Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients

Owing to the good ability of sparsely approximating piece-wise smooth functions like images, the (tight) wavelet frame has been widely investigated and applied for image restoration and other image processing problems over the past few years. Most of the variational models based on wavelet frame proposed in the past utilize 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 of frame coefficients as a sparsity prior. Very recently, the variational model which penalizes the l0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{0}$$\end{document} “norm” of frame coefficients was proposed for image restoration, and proved to outperform the commonly used 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} minimization methods in the quality of restored images. Though the l0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{0}$$\end{document} “norm” has the ability of preserving sharp edges and smooth regions, textures and small details may be mistakenly removed at the same time. Therefore, we introduce a l0-l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_0-l_2$$\end{document} regularization model which contains a nonlocal prior of frame coefficients to avoid this issue in this paper. Meanwhile, a narrow-band technique is introduced to further improve the computational efficiency of the proposed algorithm. Numerical experiments demonstrate that the propose algorithm is superior to the recently proposed algorithm for l0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{0}$$\end{document} “norm” minimization in both iterative time and recovery quality.