A Theoretically Guaranteed Quaternion Weighted Schatten p-norm
  Minimization Method for Color Image Restoration
Qing-Hua Zhang Liang-Tian He Yi-Lun Wang Liang-Jian Deng Jun Liu
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.