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Mumford–Shah and Potts Regularization for Manifold-Valued Data

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Abstract

Mumford–Shah and Potts functionals are powerful variational models for regularization which are widely used in signal and image processing; typical applications are edge-preserving denoising and segmentation. Being both non-smooth and non-convex, they are computationally challenging even for scalar data. For manifold-valued data, the problem becomes even more involved since typical features of vector spaces are not available. In this paper, we propose algorithms for Mumford–Shah and for Potts regularization of manifold-valued signals and images. For the univariate problems, we derive solvers based on dynamic programming combined with (convex) optimization techniques for manifold-valued data. For the class of Cartan–Hadamard manifolds (which includes the data space in diffusion tensor imaging (DTI)), we show that our algorithms compute global minimizers for any starting point. For the multivariate Mumford–Shah and Potts problems (for image regularization), we propose a splitting into suitable subproblems which we can solve exactly using the techniques developed for the corresponding univariate problems. Our method does not require any priori restrictions on the edge set and we do not have to discretize the data space. We apply our method to DTI as well as Q-ball imaging. Using the DTI model, we obtain a segmentation of the corpus callosum on real data.

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Acknowledgments

This work was supported by the German Federal Ministry for Education and Research under SysTec Grant 0315508. The first author acknowledges support by the Helmholtz Association within the young investigator group VH-NG-526. The third author was supported by the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement No. 267439.

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  1. Department of Mathematics, Technische Universität München, Munich, Germany

    Andreas Weinmann & Laurent Demaret

  2. Helmholtz Zentrum München, Oberschleißheim, Germany

    Andreas Weinmann

  3. Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Martin Storath

  4. Darmstadt University of Applied Sciences, Darmstadt, Germany

    Andreas Weinmann

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  1. Andreas Weinmann
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Weinmann, A., Demaret, L. & Storath, M. Mumford–Shah and Potts Regularization for Manifold-Valued Data. J Math Imaging Vis 55, 428–445 (2016). https://doi.org/10.1007/s10851-015-0628-2

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