The Moreau envelope approach for the L1/TV image denoising model
Feishe Chen, Lixin Shen, Yuesheng Xu, Xueying Zeng · Inverse Problems and Imaging · 2014
This paper presents the Moreau envelope viewpoint for the L1/TVimage denoising model. The main algorithmic difficulty for thenumerical treatment of the L1/TV model lies in thenon-differentiability of both the fidelity and regularization termsof the model. To overcome this difficulty, we propose five modifiedL1/TV models by replacing one or two non-differentiablefunctions in the L1/TV model with their corresponding Moreauenvelopes. We prove that several existing approaches for the L1/TVmodel essentially solve some of the modified models, but not theoriginal L1/TV model. Algorithms for the L1/TV model and its fivevariants are proposed under a unified framework based on fixed-point equations (via theproximity operator) which characterize the solutions of the models. Depending upon whether we smooth theregularization term or not, two different types of proximityalgorithms are presented. The convergence rates of both types of thealgorithms are improved significantly by exploring either thestrategy of the Gauss-Seidel iteration, or the FISTA, or both. Wecompare the performance of various modified L1/TV models for theproblem of impulse noise removal, and make recommendations based onour numerical experiments for using these models in applications.