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Mumford-Shah and Potts functionals are powerful variational models forregularization which are widely used in signal and image processing; typicalapplications are edge-preserving denoising and segmentation. Being bothnon-smooth and non-convex, they are computationally challenging even for scalardata. For manifold-valued data, the problem becomes even more involved sincetypical features of vector spaces are not available. In this paper, we proposealgorithms for Mumford-Shah and for Potts regularization of manifold-valuedsignals and images. For the univariate problems, we derive solvers based ondynamic programming combined with (convex) optimization techniques formanifold-valued data. For the class of Cartan-Hadamard manifolds (whichincludes the data space in diffusion tensor imaging), we show that ouralgorithms compute global minimizers for any starting point. For themultivariate Mumford-Shah and Potts problems (for image regularization) wepropose a splitting into suitable subproblems which we can solve exactly usingthe techniques developed for the corresponding univariate problems. Our methoddoes not require any a priori restrictions on the edge set and we do not haveto discretize the data space. We apply our method to diffusion tensor imaging(DTI) as well as Q-ball imaging. Using the DTI model, we obtain a segmentationof the corpus callosum.

Mumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging