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SLRQA: A Sparse Low-Rank Quaternion Model for Color Image Processing with Convergence Analysis

Zhanwang Deng, Yuqiu Su, Wen Huang

TL;DR

This paper tackles color image restoration under noise by formulating Sparse Low-rank Quaternion Approximation (SLRQA), which unifies low-rankness, sparsity, and quaternion representation without requiring initial rank estimation. It develops proximal linearized ADMM algorithms (PL-ADMM for denoising and PL-ADMM-NF for noise-free cases) and proves global convergence under a new, weaker assumption set that circumvents the traditional range constraint. The quaternion formulation preserves inter-channel correlations and yields superior denoising and inpainting performance compared to RGB and several state-of-the-art methods, with convergence to stationary points demonstrated both theoretically and empirically. This work advances color image processing by delivering robust, convergent algorithms that exploit joint priors in the quaternion domain, offering practical improvements for color image restoration tasks.

Abstract

In this paper, we propose a Sparse Low-rank Quaternion Approximation (SLRQA) model for color image processing problems with noisy observations. %Different from the existing color image processing models, The proposed SLRQA is a quaternion model that combines low-rankness and sparsity priors without an initial rank estimation. %Furthermore, it does not need an initial rank estimate. A proximal linearized ADMM (PL-ADMM) algorithm is proposed to solve SLRQA and the global convergence is guaranteed under standard assumptions. %where only one variable is linearized. When the observation is noise-free, a limiting case of the SLRQA, called SLRQA-NF, is proposed. Subsequently, a proximal linearized ADMM (PL-ADMM-NF) algorithm for SLRQA-NF is given. Since SLRQA-NF does not satisfy a widely-used assumption for global convergence of ADMM-type algorithms, we propose a novel assumption, under which the global convergence of PL-ADMM-NF is established. In numerical experiments, we verify the effectiveness of quaternion representation. Furthermore, for color image denoising and color image inpainting problems, SLRQA and SLRQA-NF demonstrate superior performance both quantitatively and visually when compared with some state-of-the-art methods.

SLRQA: A Sparse Low-Rank Quaternion Model for Color Image Processing with Convergence Analysis

TL;DR

This paper tackles color image restoration under noise by formulating Sparse Low-rank Quaternion Approximation (SLRQA), which unifies low-rankness, sparsity, and quaternion representation without requiring initial rank estimation. It develops proximal linearized ADMM algorithms (PL-ADMM for denoising and PL-ADMM-NF for noise-free cases) and proves global convergence under a new, weaker assumption set that circumvents the traditional range constraint. The quaternion formulation preserves inter-channel correlations and yields superior denoising and inpainting performance compared to RGB and several state-of-the-art methods, with convergence to stationary points demonstrated both theoretically and empirically. This work advances color image processing by delivering robust, convergent algorithms that exploit joint priors in the quaternion domain, offering practical improvements for color image restoration tasks.

Abstract

In this paper, we propose a Sparse Low-rank Quaternion Approximation (SLRQA) model for color image processing problems with noisy observations. %Different from the existing color image processing models, The proposed SLRQA is a quaternion model that combines low-rankness and sparsity priors without an initial rank estimation. %Furthermore, it does not need an initial rank estimate. A proximal linearized ADMM (PL-ADMM) algorithm is proposed to solve SLRQA and the global convergence is guaranteed under standard assumptions. %where only one variable is linearized. When the observation is noise-free, a limiting case of the SLRQA, called SLRQA-NF, is proposed. Subsequently, a proximal linearized ADMM (PL-ADMM-NF) algorithm for SLRQA-NF is given. Since SLRQA-NF does not satisfy a widely-used assumption for global convergence of ADMM-type algorithms, we propose a novel assumption, under which the global convergence of PL-ADMM-NF is established. In numerical experiments, we verify the effectiveness of quaternion representation. Furthermore, for color image denoising and color image inpainting problems, SLRQA and SLRQA-NF demonstrate superior performance both quantitatively and visually when compared with some state-of-the-art methods.
Paper Structure (22 sections, 26 theorems, 141 equations, 8 figures, 6 tables, 3 algorithms)

This paper contains 22 sections, 26 theorems, 141 equations, 8 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. A novel quaternion based model
  3. Related work
  4. Contribution
  5. Organization
  6. Preliminaries
  7. Algorithm framework
  8. An Algorithm for SLRQA
  9. An Algorithm for SLRQA-NF
  10. Convergence analysis
  11. Global Convergence for Algorithm \ref{['algorithm:denoise']}
  12. Global Convergence for Algorithm \ref{['algorithm:inpaint']}
  13. Numerical experiment
  14. Problem description, parameter setting, and testing environment
  15. Quaternion and RGB representation
  16. ...and 7 more sections

Key Result

Theorem 2.1

For any given quaternion matrix $\mathbf{\dot{A}} \in \mathbb{H}^{m \times n},$ there exist two unitary quaternion matrices $\mathbf{\dot{U}} \in \mathbb{H} ^{m \times m},\mathbf{\dot{V}} \in \mathbb{H}^{n \times n}$ such that where $\mathbf{\Sigma} \in \mathbb{R}^{r \times r}$ is a diagonal matrix with $\mathbf{\Sigma}_{1,1} \geq \mathbf{\Sigma}_{2,2} \geq \cdots \geq \mathbf{\Sigma}_{r,r} >0$

Figures (8)

  • Figure 1: The 10 color images (512 $\times$ 512 $\times$ 3) for numerical experiments.
  • Figure 2: PSNR and SSIM comparison of SLRQA-1 using quaternion and RGB representation. In the legend, the suffix (Q) means that quaternion representation is used, and (RGB) means that RGB representation is used. The numbers 10, 30, and 50 denote the variance of the noise respectively.
  • Figure 3: PSNR and SSIM comparisons of quaternion and RGB based SLRQA-NF-1. In the legend, the suffix (Q) means that quaternion representation is used, and (RGB) means that RGB representation is used. The numbers 0.3, 0.5, and 0.7 denote the missing rate of the images respectively.
  • Figure 4: Color image denoising results of different methods on "Image5" with $\tau = 10$.
  • Figure 5: The error convergence of SLRQA-1 for image denoising problems.
  • ...and 3 more figures

Theorems & Definitions (51)

  • Theorem 2.1: Quaternion singular value decomposition (QSVD)zhang1997quaternions
  • Lemma 3.1
  • proof
  • Lemma 4.1: Sufficient descent of $\mathcal{L}_\beta$ for ${\mathbf{\dot{X}}}$ update
  • proof
  • Lemma 4.2: Sufficient descent of $\mathcal{L}_\beta$ for $\mathbf{\dot{W}}$ update
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 41 more