EXISTENCE AND STABILITY OF MINIMIZERS OF MIXED ANISOTROPIC BV-L2 TIKHONOV-PHILLIPS FUNCTIONALS: APPLICATIONS TO IMAGE RESTORATION.

During the last two decades several generalizations of the traditional TikhonovPhillips regularization method for solving inverse ill -posed problems have been proposed. Many of these variants consists essentially in modifications of the penalizing term, which forces certain features in the obtained regularized solution ([8], [13]). If it is known that the regularity of the exact solution is inhomogeneous it is often desirable the use of mixed, spatially adaptive methods ([6], [9]). These methods arc also highly suitable when the preservation of borders and edges is also an important issue, since they allow for the inclusion of anisotropic penalizers for border detection ([15]). In this work, we propose the use of a penalizer resulting from the convex spatially adaptive combinations of classic penalizing L2 and anisotropic bounded variation semi norm. Results on existence and uniqueness of minimizers of the corresponding TikhonovPhillips functional arc presented. Stability results of those minimizers with respect to perturbations in the data, in the regularization parameter and in the operator arc also established. An application to image restoration problem is shown.