Characterization of minimizers of convex regularization functionals

We study variational methods of bounded variation type for the data analysis. Y. Meyer characterized minimizers of the Rudin-Osher-Fatemi functional in dependence of the G-norm of the data. These results and the follow up work on this topic are generalized to functionals defined on spaces of functions with derivatives of finite bounded variation. In order to derive a characterization of minimizers of convex regularization functionals we use the concept of generalized directional derivatives and duality. Finally we present some examples where the minimizers of convex regularization functionals are calculated analytically, repeating some recent results from the literature and adding some novel results with penalization of higher order derivatives of bounded variation.

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