A Theoretically Guaranteed Quaternion Weighted Schatten p-norm Minimization Method for Color Image Restoration

Inspired by the fact that the matrix formulated by nonlocal similar patches in a natural image is of low rank, the rank approximation issue have been extensively investigated over the past decades, among which weighted nuclear norm minimization (WNNM) and weighted Schatten $p$-norm minimization (WSNM) are two prevailing methods have shown great superiority in various image restoration (IR) problems. Due to the physical characteristic of color images, color image restoration (CIR) is often a much more difficult task than its grayscale image counterpart. However, when applied to CIR, the 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 and preserving the inherent correlation among the three color channels. Despite its empirical success, unfortunately, the convergence behavior of QWNNM has not been strictly studied yet. In this paper, on the one side, 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. On the other side, more importantly, we preliminarily provide a theoretical convergence analysis, that is, by modifying the quaternion alternating direction method of multipliers (QADMM) through a simple continuation strategy, we theoretically prove that both the solution sequences generated by the QWNNM and QWSNM have fixed-point convergence guarantees.