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

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.