Total variation regularization for manifold-valued data
This work addresses denoising for manifold-valued data, which is incremental as it extends existing TV methods to manifold settings with specific convergence guarantees.
The authors tackled the problem of total variation minimization for data on manifolds by proposing cyclic and parallel proximal point algorithms based on iterative geodesic averaging, and demonstrated their application to denoising images in various manifolds such as diffusion tensor images, with proven convergence to a global minimizer for Cartan-Hadamard manifolds.
We consider total variation minimization for manifold valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $\ell^p$-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, 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.