Based on the voted hypothetical inliers above, we then fuse these voted results with Wilson score [43] to form the high-quality hypothetical inliers $(^{\kappa})\tilde{\mathcal{M}}_{\mathbf{c}_i}$ . In detail, we denote $\mathcal{T}_{i,j}^{l} = \mathbb{I}\{\mathbf{c}_{j}\in$ $(^{\kappa})\mathcal{M}_{\mathbf{c}_{i}}^{l}\}$ to indicate whether $(^{\kappa})\mathcal{M}_{\mathbf{c}_{i}}^{l}$ contains $\mathbf{c}_j$ . The Wilson score $\mathbf{W}_{i,j}^{n}$ of accepting $\mathbf{c}_j$ into $(^{\kappa})\tilde{\mathcal{M}}_{\mathbf{c}_i}$ can be computed as:
+
+$$
+\mathbf {W} _ {i, j} ^ {n} = \frac {1}{1 + \frac {z ^ {2}}{n}} \left[ \hat {p} _ {i, j} ^ {(n)} + \frac {z ^ {2}}{2 n} - z \sqrt {\frac {\hat {p} _ {i , j} ^ {(n)} \left(1 - \hat {p} _ {i , j} ^ {(n)}\right)}{n} + \frac {z ^ {2}}{4 n ^ {2}}} \right], \tag {7}
+$$
+
+where $\hat{p}_{i,j}^{(n)} = \frac{1}{n}\sum_{\tau = 1}^{n}\mathcal{I}_{i,j}^{\tau}$ is the average acceptance ratio of top $n$ voters and $z = 1.96$ (i.e., z-score at $95\%$ confidence level). Eq. 7 indicates that the Wilson score not only considers the sample mean but also the confidence (positively related to sample number $n$ ). Finally, among the set of Wilson scores under different sample numbers $\{\mathbf{W}_{i,j}^{1},\dots,\mathbf{W}_{i,j}^{L}\}$ , we choose the largest one as the final Wilson score $\bar{\mathbf{W}}_{i,j} = \max_{n\in \{1,\dots,L\}}\mathbf{W}_{i,j}^{n}$ for $\mathbf{c}_j$ . Thus, the final hypothetical inliers set of $\mathbf{c}_i$ can be determined as
+
+$(^{\kappa})\tilde{\mathcal{M}}_{\mathbf{c}_i} = \{\mathbf{c}_i\} \cup \{\mathbf{c}_j\mid \tilde{\mathbf{W}}_{i,j}\geq \gamma_{\kappa -1}^\prime \}$ , where $\gamma_{\kappa -1}^{\prime}$ is the $(\kappa -1)$ -th largest Wilson score.
+
+Theoretical Analysis and Conservative Seed Selection. Finally, we try to theoretically analyze our deterministic inlier searching mechanism compared to RANSAC [16] and further propose a simple but effective conservative seed selection strategy for more robust 3D registration in sparse point clouds. We let $\{^{(\kappa)}\mathcal{M}_i^{sac}\}_{i=1}^J$ be the randomly sampled hypothetical inlier subset in RANSAC, and let $C_{in}, C_{out}$ and $p_{in}$ be the inlier subset, outlier subset and the inlier ratio, respectively, $\mathcal{C} = C_{in} \cup C_{out}$ , $p_{in} = |C_{in}| / |\mathcal{C}|$ . We also denote the inliers in seeds as $\tilde{C}_{in} = C_{in} \cap C_{seed}$ . Then, we can derive the following theorem (Please refer to Appendix B for the derivation process).
+
+Theorem 1 Assume the number of outliers in ${}^{(\kappa)}\tilde{\mathcal{M}}_{\mathbf{c}_i}$ ( $\mathbf{c}_i \in \tilde{\mathcal{C}}_{in}$ ) follows a Poisson distribution $Pois(\alpha \cdot \kappa)$ . Then, if $\alpha < -\frac{1}{\kappa} \cdot \log \left[1 - (1 - p_{in}^{\kappa})^{J / |\tilde{\mathcal{C}}_{in}|}\right] \triangleq \mathcal{U}$ , the probability of our method achieving the inlier subset is greater than or equal to that of RANSAC.
+
+$$
+P \left(\max _ {\mathbf {c} _ {i} \in \mathcal {C} _ {s e e d}} | ^ {(\kappa)} \tilde {\mathcal {M}} _ {\mathbf {c} _ {i}} \cap \mathcal {C} _ {i n} | = \kappa\right) \geq \tag {8}
+$$
+
+$$
+P \Big (\max _ {1 \leq i \leq J} | ^ {(\kappa)} \mathcal {M} _ {i} ^ {s a c} \cap \mathcal {C} _ {i n} | = \kappa \Big).
+$$
+
+Theorem 1 shows that as the inlier ratio $p_{in}$ degrades, the upper bound $\mathcal{U}$ will increase, indicating our method tends to be more likely to achieve the inlier subset than RANSAC in high outlier cases. Therefore, our method tends to be more robust to outliers than RANSAC. In addition, Theorem 1 also shows that the more inliers $|\tilde{\mathcal{C}}_{in}|$ in $\mathcal{C}_{\text{seed}}$ , the better for our method. It means selecting all correspondences in $\mathcal{C}$ as seeds theoretically seems the best choice since it can avoid missing any inlier. However, in real implementation, we have to perform inlier selection to accelerate the registration speed. Nevertheless, in sparse point clouds, the number of correspondences is small. If we still perform the original inlier selection, fewer seeds even may not contain any inlier, leading to registration failure. To overcome it, we develop a conservative seed selection strategy, which changes the seed number to $\max \{\lfloor v \cdot |\mathcal{C}| \rfloor, n\} (n < |\mathcal{C}|)$ , where the lower bar $n$ of the seed number can effectively avoid selecting too fewer inliers in sparse cases and in dense cases, it would degrade to the original selection strategy.
+
+# 3.4. Rigid Transformation Estimation
+
+With the voted hypothetical inliers $\{\mathbf{\Psi}^{(\kappa)}\tilde{\mathcal{M}}_{\mathbf{c}_i}\mid \mathbf{c}_i\in$ $\mathcal{C}_{seed}\}$ above, we estimate the optimal transformation parameter based on the Procrustes method [20] to minimize the least-squares errors for each hypothetical inlier group:
+
+$$
+\hat {\mathbf {R}} _ {i}, \hat {\mathbf {t}} _ {i} = \arg \min _ {\mathbf {R}, \mathbf {t}} \sum_ {\mathbf {c} _ {j} \in {} ^ {(\kappa)} \tilde {\mathcal {M}} _ {\mathbf {c} _ {i}}} \omega_ {j} \cdot \left\| \mathbf {R} ^ {\top} \mathbf {x} _ {j} + \mathbf {t} - \mathbf {y} _ {j} \right\| _ {2}, \tag {9}
+$$
+
+where $\omega_{j}$ is the error weight computed by the neural spectral matching as in [2]. Then, we select the transformation
+
+parameter that maximizes the number of overlapped correspondences as the final optimal transformation estimation:
+
+$$
+\hat {\mathbf {R}} ^ {*}, \hat {\mathbf {t}} ^ {*} = \underset {\{\hat {\mathbf {R}} _ {i}, \hat {\mathbf {t}} _ {i} \} _ {i = 1} ^ {| C |}} {\arg \max } \sum_ {\mathbf {c} _ {j} \in \mathcal {C}} v _ {i} \cdot \mathbb {I} \left\{\| \hat {\mathbf {R}} _ {i} ^ {\top} \mathbf {x} _ {j} + \hat {\mathbf {t}} _ {i} - \mathbf {y} _ {j} \| _ {2} < \varepsilon \right\}, \tag {10}
+$$
+
+where $v_{i} = 1 - \|\hat{\mathbf{R}}_{i}^{\top}\mathbf{x}_{j} + \hat{\mathbf{t}}_{i} - \mathbf{y}_{j}\|_{2}^{2} / \varepsilon^{2}$ is used to re-weight the inlier count as performed in [23]. Finally, we refine it using all recovered inliers in a least-squares optimization as a common practice in [2,4].
+
+# 4. Experiments
+
+# 4.1. Experimental Settings
+
+Implementation Details. For our variational non-local module, the number of iterations $L$ is 12, and the dimensions of the correspondence feature, random feature, and hidden feature are set to 128, 128, and 256, respectively. For our voting-based inlier sampling module, the size of hypothetical inliers $\kappa$ is 40. For seed selection, seed ratio $v$ is 0.1 and the lower bar of seed number $n$ is 1000. Our model is trained with 50 epochs using Adam optimizer with learning rate $10^{-4}$ and weight decay $10^{-6}$ . We utilize PyTorch to implement our project and perform all experiments on the server equipped with an Intel i5 2.2 GHz CPU and one Tesla V100 GPU. For simplicity, we name our Variational Bayesian-based Registration framework as VBReg.
+
+Evaluation Metric. We use three metrics to evaluate our method, including (1) Registration Recall $(RR)$ , the percent of the successful registration satisfying the error thresholds of rotation and translation at the same time, (2) average Rotation Error $(RE)$ and (3) average Translation Error $(TE)$ :
+
+$$
+\operatorname {R E} (\hat {\mathbf {R}}) = \arccos \frac {\operatorname {T r} \left(\hat {\mathbf {R}} ^ {\top} \mathbf {R} ^ {*}\right) - 1}{2}, \operatorname {T E} (\hat {\mathbf {t}}) = \left\| \hat {\mathbf {t}} - \mathbf {t} ^ {*} \right\| _ {2} ^ {2}, \tag {11}
+$$
+
+where $\hat{\mathbf{R}}$ and $\hat{\mathbf{t}}$ are the predicted rotation matrix and rotation vector, respectively, while $\mathbf{R}^*$ and $\mathbf{t}^*$ are the corresponding ground truth. The average $RE$ and $TE$ are computed only on successful aligned point cloud pairs.
+
+# 4.2. Comparison with Existing Methods
+
+Evaluation on 3DMatch. We first evaluate our method on 3DMatch benchmark [50], which contains 46 training scenes, 8 validation scenes, and 8 test scenes. We first voxelize and down-sample the point cloud with $5\mathrm{cm}$ voxel size and then leverage FCGF [11] and FPFH [37] descriptors to construct the putative correspondences based on the feature nearest neighbor. We compare our method with eight state-of-the-art (SOTA) correspondence-based methods, where FGR [52], SM [26], RANSAC (50k) [16], TEASER++ [45], and SC2_PCR [9] are representative traditional methods, while DGR [10], DHVR [25], and PointDSC [2] are advanced deep learning-based methods. As shown in Table 1,
+
+
+Figure 3. Registration visualization on 3DLoMatch [19].
+
+in the FCGF setting, our method achieves the best performance in $RR$ and $RE$ criteria while the same $TE$ with PointDSC. We need to highlight that $RR$ is a more important criterion than $RE$ and $TE$ since the rotation and translation errors are just calculated in a successful registration. It means that the higher $RR$ may include more challenging but successful registration cases, potentially increasing their errors. In the FPFH setting, it can be observed that our method can still achieve the best $RR$ score among all deep methods, but perform slightly worse than SC2_PCR. Notably, compared to PointDSC (our baseline), the precision gain on $RR$ is impressive $(5.24\% \uparrow)$ , which benefits from the effectiveness of our variational non-local feature learning and the voting-based inlier searching.
+
+Evaluation on 3DLoMatch. We further test our method on 3DLoMatch benchmark [21]. Compared to 3DMatch sharing more than $30\%$ overlap, the overlaps of point cloud pairs in 3DLoMatch just lie in $10\% \sim 30\%$ , thus presenting much more challenges. We leverage FCGF [11] and recently popular Predator [21] as the feature descriptors for putative correspondence generation. We choose six traditional methods: FGR [52], SM [26], RANSAC (50k) [16], TEASER++ [45], SC2_PCR [9], and TR_DE [8], and two deep methods: DHVR [25], and PointDSC [2] for comparison. The registration recalls $(RR)$ under different numbers of correspondences are listed in Table 2. It can be observed that regardless of FCGF or Predator descriptor, our method almost achieves the best performance on all settings, except for FCGF setting with 250 points. Notably, in more challenging cases, the performance advantage over PointDSC is further expanded $(+9.5\%)$ and $+7.8\%$ in the cases of FCGF with 500 and 250 points. These experimental results further demonstrate the outstanding robustness of our method when encountering those extremely low-overlapping cases (high outlier ratios). Some visualization results are listed in Fig. 3 and the $RR$ changes under different inlier ratios are presented in Fig. 4 (c), which suggests that our performance gains are mainly brought by model robustness in extremely high outlier situations.
+
+Evaluation on KITTI. Finally, we evaluate our method on the outdoor LIDAR-scanned driving scenarios from KITTI dataset [19]. In line with [11], we utilize sequences 0-5, 6-7, and 8-10 as the training set, validation set, and test set, respectively. Also, as the setting in [3, 11], we further refine the ground-truth rigid transformations using ICP [5] and
+
+| Model | 3DMatch | 3DLoMatch (FCGF) | 3DLoMatch (Predator) | Sec. |
| FCGF | FPFH | 5000 | 2500 | 1000 | 500 | 250 | 5000 | 2500 | 1000 | 500 | 250 |
| PointDSC w/ SCNonlocalxyz | 92.42 | 77.51 | 55.8 | 52.6 | 46.8 | 37.7 | 26.7 | 68.1 | 67.3 | 66.5 | 63.4 | 60.5 | 0.11 |
| PointDSC w/ VBNonlocalxyz | 93.04 | 80.16 | 57.7 | 55.6 | 50.2 | 39.9 | 26.1 | 69.7 | 69.6 | 67.9 | 64.9 | 61.9 | 0.17 |
| PointDSC w/ SCNonlocalfeat | 92.36 | 77.76 | 54.6 | 50.6 | 44.9 | 36.8 | 25.4 | 69.2 | 68.6 | 67.9 | 63.5 | 59.9 | 0.13 |
| PointDSC w/ VBNonlocalfeat | 93.04 | 80.53 | 56.9 | 56.5 | 50.9 | 42.2 | 28.9 | 69.2 | 68.7 | 68.0 | 64.6 | 60.6 | 0.18 |
| PointDSC w/ SCNonlocalcls | 92.98 | 78.99 | 54.1 | 52.2 | 46.0 | 38.7 | 27.7 | 67.6 | 66.9 | 67.2 | 63.7 | 60.2 | 0.11 |
| PointDSC w/ VBNonlocalfeat+Vote | 93.41 | 81.21 | 58.3 | 56.5 | 51.9 | 44.7 | 31.1 | 69.3 | 69.5 | 68.2 | 65.3 | 61.2 | 0.20 |
| PointDSC w/ VBNonlocalfeat+Vote+CS | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 68.7 | 66.4 | 63.0 | 0.22 |
| Iteration times L = 6 | 93.41 | 82.32 | 58.1 | 57.1 | 52.9 | 48.3 | 34.8 | 69.7 | 69.7 | 68.7 | 66.3 | 63.8 | 0.19 |
| Iteration times L = 9 | 93.41 | 81.58 | 58.2 | 57.0 | 53.2 | 47.6 | 32.5 | 70.0 | 69.4 | 68.5 | 66.9 | 63.2 | 0.20 |
| Iteration times L = 12* | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 68.7 | 66.4 | 63.0 | 0.22 |
| Random feat. dim. d̅ = 32 | 93.41 | 82.38 | 58.0 | 57.5 | 53.5 | 48.1 | 34.8 | 69.7 | 70.0 | 68.6 | 66.3 | 63.3 | 0.20 |
| Random feat. dim. d̅ = 64 | 93.41 | 81.45 | 57.9 | 56.9 | 52.8 | 48.6 | 35.0 | 69.6 | 69.7 | 68.4 | 66.5 | 62.3 | 0.21 |
| Random feat. dim. d̅ = 128* | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 69.3 | 66.4 | 63.3 | 0.22 |
+
+Table 3. Ablation studies on 3DMatch [50] and 3DLoMatch [21] datasets. SCNonlocal: Spatial consistency-guided non-local network; VBNonlocal: Variational Bayesian-based non-local network; Vote: Voting-based inlier searching; CS: Conservative seed selection.
+
+eling can effectively learn the discriminative inlier/outlier features for reliable inlier search. (2): Then, to further highlight the superiority of our variational inference-guided feature learning, we also try to add classification loss on the features produced by each iteration in SCNonlocal to guide their learning (denoted as $SCNonlocal^{cls}$ ). As shown in the fifth row in Table 3, such loss-based label-propagation way just can achieve very limited performance gain and even sometimes degrades score. It demonstrates the superiority of our label-dependent posterior guidance for (prior) feature learning. Also, owing to such posterior guidance, the training curves in Fig. 4 (d) show that our method can achieve significantly faster convergence speed than SCNonlocal.
+
+Discriminative Feature Learning? In order to verify whether VBNonlocal can learn more discriminative features than SCNonlocal, we visualize the distribution of the feature similarity of inliers on 3DMatch (Fig. 4 (a)) and 3DLoMatch (Fig. 4 (b)). As we can see, our learned inlier features own much higher similarities (approximate to 1) than SCNonlocal on both datasets, which demonstrates that our proposed Bayesian-inspired non-local network truly can promote more discriminative correspondence-feature learning.
+
+Variational Non-local Setting. We further test the performance changes under different VBNonlocal settings. (1) We first test the model robustness under different numbers of non-local iterations. The second block in Table 3 verifies that our method is robust to the iteration time and tends to consistently achieve outstanding $RR$ score. (2) Then, the bottom block in Table 3 further shows our model stability to different dimensions of random features.
+
+Voting-based Inlier Searching. Furthermore, we evaluate the performance contribution of the proposed voting-based inlier searching strategy (Vote). As we can see in the sixth row of Table 3, voting strategy can consistently achieve performance improvement regardless in 3DMatch or in more
+
+challenging 3DLoMatch. It mainly benefits from the high-quality hypothetical inliers sampled by our voting policy.
+
+Conservative Seed Selection. Finally, we test the effectiveness of the proposed conservative seed selection strategy (CS) motivated by Theorem 1. As we can see in the seventh row of Table 3, CS can achieve consistent performance gain in each setting. Especially, in the cases of fewer points (e.g., FCGF setting with 250 and 500 points), the improvement is much more significant $(+3.4\%, +2.5\%)$ . As the analysis before, in sparse point clouds, the original inlier selection strategy like in [2] is aggressive and prone to miss too many inlier seeds. Instead, our conservative selection strategy can effectively mitigate it as well as keep registration efficiency.
+
+# 5. Conclusion
+
+In this paper, we adapted the variational Bayesian inference into the non-local network and developed the effective Bayesian-guided long-term dependencies for discriminative correspondence-feature learning. To achieve effective variational inference, a probabilistic graphical model was customized over our non-local network, and the variational low bound was derived as the optimization objective for model training. In addition, we proposed a Wilson score-based voting mechanism for high-quality inlier sampling and theoretically verified its superiority over RANSAC. Extensive experiments on indoor/outdoor datasets demonstrated its promising performance.
+
+# 6. Acknowledgments
+
+This work was supported by the National Science Fund of China (Grant Nos. U1713208, U62276144).
+
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+ "content": "Learning-based outlier (mismatched correspondence) rejection for robust 3D registration generally formulates the outlier removal as an inlier/outlier classification problem. The core for this to be successful is to learn the discriminative inlier/outlier feature representations. In this paper, we develop a novel variational non-local network-based outlier rejection framework for robust alignment. By reformulating the non-local feature learning with variational Bayesian inference, the Bayesian-driven long-range dependencies can be modeled to aggregate discriminative geometric context information for inlier/outlier distinction. Specifically, to achieve such Bayesian-driven contextual dependencies, each query/value component in our nonlocal network predicts a prior feature distribution and a posterior one. Embedded with the inlier/outlier label, the posterior feature distribution is label-dependent and discriminative. Thus, pushing the prior to be close to the discriminative posterior in the training step enables the features sampled from this prior at test time to model high-quality long-range dependencies. Notably, to achieve effective posterior feature guidance, a specific probabilistic graphical model is designed over our non-local model, which lets us derive a variational low bound as our optimization objective for model training. Finally, we propose a voting-based inlier searching strategy to cluster the high-quality hypothetical inliers for transformation estimation. Extensive experiments on 3DMatch, 3DLoMatch, and KITTI datasets verify the effectiveness of our method. Code is available at https://github.com/Jiang-HB/VBReg."
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+ "content": "ing scenes (e.g., the low-overlapping case). Two essential reasons are: (i) Wrong geometric compatibility. As shown in Fig. 1, for 3DLoMatch dataset with Predator and FCGF descriptors, almost "
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+ "content": " of inlier-outlier pairs own the positive compatibility values, respectively, which potentially misleads the attention weight for wrong feature clustering. (ii) Lack of uncertainty modeling. In symmetric or repetitive scenes, the inlier/outlier prediction contains significant uncertainty. Therefore, it's necessary to design a robust feature representation to capture such uncertainty."
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+ "content": "To overcome them, we develop a variational non-local network, a probabilistic feature learning framework, for discriminative correspondence embedding. Our core idea is to inject random features into our model to capture the ambiguous nature of long-range dependencies, and then leverage the variational Bayesian inference to model the Bayesian-driven long-range dependencies for discriminative feature aggregation. Specifically, we first introduce the random feature variables "
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+ "type": "text",
+ "content": " components in our non-local module to capture their potential uncertainty in the long-range dependency modeling. Then, the prior/posterior encoders are constructed to predict their prior feature distribution and the posterior one, respectively. Embedded with the inlier/outlier label, the posterior feature distribution is label-dependent and discriminative. Thus, by pushing the prior close to the discriminative posterior in the training phase, this prior at test time also tends to sample discriminative query, key, and value features for high-quality long-range dependency modeling."
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+ "content": "Probabilistic Graphical Model over Variational Nonlocal Network. To achieve effective variational Bayesian inference, we need to first characterize the conditional dependencies of the injected random features so that the variational lower bound can be derived as the optimization objective for model training. As shown in Fig. 2, we customize the probabilistic graphical model over our non-local network to clarify the dependencies of random features (the circles). The solid line denotes the label prediction process, while the dashed line represents our label-dependent posterior encoder (i.e., inference model). Notably, the deterministic hidden query/key/value features "
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+ "content": " times. The output of each posterior encoder is a diagonal Gaussian distribution with parameterized mean and standard deviation."
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+ "content": "\\operatorname {E L B O} (\\theta , \\phi) = \\mathbb {E} _ {\\prod_ {l = 0} ^ {L - 1} q _ {\\phi} (z _ {q, k, v} ^ {l} | \\mathbf {h} _ {q, k, v} ^ {l}, \\mathbf {b})} \\left[ \\ln y _ {\\theta} (\\mathbf {b} \\mid \\tilde {\\mathbf {F}} ^ {L}) \\right] -",
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+ "content": "where for clarity, we utilize the subscript "
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+ "content": " denotes the Kullback-Leibler (KL) divergence between two distributions. By maximizing the variational lower bound above, we can optimize the network parameters to indirectly maximize the log-likelihood value of correspondence labels. Eq. 6 indicates that the discriminative, label-dependent feature posterior explicitly constrains the prior by reducing their KL divergence in the training phase, which promotes the query, key, and value features sampled from the prior to model the high-quality long-term dependencies at test time."
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+ "content": "3.3. Voting-based Inlier Searching"
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+ "content": " based on their inlier confidence (predicted in § 3.2) and the Non-Maximum Suppression (as performed in [2]), where the number of seeds "
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+ "content": "|\\mathcal{C}_{seed}| = \\lfloor |C|*v\\rfloor"
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+ "type": "text",
+ "content": " is a fixed seed ratio). Then, for each seed, we cluster its most compatible correspondences into it to form the hypothetical inliers. Ideally, if the seed is an inlier and the compatible value is correct (without ambiguity problem as in Fig. 2), its most compatible correspondences are also inliers, thus we can cluster the desired inlier subset successfully. To cluster high-confidence hypothetical inliers, we perform a coarse-to-fine clustering strategy, including the voting-based coarse clustering and the Wilson score-based fine clustering."
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+ "type": "inline_equation",
+ "content": "\\{\\tilde{\\mathbf{F}}^1,\\dots,\\tilde{\\mathbf{F}}^L\\}"
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+ "content": " learned from all non-local iterations as "
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+ "type": "inline_equation",
+ "content": "(\\kappa - 1)\\mathcal{A}_{\\mathbf{c}_i}^l"
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+ "content": " for each seed "
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+ "content": "\\mathbf{c}_i \\in \\mathcal{C}_{\\text{seed}}"
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+ "type": "inline_equation",
+ "content": "(\\kappa)\\mathcal{M}_{\\mathbf{c}_i}^l = \\{\\mathbf{c}_i\\} \\cup (\\kappa - 1)\\mathcal{A}_{\\mathbf{c}_i}^l"
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+ "type": "inline_equation",
+ "content": "\\mathbf{S}^l \\in \\mathbb{R}^{|\\mathcal{C}| \\times |\\mathcal{C}|}"
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+ "type": "text",
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+ "type": "inline_equation",
+ "content": "\\mathbf{S}_{i,j}^l = \\mathrm{clip}(1 - (1 - \\cos(\\tilde{\\mathbf{F}}_i^l, \\tilde{\\mathbf{F}}_j^l) / \\sigma^2), 0, 1) \\cdot \\beta_{i,j}"
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+ "content": " is defined in Eq. 2. Thus, "
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+ "content": "\\{\\mathbf{c}_j | \\mathbf{S}_{i,j}^l \\geq \\gamma_{\\kappa - 1}\\}"
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+ "type": "text",
+ "content": "-th largest compatibility. Benefiting from our discriminative feature representations, "
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+ "content": "\\mathbf{S}^l"
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+ "type": "text",
+ "content": " can effectively handle the ambiguity problem in Fig. 2 and promote robust inlier clustering. After the voting process above, each seed "
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+ "type": "inline_equation",
+ "content": "\\mathbf{c}_i"
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+ "type": "text",
+ "content": " can achieve "
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+ "content": " candidate hypothetical inliers "
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+ "type": "inline_equation",
+ "content": "\\left\\{\\left(\\kappa\\right)\\mathcal{M}_{\\mathbf{c}_i}^1, \\dots, \\left(\\kappa\\right)\\mathcal{M}_{\\mathbf{c}_i}^L\\right\\}"
+ },
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+ "type": "text",
+ "content": "."
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+ "type": "text",
+ "content": "Wilson score-based Fine Clustering. Based on the voted hypothetical inliers above, we then fuse these voted results with Wilson score [43] to form the high-quality hypothetical inliers "
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+ "content": ". In detail, we denote "
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+ "content": "\\mathcal{T}_{i,j}^{l} = \\mathbb{I}\\{\\mathbf{c}_{j}\\in"
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+ "content": " confidence level). Eq. 7 indicates that the Wilson score not only considers the sample mean but also the confidence (positively related to sample number "
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+ "type": "text",
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+ "content": "(^{\\kappa})\\tilde{\\mathcal{M}}_{\\mathbf{c}_i} = \\{\\mathbf{c}_i\\} \\cup \\{\\mathbf{c}_j\\mid \\tilde{\\mathbf{W}}_{i,j}\\geq \\gamma_{\\kappa -1}^\\prime \\}"
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+ "type": "text",
+ "content": "Theoretical Analysis and Conservative Seed Selection. Finally, we try to theoretically analyze our deterministic inlier searching mechanism compared to RANSAC [16] and further propose a simple but effective conservative seed selection strategy for more robust 3D registration in sparse point clouds. We let "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "\\{^{(\\kappa)}\\mathcal{M}_i^{sac}\\}_{i=1}^J"
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+ "type": "text",
+ "content": " be the randomly sampled hypothetical inlier subset in RANSAC, and let "
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+ "type": "inline_equation",
+ "content": "C_{in}, C_{out}"
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+ "type": "text",
+ "content": " and "
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+ "content": "p_{in}"
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+ "type": "text",
+ "content": " be the inlier subset, outlier subset and the inlier ratio, respectively, "
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+ "type": "inline_equation",
+ "content": "\\mathcal{C} = C_{in} \\cup C_{out}"
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+ "type": "text",
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+ "content": "p_{in} = |C_{in}| / |\\mathcal{C}|"
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+ "type": "text",
+ "content": ". We also denote the inliers in seeds as "
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+ "type": "inline_equation",
+ "content": "\\tilde{C}_{in} = C_{in} \\cap C_{seed}"
+ },
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+ "type": "text",
+ "content": ". Then, we can derive the following theorem (Please refer to Appendix B for the derivation process)."
+ }
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+ }
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+ },
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+ "type": "text",
+ "content": "Theorem 1 Assume the number of outliers in "
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+ "type": "inline_equation",
+ "content": "{}^{(\\kappa)}\\tilde{\\mathcal{M}}_{\\mathbf{c}_i}"
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+ "type": "text",
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+ "content": "\\mathbf{c}_i \\in \\tilde{\\mathcal{C}}_{in}"
+ },
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+ "type": "text",
+ "content": ") follows a Poisson distribution "
+ },
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+ "type": "inline_equation",
+ "content": "Pois(\\alpha \\cdot \\kappa)"
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+ "type": "text",
+ "content": ". Then, if "
+ },
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+ "type": "inline_equation",
+ "content": "\\alpha < -\\frac{1}{\\kappa} \\cdot \\log \\left[1 - (1 - p_{in}^{\\kappa})^{J / |\\tilde{\\mathcal{C}}_{in}|}\\right] \\triangleq \\mathcal{U}"
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+ "type": "text",
+ "content": ", the probability of our method achieving the inlier subset is greater than or equal to that of RANSAC."
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+ }
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+ "content": "P \\left(\\max _ {\\mathbf {c} _ {i} \\in \\mathcal {C} _ {s e e d}} | ^ {(\\kappa)} \\tilde {\\mathcal {M}} _ {\\mathbf {c} _ {i}} \\cap \\mathcal {C} _ {i n} | = \\kappa\\right) \\geq \\tag {8}",
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+ "content": "P \\Big (\\max _ {1 \\leq i \\leq J} | ^ {(\\kappa)} \\mathcal {M} _ {i} ^ {s a c} \\cap \\mathcal {C} _ {i n} | = \\kappa \\Big).",
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+ "content": "p_{in}"
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+ "content": " degrades, the upper bound "
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+ "type": "text",
+ "content": " will increase, indicating our method tends to be more likely to achieve the inlier subset than RANSAC in high outlier cases. Therefore, our method tends to be more robust to outliers than RANSAC. In addition, Theorem 1 also shows that the more inliers "
+ },
+ {
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+ "content": "|\\tilde{\\mathcal{C}}_{in}|"
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+ "type": "text",
+ "content": " in "
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+ "type": "inline_equation",
+ "content": "\\mathcal{C}_{\\text{seed}}"
+ },
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+ "type": "text",
+ "content": ", the better for our method. It means selecting all correspondences in "
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+ "type": "inline_equation",
+ "content": "\\mathcal{C}"
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+ {
+ "bbox": [
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+ "type": "text",
+ "content": " as seeds theoretically seems the best choice since it can avoid missing any inlier. However, in real implementation, we have to perform inlier selection to accelerate the registration speed. Nevertheless, in sparse point clouds, the number of correspondences is small. If we still perform the original inlier selection, fewer seeds even may not contain any inlier, leading to registration failure. To overcome it, we develop a conservative seed selection strategy, which changes the seed number to "
+ },
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+ "type": "inline_equation",
+ "content": "\\max \\{\\lfloor v \\cdot |\\mathcal{C}| \\rfloor, n\\} (n < |\\mathcal{C}|)"
+ },
+ {
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+ "type": "text",
+ "content": ", where the lower bar "
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+ "type": "inline_equation",
+ "content": "n"
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+ {
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+ "type": "text",
+ "content": " of the seed number can effectively avoid selecting too fewer inliers in sparse cases and in dense cases, it would degrade to the original selection strategy."
+ }
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+ }
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+ "type": "text",
+ "content": "3.4. Rigid Transformation Estimation"
+ }
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+ }
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+ "content": "With the voted hypothetical inliers "
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+ "type": "inline_equation",
+ "content": "\\{\\mathbf{\\Psi}^{(\\kappa)}\\tilde{\\mathcal{M}}_{\\mathbf{c}_i}\\mid \\mathbf{c}_i\\in"
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+ "type": "text",
+ "content": " above, we estimate the optimal transformation parameter based on the Procrustes method [20] to minimize the least-squares errors for each hypothetical inlier group:"
+ }
+ ]
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+ "type": "interline_equation",
+ "content": "\\hat {\\mathbf {R}} _ {i}, \\hat {\\mathbf {t}} _ {i} = \\arg \\min _ {\\mathbf {R}, \\mathbf {t}} \\sum_ {\\mathbf {c} _ {j} \\in {} ^ {(\\kappa)} \\tilde {\\mathcal {M}} _ {\\mathbf {c} _ {i}}} \\omega_ {j} \\cdot \\left\\| \\mathbf {R} ^ {\\top} \\mathbf {x} _ {j} + \\mathbf {t} - \\mathbf {y} _ {j} \\right\\| _ {2}, \\tag {9}",
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+ "type": "text",
+ "content": " is the error weight computed by the neural spectral matching as in [2]. Then, we select the transformation"
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+ "content": ". We utilize PyTorch to implement our project and perform all experiments on the server equipped with an Intel i5 2.2 GHz CPU and one Tesla V100 GPU. For simplicity, we name our Variational Bayesian-based Registration framework as VBReg."
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+ "content": ", thus presenting much more challenges. We leverage FCGF [11] and recently popular Predator [21] as the feature descriptors for putative correspondence generation. We choose six traditional methods: FGR [52], SM [26], RANSAC (50k) [16], TEASER++ [45], SC2_PCR [9], and TR_DE [8], and two deep methods: DHVR [25], and PointDSC [2] for comparison. The registration recalls "
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+ "content": " under different numbers of correspondences are listed in Table 2. It can be observed that regardless of FCGF or Predator descriptor, our method almost achieves the best performance on all settings, except for FCGF setting with 250 points. Notably, in more challenging cases, the performance advantage over PointDSC is further expanded "
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+ "content": " in the cases of FCGF with 500 and 250 points. These experimental results further demonstrate the outstanding robustness of our method when encountering those extremely low-overlapping cases (high outlier ratios). Some visualization results are listed in Fig. 3 and the "
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+ "content": " changes under different inlier ratios are presented in Fig. 4 (c), which suggests that our performance gains are mainly brought by model robustness in extremely high outlier situations."
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| FCGF | FPFH | 5000 | 2500 | 1000 | 500 | 250 | 5000 | 2500 | 1000 | 500 | 250 |
| PointDSC w/ SCNonlocalxyz | 92.42 | 77.51 | 55.8 | 52.6 | 46.8 | 37.7 | 26.7 | 68.1 | 67.3 | 66.5 | 63.4 | 60.5 | 0.11 |
| PointDSC w/ VBNonlocalxyz | 93.04 | 80.16 | 57.7 | 55.6 | 50.2 | 39.9 | 26.1 | 69.7 | 69.6 | 67.9 | 64.9 | 61.9 | 0.17 |
| PointDSC w/ SCNonlocalfeat | 92.36 | 77.76 | 54.6 | 50.6 | 44.9 | 36.8 | 25.4 | 69.2 | 68.6 | 67.9 | 63.5 | 59.9 | 0.13 |
| PointDSC w/ VBNonlocalfeat | 93.04 | 80.53 | 56.9 | 56.5 | 50.9 | 42.2 | 28.9 | 69.2 | 68.7 | 68.0 | 64.6 | 60.6 | 0.18 |
| PointDSC w/ SCNonlocalcls | 92.98 | 78.99 | 54.1 | 52.2 | 46.0 | 38.7 | 27.7 | 67.6 | 66.9 | 67.2 | 63.7 | 60.2 | 0.11 |
| PointDSC w/ VBNonlocalfeat+Vote | 93.41 | 81.21 | 58.3 | 56.5 | 51.9 | 44.7 | 31.1 | 69.3 | 69.5 | 68.2 | 65.3 | 61.2 | 0.20 |
| PointDSC w/ VBNonlocalfeat+Vote+CS | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 68.7 | 66.4 | 63.0 | 0.22 |
| Iteration times L = 6 | 93.41 | 82.32 | 58.1 | 57.1 | 52.9 | 48.3 | 34.8 | 69.7 | 69.7 | 68.7 | 66.3 | 63.8 | 0.19 |
| Iteration times L = 9 | 93.41 | 81.58 | 58.2 | 57.0 | 53.2 | 47.6 | 32.5 | 70.0 | 69.4 | 68.5 | 66.9 | 63.2 | 0.20 |
| Iteration times L = 12* | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 68.7 | 66.4 | 63.0 | 0.22 |
| Random feat. dim. d̅ = 32 | 93.41 | 82.38 | 58.0 | 57.5 | 53.5 | 48.1 | 34.8 | 69.7 | 70.0 | 68.6 | 66.3 | 63.3 | 0.20 |
| Random feat. dim. d̅ = 64 | 93.41 | 81.45 | 57.9 | 56.9 | 52.8 | 48.6 | 35.0 | 69.6 | 69.7 | 68.4 | 66.5 | 62.3 | 0.21 |
| Random feat. dim. d̅ = 128* | 93.53 | 82.75 | 58.3 | 56.8 | 52.9 | 47.2 | 34.5 | 69.9 | 69.8 | 69.3 | 66.4 | 63.3 | 0.22 |
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+ "content": "). As shown in the fifth row in Table 3, such loss-based label-propagation way just can achieve very limited performance gain and even sometimes degrades score. It demonstrates the superiority of our label-dependent posterior guidance for (prior) feature learning. Also, owing to such posterior guidance, the training curves in Fig. 4 (d) show that our method can achieve significantly faster convergence speed than SCNonlocal."
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+ "content": "Discriminative Feature Learning? In order to verify whether VBNonlocal can learn more discriminative features than SCNonlocal, we visualize the distribution of the feature similarity of inliers on 3DMatch (Fig. 4 (a)) and 3DLoMatch (Fig. 4 (b)). As we can see, our learned inlier features own much higher similarities (approximate to 1) than SCNonlocal on both datasets, which demonstrates that our proposed Bayesian-inspired non-local network truly can promote more discriminative correspondence-feature learning."
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+ "content": "challenging 3DLoMatch. It mainly benefits from the high-quality hypothetical inliers sampled by our voting policy."
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+ "content": "Conservative Seed Selection. Finally, we test the effectiveness of the proposed conservative seed selection strategy (CS) motivated by Theorem 1. As we can see in the seventh row of Table 3, CS can achieve consistent performance gain in each setting. Especially, in the cases of fewer points (e.g., FCGF setting with 250 and 500 points), the improvement is much more significant "
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+ "content": ". As the analysis before, in sparse point clouds, the original inlier selection strategy like in [2] is aggressive and prone to miss too many inlier seeds. Instead, our conservative selection strategy can effectively mitigate it as well as keep registration efficiency."
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+ "content": "In this paper, we adapted the variational Bayesian inference into the non-local network and developed the effective Bayesian-guided long-term dependencies for discriminative correspondence-feature learning. To achieve effective variational inference, a probabilistic graphical model was customized over our non-local network, and the variational low bound was derived as the optimization objective for model training. In addition, we proposed a Wilson score-based voting mechanism for high-quality inlier sampling and theoretically verified its superiority over RANSAC. Extensive experiments on indoor/outdoor datasets demonstrated its promising performance."
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diff --git a/2023/Robust Single Image Reflection Removal Against Adversarial Attacks/9967a3ae-116f-48d0-8a30-d1102ebae7fe_content_list.json b/2023/Robust Single Image Reflection Removal Against Adversarial Attacks/9967a3ae-116f-48d0-8a30-d1102ebae7fe_content_list.json
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+ "type": "text",
+ "text": "Robust Single Image Reflection Removal Against Adversarial Attacks",
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+ "type": "text",
+ "text": "Zhenbo Song $^{1}$ Zhenyuan Zhang $^{1*}$ Kaihao Zhang $^{2}$ Wenhan Luo $^{3\\boxtimes}$ Zhaoxin Fan $^{4}$ Wenqi Ren $^{3}$ Jianfeng Lu $^{1}$",
+ "bbox": [
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+ {
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+ "text": "$^{1}$ Nanjing University of Science and Technology $^{2}$ Australian National University $^{3}$ Shenzhen Campus of Sun Yat-sen University $^{4}$ Renmin University of China",
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+ "page_idx": 0
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+ "text": "Abstract",
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+ {
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+ "text": "This paper addresses the problem of robust deep single-image reflection removal (SIRR) against adversarial attacks. Current deep learning based SIRR methods have shown significant performance degradation due to unnoticeable distortions and perturbations on input images. For a comprehensive robustness study, we first conduct diverse adversarial attacks specifically for the SIRR problem, i.e. towards different attacking targets and regions. Then we propose a robust SIRR model, which integrates the cross-scale attention module, the multi-scale fusion module, and the adversarial image discriminator. By exploiting the multi-scale mechanism, the model narrows the gap between features from clean and adversarial images. The image discriminator adaptively distinguishes clean or noisy inputs, and thus further gains reliable robustness. Extensive experiments on Nature, $\\mathrm{SIR}^2$ , and Real datasets demonstrate that our model remarkably improves the robustness of SIRR across disparate scenes.",
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+ "type": "text",
+ "text": "1. Introduction",
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Single image reflection removal (SIRR) is a classic topic in the low-level image processing area, namely a kind of image restoration. When taking an image through a transparent surface, a reflection layer would be blended with the original photography (i.e. the transmission layer), resulting in imaging corruptions. The SIRR is devoted to recovering a clear transmission image by removing the reflection layer. However, the SIRR is fundamentally ill-posed [42] that there could be an infinite number of transmission and reflection decompositions from a blended image. Therefore, traditional methods often exploit manual priors to optimize the layer separation, such as gradient sparsity prior [25] and",
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+ "img_path": "images/68f5ff40cfa5edd62046af01835fdda4a786d65011c62be388b2235f82c84bb8.jpg",
+ "image_caption": [
+ "Figure 1. The PSNR measurements of our approach under different kinds of adversarial attacks. 'Clean' indicates no-attacks, 'MSE_FR' represents attacking on Full Region with MSE objective, and so on. The testing result is from Nature dataset [26]."
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "relative smoothness prior [27]. These priors are often violated when facing complex scenes. Recently, deep learning based methods [10, 16, 55] have attracted considerable attention to tackle the SIRR problem. By learning semantic and contextual features, deep SIRR methods have achieved much better quality of the recovered images.",
+ "bbox": [
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+ "page_idx": 0
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+ {
+ "type": "text",
+ "text": "However, deep neural networks are often vulnerable to visually imperceptible adversarial perturbations [14, 29]. The prediction can be totally invalid even with slight and unnoticeable attacks on inputs. Similarly, such vulnerability is also an important issue for the deep SIRR problem, and the robustness of current methods has not been thoroughly studied. There have been no benchmarks and evaluations for the robustness of deep SIRR models against intended attacks. Meanwhile, general defense methods [44] have not been applied to SIRR models. Accordingly, the robust SIRR model is still a crucial and desiderate research problem.",
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+ "text": "In this paper, we first investigate the robustness of deep SIRR methods. We apply the widely-used and powerful attack method PGD [29] to generate adversarial samples. For completeness of the robustness evaluation, we present various attack modes by referring [3, 48]. Specifically, we employ different attack objectives i.e. mean squared error (MSE) and learned perceptual image patch similarity (LPIPS) [52], as well as different attack regions, i.e. the",
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+ "page_idx": 0
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+ {
+ "type": "header",
+ "text": "This CVPR 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": "* Equal contribution",
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+ {
+ "type": "page_footnote",
+ "text": "Corresponding Author: