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A Tunable Despeckling Neural Network Stabilized via Diffusion Equation

Yi Ran, Zhichang Guo, Jia Li, Yao Li, Martin Burger, Boying Wu

TL;DR

This work tackles despeckling of SAR imagery corrupted by multiplicative Gamma noise and the vulnerability of neural denoisers to adversarial perturbations. It introduces a tunable unrolled neural network that embeds a diffusion regularity block based on the linear heat equation, with a single time-step parameter $\tau$ to control output smoothness, enabling post-training adaptability. Theoretical analysis establishes global convergence and stability, while FFT-based implicit schemes ensure efficient diffusion updates. Empirical results on simulated, adversarial, and real SAR data show improved denoising performance, enhanced robustness to adversarial attacks, and flexible trade-offs between noise removal and detail preservation.

Abstract

The removal of multiplicative Gamma noise is a critical research area in the application of synthetic aperture radar (SAR) imaging, where neural networks serve as a potent tool. However, real-world data often diverges from theoretical models, exhibiting various disturbances, which makes the neural network less effective. Adversarial attacks can be used as a criterion for judging the adaptability of neural networks to real data, since adversarial attacks can find the most extreme perturbations that make neural networks ineffective. In this work, the diffusion equation is designed as a regularization block to provide sufficient regularity to the whole neural network, due to its spontaneous dissipative nature. We propose a tunable, regularized neural network framework that unrolls a shallow denoising neural network block and a diffusion regularity block into a single network for end-to-end training. The linear heat equation, known for its inherent smoothness and low-pass filtering properties, is adopted as the diffusion regularization block. In our model, a single time step hyperparameter governs the smoothness of the outputs and can be adjusted dynamically, significantly enhancing flexibility. The stability and convergence of our model are theoretically proven. Experimental results demonstrate that the proposed model effectively eliminates high-frequency oscillations induced by adversarial attacks. Finally, the proposed model is benchmarked against several state-of-the-art denoising methods on simulated images, adversarial samples, and real SAR images, achieving superior performance in both quantitative and visual evaluations.

A Tunable Despeckling Neural Network Stabilized via Diffusion Equation

TL;DR

This work tackles despeckling of SAR imagery corrupted by multiplicative Gamma noise and the vulnerability of neural denoisers to adversarial perturbations. It introduces a tunable unrolled neural network that embeds a diffusion regularity block based on the linear heat equation, with a single time-step parameter to control output smoothness, enabling post-training adaptability. Theoretical analysis establishes global convergence and stability, while FFT-based implicit schemes ensure efficient diffusion updates. Empirical results on simulated, adversarial, and real SAR data show improved denoising performance, enhanced robustness to adversarial attacks, and flexible trade-offs between noise removal and detail preservation.

Abstract

The removal of multiplicative Gamma noise is a critical research area in the application of synthetic aperture radar (SAR) imaging, where neural networks serve as a potent tool. However, real-world data often diverges from theoretical models, exhibiting various disturbances, which makes the neural network less effective. Adversarial attacks can be used as a criterion for judging the adaptability of neural networks to real data, since adversarial attacks can find the most extreme perturbations that make neural networks ineffective. In this work, the diffusion equation is designed as a regularization block to provide sufficient regularity to the whole neural network, due to its spontaneous dissipative nature. We propose a tunable, regularized neural network framework that unrolls a shallow denoising neural network block and a diffusion regularity block into a single network for end-to-end training. The linear heat equation, known for its inherent smoothness and low-pass filtering properties, is adopted as the diffusion regularization block. In our model, a single time step hyperparameter governs the smoothness of the outputs and can be adjusted dynamically, significantly enhancing flexibility. The stability and convergence of our model are theoretically proven. Experimental results demonstrate that the proposed model effectively eliminates high-frequency oscillations induced by adversarial attacks. Finally, the proposed model is benchmarked against several state-of-the-art denoising methods on simulated images, adversarial samples, and real SAR images, achieving superior performance in both quantitative and visual evaluations.

Paper Structure

This paper contains 20 sections, 2 theorems, 21 equations, 10 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Adversarial attack and network instability
  4. Regularity of Heat equation
  5. Methodology
  6. Framework of our model
  7. Diffusion regularity block
  8. Denoising neural network block
  9. Theoretical analysis
  10. Numerical scheme of diffusion block
  11. Controllable smoothness
  12. Experiment
  13. Data Description
  14. Experimental Setup and training
  15. Evaluation Index
  16. ...and 5 more sections

Key Result

Theorem 1

Assume $f \in C(\mathbb{R}^n) \cap L^\infty(\mathbb{R}^n)$, then the solution of the following initial-value is where $\Phi(x, t)= \frac{1}{(4 \pi t)^{n / 2}} e^{-\frac{|x|^2}{4 t}}$ for $x \in \mathbb{R}^n$ and $t>0$ is called the fundamental solution and $u \in C^\infty(\mathbb{R}^n \times (0,\infty))$.

Figures (10)

  • Figure 1: High frequency oscillations generated by adversarial attack. The 70th column of "House" is shown in the center. The restored results on noisy image and adversarial sample are placed in the left and right respectively.
  • Figure 2: The results of the heat equation for different frequency signals as the time increases.
  • Figure 3: framework of our model.
  • Figure 4: The performance of heat equation to different noise. The "Noisy" signals are constructed by multiplying Gamma noise with $L=10$ to the "Clean Signal". Adding "Gaussian noise", "Uniform noise", and "Rayleigh noise" to 0-100, 100-200 and 200-300 respectively to simulate three different attacks. The processed "Noisy" signal is "Denoised Signal".
  • Figure 5: The structure of denoising block.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2: Convergence
  • proof
  • Remark : Stability