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An Algorithmic Framework for Mumford-Shah Regularization of Inverse Problems in Imaging

K. Hohm, M. Storath, A. Weinmann

Inverse Problems, vol. 31, no. 11, paper no. 115011, November 2015.

The Mumford-Shah model is a very powerful variational approach for edge preserving regularization of image reconstruction processes. However, it is algorithmically challenging because one has to deal with a non-smooth and non-convex functional. In this paper, we propose a new efficient algorithmic framework for Mumford-Shah regularization of inverse problems in imaging. It is based on a splitting into specific subproblems that can be solved exactly. We derive fast solvers for the subproblems which are key for an efficient overall algorithm. Our method neither requires a priori knowledge of the gray or color levels nor of the shape of the discontinuity set. We demonstrate the wide applicability of the method for different modalities. In particular, we consider the reconstruction from Radon data, inpainting, and deconvolution. Our method can be easily adapted to many further imaging setups. The relevant condition is that the proximal mapping of the data fidelity can be evaluated a within reasonable time. In other words, it can be used whenever classical Tikhonov regularization is possible.

Selected by the Editorial Board as a highlight paper from Inverse Problems in 2015.

AUTHOR="Hohm, K. and Storath, M. and Weinmann, A.",
TITLE="An Algorithmic Framework for {M}umford-{S}hah Regularization of
        Inverse Problems in Imaging",
JOURNAL="Inverse Problems",
note="paper no.\ 115011")

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