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Coupling local and nonlocal total variation flow for image despeckling

Yi Ran, Zhichang Guo, Kehan Shi, Qirui Zhou, Jingfeng Shao, Martin Burger, Boying Wu

TL;DR

This work proposes a coupled local-nonlocal total variation flow for image despeckling to address the trade-off between texture preservation and regularization in SAR speckle removal. It provides a rigorous analysis of the weak solution, including existence, uniqueness, and equivalent formulations, and proves that the coupled model converges to the classical TV flow under kernel rescaling. A key theoretical result shows the NLTV-to-TV limit as the nonlocal scale vanishes, while numerical experiments demonstrate that coupling yields superior denoising and texture retention compared with purely local or nonlocal approaches. The combination of BV framework, kernel-guided diffusion via a grayscale indicator, and nonlocal integration-by-parts underpins both the theoretical and practical robustness of the approach.

Abstract

Nonlocal equations effectively preserve textures but exhibit weak regularization effects in image denoising, whereas local equations offer strong denoising capabilities yet fail to protect textures. To integrate the advantages of both approaches, this paper investigates a coupled local-nonlocal total variation flow for image despeckling. We establish the existence and uniqueness of the weak solution for the proposed equation. Several properties, including the equivalent forms of the weak solution and its asymptotic behavior, are derived. Furthermore, we demonstrate that the weak solutions of the proposed equation converge to the weak solution of the classical total variation flow under kernel rescaling. The importance of coupling is highlighted through comparisons with local and nonlocal models for image despeckling.

Coupling local and nonlocal total variation flow for image despeckling

TL;DR

This work proposes a coupled local-nonlocal total variation flow for image despeckling to address the trade-off between texture preservation and regularization in SAR speckle removal. It provides a rigorous analysis of the weak solution, including existence, uniqueness, and equivalent formulations, and proves that the coupled model converges to the classical TV flow under kernel rescaling. A key theoretical result shows the NLTV-to-TV limit as the nonlocal scale vanishes, while numerical experiments demonstrate that coupling yields superior denoising and texture retention compared with purely local or nonlocal approaches. The combination of BV framework, kernel-guided diffusion via a grayscale indicator, and nonlocal integration-by-parts underpins both the theoretical and practical robustness of the approach.

Abstract

Nonlocal equations effectively preserve textures but exhibit weak regularization effects in image denoising, whereas local equations offer strong denoising capabilities yet fail to protect textures. To integrate the advantages of both approaches, this paper investigates a coupled local-nonlocal total variation flow for image despeckling. We establish the existence and uniqueness of the weak solution for the proposed equation. Several properties, including the equivalent forms of the weak solution and its asymptotic behavior, are derived. Furthermore, we demonstrate that the weak solutions of the proposed equation converge to the weak solution of the classical total variation flow under kernel rescaling. The importance of coupling is highlighted through comparisons with local and nonlocal models for image despeckling.

Paper Structure

This paper contains 15 sections, 12 theorems, 118 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic assumptions
  4. The solution space
  5. The results of coupling local and nonlocal $p$-laplacian equation
  6. The results of coupling local and nonlocal $p$-laplacian equation
  7. The main results
  8. Proof of main results
  9. Proof of Theorem \ref{['th:exit_unique']}
  10. Proof of propositions
  11. Proof of Theorem \ref{['th:nl2local']}
  12. Experiments
  13. Numerical scheme
  14. Experimental results
  15. Conclusion

Key Result

Lemma 1

Assume $\Omega \subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partial \Omega$. If $\mathbf{z} \in X_p(\Omega)$ and $u \in B V(\Omega) \cap L^{p^{\prime}}(\Omega)$, then we have where $[\mathbf{z}, \nu]$ is the weak trace on $\partial \Omega$ of the normal component of $\mathbf{z} \in X_p(\Omega)$.

Figures (4)

  • Figure 1: Test images used in experiments.
  • Figure 2: The restoration results $u$, difference images ($f - u$), PSNR values, and SSIM values for a synthesis image under different $\lambda$ in Eq.\ref{['eq:tv_case']}.
  • Figure 3: Restoration results of four methods for test images with noise level $L=10$.
  • Figure 4: Restoration results of four methods for test images with noise level $L=4$.

Theorems & Definitions (25)

  • Lemma 1: Green's formula andreu2011local
  • Definition 1: chen2003minimization
  • Remark 1: chen2003minimization
  • Lemma 2: lower semicontinuity chen2003minimization
  • Definition 2
  • Lemma 3: shi2021coupling
  • Lemma 4
  • proof
  • Definition 3
  • Remark 2
  • ...and 15 more