Quaternion weighted Schatten p-norm minimization for color image restoration with convergence guarantee

Over the past decades, the rank approximation issue have been extensively investigated, among which weighted nuclear norm minimization (WNNM) and weighted Schatten p-norm minimization (WSNM) are two prevailing methods and have shown great superiority in various image restoration (IR) problems. However, for the complicated color image restoration (CIR) problems, traditional WNNM/WSNM method only processes three color channels individually and fails to consider their cross-channel correlations. Very recently, a quaternion-based WNNM approach (QWNNM) has been developed to mitigate this issue, which is capable of representing the color image as a whole in the quaternion domain. Despite QWNNM's empirical success, its convergence behavior has not been rigorously studied. The main contributions of this paper are twofold. Firstly, we extend the WSNM into quaternion domain and correspondingly propose a novel quaternion-based WSNM model (QWSNM) for tackling the CIR problems. Extensive experiments on two representative CIR tasks, including color image denoising and deblurring, demonstrate that the proposed QWSNM method performs favorably against many state-of-the-art alternatives, in both quantitative and qualitative evaluations. Secondly, we provide a preliminary theoretical convergence analysis. By modifying the quaternion alternating direction method of multipliers (QADMM) through a simple continuation strategy, we theoretically prove the fixed-point convergence property of the iterative sequences generated by QWNNM and QWSNM. The source code of our algorithm can be available at the website: https://github.com/qiuxuanzhizi/QWSNM.