Mumford-Shah and Potts regularization for manifold-valued data. (English) Zbl 1344.49055
Summary: 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 nonsmooth and nonconvex, 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 a-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.
MSC:
| 49M37 | Numerical methods based on nonlinear programming |
| 49L20 | Dynamic programming in optimal control and differential games |
| 90C25 | Convex programming |
| 90C39 | Dynamic programming |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 68U10 | Computing methodologies for image processing |
Keywords:
Mumford-Shah functional; Potts functional; regularization; dynamic programming; convex optimization; diffusion tensor imaging; Q-ball imaging; jump sparsity; Hadamard manifold; proximal methodsReferences:
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