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2025/Generalizable Multi-Camera 3D Object Detection from a Single Source via Fourier Cross-View Learning/full.md

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+# Generalizable Multi-Camera 3D Object Detection from a Single Source via Fourier Cross-View Learning
+
+Xue Zhao<sup>1</sup> Qinying Gu<sup>2</sup> Xinbing Wang<sup>1</sup> Chenghu Zhou<sup>1</sup> Nanyang Ye<sup>1</sup>
+
+# Abstract
+
+Improving the generalization of multi-camera 3D object detection is essential for safe autonomous driving in the real world. In this paper, we consider a realistic yet more challenging scenario, which aims to improve the generalization when only single source data available for training, as gathering diverse domains of data and collecting annotations is time-consuming and labor-intensive. To this end, we propose the Fourier Cross-View Learning (FCVL) framework including Fourier Hierarchical Augmentation (FHiAug), an augmentation strategy in the frequency domain to boost domain diversity, and Fourier Cross-View Semantic Consistency Loss to facilitate the model to learn more domain-invariant features from adjacent perspectives. Furthermore, we provide theoretical guarantees via augmentation graph theory. To the best of our knowledge, this is the first study to explore generalizable multi-camera 3D object detection with a single source. Extensive experiments on various testing domains have demonstrated that our approach achieves the best performance across various domain generalization methods.
+
+# 1. Introduction
+
+Multi-camera 3D detection based on Bird's Eye View (BEV) representations has achieved rapid development, as BEV captures both spatial locations and semantic features without being heavily affected by occlusions. While existing models (Philion & Fidler, 2020; Huang et al., 2022; Li et al., 2023; 2022b) have achieved excellent performance on in-distribution datasets like nuScenes (Caesar et al., 2020), they struggle in real-world settings where the environment
+
+<sup>1</sup>Shanghai Jiao Tong University, Shanghai, China, <sup>2</sup>Shanghai Artificial Intelligence Laboratory, Shanghai, China. Correspondence to: Nanyang Ye <nylincoln@sjtu.edu.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+and conditions vary widely. This performance drop occurs because camera data in practical applications often has different distributions compared to the limited training data. As a result, enhancing the generalization of these models is critical for their safe deployment. Domain generalization (DG) aims to generalize a model to an unseen target domain by learning from multiple source domains. However, gathering diverse sources of data for training is time-consuming and labor-intensive, especially in autonomous driving scenarios, and cannot always guarantee improved performance. In this paper, we tackle a more practical yet challenging problem: boosting the generalization of 3D object detectors trained on a single source domain. Focusing on single-domain generalization (SDG) not only addresses practical constraints but also provides a more rigorous assessment of model adaptability.
+
+In SDG for 2D image classification, previous works (Zhao et al., 2024; Qiao et al., 2020) focus on enhancing data diversity through common 2D data augmentation techniques, such as geometric transformations, style transfer, or adversarial data generation. However, directly applying these approaches to BEV-based tasks introduces several challenges. First, BEV representations are generated by projecting multi-view 2D features using real-world physical constraints, which limits the use of strong geometric transformations. For example, large rotation of the input images will disrupt the spatial relationships, thereby affecting the spatial consistency of the BEV features. Second, style transfer techniques (Zhao et al., 2024) replace the original image statistics with those from the target style, but this often blurs the boundary between style and content (Lee et al., 2023), distorting important features and ultimately harming model accuracy and generalization. Third, adversarial generation methods (Goodfellow et al., 2020) suffer from unstable training and mode collapse. While diffusion-based techniques (Ho et al., 2020) are more stable, they add significant computational and storage overhead, making them impractical for complex 3D detection models. Therefore, common 2D data augmentations cannot be effectively leveraged to create diverse training samples for BEV-based tasks. More importantly, for multi-camera 3D object detection, the natural availability of cross-view data offers a unique opportunity to learn domain-invariant features, a potential that remains
+
+![](images/4bef44aecf743a7b5a20d41ffac619c4b3f990ae90a44421020baba840eebc6c.jpg)  
+Figure 1. (a) Detection results of different models: the proposed FCVL can improve the generalization of 3D detection on multiple target domains with single source training data. (b) Cross-View Learning: make the most of the natural cross-view input to improve the generalization. (c) Augmentation graph connectivity: augmentations of data from the same classes are assumed to be connected. FHiAug increases the augmentation graph connectivity between source and unseen domains.
+
+![](images/daa6922eb469147616917a45739559e4da9aa8cc5ee42fe920358f60ad757a7d.jpg)
+
+![](images/49b71b27104df7c737cb6a92a99c93870d922e079e3cf2e23fe6fe8f96d375cc.jpg)
+
+underexplored in scenarios with limited training data.
+
+In response to these limitations and challenges, we propose the Fourier Cross-View Learning (FCVL) framework including Fourier Hierarchical Augmentation (FHiAug), an augmentation strategy in the frequency domain to boost domain diversity and Fourier Cross-View Semantic Consistency Loss to facilitate the model to learn more domain-invariant features. Different from work (Zhao et al., 2024) expanding style statistics in the pixel domain, we utilize the Fourier transform to introduce style variations while minimizing content distortion. This is motivated by the well-known property of the Fourier transformation: the phase component encodes high-level semantic information, while the amplitude component captures low-level image statistics (Xu et al., 2021). This separation allows us to independently manipulate style (low-level statistics) and content (high-level semantics) in the frequency domain. At the image level, we introduce Frequency Jitter, which perturbs both amplitude and phase components to create diverse samples that complement the single source domain. At the feature level, we propose Amplitude Transfer, a novel method for generating fine-grained style variations, ensuring domain diversity in the latent space. To leverage the natural cross-view input, we design the Fourier Cross-View Semantic Consistency Loss to help the model acquire more robust features from adjacent perspectives. Furthermore, by leveraging augmentation graph theory (HaoChen et al., 2021; Wang et al., 2024), we provide a unique theoretical perspective on FCVL and establish its theoretical soundness.
+
+In summary, our major contributions are as follows:
+
+Towards SDG for multi-camera 3D object detection, we present the Fourier Cross-View Learning framework to fully exploit natural cross-view input.  
+- We propose FHiAug, a novel, efficient, plug-and-play augmentation strategy that operates on both image and
+
+feature levels, to enhance domain diversity without requiring additional modules or specialized training strategies.
+
+- We propose Fourier Cross-View Semantic Consistency Loss to facilitate the model to learn more domain-invariant features from adjacent perspectives.  
+- Leveraging augmentation graph theory, we provide a valid theoretical foundation for the effectiveness of FCVL (See Appendix A).  
+- To the best of our knowledge, this is the first work to address generalizable multi-camera 3D object detection using single-source data. Extensive experiments across various test domains demonstrate that our approach achieves superior performance compared to existing domain generalization methods (See Fig.1(a)).
+
+# 2. Related Work
+
+# 2.1. Multi-view 3D Object Detection
+
+The recent advances in bird's eye view (BEV) representation exhibit great potential for MC3D-Det. The camera-based BEV models typically project 2D image features from six cameras to explicit BEV feature maps in the 3D space and make predictions based on BEV features. In terms of the BEV projection, Lift-splat-shoot (LSS) (Philion & Fidler, 2020), BEVDet (Huang et al., 2022) and BEVDepth (Li et al., 2023) distribute the 2D features into 3D space according to the depth information while BEVFormer (Li et al., 2022b) adopts cross attention to query BEV features from 2D images. LSS-based method and transformer-based methods have achieved excellent performance on the indistribution datasets, like nuScenes (Caesar et al., 2020). Lift-splat-shoot (LSS) proposes a view transform method that explicitly predicts depth distribution and projects image features onto a bird's-eye view (BEV), which has been
+
+proved practical for 3D object detection. Based on LSS, BEVDepth designs a novel Depth Refinement Module to refine feature projection process (Li et al., 2023). BEV-Former further designs a transformer structure to automatically extract and fuse BEV features, leading to excellent performance on 3D detection (Li et al., 2022b).
+
+# 2.2. Single Domain Generalization
+
+Domain Generalization (DG) aims to generalize a model trained on multiple source domains to a target domain which is distributionally different (Muandet et al., 2013). Data augmentation has been a common strategy to regularize the training process to avoid overfitting and improve generalization (Zhou et al., 2023). CIRL (Lv et al., 2022) generates augmented images by a causal intervention module with intervention upon non-causal factors. AGFA (Kim et al., 2023) trains the classifier and the amplitude generator adversarially to synthesize a worst-case domain for adaptation. Compared with these methods, our proposed method is simple, stable, yet effective, without extra module design or special training strategy.
+
+This paper focuses on Single Domain Generalization (SDG) (Wang et al., 2023b) which is a more challenging yet realistic setting. Existing works enhance data diversity first using 2D data augmentation techniques, including three categories of data augmentation methods: photometric and geometric augmentations, style transfer, and data generation. Chen et al. (2023) use photometric augmentations (e.g., Invert) and geometric augmentations (e.g., Rotation) to create diverse domains, then learn to analyze the causes of domain shift, and finally learn to reduce the domain shift for model adaptation. BEV representations are generated by projecting multi-view 2D features using real-world physical constraints, which limits the use of geometric transformations. For example, large rotation of the input images will disrupt the spatial relationships, thereby affecting the spatial consistency of the BEV features<sup>1</sup>. Zhao et al. (2024) propose CPerb, a simple yet effective cross-perturbation method with style transfer to enhance the diversity of the training data and introduce multi-route perturbation to learn domain-invariant features. Style transfer techniques (Nuriel et al., 2021) replace the original image statistics with those from the target style, but this often blurs the boundary between style and content (Lee et al., 2023), distorting important features and ultimately harming model generalization. Wang et al. (2023b) propose a style-complement module with generative adversarial network to enhance the generalization power of the model by synthesizing images from diverse distributions that are complementary to the source ones. Qiao et al. (2020) leverage adversarial training to create "fictitious" yet "challenging" populations and use a
+
+Wasserstein Auto-Encoder (WAE) to relax the widely used worst-case constraint in a meta-learning scheme. Adversarial generation can suffer from unstable training and mode collapse. It often requires a lot of experiments and fine-tuning to get a GAN to work well, making them impractical for complex 3D detection models.
+
+# 3. Methodology
+
+# 3.1. Overview of Fourier Cross-View Learning Framework
+
+The overall structure of our framework is depicted in Fig.2 and the process is outlined in Algorithm 1 in Appendix B.
+
+The FCVL framework is motivated by the cross-view consistency in multi-camera 3D object detection. For example, as shown in Fig.1(b), objects such as cars or pedestrians are often visible across multiple adjacent camera views. To capture this cross-view relationship, we implement a Fourier Cross-View Semantic Consistency Loss, where features from nearby camera views are considered positive samples, while those from distant views are treated as negative samples. However, its effectiveness is limited in the single-domain setting due to restricted feature diversity. Therefore, we propose Fourier Hierarchical Augmentation (FHiAug) to alleviate the bias in single-domain representations. FHiAug, on one hand, expands the domain diversity to force the model to learn from different feature distributions. On the other hand, it expands the quantity and diversity of the cross-view sample pairs, enabling the consistency loss to more effectively explore semantic alignments between adjacent perspectives.
+
+In the following sections, we provide an in-depth explanation of both the Fourier Hierarchical Augmentation and the Fourier Cross-View Semantic Consistency Loss.
+
+# 3.2. Fourier Hierarchical Augmentation
+
+Fourier Hierarchical Augmentation includes data augmentation at image level (Frequency Jitter) and domain perturbation at feature level (Amplitude Transfer), which is a plug-and-play method to boost domain diversity without extra module designing or special training strategies.
+
+Frequency Jitter at image level For a single channel image  $x \in \mathcal{R}^{d_1 \times d_2}$ , the 2D Fourier transformation is defined as follows,
+
+$$
+F (x) (u, v) = \sum_ {m = 0} ^ {d _ {1} - 1} \sum_ {n = 0} ^ {d _ {2} - 1} x (m, n) \exp^ {- 2 \pi i \left(\frac {m u}{d _ {1}} + \frac {n v}{d _ {2}}\right)}, \tag {1}
+$$
+
+where  $F$  denotes Fourier Transform;  $d_{1}$  and  $d_{2}$  denote height and width of the image;  $u$  and  $v$  denote frequency coordinates;  $m$  and  $n$  denote spatial coordinates.
+
+![](images/0e21f028a7e3379ac289f15e650858d13949f582bf2d8d24b4539f3a57c3a195.jpg)  
+Figure 2. Overview of our FCVL framework. FCVL includes two major parts: FHiAug to boost domain diversity and Fourier Cross-View Semantic Consistency Loss to ensure domain-invariant BEV features. FHiAug consists of two stages. One is Frequency Jitter at image level. The other is Amplitude Transfer at feature level. Notably, we achieve cross-camera augmentation via FHiAug, which means a set of surrounding views have different "styles". This forces the model to learn from diversified domains. Besides, on cross-view features, we calculate Fourier Cross-View Semantic Consistency Loss to learn more domain-invariant BEV features from adjacent perspectives.
+
+The amplitude components  $\mathcal{A}$  and phase components  $\mathcal{P}$  are then respectively expressed as:
+
+$$
+\mathcal {A} (x) (u, v) = \left[ R ^ {2} (x) (u, v) + I ^ {2} (x) (u, v) \right] ^ {1 / 2},
+$$
+
+$$
+\mathcal {P} (x) (u, v) = \arctan \left[ \frac {I (x) (u , v)}{R (x) (u , v)} \right], \tag {2}
+$$
+
+where  $R(x)$  and  $I(x)$  represent the real and imaginary parts of  $F(x)$ , respectively.
+
+To generate diverse samples that complement the single source domain, we employ two strategies. First, we perturb the amplitude component using a hyperparameter,  $\alpha$ , to create variations in low-level statistics. Second, we modify the intensity of the phase component with a hyperparameter,  $\beta$ , to expose the model to previously less emphasized features (Chen et al., 2020).
+
+$$
+\hat {\mathcal {A}} (x) (u, v) = \alpha \mathcal {A} (x) (u, v),
+$$
+
+$$
+\hat {\mathcal {P}} (x) (u, v) = \beta \mathcal {P} (x) (u, v), \tag {3}
+$$
+
+where  $\alpha \sim U(\eta, 1)$  and the hyperparameter  $\eta$  control the strength of the augmentation on amplitude;  $\beta \sim U(\lambda, 1)$  and the hyperparameter  $\lambda$  control the strength of the augmentation on phase.
+
+With new amplitude and phase component, we can form a new Fourier representation and use inverse Fourier transform
+
+mation to generate the augmented image  $\hat{x}$
+
+$$
+F (\hat {x}) (u, v) = \hat {\mathcal {A}} (x) (u, v) * e ^ {- j * \hat {\mathcal {P}} (x) (u, v)},
+$$
+
+$$
+\hat {x} = F ^ {- 1} \left[ F (\hat {x}) (u, v) \right].
+$$
+
+(4)
+
+In the training phase, we set  $p_i$  as the calling probability of Frequency Jitter and sample  $p \sim U(0,1)$ . For image input  $x$ , we acquire the augmented  $x_{aug}$  as:
+
+$$
+x _ {a u g} = \text {F r e q u e n c y J i t t e r} (x), \text {i f} p \leq p _ {i}. \tag {5}
+$$
+
+This Fourier-based augmentation strategy, termed Frequency Jitter, manipulates both amplitude and phase components, as shown in Fig.7. The top row demonstrates adjustments to the amplitude, primarily affecting image brightness, which helps the model become robust to varying lighting conditions. The bottom row shows modifications to the phase component, creating samples with varying levels of semantic detail while preserving the overall structure. This controlled manipulation of semantic strength encourages the model to learn more domain-invariant and robust features. Additional examples highlighting the effect of phase adjustments are provided in Fig.10.
+
+Amplitude Transfer at feature level To implement domain perturbation and create diverse virtual styles during training, we apply Amplitude Transfer based on the style
+
+statistics of intermediate features. This approach aims to improve model robustness and generalization.
+
+Given an intermediate feature map  $\mathbf{X} \in \mathcal{R}^{B \times C \times H \times W}$ , where  $B, C, H$ , and  $W$  denote batch size, number of channels, height, and width, respectively, we first perform a Fourier transformation and extract its amplitude component  $\mathcal{A}(\mathbf{X}) \in \mathcal{R}^{B \times C \times H \times W}$ . We then compute the channel-wise mean  $(\mu)$  and standard deviation  $(\sigma)$  for each instance's amplitude as follows:
+
+$$
+\mu (\mathcal {A} (\mathbf {X})) = \frac {1}{H W} \sum_ {\substack {h = 1 \\ W}} ^ {H} \sum_ {w = 1} ^ {W} \mathcal {A} (\mathbf {X}), \tag{6}
+$$
+
+$$
+\sigma^ {2} (\mathcal {A} (\mathbf {X})) = \frac {1}{H W} \sum_ {h = 1} ^ {H} \sum_ {w = 1} ^ {W} [ \mathcal {A} (\mathbf {X}) - \mu (\mathcal {A} (\mathbf {X})) ] ^ {2}.
+$$
+
+Now, we acquire the style statistics  $\mu (\mathcal{A}(\mathbf{X}))$  and  $\sigma^2 (\mathcal{A}(\mathbf{X}))$  of the features. To achieve feature-level perturbation, different from (Xu et al., 2021) and (Zhou et al., 2021) to mix up different domains' style information directly, inspired by (Li et al., 2022a) we make uncertainty estimation on  $\mu (\mathcal{A}(\mathbf{X}))$  and  $\sigma (\mathcal{A}(\mathbf{X}))$  with the variance as follows:
+
+$$
+\operatorname {V a r} (\mu (\mathcal {A} (\mathbf {X}))) = \frac {1}{B} \sum_ {b = 1} ^ {B} [ \mu (\mathcal {A} (\mathbf {X})) - \mathbb {E} (\mu (\mathcal {A} (\mathbf {X}))) ] ^ {2},
+$$
+
+$$
+\operatorname {V a r} (\sigma (\mathcal {A} (\mathbf {X}))) = \frac {1}{B} \sum_ {b = 1} ^ {B} [ \sigma (\mathcal {A} (\mathbf {X})) - \mathbb {E} (\sigma (\mathcal {A} (\mathbf {X}))) ] ^ {2}, \tag {7}
+$$
+
+where  $B$  is the batch size and  $\mathbb{E}$  denotes the mathematical expectations.
+
+Next, we obtain new style statistics  $\beta$  and  $\gamma$  by random sampling from the Gaussian distributions:
+
+$$
+\beta (\mathcal {A} (\mathbf {X})) = \mu (\mathcal {A} (\mathbf {X})) + \epsilon_ {\mu} \sqrt {\operatorname {V a r} (\mu (\mathcal {A} (\mathbf {X})))}, \epsilon_ {\mu} \sim \mathcal {N} (0, 1),
+$$
+
+$$
+\gamma (\mathcal {A} (\mathbf {X})) = \sigma (\mathcal {A} (\mathbf {X})) + \epsilon_ {\sigma} \sqrt {\operatorname {V a r} (\sigma (\mathcal {A} (\mathbf {X})))}, \epsilon_ {\sigma} \sim \mathcal {N} (0, 1). \tag {8}
+$$
+
+Finally, we replace the original style statistics with the perturbed values and perform an inverse Fourier transform to obtain the augmented feature map  $\hat{\mathbf{X}}$ :
+
+$$
+\hat {\mathcal {A}} (\mathbf {X}) = \gamma (\mathcal {A} (\mathbf {X})) \times \frac {\mathcal {A} (\mathbf {X}) - \mu (\mathcal {A} (\mathbf {X}))}{\sigma (\mathcal {A} (\mathbf {X}))} + \beta (\mathcal {A} (\mathbf {X})). \tag {9}
+$$
+
+This allows us to create diverse styled features in each training iteration without explicitly defining content and style. During training, we set  $p_f$  as the probability of applying Amplitude Transfer and sample  $p \sim U(0,1)$ . For a given feature input  $\mathbf{X}$ , the augmented feature  $\mathbf{X}_{\text{aug}}$  is generated as follows:
+
+$$
+\mathbf {X} _ {a u g} = \text {A m p l i t u d e} \operatorname {T r a n s f e r} (\mathbf {X}), \text {i f} p \leq p _ {f}. \tag {10}
+$$
+
+We visualize the style variations of some pictures via Amplitude Transfer in Fig.8. The left column shows the original
+
+images, while the adjacent columns display styled variations. As observed, the augmented images exhibit different colors and textures, showcasing the effectiveness of the proposed method in generating diverse feature styles. The visualized results of FHiAug are shown in Fig.9. For the same image, via FHiAug, we can generate multiple styles.
+
+# 3.3. Fourier Cross-View Semantic Consistency Loss
+
+For multi-camera 3D object detection, the input inherently includes cross-view data, which is beneficial for learning domain-invariant features. This has not yet been harnessed to improve generalization. As illustrated in Fig.3, consider a car appearing in both the front and front-right views. Such cross-view targets are common in multi-camera inputs, providing natural opportunities to observe the same object from different perspectives.
+
+To exploit this, we propose the Fourier Cross-View Semantic Consistency Loss to help the model learn more robust features from adjacent views. Unlike conventional consistency losses that operate in the pixel domain, we minimize the distance between the phase distributions of the targets with the same semantics, as the phase component usually encodes high-level semantic information. Concretely, for adjacent views, we split the features into halves as shown in Fig.3. There is a cross-view instance binding mechanism that the identical instance labels are assigned to cross-view instances of the same object. When taking a target object in one view as the anchor, we search for objects with the same instance label in adjacent views as positive samples. For negative samples, we choose samples of different categories from other views. Next, we calculate triplet loss (Schroff et al., 2015) in the frequency domain to explore potential semantic similarity as follows:
+
+$$
+v _ {\text {a u g}} ^ {\text {p o s}} = \mathrm {F H i A u g} (v ^ {\text {p o s}}), \tag {11}
+$$
+
+$$
+v _ {\mathrm {a u g}} ^ {\mathrm {n e g}} = \mathrm {F H i A u g} (v ^ {\mathrm {n e g}}),
+$$
+
+$$
+a = \mathcal {P} (v ^ {\text {a n c h o r}}), p = \mathcal {P} (v _ {\text {a u g}} ^ {\text {p o s}}), n = \mathcal {P} (v _ {\text {a u g}} ^ {\text {n e g}}), \tag {12}
+$$
+
+$$
+\mathcal {L} _ {\text {c r o s s}} = \max  (\operatorname {d i s t} (a, p) - \operatorname {d i s t} (a, n) + \operatorname {m a r g i n}, 0), \tag {13}
+$$
+
+where FHiAug is the proposed augmentation method;  $v^{\text{anchor}}$  is the anchor example;  $v^{\text{pos}}$  is the sample with the same category as anchor;  $v^{\text{neg}}$  is the sample with different categories;  $\mathcal{P}$  denotes calculating the phase components of different views after Fourier transformation; dist is the distance measurement; margin is a constant greater than zero.
+
+Overall, the training objective loss including detection loss  $\mathcal{L}_{\mathrm{det}}$  and consistency loss  $\mathcal{L}_{\mathrm{cross}}$  can be written as:
+
+$$
+\mathcal {L} _ {\text {t r a i n}} = \mathcal {L} _ {\det } + \gamma \mathcal {L} _ {\text {c r o s s}}, \tag {14}
+$$
+
+where  $\gamma$  is the weighting parameter to balance different loss terms.
+
+![](images/50a5cada27789cdc181b3c05e9e364a316e9832eb58b1e1829439c3a189028c7.jpg)  
+Figure 3. Illustration of Fourier Cross-View Semantic Consistency Loss.
+
+# 4. Experiments
+
+In this section, we showcase the overall performance of our method. To comprehensively evaluate our method, we conduct extensive experiments across four distinct baseline models, four varied domain generalization methods, and three different datasets. To gauge the practical viability and performance of our algorithm, we also evaluate it within real-world application scenarios.
+
+# 4.1. Implementation details
+
+We first choose three baselines BEVFormer, BEVDet and experiment on nuScenes and nuScenes-C (Xie et al., 2023). Besides, we choose a new SOTA Far3D (Jiang et al., 2024) as another baseline and experiment on dataset Argoverse 2 (Wilson et al., 2023). More details of datasets can be found in Appendix D.1. We use ResNet50 as the backbone and extend our method to different baselines respectively. All models are trained until full convergence for 24 epochs. All parameters in our framework are initialized from ImageNet. We apply an AdamW optimizer with the learning rate set to 0.0002 and set the batch size to 2 per GPU. All experiments are conducted with 4 GPUs. After training, we mainly report mean Average Precision (mAP) and nuScenes Detection Score (NDS) (Caesar et al., 2020).
+
+# 4.2. Comparison with SOTA methods
+
+We compare our method with some SOTA SDG and DG methods which involve frequency-domain data augmentation (CPerb (Zhao et al., 2024), FACT (Xu et al., 2021)) and style transformation (DAC-SC (Lee et al., 2023)). PDBEV (Lu et al., 2025) working on BEVDepth, is proposed to ensure consistent and accurate detection and improve generalization via perspective debiasing.
+
+The results on nuScenes and nuScenes-C are shown in Table 1. On transformer-based framework, our FCVL can greatly improve the generalization of BEVFormer as shown in Table 1. The average NDS of eight testing domains is in-
+
+creasing from 0.3028 to 0.3567 ( $\uparrow$  5.39%). FCVL achieves SOTA out-of-domain performance across different SDG or DG methods. On LLS-based framework, FCVL achieves SOTA performance on both BEVDepth and BEVDet as well. In terms of the average NDS of eight testing domains, our method achieves much better performance than BEVDepth ( $\uparrow$  5.05%) and BEVDet ( $\uparrow$  6.07%). Especially, FCVL improves the performance of BEVDepth by 10.87% for adverse weather conditions Snow and 10.19% for Low Light. Similarly, FCVL improves the performance of BEVDet by 8.07% for adverse weather conditions Snow and 12.50% for Low Light. FCVL has more stable generalization ability for adverse weather and light conditions on different 3D detection frameworks. For worst cases Low Light and Snow, as shown in Fig.4(b), FCVL has shown significant improvement. Overall, as is shown in Fig.4(a), the proposed FCVL outperforms other methods with great margin on average of three frameworks ( $\uparrow$  2.08%). Besides, for both transformer-based framework and LLS-based framework, FCVL has the superiority in stable maintenance of better generalization ability in eight testing domains, especially in Low Light, Motion Blur and Snow.
+
+To further evaluate our method, we extend our methods to the 3D detectors without explicit BEV features, such as Sparse4D (Lin et al., 2023) and multi-modal method, such as BEVFusion (Liu et al., 2023) as well. We list the experimental results including different 3D detection schemes (explicit BEV or not, multi frames or not, etc.) in the Table 2. Our method can improve the out-of-distribution performance in all the settings, while maintaining the in-distribution performances as far as possible.
+
+More results on Argoverse 2 are shown in Table 3. We experiment on a new SOTA Far3D, which presents a sparse query-based method for multi-view 3D long-range detection without explicit BEV features. To achieve training on one domain and testing on unseen domains, we sample data from sunny weather in Miami scenarios as the training data and data from cloudy weather or other five cities as the ood test set. As is shown, our method improves the generalization for long-range detection as well. In addition, we have visualized some detection results in Fig.12. As can be seen, after being optimized by FCVL, Far3D exhibits great performance when the lighting conditions are poor.
+
+# 4.3. Ablation Study on nuScenes
+
+Effects of different components of FCVL Firstly, we analyze the effects of different components of Frequency Jitter at image level, as shown in Table 4. As is shown, jittering amplitude only(#1) or jittering phase of the input only (#2) has impressive performance. When jittering both phase and amplitude (#3), Frequency Jitter improves the performance further. Next, we separately examine the ef
+
+Table 1. Comparison with baseline methods on nuScenes and nuScenes-C. The table represents the results of NDS  $\uparrow$  with ResNet50 as backbone. "Clean" denotes the normal validation set of nuScenes. "OoD Avg." is the average performance of eight testing domains. FCVL achieves SOTA out-of-distribution performance on three frameworks. PD-BEV  $\dagger$  (Lu et al., 2025) has released the code for BEVDepth. Thus, we mainly compare our method with PDBEV on BEVDepth for fair comparison. The best and second-best results are highlighted in Red and Blue, respectively.  
+
+<table><tr><td>Model</td><td>Clean</td><td>CamCrash</td><td>FrameLost</td><td>ColorQuant</td><td>MotionBlur</td><td>Bright</td><td>LowLight</td><td>Fog</td><td>Snow</td><td>OoD Avg.</td></tr><tr><td>BEVFormer (Li et al., 2022b)</td><td>0.4362</td><td>0.3175</td><td>0.3246</td><td>0.3410</td><td>0.2549</td><td>0.4022</td><td>0.2461</td><td>0.3853</td><td>0.1510</td><td>0.3028</td></tr><tr><td>+CPerb (Zhao et al., 2024)</td><td>0.4356</td><td>0.3199</td><td>0.3292</td><td>0.3372</td><td>0.2548</td><td>0.4096</td><td>0.2420</td><td>0.3907</td><td>0.1661</td><td>0.3062</td></tr><tr><td>+DSU (Li et al., 2022a)</td><td>0.4359</td><td>0.3206</td><td>0.3322</td><td>0.3609</td><td>0.3425</td><td>0.4083</td><td>0.2458</td><td>0.3937</td><td>0.2601</td><td>0.3330</td></tr><tr><td>+DAC-SC (Lee et al., 2023)</td><td>0.4332</td><td>0.3085</td><td>0.2872</td><td>0.3703</td><td>0.3691</td><td>0.4161</td><td>0.3155</td><td>0.4093</td><td>0.3086</td><td>0.3481</td></tr><tr><td>+FACT (Xu et al., 2021)</td><td>0.4379</td><td>0.3181</td><td>0.3285</td><td>0.3436</td><td>0.2585</td><td>0.4100</td><td>0.2494</td><td>0.3916</td><td>0.1486</td><td>0.3060</td></tr><tr><td>+FCVL(Ours)</td><td>0.4375</td><td>0.3244</td><td>0.3374</td><td>0.3751</td><td>0.3748</td><td>0.4202</td><td>0.3078</td><td>0.4170</td><td>0.2969</td><td>0.3567</td></tr><tr><td>BEVDepth (Li et al., 2023)</td><td>0.4028</td><td>0.2654</td><td>0.2178</td><td>0.2801</td><td>0.2697</td><td>0.3072</td><td>0.1558</td><td>0.3080</td><td>0.0881</td><td>0.2365</td></tr><tr><td>PD-BEV † (Lu et al., 2025)</td><td>0.4094</td><td>0.2822</td><td>0.2316</td><td>0.3102</td><td>0.2842</td><td>0.3011</td><td>0.1411</td><td>0.3151</td><td>0.1091</td><td>0.2468</td></tr><tr><td>+CPerb (Zhao et al., 2024)</td><td>0.4034</td><td>0.2698</td><td>0.2294</td><td>0.2847</td><td>0.2873</td><td>0.3180</td><td>0.1616</td><td>0.3164</td><td>0.1054</td><td>0.2466</td></tr><tr><td>+DSU (Li et al., 2022a)</td><td>0.4057</td><td>0.2722</td><td>0.2330</td><td>0.3065</td><td>0.3270</td><td>0.3462</td><td>0.2165</td><td>0.3249</td><td>0.1565</td><td>0.2729</td></tr><tr><td>+DAC-SC (Lee et al., 2023)</td><td>0.4007</td><td>0.2714</td><td>0.2200</td><td>0.2846</td><td>0.2861</td><td>0.3284</td><td>0.1586</td><td>0.3172</td><td>0.1299</td><td>0.2495</td></tr><tr><td>+FACT (Xu et al., 2021)</td><td>0.4026</td><td>0.2670</td><td>0.2224</td><td>0.2872</td><td>0.2749</td><td>0.3276</td><td>0.1611</td><td>0.3141</td><td>0.0957</td><td>0.2438</td></tr><tr><td>+FCVL(Ours)</td><td>0.4050</td><td>0.2722</td><td>0.2346</td><td>0.3106</td><td>0.3318</td><td>0.3539</td><td>0.2577</td><td>0.3380</td><td>0.1968</td><td>0.2870</td></tr><tr><td>BEVDet (Huang et al., 2022)</td><td>0.3880</td><td>0.2508</td><td>0.1955</td><td>0.2409</td><td>0.2201</td><td>0.2591</td><td>0.1112</td><td>0.2633</td><td>0.0728</td><td>0.2017</td></tr><tr><td>+CPerb (Zhao et al., 2024)</td><td>0.3908</td><td>0.2590</td><td>0.2065</td><td>0.2479</td><td>0.2325</td><td>0.2643</td><td>0.1322</td><td>0.2752</td><td>0.0782</td><td>0.2120</td></tr><tr><td>+DSU (Li et al., 2022a)</td><td>0.3835</td><td>0.2582</td><td>0.2061</td><td>0.2814</td><td>0.3019</td><td>0.3128</td><td>0.1806</td><td>0.2961</td><td>0.1065</td><td>0.2430</td></tr><tr><td>+DAC-SC (Lee et al., 2023)</td><td>0.3884</td><td>0.2574</td><td>0.2046</td><td>0.2688</td><td>0.2644</td><td>0.2986</td><td>0.1450</td><td>0.2926</td><td>0.1028</td><td>0.2293</td></tr><tr><td>+FACT (Xu et al., 2021)</td><td>0.3907</td><td>0.2581</td><td>0.2054</td><td>0.2430</td><td>0.2277</td><td>0.2708</td><td>0.1230</td><td>0.2727</td><td>0.0773</td><td>0.2098</td></tr><tr><td>+FCVL(Ours)</td><td>0.3848</td><td>0.2579</td><td>0.2064</td><td>0.2928</td><td>0.3204</td><td>0.3244</td><td>0.2393</td><td>0.3156</td><td>0.1848</td><td>0.2677</td></tr></table>
+
+Table 2. The table represents the effectiveness of our proposed method under different settings on nuScenes and nuScenes-C. "C" denotes camera input. "L" denotes lidar input. "Explicit BEV" means 3D detectors generate explicit BEV features. "Temporal" denotes whether utilizing multi frames. "Depth" denotes whether utilizing depth information. Bold fonts indicate the best results.  
+
+<table><tr><td>Model</td><td>Modality</td><td>Temporal</td><td>Depth</td><td>Explicit BEV</td><td>Clean</td><td>OoD Avg.</td></tr><tr><td>BEVFormer</td><td>C</td><td>✓</td><td></td><td>✓</td><td>0.4362</td><td>0.3028</td></tr><tr><td>+FCVL(Ours)</td><td>C</td><td>✓</td><td></td><td>✓</td><td>0.4375</td><td>0.3567</td></tr><tr><td>BEVDepth</td><td>C</td><td></td><td>✓</td><td>✓</td><td>0.4028</td><td>0.2365</td></tr><tr><td>+FCVL(Ours)</td><td>C</td><td></td><td>✓</td><td>✓</td><td>0.4050</td><td>0.2870</td></tr><tr><td>BEVDepth</td><td>C</td><td>✓</td><td>✓</td><td>✓</td><td>0.4828</td><td>0.4128</td></tr><tr><td>+FCVL(Ours)</td><td>C</td><td>✓</td><td>✓</td><td>✓</td><td>0.4827</td><td>0.4291</td></tr><tr><td>BEVDet</td><td>C</td><td></td><td></td><td>✓</td><td>0.3880</td><td>0.2017</td></tr><tr><td>+FCVL(Ours)</td><td>C</td><td></td><td></td><td>✓</td><td>0.3848</td><td>0.2677</td></tr><tr><td>Sparse4Dv3</td><td>C</td><td>✓</td><td>✓</td><td></td><td>0.5590</td><td>0.4431</td></tr><tr><td>+FCVL(Ours)</td><td>C</td><td>✓</td><td>✓</td><td></td><td>0.5592</td><td>0.4492</td></tr><tr><td>BEVFusion</td><td>L+C</td><td></td><td></td><td>✓</td><td>0.7074</td><td>0.6865</td></tr><tr><td>+FCVL(Ours)</td><td>L+C</td><td></td><td></td><td>✓</td><td>0.7123</td><td>0.6948</td></tr></table>
+
+fect of the Amplitude Transfer at feature level (#4). It can be observed that enhancing at feature level alone can also lead to great improvement. Then, we combine Frequency Jitter and Amplitude Transfer (#5). Importantly, the combination of image and feature levels augmentations yields more significant performance gains. The image level augmentation expands the data distribution first, while feature level augmentation capitalizes on this augmented input to derive fine-grained style variations, further enhancing representation diversity. This hierarchical augmentation pipeline increases domain diversity collectively, thereby enhancing the model's ability to generalize to unseen scenarios. At
+
+Table 3. Comparison with baseline methods on Argoverse 2 based on Far3D (Jiang et al., 2024). The table represents the results of mAP ↑. "Clean" denotes the in-domain set.  
+
+<table><tr><td>Model</td><td>Clean</td><td>City</td><td>Cloudy</td><td>OoD Avg.</td></tr><tr><td>Far3D</td><td>0.1964</td><td>0.1140</td><td>0.0800</td><td>0.0970</td></tr><tr><td>+CPerb</td><td>0.2044</td><td>0.1172</td><td>0.0820</td><td>0.0996</td></tr><tr><td>+DSU</td><td>0.2072</td><td>0.1352</td><td>0.0904</td><td>0.1128</td></tr><tr><td>+DAC-SC</td><td>0.1904</td><td>0.1152</td><td>0.0830</td><td>0.0991</td></tr><tr><td>+FACT</td><td>0.2052</td><td>0.1302</td><td>0.0898</td><td>0.1100</td></tr><tr><td>+FCVL(Ours)</td><td>0.2170</td><td>0.1610</td><td>0.1082</td><td>0.1346 (↑ 2.18%)</td></tr></table>
+
+last, we add our  $\mathcal{L}_{\mathrm{cross}}$  in the training (#6). This further enables the model to learn more domain-invariant features. Notably, the consistency loss is not only beneficial for the in-domain performance, but also boosts the out-of-domain performance.
+
+Effects of different inserted positions of Amplitude Transfer We evaluate the impact of different inserted positions of Amplitude Transfer, as shown in Table 5. Inserted position of ResNet is numbered as follows: after first Conv 0, after Max Pooling layer 1, after first Resblock 2, after second Resblock 3, after third Resblock 4 and after fourth Resblock 5, respectively. According to (Zhou et al., 2021), Resblock 1 to Resblock 3 contain domain-related information, which means domain-related information usually lies in shallow layers. Thus, in our method, Amplitude Transfer is inserted in Position 0-2. We make more experiments by increasing Position 3-5 gradually to find more suitable
+
+![](images/af8076463d455a1b446eabff82db50b320f03bd71891f4ab467e93ec312eacf2.jpg)  
+(a)
+
+![](images/3ccdfc8136ff690920256e1b34c1a0142b030ec817f87aea138c71fbf1d47667.jpg)  
+(b)  
+Figure 4. (a) The average detection results of different methods including eight OoD domains under three baseline frameworks. As is shown, the proposed FCVL outperforms other methods with great margin on average. (b) Worst cases analysis. Our method has shown significant improvement in the worst cases, Low Light and Snow. (c) Hyperparameters analysis of FHiAug. The strength of augmentation  $\eta$  and  $\lambda$  for Frequency Jitter; the probability  $p_i$  for Frequency Jitter and  $p_f$  for Amplitude Transfer.
+
+![](images/4e0a6a153c85d1a7c8de5d79b907c452c41a201957f7a7bb4cd19ee13ad0cadf.jpg)  
+(c)
+
+Table 4. Ablation Study on different components of FCVL on BEVDepth (Li et al., 2023). Amp. denotes Amplitude. Amp. Jitter means only jittering on amplitude component. Phase Jitter means only jittering on phase component. Jittering on both is the Frequency Jitter operated at image level. Amplitude Transfer is operating at feature level.  $\mathcal{L}_{\mathrm{cross}}$  denotes consistency loss.  $\checkmark$  denotes performing corresponding operation. Bold fonts indicate the best results.  
+
+<table><tr><td rowspan="2">#</td><td colspan="2">Image Level</td><td>Feature Level</td><td rowspan="2">\( {\mathcal{L}}_{\text{cross }} \)</td><td rowspan="2">Clean</td><td rowspan="2">OoD Avg.</td></tr><tr><td>Amp. Jitter</td><td>Phase Jitter</td><td>Amp. Transfer</td></tr><tr><td>#0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.4028</td><td>0.2365</td></tr><tr><td>#1</td><td>✓</td><td>-</td><td>-</td><td>-</td><td>0.4037</td><td>0.2735</td></tr><tr><td>#2</td><td>-</td><td>✓</td><td>-</td><td>-</td><td>0.4021</td><td>0.2690</td></tr><tr><td>#3</td><td>✓</td><td>✓</td><td>-</td><td>-</td><td>0.4037</td><td>0.2767</td></tr><tr><td>#4</td><td>-</td><td>-</td><td>✓</td><td>-</td><td>0.4022</td><td>0.2570</td></tr><tr><td>#5</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>0.4004</td><td>0.2843</td></tr><tr><td>#6</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>0.4050</td><td>0.2870</td></tr></table>
+
+positions. As shown in Table 5, in terms of in- and out- of distribution performance, inserting Amplitude Transfer in Position 0-3 achieves both the best performance.
+
+Table 5. Effects of different inserted positions of Amplitude Transfer. Inserted position of ResNet is numbered as: after first Conv 0, after Max Pooling layer 1, after first Resblock 2, after second Resblock 3, after third Resblock 4 and after fourth Resblock 5. "0-5" means inserting Amplitude Transfer from Position 0 to Position 5.  
+
+<table><tr><td>Model</td><td>Clean</td><td>OoD Avg.</td></tr><tr><td>BEVFormer</td><td>0.4362</td><td>0.3028</td></tr><tr><td>0-5</td><td>0.4404</td><td>0.3267</td></tr><tr><td>0-4</td><td>0.4393</td><td>0.3289</td></tr><tr><td>0-3</td><td>0.4421</td><td>0.3294</td></tr><tr><td>0-2</td><td>0.4404</td><td>0.3280</td></tr></table>
+
+# 4.4. Hyperparameters Analysis
+
+In Frequency Jitter, there are three hyper-parameters. The hyperparameter  $\eta$  controls the strength of Amplitude augmentation;  $\lambda$  controls the strength of Phase augmentation and  $p_i$  is the provability of implementing Frequency Jitter. For Amplitude Transfer, we experiment on the probability  $p_f$  of implementing Amplitude Transfer. As shown in Fig.4(c), initially, as the probability and intensity increase,
+
+the out-of-domain performance gradually improves. After reaching a certain level of probability and intensity, further changes in the parameters will no longer cause drastic changes in ood performance, indicating that the model is stable against hyper-parameter misspecifications as long as the hyper-parameters are within reasonable ranges. We set  $\eta = 0.25$ ,  $\lambda = 0.5$ ,  $p_i = 0.25$  and  $p_f = 0.75$  as the final setting.
+
+The effect of Fourier Cross-View Semantic Consistency Loss alone are shown in Table 6.
+
+Table 6. The effect of Fourier Cross-View Semantic Consistency Loss alone.  $\gamma$  is the weight of  $\mathcal{L}_{\mathrm{cross}}$ .  
+
+<table><tr><td>Model</td><td>Cam Crash</td><td>Frame Lost</td><td>Color Quant</td><td>Motion Blur</td><td>Bright</td><td>Low Light</td><td>Fog</td><td>Snow</td><td>OoD Avg.</td></tr><tr><td>BEVDet</td><td>0.2508</td><td>0.1955</td><td>0.2409</td><td>0.2201</td><td>0.2591</td><td>0.1112</td><td>0.2633</td><td>0.0728</td><td>0.2017</td></tr><tr><td>γ = 0.5</td><td>0.2487</td><td>0.1942</td><td>0.2444</td><td>0.2132</td><td>0.2583</td><td>0.1328</td><td>0.2635</td><td>0.0641</td><td>0.2024</td></tr><tr><td>γ = 1.0</td><td>0.2501</td><td>0.1952</td><td>0.2785</td><td>0.2882</td><td>0.2890</td><td>0.1407</td><td>0.2807</td><td>0.1140</td><td>0.2296</td></tr><tr><td>γ = 2.0</td><td>0.2462</td><td>0.1932</td><td>0.2806</td><td>0.2863</td><td>0.2872</td><td>0.1340</td><td>0.2803</td><td>0.1147</td><td>0.2278</td></tr></table>
+
+# 4.5. Efficiency Analysis
+
+In this part, we make efficiency analysis to delve into the proposed FCVL. We investigate how the method scales with increasing image resolution and computational complexity. The results are listed in Table 7. As it can seen, at larger image scales, FCVL can still significantly improve the model's generalization performance. Though there will be a slight increase in training time (+0.11s per training step) during the training phase, as the proposed FCVL is only used during the training phase, it introduces no latency in inference phase. Without the need for more time-consuming or costly data collections, the proposed FCVL can improve the generalization performance almost for free, which is highly valuable for practical applications.
+
+# 4.6. Evaluation in Real-world Autonomous Driving Scenarios
+
+To further evaluate the performance of our algorithm in practical application scenarios, we test our method in the night with the model training on daylight samples only.
+
+Table 7. Efficiency analysis of FCVL. "Train" refers to the time it takes for one training step when the batch size is 1. "Inference" refers to the time for inferring a single sample. "Memory" is the consumed GPU memory during training with batch size 1.  
+
+<table><tr><td>Model</td><td>Resolution</td><td>Train(s)</td><td>Inference(s)</td><td>Memory(MB)</td><td>OoD Avg.</td></tr><tr><td>BEVDet</td><td>256 × 704</td><td>0.257</td><td>0.073</td><td>5498</td><td>0.2017</td></tr><tr><td>+FCVL</td><td>256 × 704</td><td>0.364</td><td>0.073</td><td>7383</td><td>0.2677(↑ 6.60%)</td></tr><tr><td>BEVDet</td><td>512 × 1408</td><td>0.482</td><td>0.143</td><td>11698</td><td>0.2006</td></tr><tr><td>+FCVL</td><td>512 × 1408</td><td>0.605</td><td>0.143</td><td>20094</td><td>0.2394(↑ 3.88%)</td></tr></table>
+
+More results can be found in Appendix D.3. This example in Fig.5 well demonstrates that our model can robustly deal with rapid environmental changes, such as variations in lighting conditions. As it can be seen, in the distance where vehicles are dense and the lighting is very strong, the model can stably detect the targets. Although the model has only seen normal daylight samples, with our proposed FCVL, it also performs well under the extreme changes in light condition at night.
+
+![](images/b41331c2e56b2d5c2627397098dd2231e799c8887f201db9399383d675672d66.jpg)  
+Figure 5. Visualized detection results of FCVL at night with light variations. The model is trained on only daylight samples with the proposed FCVL.
+
+# 4.7. Visualization Analysis with t-SNE
+
+We use t-SNE to visualize the BEV features from different domains of BEVDet and FCVL. In the Fig.6, source domain is represented in red and other colors represent different target domains. We can find that the features of different domains extracted from BEVDet are distant from each other and loosely distributed in the feature space. While, after optimization with FCVL, the distribution of four domains becomes more compact and connected, which is in line with augmentation graph theory. FCVL increases the augmentation graph connectivity between source and unseen domains and improve the generalization a lot.
+
+![](images/8b81a602148a5a89306dc6d7de89c6e877f452f7c3ec5e80aec026bf7cc9d5ae.jpg)  
+Figure 6. t-SNE Visualization of FCVL.
+
+# 5. Conclusion
+
+In conclusion, this paper addresses the challenge of Single Domain Generalization in multi-camera 3D object detection via Fourier Cross-View Learning framework. We propose a non-parametric Fourier Hierarchical Augmentation at both image and feature levels to enhance data diversity and Semantic Consistency Loss to facilitate model to learn more domain-invariant features from adjacent perspectives. Besides, via augmentation graph theory, we make valid theoretical guarantees. Extensive experiments have demonstrated that our approach achieves the best performance across various domain generalization methods.
+
+Limitations There are several hyperparameters to be tuned in FCVL. In the future work, we can explore additional techniques to avoid spending too much time on tuning hyperparameters. Additionally, for Snow, we have already improved by  $10\%$ , but the performance is still much worse compared to the performance in other scenarios such as Low Light. Consequently, there is a substantial potential for enhancement in adverse weather conditions.
+
+# Acknowledgements
+
+Thanks to all reviewers for your professional and insightful review comments. This work is supported by National Natural Science Foundation of China (Grant No.62106139).
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning and Autonomous Driving. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
+# References
+
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+
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+
+# A. Theoretical Analysis
+
+To analyze the influence of data augmentation, we adopt the standard augmentation graph framework (HaoChen et al., 2021; Wang et al., 2024), where data augmentations induce interactions (as edges) between training samples (as nodes). Given a natural data sample  $\overline{x} \in \overline{\mathcal{X}}$ , we use  $\mathcal{A}(\cdot|\overline{x})$  to denote the distribution of its augmentations. For any two augmented data  $x, x' \in \mathcal{X}$ , define the adjacency matrix  $W_{xx'}$  as the marginal probability of  $x$  and  $x'$  from a random natural data  $\overline{x} \sim \mathcal{P}_{\overline{\mathcal{X}}}$ :
+
+$$
+W _ {x x ^ {\prime}} = \mathbb {E} _ {\bar {x} \sim \mathcal {P} _ {\bar {\chi}}} [ \mathcal {A} (x | \bar {x}) \mathcal {A} \left(x ^ {\prime} | \bar {x}\right) ]. \tag {15}
+$$
+
+Let  $\mathbb{L} = I - D^{-\frac{1}{2}}WD^{-\frac{1}{2}}$  be the normalized graph Laplacian matrix, where  $D$  is a diagonal degree matrix with the  $(x,x)$ -th diagonal element as  $D_{xx} = \sum_{x'}W_{xx'}$ .
+
+Based on the above augmentation graph framework, we construct the augmentation graph  $G(\mathcal{X}, \overline{\mathcal{X}}, W)$  in the feature space for single source domain and augmented domains as shown in Fig.1(c). Then, we have the following theorem:
+
+Theorem A.1. For the optimal encoder  $f^{*}$ , BEV projection module  $P_{BEV}^{*}$ , a learned classification head  $C^{*}$  and regression head  $R^{*}$  on augmented data  $\mathcal{X}$ , its linear probing error has the following generalization upper bound,
+
+$$
+\mathcal {E} \left(f ^ {*}, P _ {B E V} ^ {*}, C ^ {*}, R ^ {*}\right) \leq \frac {2 \alpha}{\lambda_ {k + 1}} + 4 \Delta \left(y _ {c}, \hat {y _ {c}}\right) + 4 \Delta \left(y _ {r}, \hat {y _ {r}}\right), \tag {16}
+$$
+
+where  $\alpha$  denotes the labeling error caused by data augmentation;  $\lambda_{k+1}$  denotes the  $k+1$ -th smallest eigenvalue of the Laplacian matrix  $\mathbb{L}$ ;  $\Delta(y_c, \hat{y}_c)$  denotes the average disagreement between  $\hat{y}_c$  and the ground-truth labeling  $y_c$  for classification;  $\Delta(y_r, \hat{y}_r)$  denotes the average disagreement between  $\hat{y}_r$  and the ground-truth labeling  $y_r$  for regression.
+
+Based on the generalization upper bound in Eq.16 (proof in AppendixA.1), we can provide rigorous explanations to show that our method can increase graph connectivity  $\lambda_{k + 1}$  and reduce label error  $\alpha$  to decrease the generalization loss.
+
+First, as shown in Fig.1(c), as we can only get access to the single source data, the connectivity of the graph is poor and only a few feature points are connected. There is a large margin between the source and target domains. The proposed FHiAug plays a positive role in expanding graph connectivity  $\lambda_{k + 1}$ , since it creates more diverse "middle" domains between single source data and unseen target domains. According to augmentation graph theory, with the increase of augmentation strength, the graph connectivity  $\lambda_{k + 1}$  can be increased. Via increasing  $\lambda_{k + 1}$  in Eq.16, the generalization upper bound can be tighter and the generalization ability can be improved.
+
+However, common strong augmentation, such as strong geometric enhancement, also causes label error (larger  $\alpha$  in Eq.16) and increases the generalization loss. The proposed FHiAug augmenting in the frequency domain can effectively alleviate this issue. Next, we will provide a theoretical analysis and show that FHiAug can ensure semantic consistency under strong augmentation strength to increase connectivity. As mentioned in Sec.3, input data  $\mathcal{X}$  can be decomposed into two components: phase  $\mathcal{X}_p$ , and amplitude  $\mathcal{X}_a$ , where  $\mathcal{X}_p$  contains semantic information about the label  $y$ , denoting the causal component, and  $\mathcal{X}_a$  contains more low-level information, denoting the non-causal components.
+
+Assumption A.2. We assume the linear relationship between  $\mathcal{X}_p$  and  $y$ ,
+
+$$
+y = \mathcal {X} _ {p} \phi + \epsilon , \tag {17}
+$$
+
+where  $\epsilon$  is the noise,  $\operatorname{Cov}(\mathcal{X}_p,\epsilon) = 0$ ,  $\mathbf{E}[\mathcal{X}_p] = 0$ .
+
+Theorem A.3. If input data  $\mathcal{X}$  consists of all the phase components,  $\mathcal{X} = \mathcal{X}_p$ , the optimal linear predictor  $\phi$  can be estimated without bias. Otherwise, the predictor  $\phi$  is biased.
+
+For some style transferring methods in pixel domain, both phase and amplitude components are modified. In this situation, the predictor  $\phi$  is biased, which means that the predictor probably gives wrong prediction of label. While the proposed method FHiAug augments in the frequency domain and retains the phase congruency, avoiding label error effectively. At image level, Frequency Jitter only adjusts the intensity of phase component in the global. The distribution of semantic information is not changed. At feature level, we achieve style transfer with operating on amplitude component only. More proof for Theorem A.3 is in Appendix A.2.
+
+# A.1. Proof for TheoremA.1
+
+Lemma A.4. Let  $G = (\mathcal{X}, W)$  be the augmentation graph,  $r$  be the number of underlying classes. There exists an extended labeling function  $\hat{y}$  such that
+
+$$
+\phi^ {\hat {y}} = \sum_ {x, x ^ {\prime} \in \mathcal {X}} W _ {x x ^ {\prime}} \cdot \mathbb {I} [ \hat {y} (x), \hat {y} \left(x ^ {\prime}\right) ] \leq 2 \alpha . \tag {18}
+$$
+
+Lemma A.5. (Theorem B.3 (HaoChen et al., 2021)). Assume the set of augmented data  $\mathcal{X}$  is finite. Let  $f^{*}$  be the optimal encoder. Then, for any labeling function  $\hat{y}:\mathcal{X}\gets [r]$ , there exists a linear probe  $B^{*}$  such that
+
+$$
+\mathbb {E} _ {\bar {x} \sim \mathcal {P} _ {x \sim \mathcal {A} (\cdot | \bar {x})}} = \left[ \| y (\bar {x}) - B ^ {*} f ^ {*} (x) \| _ {2} ^ {2} \right] \leq \frac {\phi^ {y}}{\lambda_ {k + 1}} + 4 \Delta (y, \hat {y}), \tag {19}
+$$
+
+where  $\lambda_{k + 1}$  denotes the  $k + 1$ -th smallest eigenvalue of the Laplacian matrix  $\mathbb{L}$ ;  $\Delta(y, \hat{y})$  denotes the average disagreement between  $\hat{y}$  and the ground-truth labeling  $y$ .
+
+According to above lemmas, for detection task, the optimal encoder  $f^{*}$ , BEV projection module  $P_{\mathrm{BEV}}^{*}$ , a learned classification head  $C^{*}$  and regression head  $R^{*}$  on augmented data  $\mathcal{X}$ , its linear probing error has the following generalization upper bound,
+
+$$
+\begin{array}{l} \mathcal {E} \left(f ^ {*}, P _ {\mathrm {B E V}} ^ {*}, C ^ {*}, R ^ {*}\right) \leq \frac {\phi^ {y _ {c}}}{\lambda_ {k + 1}} + 4 \Delta \left(y _ {c}, \hat {y} _ {c}\right) + 4 \Delta \left(y _ {r}, \hat {y} _ {r}\right) \tag {20} \\ \leq \frac {2 \alpha}{\lambda_ {k + 1}} + 4 \Delta \left(y _ {c}, \hat {y _ {c}}\right) + 4 \Delta \left(y _ {r}, \hat {y _ {r}}\right), \\ \end{array}
+$$
+
+where  $\alpha$  denotes the labeling error caused by data augmentation;  $\Delta(y_c, \hat{y}_c)$  denotes the average disagreement between  $\hat{y}_c$  and the ground-truth labeling  $y_c$  for classification;  $\Delta(y_r, \hat{y}_r)$  denotes the average disagreement between  $\hat{y}_r$  and the ground-truth labeling  $y_r$  for regression.
+
+# A.2. Proof for Theorem A.3
+
+The optimal linear predictor for  $y = \mathcal{X}_p\phi + \epsilon$  is
+
+$$
+\begin{array}{l} \phi^ {*} = \operatorname {a r g m i n} \left[ \left(y - \mathcal {X} _ {p} \phi\right) ^ {T} \left(y - \mathcal {X} _ {p} \phi\right) \right] (21) \\ = \left(\mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \mathcal {X} _ {p} \right]\right) ^ {- 1} \mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} y \right] (22) \\ = \phi + \left(\mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \mathcal {X} _ {p} \right]\right) ^ {- 1} \left(\mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} y \right] - \mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \mathcal {X} _ {p} \right] \phi\right) (23) \\ = \phi + \left(\mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \mathcal {X} _ {p} \right]\right) ^ {- 1} \mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \left(y - \phi^ {T} \mathcal {X} _ {p}\right) \right] (24) \\ = \phi + \left(\mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \mathcal {X} _ {p} \right]\right) ^ {- 1} \mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \epsilon \right] (25) \\ = \phi + \left(\mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \mathcal {X} _ {p} \right]\right) ^ {- 1} \left(\mathbf {E} \left[ \mathcal {X} _ {p} ^ {T} \right] \mathbf {E} [ \epsilon ] + \operatorname {C o v} \left(\mathcal {X} _ {p}, \epsilon\right)\right) (26) \\ = \phi (27) \\ \end{array}
+$$
+
+If input data  $\mathcal{X}$  is deteriorated due to data augmentation in pixel domain, the phase components,  $\mathcal{X}_p = \mathcal{X}_{p^-}$ , is no longer a distribution with  $\mathbf{E}[\mathcal{X}_{p^{-}}] = 0$ . Then, the predictor  $\phi$  is biased:
+
+$$
+\phi^ {*} = \phi + \left(\mathbf {E} \left[ \mathcal {X} _ {p ^ {-}} ^ {T} \mathcal {X} _ {p ^ {-}} \right]\right) ^ {- 1} \left(\mathbf {E} \left[ \mathcal {X} _ {p ^ {-}} ^ {T} \right] \mathbf {E} [ \epsilon ]\right) \tag {28}
+$$
+
+# B. Algorithm
+
+The algorithm of the proposed method is illustrated in 1.
+
+Algorithm 1 The proposed algorithm (FCVL)  
+Input: Training data  $(x,y)$ , detector network  $f$  with parameter  $\theta$ , learning rate  $\beta$ , probability  $p_i$  to do Frequency Jitter, probability  $p_f$  to do Amplitude Transfer.  
+Output: The optimized network parameter  $\theta^*$ .  
+1: while  $t \leq T$  do  
+2: # Fourier-based data augmentation at image level.  
+3: Sample  $p_0 \sim U(0,1)$   
+4: for  $(x,y)$  do  
+5: if  $p_0 \leq p_i$  then  
+6: Perform Frequency Jitter according to Eq. 3.  
+7: Obtain augmented image  $\hat{x}$  according to Eq. 4.  
+8: else  
+9:  $\hat{x} \gets x$   
+10: end if  
+11: end for  
+12: # Fourier-based domain perturbation at feature level.  
+13: Sample  $p_1 \sim U(0,1)$   
+14: for intermediate features  $X$  do  
+15: if  $p_1 \leq p_f$  then  
+16: Perform Amplitude Transfer according to Eq.6 - Eq.9.  
+17: Obtain perturbed features  $\hat{X}$  according to Eq.4.  
+18: else  
+19:  $\hat{X} \gets X$   
+20: end if  
+21: end for  
+22: #Cross-view Semantic Consistency Loss.  
+23:  $\theta \gets \theta - \beta \cdot \nabla_{\theta} \mathcal{L}_{train}((\hat{x},y),\hat{X};\theta)$ ;  
+24: end while  
+25: return Save the optimized network  $f(\theta^*)$ .
+
+# C. More Analysis of Common Data Augmentation Techniques for Single Domain Generalization
+
+In SDG for 2D image classification, previous works (Zhao et al., 2024; Qiao et al., 2020) aim to enhance data diversity using common 2D data augmentation techniques, such as geometric transformations, style transfer, or data generation. However, directly applying these approaches to BEV-based tasks introduces several challenges.
+
+First, BEV representations are generated by projecting multi-view 2D features using real-world physical constraints, which limits the use of strong geometric transformations, such as 270-degree rotations, as they would disrupt the spatial consistency of the BEV space. We add a strong geometric enhancement experiment including significant rotation and translation on the image and the results are as follows in the Table 8. For one thing, it shows that strong geometric enhancement hurts in-domain performance, as large scale rotation or translation may destroy physical restraints in the real driving scenario. For another, geometric enhancement is not very effective in improving OoD performance.
+
+Table 8. Strong geometric enhancement experiments including significant rotation and translation on the images.  
+
+<table><tr><td>Model</td><td>Clean</td><td>Cam Crash</td><td>Frame Lost</td><td>Color Quant</td><td>Motion Blur</td><td>Bright</td><td>Low Light</td><td>Fog</td><td>Snow</td><td>OoD Avg.</td></tr><tr><td>BEVDet</td><td>0.3880</td><td>0.2508</td><td>0.1955</td><td>0.2409</td><td>0.2201</td><td>0.2591</td><td>0.1112</td><td>0.2633</td><td>0.0728</td><td>0.2017</td></tr><tr><td>strong geo</td><td>0.3505</td><td>0.2338</td><td>0.1875</td><td>0.2249</td><td>0.2030</td><td>0.2371</td><td>0.1188</td><td>0.2511</td><td>0.0639</td><td>0.1900</td></tr></table>
+
+Second, some style transfer techniques (Zhao et al., 2024; Nuriel et al., 2021) replace the original image statistics with those from the target style, but this often blurs the boundary between style and content (Lee et al., 2023), distorting important features and ultimately harming model generalization. These methods need to remove the "style" in the pixel domain first.
+
+Some content cues will be removed inevitably.
+
+Third, data generation methods including adversarial generation and diffusion-based techniques. Training a Generative Adversarial Network (GAN) (Goodfellow et al., 2020) involves a competitive process between two neural networks: the generator and the discriminator. Adversarial generation can suffer from unstable training and mode collapse. It often requires a lot of experiments and fine-tuning to get a GAN to work well. While diffusion-based techniques (Ho et al., 2020) are more stable, they need significant computational and storage overhead, making them impractical for complex 3D detection models. Additionally, although we can spend much time generating a large number of samples, we would also require extra storage space. However, our method involves online augmentation and does not require any additional storage space.
+
+Therefore, common 2D data augmentations cannot be effectively leveraged to create diverse training samples for BEV-based tasks.
+
+Besides, we further clarify the differences between our method and other frequency-domain approaches. Compared with these methods (Xu et al., 2021; Lv et al., 2022; Kim et al., 2023), our method has major strengths in two aspects including accuracy and efficiency. Firstly, in the setting of single source data, our proposed method can enhance the generalization ability of the detectors by large margin. FACT (Xu et al., 2021) mixes up two different domains' data in frequency, e.g. Cartoon and Photo from dataset PACS and achieves great OOD performance in the paper. But when training with only single domain, FACT can only mix the samples within the single domain and it indeed improves the in-domain clean set a little, but the improvement of OOD sets is very slim in the single domain setting. Different from FACT, we first propose Frequency Jitter at image level to create diverse samples that are complementary to the single source domain. Then, at feature level, we introduce a novel method Amplitude Transfer to achieve style transfer without content distortion. Through uncertainty estimation, we can obtain uncertain feature statistics, which can gradually shift the features to more diverse domains through continuous training. Secondly, due to the high complexity of BEV-based 3D object detection models, our plug-and-play and non-parameter data augmentation method can achieve better generalization results more efficiently. CIRL (Lv et al., 2022) generates augmented images by a causal intervention module with intervention upon non-causal factors. AGFA (Kim et al., 2023) trains the classifier and the amplitude generator adversially to synthesise a worst-case domain for adaptation. Compared with these methods, our proposed method is simple, stable, yet effective without extra module designing or special training strategies.
+
+# D. More Experimental Results
+
+# D.1. Datasets
+
+To verify different methods' single domain generalization ability, we first utilize nuScenes (Caesar et al., 2020) as the single training source and nuScenes-C (Xie et al., 2023) as the testing sets. NuScenes-C is a comprehensive dataset that encompasses eight distinct corruptions, including Bright, Dark, Fog, Snow, Motion Blur, Color Quant, Camera Crash, and Frame Lost. Each type of corruption has three different levels of corruption intensity (i.e., easy, moderate, and hard). These eight corruptions include different weather conditions, different light conditions, potential equipment damage situations. These scenarios are common out-of-distribution problems in real-world application. We use eight distinct corruptions as our multi testing domains to evaluate the effectiveness of different DG methods.
+
+In addition, we experiment on public dataset Argoverse 2 (Wilson et al., 2023). The Argoverse contains different driving scenarios across six major U.S. cities (Miami, Washington D.C. and so on), including various weather conditions such as sunny days and cloudy conditions. To adhere to the single domain to multi domain generalization paradigm, we take sunny-day data from Miami as the single-domain training set, while sunny-day data from other cities (with diverse urban road structures) as the first ood test set (City), and cloudy (dim lighting) data from other cities as the second ood test set (Cloudy).
+
+# D.2. Experiments with Random Seed
+
+In early experiments, we find that the effect of random seeds on BEVDepth or BEVFormer is relatively small. We run our method FCVL three times on BEVDepth and the average NDS on clean testing set is  $0.4004 \pm 0.0002$ ; the average NDS of OoD sets is  $0.2845 \pm 0.0003$ . The standard deviation for three trials is 0.0002 or 0.0003, which means the method is quite robust to different seeds. Thus, in later experiments, we run our method with random seed.
+
+# D.3. Evaluation in Practical Application Scenarios
+
+To further evaluate the performance of our algorithm in practical application scenarios, we collect a large dataset consisting of sunny daytime and nighttime. We train the detection model with our method on 61716 samples of sunny daytime and test on daytime (6169) and night (8200) samples. More results can be found in Table 9. Notably, on the night testing set, FCVL can improve the mAP from 0.0420 to  $0.1004(\uparrow 5.84\%)$ .
+
+Table 9. Evaluation results (mAP ↑) in the real-world autonomous driving scenarios.  
+
+<table><tr><td>Model</td><td>Daytime</td><td>Night</td></tr><tr><td>Baseline</td><td>0.2690</td><td>0.0420</td></tr><tr><td>+FCVL(Ours)</td><td>0.2687</td><td>0.1004(↑ 5.84%)</td></tr></table>
+
+# E. Visualization Analysis
+
+# E.1. Visualization of Proposed FHiAug
+
+In this section, we visualize diverse styles generated via FHiAug. Visualization of Frequency Jitter at image level and visualization of style variations via Amplitude Transfer are shown in Fig.7, 8 and 9. To better illustrate the phase component, we provide more examples of changing phase components in Fig.10.
+
+![](images/647a397e67b17245589ed8e35657c6d06c024d84e17d4883b5ee1bc7de8391b5.jpg)  
+Figure 7. Frequency Jitter at image level. The top line is adjusting the amplitude. It mainly influences the image brightness. The bottom line is adjusting the phase. As it can be seen, main structures of different targets have been retained.
+
+![](images/7cad8963c606178f637a18456bb7fb0409f4386018cc10f9999150fb78772a26.jpg)  
+Figure 8. Amplitude Transfer at feature level. The left cols are original pictures. The other two cols are styled pictures.
+
+![](images/1489a084b71a5518f9b8ef47b985e2ef78671bf42a04f250fd40bd8cfac08167.jpg)
+
+![](images/446476c1a574dfbc6ae03ad456402c0ee29d9a1698e9bd197546e3272ac2a2ee.jpg)
+
+![](images/341c4de449e567008c3a8f3103d7769fb8783936061bbd9a56398c20c90a310b.jpg)
+
+![](images/962aa9cdf1803c548a532cfae53910ac65c9923e88d84525257653775b05e47d.jpg)
+
+![](images/3cf31c722e1c956b2ad80f2f00c5022aeb6dc73c02cd4fe56101977939a7f500.jpg)
+
+![](images/9af5c6cef30ee6e210ed328fbd60cf4880a82c7b1f73d38d1dcb45e6b8e3aec1.jpg)
+
+![](images/deb7c73f9c7762195b21c506a3b6add43104defc82488926d774e5d5c8f45d84.jpg)  
+Original Img
+
+![](images/17eaed9dc32b3db06a7367c078ac74920838e620bf911a1c798a07a627c2e8b8.jpg)  
+FHi Aug
+
+![](images/34062eff67acfe8623345a398f1bf9666bc79bf8603444e7c6d685348d204a36.jpg)
+
+![](images/d21020237fbd841a5142d95763bd5c996a1d9ea535be5ccf38e4f916e7bc3856.jpg)  
+Figure 9. Visualized results of FHiAug. For the same image, via FHiAug, we can generate multiple samples, which can facilitate the model to learn more domain invariant feature.  
+Original Img
+
+![](images/025a3fbde7def7b7b4ea073242b7079c241ad0d71674b2407bd8b0c19c940d4b.jpg)  
+Image reconstructed with phase component only
+
+![](images/ec04a1c475c07ee5b177683764d0c9ffb29b0c6bf36c820e522c4f87ccff1a7c.jpg)  
+Phase \*0.5  
+Figure 10. The sample on the top right is the image reconstructed with the phase only. As it can be seen, the phase components mainly contain the semantic information. The images in the bottom show how the image changing when adjusting the strength of phase components only. The image after the phase changing is similar to dirty lenses and weather changes in real world, still preserving key BEV prediction information.
+
+![](images/9a61e381af1ee1b3ca70e025fdec3156d3bc02c196574241c7c75bf546514775.jpg)  
+Phase \*0.75
+
+# E.2. More Visualized Detection Results
+
+Notably, FCVL greatly improves the performance for Snow. We visualize some detection results of these samples to compare the performance between baseline models and FCVL in Fig.11. Under the condition of Snow, baseline model misses detecting the small targets severely, while FCVL can greatly alleviate the problem of missing detection. Compared with CPerb, FCVL still shows more stable and more accurate localization and recognition ability.
+
+![](images/98b274bae3ca94c25573233f1bcccd5cb99389b6a71587dd87feae1267273fe6.jpg)
+
+![](images/2bbe3a8588469f7680d359c80293c5ac0697c2e0e4b81f4891e0f695ed42810f.jpg)  
+Ground Truth
+
+![](images/00e7e76d63b9349553787fb637857eecea12ab5d572b30fd94d197f998b6f548.jpg)
+
+![](images/ce5890b0c203127c2c7da46b635afbacde353d618d68dc4a2ff7e8cd4e657b67.jpg)  
+Baseline
+
+![](images/e9e9232fbb09d4c39fe3f21c44b62fb0a299c0311d3d75a4b6dfde0cb49fa497.jpg)
+
+![](images/14022bb73d88a9ed83e9f10b7e65f9c83f6f86f90ea401bac3d7225dfb6799b9.jpg)  
+CPerb  
+Figure 11. Visualized detection results of baseline and FCVL from Snow set.
+
+![](images/fd7de2ca04620d708113a661f0f55f7a1ddca034b86448461fc316e50b5649a2.jpg)
+
+![](images/e38a3ac7ce3426501444ae0f18d0702ecb4b558cff02e756e0e3b70ea95a7d2d.jpg)  
+FCVL
+
+![](images/5378b2cbc8d70e5a4ad3d321a871cf7469237c382164ba42b75f8592ffae78a3.jpg)
+
+![](images/e5fe307f2feb64910cd75ab9af4ed6900bdb621e3ade2af62b8726ba245cba6a.jpg)  
+Baseline  
+Figure 12. Visualized detection results of baseline and FCVL on Argoverse 2.
+
+![](images/c13887ab1266e9d405cbc9d81c6b44d10edc6c0bf9b1fb12e03e7e3d95897d3e.jpg)
+
+![](images/cf775745966859bacc3fedd64878981eba911f9d61a8d4cc046954c223a17e1e.jpg)  
+Baseline+FCVL

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