Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robust SAR Target Recognition

Comment1. The experimental validation is limited to a single dataset (MSTAR).

Answer1: Thanks for pointing out this important concern. We acknowledge this limitation. Unlike optical imagery, SAR data is challenging to obtain at scale due to sensor costs, operational constraints, and the factor of security sensitivities. Therefore, most studies [R3-R6] rely solely on MSTAR for evaluation. To enhance experimental validation, we construct a dataset based on SAR-AIRcraft-1.0, a widely used public SAR aircraft detection dataset. Compared to MSTAR, SAR-AIRcraft-1.0 offers richer data diversity and more complex operational scenarios. We extract individual targets from the original imagery to create a dedicated SAR ATR dataset. Below, we provide a detailed description of its composition.

Since SAR-AIRcraft-1.0 lacks diverse and separable imaging angles [R7-R9], a standard target recognition strategy and conducted experiments are implemented. Specifically, 80% of the data was used for training, and the remaining 20% for testing. The experimental results are outlined below:

Compared to the MSTAR dataset, SAR-AIRcraft-1.0 is lack of imaging angle annotations, and thus the angle-related constraints are eliminated in fact. Consequently, in this experiment on SAR-AIRcraft-1.0, most of the evaluated networks exhibit superior performance. But notably, existing mainstream networks incorporating our ΓPCA consistently outperform their original backbones, even under this less constrained evaluation scenario. These results validate the generalizability of our method across diverse data characteristics and scenarios.

Reference:

[R3] Huang, Z., et al. Physics inspired hybrid attention for SAR target recognition. ISPRS Journal of Photogrammetry and Remote Sensing, 2024.
[R4] Zhang, L., et al. Optimal azimuth angle selection for limited SAR vehicle target recognition. International Journal of Applied Earth Observation and Geoinformation, 2024.
[R5] Wang, R., et al. MIGA-Net: Multi-view image information learning based on graph attention network for SAR target recognition. IEEE Transactions on Circuits and Systems for Video Technology, 2024.
[R6] Li, W., et al. Hierarchical disentanglement-alignment network for robust SAR vehicle recognition. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 2023.
[R7] Zhirui, W., et al. SAR-AIRcraft-1.0: High-resolution SAR Aircraft Detection and Recognition Dataset. Journal of Radars, 2023.
[R8] Huang, B., et al. Scattering Enhancement and Feature Fusion Network for Aircraft Detection in SAR Images. IEEE Transactions on Circuits and Systems for Video Technology, 2024.
[R9] Zhou, J., et al. DiffDet4SAR: Diffusion-based aircraft target detection network for SAR images. IEEE Geoscience and Remote Sensing Letters, 2024.

Suggestions and writing issues:

Answer2: Thank you for your careful review and valuable suggestions. We will thoroughly proofread and incorporate these revisions into the final version of the paper.

We sincerely appreciate the reviewer’s valuable comments. We will enhance the related work and include more experimental results in our final version. Below are our detailed responses to each point.

Q1. MSTAR is small.

A1: We constructed a new dataset based on SAR-AIRcraft-1.0. A detailed description of the dataset and the experimental results can be found in Answer1 for Reviewer Otus.

Q2. SAR-specific baselines.

A2: We incorporat the ASC-based method MSNet-PIHA [R10] into our experiments. The comparison results are shown below.

MSNet-PIHA performs well when trained on diverse azimuth angles [R10] but struggles with limited-angle data, which can lead to overfitting and suboptimal predictions. While ASC information enhances feature extraction with limited data, severe azimuth constraints remain challenging. In contrast, ΓPCA demonstrates better angle robustness.

[R10] Huang, Z., et al. Physics inspired hybrid attention for SAR target recognition. ISPRS, 2024.

Q3. Statistical significance tests.

A3: We conduct significance tests by repeating all experiments in our paper three times and reported the mean and variance to reduce the influence of randomness. Part of the updated experimental results are presented below.

Table R1. Partially updated recognition results of Table 1.

More results can be found at: https://anonymous.4open.science/r/ICML2025-E1F8.

Q4. Theoretical proof of ΓPCA is better than PCA.

A4: Previous work in [R11] has proven that generalized PCA based on exponential family distributions theoretically outperforms traditional PCA. As a variant, our ΓPCA, based on the Gamma distribution, inherits this theoretical advantage.

[R11] Landgraf, A. J., et al. Generalized principal component analysis: Projection of saturated model parameters. Technometrics, 2020.

Q5. Theoretical link between angle-robustness and ΓPCA projections.

A5: Under different orientation observation angles, the statistics of the same class of SAR targets are similar. Base on this characteristic, ΓPCA assumes that the SAR target image follows the Gamma distribution, and then derives ΓPCA projection which can extract the principal component information, i.e., the low-rank information, of these SAR targets. Then the projection matrix is used to construct the convolution kernel to extract the SAR target features, which are characterized by a low sensitivity to angle.

Q6. Comparison of different components.

A6: We provide an isolated comparison of different components (see Table 3, page 8, PCA-ViT vs. ΓPCA-ViT). As demonstrated in Table 3, ΓPCA consistently outperforms conventional PCA with ViT backbone for SAR data. For example, with a kernel size of 17, PCA-ViT and ΓPCA-ViT achieve accuracies of 73.52% and 76.01%, respectively, demonstrating its effectiveness in non-Gaussian SAR data analysis.

Q7. Effects of different settings.

A7: Kernel size analysis is presented in Sec. 4.3 (Page 8, Left Column, Lines 410–434), with quantitative comparisons in Table 3. To isolate the impact of ΓPCA, all experiments exploit the same training settings (optimizer and learning rate: the default configurations of the original backbones, batch size: 32), ensuring a fair comparison within the scope of the paper on fundamental ΓPCA theory for SAR data.

Q8. Real-World Relevance.

A8: We conducted experiments under various noise conditions, focusing on multiplicative noise, i.e. speckle, the most common noise type in SAR image. Noise levels are quantified by the Equivalent Number of Looks (ENL), defined as: μ^2/σ^2, where μ and σ are mean and standard deviation of pixel values. A higher ENL value indicates lower noise.

More results are available at: https://anonymous.4open.science/r/ICML2025-E1F8. These results show the anti-noise performance of ΓPCA, attributed to its ability to extract principal components while filtering out noise-dominated ones.

Our study primarily addresses the challenge of angle-induced performance degradation in SAR ATR. Extensive experiments validate the dual robustness of ΓPCA to angular variations and speckle noise. While sensor types and imaging artifacts are also important for SAR ATR, these aspects fall beyond the scope of our current work, as they pose distinct research challenges. We will further clarify the influence of these two aspects in the final version.

We sincerely appreciate your positive feedback on our work and are truly grateful for the time and effort you have dedicated to reviewing our manuscript. Should you have any additional comments or suggestions, we would be delighted to engage in further discussion and improvement.

Comment1. Dependence on Gamma distribution assumptions (may not hold for all targets/clutter).

Answer1: We appreciate this insightful observation regarding the distributional assumption. To provide clarification:

1. Theoretical and empirical validation: The applicability of Gamma distribution to SAR image statistics is theoretically grounded and empirically validated, as evidenced by [R1, R2]. These works demonstrate its effectiveness in capturing the multiplicative speckle noise and scattering characteristics inherent to SAR systems.
2. Scope of the Gamma assumption: Our method adopts the Gamma distribution to model the reflection intensity of the entire SAR image (encompassing targets and clutter collectively). Of course, statistical distributions such as generalized Gaussian, fisher, etc. can be used, but these distributions are complex, and the corresponding parameter estimation and derivation of matrix $U$ are challenging to realize in practice.
3. Robustness of approach: By operating at this holistic statistical level, our method maintains robustness across diverse imaging scenarios, even when local target/clutter distributions vary. This aligns with established practices in SAR image analysis, where system-level statistical modeling often supersedes component-specific assumptions.

Reference:

[R1] Li, H. C., et al. On the empirical-statistical modeling of SAR images with generalized gamma distribution. IEEE Journal of selected topics in signal processing, 2011.
[R2] Nascimento, A. D. C., et al. Compound truncated Poisson gamma distribution for understanding multimodal SAR intensities. Journal of Applied Statistics, 2023.

Comment2. Limited discussion on computational efficiency vs. performance trade-offs.

Answer2: Thanks for your detailed comment. Below, we analyze the computational complexity of our proposed ΓPCA method and compare it with standard PCA.

The primary computational cost of ΓPCA mainly arises from the derivation using the MM.

1. Update the mean vector $\mu^{(t+1)}$ by equation (13) in our manuscript:

The update involves matrix multiplications and a linear system solve. Dominant operations include:

• $O(d^2k)$ for $U^{(t)}(U^{(t)})^T$

• $O(nd^2)$ for matrix multiplication of $H(·)(·)^T$

Here, $n$ is the number of samples, $d$ is the original feature dimension and $k$ is the reduced dimension. Therefore, the total complexity for $\mu^{(t+1)}$ is $O(nd^2+d^2k)$.

2. Calculate the deviance $M(\boldsymbol{\Theta}|\boldsymbol{\Theta}^{(t)})$ by equation (27):

Computes the Frobenius norm of an $n \times d$ matrix after rank-$k$ projections. Dominate terms:

• $O(ndk)$ for matrix products with $\mathrm{H}+1_n\mu^{(t)}$ with $U^{(t)}$.

• $O(d^{2}k)$ for $U^{(t)}(U^{(t)})^T$ operations.

The total complexity for $M(\boldsymbol{\Theta}|\boldsymbol{\Theta}^{(t)})$ is $O(ndk+d^2k)$.

3. Calculate the matrix $F^{(t)}$ by equation (30) and update $U^{(t+1)}$ to the first $k$ eigenvectors of $F^{(t)}$:

• $O(nd^{2})$ for the matrix conjunction of $(·)^{T}Q^{(t)}(·)$.
• $O(d^{3})$ for singular value decomposition (SVD).

Therefore, the total complexity for $U^{(t+1)}$ is $O(nd^{2}+d^3)$.

From the above discussion, the overall computational complexity of ΓPCA algorithm is approximately $O(nd^2+d^2k)+O(ndk+d^2k)+O(nd^2+d^3)\approx O(nd^2+d^2k+ndk+d^3).$

In practical applications, parameters typically exhibit the following relationships: $n \gg d \gg k$. Therefore, the overall computational complexity of ΓPCA algorithm is $O(nd^2+d^3)$.

In contrast, the computational complexity of the standard PCA method is mainly determined by two key operations:

1. Computation of covariance matrix: $\Sigma=\frac{1}{n}\mathbf{X}^T\mathbf{X}$, which has a computational complexity of $O(nd^2)$.

2. Eigenvalue decomposition of covariance matrix: $\Sigma=\mathrm{U}\Lambda\mathrm{U}^T$, with a computational complexity of $O(d^3)$, equivalent to the SVD operation in ΓPCA .

Here, $\text{X}\in\mathbb{R}^{n\times d}$ is the data matrix, $U$ is the eigenvector matrix, and $\Lambda$ is the eigenvalue diagonal matrix. Consequently, the overall computational complexity of standard PCA is approximately $O(nd^2+d^3)$.

The computational complexity is identical to that of standard PCA under the condition of $n \gg d \gg k$. The overall computational cost of our method is on the same order of magnitude as standard PCA, making the new algorithm computationally acceptable.