[hal-00826121, v1] Parallel ProXimal Algorithm for image restoration using hybrid regularization

Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet domain regularization brings other artefacts, e.g. ringing. However, a trade-off can be made by introducing a hybrid regularization including several terms non necessarily acting in the same domain (e.g. spatial and wavelet transform domains). While this approach was shown to provide good results for solving deconvolution problems in the presence of additive Gaussian noise, an important issue is to efficiently deal with this hybrid regularization for more general noise models. To solve this problem, we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms. For spatial domain regularization, isotropic or anisotropic total variation definitions using various gradient filters are considered. An accelerated version of the Parallel Proximal Algorithm is proposed to perform the minimization. Some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper. Numerical experiments performed in the context of Poisson data recovery, show the good behaviour of the algorithm as well as promising results concerning the use of hybrid regularization techniques. Part of this work appeared in the conference proceedings of EUSIPCO 2009 [1]. This work was supported by the Agence Nationale de la Recherche under grant ANR-09-EMER-004-03. N. Pustelnik (Corresponding Author), C. Chaux and J.-C. Pesquet are with the Université Paris-Est, Laboratoire d’Informatique Gaspard Monge, CNRS-UMR 8049, 77454 Marne-la-Vallée Cedex 2, France. Phone: +33 1 60 95 77 39, E-mail: {nelly.pustelnik,caroline.chaux,jean-christophe.pesquet}@univ-paris-est.fr 1 ha l-0 08 26 12 1, v er si on 1 26 M ay 2 01 3 Author manuscript, published in "IEEE Transactions on Image Processing 20, 6 (2011) 2450-2462" DOI : 10.1109/TIP.2011.2128335