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In this work, we introduce a function space setting for a wide class ofstructural/weighted total variation (TV) regularization methods motivated bytheir applications in inverse problems. In particular, we consider aregularizer that is the appropriate lower semi-continuous envelope (relaxation)of a suitable total variation type functional initially defined forsufficiently smooth functions. We study examples where this relaxation can beexpressed explicitly, and we also provide refinements for weighted totalvariation for a wide range of weights. Since an integral characterization ofthe relaxation in function space is, in general, not always available, we showthat, for a rather general linear inverse problems setting, instead of theclassical Tikhonov regularization problem, one can equivalently solve asaddle-point problem where no a priori knowledge of an explicit formulation ofthe structural TV functional is needed. In particular, motivated by concreteapplications, we deduce corresponding results for linear inverse problems withnorm and Poisson log-likelihood data discrepancy terms. Finally, we provideproof-of-concept numerical examples where we solve the saddle-point problem forweighted TV denoising as well as for MR guided PET image reconstruction.

A function space framework for structural total variation regularization with applications in inverse problems