Inverse Problems and Imaging (IPI)

The Moreau envelope approach for the L1/TV image denoising model

Pages: 53 - 77, Volume 8, Issue 1, February 2014      doi:10.3934/ipi.2014.8.53

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Feishe Chen - Department of Mathematics, Syracuse University, Syracuse, NY 13244, United States (email)
Lixin Shen - Department of Mathematics, Syracuse University, Syracuse, NY 13244, United States (email)
Yuesheng Xu - Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, United States (email)
Xueying Zeng - School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China (email)

Abstract: This paper presents the Moreau envelope viewpoint for the L1/TV image denoising model. The main algorithmic difficulty for the numerical treatment of the L1/TV model lies in the non-differentiability of both the fidelity and regularization terms of the model. To overcome this difficulty, we propose five modified L1/TV models by replacing one or two non-differentiable functions in the L1/TV model with their corresponding Moreau envelopes. We prove that several existing approaches for the L1/TV model essentially solve some of the modified models, but not the original L1/TV model. Algorithms for the L1/TV model and its five variants are proposed under a unified framework based on fixed-point equations (via the proximity operator) which characterize the solutions of the models. Depending upon whether we smooth the regularization term or not, two different types of proximity algorithms are presented. The convergence rates of both types of the algorithms are improved significantly by exploring either the strategy of the Gauss-Seidel iteration, or the FISTA, or both. We compare the performance of various modified L1/TV models for the problem of impulse noise removal, and make recommendations based on our numerical experiments for using these models in applications.

Keywords:  Impulse noise, Moreau envelope, L1/TV denoising model, proximity operator.
Mathematics Subject Classification:  Primary: 94A08; Secondary: 49N45, 68U10.

Received: October 2012;      Revised: August 2013;      Available Online: March 2014.