Anisotropic BV–L2 regularization of linear inverse ill-posed problems

Abstract

During the last two decades several generalizations of the traditional Tikhonov-Phillips regularization method for solving inverse ill-posed problems have been proposed. Many of these variants consist essentially of modifications on the penalizing term, which force certain features in the obtained regularized solution ([11,18]). If it is known that the regularity of the exact solution is inhomogeneous it is often desirable the use of mixed, spatially adaptive methods ([7,12]). These methods are also highly suitable when the preservation of edges is an important issue, since they allow for the inclusion of anisotropic penalizers for border detection ([20]). In this work we propose the use of a penalizer resulting from the convex spatially-adaptive combination of a classic L2penalizer and an anisotropic bounded variation seminorm. Results on existence and uniqueness of minimizers of the corresponding Tikhonov-Phillips functional are presented. Results on the stability of those minimizers with respect to perturbations in the data, in the regularization parameter and in the operator are also established. Applications to image restoration problems are shown.