Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration

Abstract

The Tikhonov-Phillips method is widely used for regularizing ill-posed problems due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a few other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method of order zero. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method, etc. The purpose of this article is twofold. First we study the problem of determining general sufficient conditions on the penalizers in generalized Tikhonov-Phillips functionals which guarantee existence and uniqueness of minimizers, in such a way that finding such minimizers constitutes a regularization method, that is, in such a way that these minimizers approximate, as the regularization parameter tend to 0+, a least squares solution of the problem. Secondly we also study the problem of characterizing those limiting least square solutions in terms of properties of the penalizers and finnd conditions which guarantee that the regularization method thus defined is stable under different types of perturbations. Finally, several examples with different penalizers are presented and a few numerical results in an image restoration problem are shown which better illustrate the results.