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

arXiv: https://arxiv.org/abs/1410.1699 · HTML: https://mlsanity.com/paper/1410.1699/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2014.

## Built on by (most cited first, 2 of 2)

- [Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields](https://mlsanity.com/paper/2107.06028/) (2021)
- [Tikhonov Regularization of Circle-Valued Signals](https://mlsanity.com/paper/2108.02602/) (2021)

## Abstract (arXiv)

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), 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 diffusion tensor imaging (DTI) as well as Q-ball imaging. Using the DTI model, we obtain a segmentation of the corpus callosum.