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A Dual Alternating Direction Method of Multipliers for Image Decomposition and Restoration

Qingsong Wang, Chengjing Wang, Peipei Tang, Dunbiao Niu

TL;DR

The paper proposes a dual ADMM (dADMM) framework for simultaneous image decomposition into cartoon $u$ and texture $v$ and restoration under degradation modeled by $b \approx H(u+v)$. It reformulates the problem as $\min_{x,y} \frac{1}{2}\|Ax+By-b\|_2^2 + p(x) + q(y)$ with $p(x)=\tau\||\nabla x|\|_1$ and $q(y)=\mu\| |y| \|_s$, derives its dual, and develops a three-step dADMM algorithm that updates $(u,v,w)$ via a linear system and proximal operators, followed by $(x,y)$ updates with a relaxation parameter $\tau$. The authors prove global convergence for $\tau$ in $(0,(1+\sqrt{5})/2)$ and a local linear convergence rate, and validate the method through extensive experiments across four observation models ($A=I,S,K,KS$), showing higher PSNR and faster convergence than competing ADMM-based approaches in denoising, deblurring, inpainting, and joint blur-missing data scenarios. The results highlight the practical significance of algorithm choice in image processing tasks and demonstrate the robustness and efficiency of dADMM for complex degradation settings.

Abstract

In this paper, we develop a dual alternating direction method of multipliers (ADMM) for an image decomposition model. In this model, an image is divided into two meaningful components, i.e., a cartoon part and a texture part. The optimization algorithm that we develop not only gives the cartoon part and the texture part of an image but also gives the restored image (cartoon part + texture part). We also present the global convergence and the local linear convergence rate for the algorithm under some mild conditions. Numerical experiments demonstrate the efficiency and robustness of the dual ADMM (dADMM). Furthermore, we can obtain relatively higher signalto-noise ratio (SNR) comparing to other algorithms. It shows that the choice of the algorithm is also important even for the same model.

A Dual Alternating Direction Method of Multipliers for Image Decomposition and Restoration

TL;DR

The paper proposes a dual ADMM (dADMM) framework for simultaneous image decomposition into cartoon and texture and restoration under degradation modeled by . It reformulates the problem as with and , derives its dual, and develops a three-step dADMM algorithm that updates via a linear system and proximal operators, followed by updates with a relaxation parameter . The authors prove global convergence for in and a local linear convergence rate, and validate the method through extensive experiments across four observation models (), showing higher PSNR and faster convergence than competing ADMM-based approaches in denoising, deblurring, inpainting, and joint blur-missing data scenarios. The results highlight the practical significance of algorithm choice in image processing tasks and demonstrate the robustness and efficiency of dADMM for complex degradation settings.

Abstract

In this paper, we develop a dual alternating direction method of multipliers (ADMM) for an image decomposition model. In this model, an image is divided into two meaningful components, i.e., a cartoon part and a texture part. The optimization algorithm that we develop not only gives the cartoon part and the texture part of an image but also gives the restored image (cartoon part + texture part). We also present the global convergence and the local linear convergence rate for the algorithm under some mild conditions. Numerical experiments demonstrate the efficiency and robustness of the dual ADMM (dADMM). Furthermore, we can obtain relatively higher signalto-noise ratio (SNR) comparing to other algorithms. It shows that the choice of the algorithm is also important even for the same model.

Paper Structure

This paper contains 9 sections, 2 theorems, 38 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Algorithm
  4. Numerical Experiments
  5. Case 1: $A=I$
  6. Case 2: $A=S$
  7. Case 3: $A=K$
  8. Case 4: $A=KS$
  9. Conclusion

Key Result

Theorem 2.1

We consider the situation of $\mathcal{S}=0$ and $\mathcal{T}=0$ in (Fazel_Pong_Sun_Tseng, Theorem B.1). Assume that the solution set of $(D)$ is nonempty and that the constraint qualification holds. Let the sequence $\{(u^{k},v^{k},w^{k},x^{k},y^{k})\}$ be generated from the dADMM. If $\tau \in (0,

Figures (13)

  • Figure 1: Image decomposition on Weave image (c) in Fig. \ref{['test_jpg']}.
  • Figure 2: Testing images: (a) 512 $\times$ 512 Lena image, (b) 512 $\times$ 512 Wool image, (c) combined 512 $\times$ 512 Lena and Wool image (denoted Mixed image), (d) 768 $\times$ 1024 $\times$ 3 Weave image, (e) 576 $\times$ 787 $\times$ 3 Barbara_RGB image, (f) 512 $\times$ 512 a part of Barbara image, (g) 393 $\times$ 635 $\times$ 3 Brick image, (h) 1024 $\times$ 1024 $\times$ 3 Wood image.
  • Figure 3: Image decomposition on clean images ((f) and (g) in Fig. \ref{['test_jpg']}, respectively) with different values of $s$. From left to right: the cartoon part, the texture part of (f) and (g), respectively. The top row: $s=1$. The center row: $s=2$. The bottom row: $s=\infty$.
  • Figure 4: Variations of Corr$(u,v)$ with respect to the iterations for the image (f) Barbara in Fig. \ref{['test_jpg']} with $s=1$, $s=2$, and $s=\infty$.
  • Figure 5: Image decomposition on the Mixed image (c) in Fig. \ref{['test_jpg']} with $A=I$. The first column: the cartoon part. The second column: the texture part. From top to bottom are the decomposed results by the ADME, ADMGB, and dADMM.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 1