Existence, Uniqueness and Stability of Solutions of Generalized Tikhonov-Phillips Functionals

摘要

The Tikhonov-Phillips method is widely used for regularizing ill-posedinverse problems mainly due to the simplicity of its formulation as anoptimization problem. The use of different penalizers in the functionalsassociated to the corresponding optimization problems has originated a varietyother methods which can be considered as "variants" of the traditionalTikhonov-Phillips method of order zero. Such is the case for instance of theTikhonov-Phillips method of order one, the total variation regularizationmethod, etc. In this article we find sufficient conditions on the penalizers ingeneralized Tikhonov-Phillips functionals which guarantee existence anduniqueness and stability of the minimizers. The particular cases in which thepenalizers are given by the bounded variation norm, by powers of seminorms andby linear combinations of powers of seminorms associated to closed operators,are studied. Several examples are presented and a few results on imagerestoration are shown.