Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Doi
Abstract
In this work, we introduce a function space setting for a wide class of
structural/weighted total variation (TV) regularization methods motivated by
their applications in inverse problems. In particular, we consider a
regularizer that is the appropriate lower semi-continuous envelope
(relaxation) of a suitable total variation type functional initially defined
for sufficiently smooth functions. We study examples where this relaxation
can be expressed explicitly, and we also provide refinements for weighted
total variation for a wide range of weights. Since an integral
characterization of the relaxation in function space is, in general, not
always available, we show that, for a rather general linear inverse problems
setting, instead of the classical Tikhonov regularization problem, one can
equivalently solve a saddle-point problem where no a priori knowledge of an
explicit formulation of the structural TV functional is needed. In
particular, motivated by concrete applications, we deduce corresponding
results for linear inverse problems with norm and Poisson log-likelihood data
discrepancy terms. Finally, we provide proof-of-concept numerical examples
where we solve the saddle-point problem for weighted TV denoising as well as
for MR guided PET image reconstruction