Quaternion-based weighted nuclear norm minimization for color image restoration

Color image restoration is one of the basic tasks in pattern recognition. Unlike grayscale image, each color image has three channels in the RGB color space. Due to the inner-relationship within the three channels, color image restoration is usually much more difficult than its grayscale counterpart. Indeed, new problems such as color artifacts could emerge when the grayscale image processing methods are extended to color images directly. Note that one of the most effective gray image restoration methods is the weighted nuclear norm minimization (WNNM) approach. However, when applied to color images, the results of WNNM are usually not as promising as that of grayscale images. In order to solve this problem, in this paper, we propose to restore color images with the quaternion-based WNNM method (QWNNM) since the structure of color channels can be well preserved with quaternion representation. The proposed model can be solved efficiently by the alternating direction method of multipliers (ADMM). The theoretical analysis of the optimal solution is also presented. Numerical experiments are carefully conducted with different kinds of degradation to illustrate the superior performance of our proposed QWNNM over the state-of-the-art methods, including a celebrated deep learning approach, in both visual quality and numerical results.