Results on the Simultaneous Use of Classical Tikhonov-Phillips and Bounded-Variation Regularization Methods for Inverse Ill-Posed Problems

Abstract

Several generalizations of the traditional Tikhonov-Phillips regularization method for inverse ill-posed problems have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers which allow the capturing of diverse properties of the exact solution (e.g. edges, discontinuities, borders, etc.). However, in some problems in which it is known that the regularity of the exact solution is heterogeneous and/or anisotropic, it is reasonable to think that a much better option could be the simultaneous use of two or more penalizers of different nature. Such is the case, for instance, in some image restoration problems in which preservation of edges, borders or discontinuities is an important matter. In this work we present some new results on the simultaneous use of penalizers of L2 and of bounded-variation (BV) type. For particular cases, existence and uniqueness results are shown. Open problems are discussed and some results in applications to signal and image restoration problems are presented.