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<p>In this paper, we propose a Sparse Low-rank Quaternion Approximation (SLRQA) model for color image processing problems with noisy observations. The proposed SLRQA is a quaternion model that combines low-rankness and sparsity priors without requiring an initial rank estimation. A proximal linearized ADMM (PL-ADMM) algorithm is proposed to solve SLRQA, and the global convergence is guaranteed under standard assumptions. 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 the 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年8月20日

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  • Zhanwang Deng,
  • Yuqiu Su,
  • Wen Huang

摘要

In this paper, we propose a Sparse Low-rank Quaternion Approximation (SLRQA) model for color image processing problems with noisy observations. The proposed SLRQA is a quaternion model that combines low-rankness and sparsity priors without requiring an initial rank estimation. A proximal linearized ADMM (PL-ADMM) algorithm is proposed to solve SLRQA, and the global convergence is guaranteed under standard assumptions. 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 the 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.