Total Variation Regularization for Manifold-Valued Data

We consider total variation (TV) minimization for manifold-valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $\ell pi $-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images and interferometric SAR images as well as sphere- and cylinder-valued images. For the class of Cartan--Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer.