Efficient Nonsmooth Nonconvex Optimization for Image Restoration and Segmentation

In this article, we introduce variational image restoration and segmentation models that incorporate 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} data-fidelity measure and a nonsmooth, nonconvex regularizer. 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} fidelity term allows us to restore or segment an image with low contrast or outliers, and the nonconvex regularizer enables homogeneous regions of the objective function (a restored image or an indicator function of a segmented region) to be efficiently smoothed while edges are well preserved. To handle the nonconvexity of the regularizer, a multistage convex relaxation method is adopted. This provides a better solution than the classical convex total variation regularizer, or than the standard 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} convex relaxation method. Furthermore, we design fast and efficient optimization algorithms that can handle the non-differentiability of both the fidelity and regularization terms. The proposed iterative algorithms asymptotically solve the original nonconvex problems. Our algorithms output a restored image or segmented regions in the image, as well as an edge indicator that characterizes the edges of the output, similar to Mumford–Shah-like regularizing functionals. Numerical examples demonstrate the promising results of the proposed restoration and segmentation models.