Nested Iterative Algorithms for Convex Constrained Image Recovery Problems

作     者：Chaux, Caroline Pesquet, Jean-Christophe Pustelnik, Nelly 

作者机构：Univ Paris Est CNRS UMR 8049 Lab Informat Gaspard Monge F-77454 Marne La Vallee 2 France 

出 版 物：《SIAM JOURNAL ON IMAGING SCIENCES》 (SIAM成像科学杂志)

年 卷 期：2009年第2卷第2期

页      面：730-762页

核心收录：

学科分类：1002[医学-临床医学] 070207[理学-光学] 07[理学] 08[工学] 0835[工学-软件工程] 0803[工学-光学工程] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 0702[理学-物理学] 

基　　金：Agence Nationale de la Recherche [ANR-05-MMSA-0014-01] 

主　　题：wavelets dual-trees restoration deconvolution optimization convex analysis iterative algorithms forward-backward Douglas-Rachford variational methods Bayesian approaches maximum a posteriori Poisson noise 

摘      要：The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be nonsmooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas-Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.