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Perception-based multiplicative noise removal using SDEs

An Vuong, Thinh Nguyen

TL;DR

The paper tackles despeckling under multiplicative noise, a problem common in SAR and laser imaging. It proposes a diffusion-based approach by modeling multiplicative noise as a Geometric Brownian Motion in the log domain and deriving a reverse SDE via the Fokker-Planck equation. Training uses a DDPM-like score-matching objective in the log domain, and sampling can be performed with deterministic ODE/DDIM schemes. Experiments on CelebA and UC Merced show superior perception-based metrics (FID and LPIPS) while keeping competitive PSNR/SSIM, and the method generalizes to out-of-distribution data. This work provides a flexible, noise-level-agnostic framework for despeckling with strong perceptual fidelity.

Abstract

Multiplicative noise, also known as speckle or pepper noise, commonly affects images produced by synthetic aperture radar (SAR), lasers, or optical lenses. Unlike additive noise, which typically arises from thermal processes or external factors, multiplicative noise is inherent to the system, originating from the fluctuation in diffuse reflections. These fluctuations result in multiple copies of the same signal with varying magnitudes being combined. Consequently, despeckling, or removing multiplicative noise, necessitates different techniques compared to those used for additive noise removal. In this paper, we propose a novel approach using Stochastic Differential Equations based diffusion models to address multiplicative noise. We demonstrate that multiplicative noise can be effectively modeled as a Geometric Brownian Motion process in the logarithmic domain. Utilizing the Fokker-Planck equation, we derive the corresponding reverse process for image denoising. To validate our method, we conduct extensive experiments on two different datasets, comparing our approach to both classical signal processing techniques and contemporary CNN-based noise removal models. Our results indicate that the proposed method significantly outperforms existing methods on perception-based metrics such as FID and LPIPS, while maintaining competitive performance on traditional metrics like PSNR and SSIM.

Perception-based multiplicative noise removal using SDEs

TL;DR

The paper tackles despeckling under multiplicative noise, a problem common in SAR and laser imaging. It proposes a diffusion-based approach by modeling multiplicative noise as a Geometric Brownian Motion in the log domain and deriving a reverse SDE via the Fokker-Planck equation. Training uses a DDPM-like score-matching objective in the log domain, and sampling can be performed with deterministic ODE/DDIM schemes. Experiments on CelebA and UC Merced show superior perception-based metrics (FID and LPIPS) while keeping competitive PSNR/SSIM, and the method generalizes to out-of-distribution data. This work provides a flexible, noise-level-agnostic framework for despeckling with strong perceptual fidelity.

Abstract

Multiplicative noise, also known as speckle or pepper noise, commonly affects images produced by synthetic aperture radar (SAR), lasers, or optical lenses. Unlike additive noise, which typically arises from thermal processes or external factors, multiplicative noise is inherent to the system, originating from the fluctuation in diffuse reflections. These fluctuations result in multiple copies of the same signal with varying magnitudes being combined. Consequently, despeckling, or removing multiplicative noise, necessitates different techniques compared to those used for additive noise removal. In this paper, we propose a novel approach using Stochastic Differential Equations based diffusion models to address multiplicative noise. We demonstrate that multiplicative noise can be effectively modeled as a Geometric Brownian Motion process in the logarithmic domain. Utilizing the Fokker-Planck equation, we derive the corresponding reverse process for image denoising. To validate our method, we conduct extensive experiments on two different datasets, comparing our approach to both classical signal processing techniques and contemporary CNN-based noise removal models. Our results indicate that the proposed method significantly outperforms existing methods on perception-based metrics such as FID and LPIPS, while maintaining competitive performance on traditional metrics like PSNR and SSIM.
Paper Structure (17 sections, 1 theorem, 36 equations, 6 figures, 2 tables, 4 algorithms)

This paper contains 17 sections, 1 theorem, 36 equations, 6 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Methods
  4. SDE and Diffusion models
  5. Noise models
  6. Loss function in the logarithmic domain
  7. Enhancing image quality by deterministic sampling
  8. Experiments
  9. Experiment settings
  10. Experiment results
  11. Conclusion
  12. Solutions to the forward SDE
  13. Derivations of the reverse SDEs
  14. In pixel domain
  15. In logarithmic domain
  16. ...and 2 more sections

Key Result

Lemma 1

Let $f(t)$ be some function that is square-integrable, i.e. $\int_{0}^{t}f^{2}(s)ds <\infty$, and $\beta(t)$ be a some Brownian motion, then Proof. Since Riemann's sum of $\int_{0}^{t}f(s)d\beta$ exists if we fix the midpoints, let $t_{k}=\frac{k}{2^{n}}t$, then and since increment of Brownian motion follows $\mathcal{N}(0,\Delta t)$, where $\Delta t = 2^{-n}t$, thus Now we can take the limit

Figures (6)

  • Figure 1: Samples generated by our methods, on images randomly selected from CelebA dataset. From left to right are the original, corrupted by multiplicative noise (noise level $0.08$), and denoised versions.
  • Figure 2: Comparing between different sampling techniques on randomly selected CelebA images, at noise level $0.12$. The first two columns include the original images and their noised versions, respectively. These are followed by the results generated by our model using ODE, DDIM, and stochastic samplers, respectively. High resolution version is available in the appendix.
  • Figure 3: Comparing between different denoising models on randomly selected CelebA images, at noise level $0.08$. The first two columns include the original images and their noised versions, respectively. These are followed by the results generated by our method and other popular techniques. Last two rows present a zoomed in example.
  • Figure 4: Comparing between different denoising models on randomly selected UC Merced Land Use images, at noise level $0.12$. The first two columns include the original images and their noised versions, respectively. These are followed by the results generated by our method and other popular techniques.
  • Figure 5: Comparing between different denoising models on randomly selected CelebA images, at noise level $0.12$. The first two columns include the original images and their noised versions, respectively. These are followed by the results generated by our method and other popular techniques.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Lemma 1