Regularization of Ill-Posed Problems with Combined Quadratic and Anisotropic Bounded-Variation Penalization

Abstract

From the early works of Tikhonov and Phillips in 1962 and 1963, the treatment of inverse ill-posed problems has seen an enormous growth. Many methods, tools and ad-hoc algorithms bound to extract as much information as possible about the exact solution of the problem have been developed. In particular, during the last two decades a wide variety of new mathematical tools ranging from the use of variable Lp spaces to bounded-variation (BV ) penalization, anisotropic diffusion methods and Bayesian models and hypermodels has arisen. Although it cannot be expected that a single method be better than all others for all type of problems, the ability of “detection” of discontinuities and borders and subsequent “self-adaptation” to different types of patterns, structures and degrees of regularity is a highly desired property of a regularization method. In this work we present some mathematical results on the existence and uniqueness of global minimizers of generalized Tikhonov-Phillips functionals with penalizers given by convex spatially-adaptive combinations of L2 and isotropic and anisotropic BV type. Open problems are discussed and results to signal and image restoration problems are presented.