Corrected Group Sparse Residual Constraint Model for Image Denoising and Inpainting

Group sparse residual constraint model with nonlocal priors (GSRC-NLP) has made great success in image restoration and produce state-of-the-art performance, realized through reducing the group sparsity residual. In the GSRC-NLP model, L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} norm is used to reduce the group sparsity residual. However, the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} norm penalty has been known to lead to the over-shrinkage of the large sparse coefficients. In this paper, we utilize the adaptive correction procedure to reduce excessive penalties on large coefficient values while improving group sparsity. The proposed corrected group sparse residual constraint model (CGSRC) can improve the sparsity of the group sparsity residual, which lead to better performance in image restoration. We apply the iterative shrinkage/thresholding and the alternating direction method of multipliers to solve the proposed models. In addition, we study the properties of the proposed CGSRC model including the existence and uniqueness of the solution as well as the convergence of the proposed algorithms. Experimental results on image denoising and image inpainting show that the proposed models outperform several state-of-the-art image denoising and image inpainting methods in terms of both objective and perceptual quality metrics.