I will review the Mumford-Shah functional, some special cases and generalizations including the piecewise constant limit, anisotropic versions and vectorial formulations. I will discuss how respective functionals can be relaxed to convex problems which can be solved efficiently using provably convergent primal-dual algorithms. Applications to computer vision problems such as segmentation, denoising and semantic labeling demonstrate that the convex relaxations allow to compute near-optimal solutions which are independent of initialization. This is joint work with E. Strekalovskiy, A. Chambolle, T. Pock, C. Nieuwenhuis and M. Souiai.
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