On Convex Envelopes and Regularization of Non-convex Functionals Without Moving Global Minima

Supplementary Material: Non-Convex Rank/Sparsity Regularization and Local Minima

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 minimiz...

Non-convex Sparse Regularization

We study the regularising properties of Tikhonov regularisation on the sequence space l with weighted, non-quadratic penalty term acting separately on the coefficients of a given sequence. We derive sufficient conditions for the penalty term that guarantee the well-posedness of the method, and investigate to which extent the same conditions are also necessary. A particular interest of this pape...

Convex Representation for Lower Semicontinuous Envelopes of Functionals in L

G. Alberti, G. Bouchitté and G. Dal Maso recently found sufficient conditions for the minimizers of the (nonconvex) Mumford-Shah functional. Their method consists in an extension of the calibration method (that is used for the characterization of minimal surfaces), adapted to this functional. The existence of a calibration, given a minimizer of the functional, remains an open problem. We introd...

Evolution by Non-Convex Functionals

We establish a semi-group solution concept for morphological differential equations, such as the mean curvature flow equation. The proposed method consists in generating flows from generalized minimizers of nonconvex energy functionals. We use relaxation and convexification to define generalized minimizers. The main part of this work consists in verification of the solution concept by comparing...