Abstract

Next to traditional processing tasks there exist real applications in which measured data are not in a Euclidean vector space but rather are given on a Riemannian manifold. This is the case, e.g., when dealing with Interferometric Synthetic Aperture Radar (InSAR) data consisting of phase values or data obtained in Diffusion Tensor Magnetic Resonance Imaging (DT-MRI). In this talk we present a framework for processing discrete manifold-valued data, for which the underlying (sampling) topology is modeled by a graph. We introduce the notion of a manifold-valued derivative on a graph and based on this deduce a family of manifold-valued graph operators. In particular, we introduce the graph p-Laplacian and graph infinity-Laplacian for manifold-valued data. We discuss a simple numerical scheme to compute a solution to the corresponding parabolic PDEs and apply this algorithm to different manifold-valued data, illustrating the diversity and flexibility of the proposed framework in denoising and inpainting applications. This is joint work with Dr. Ronny Bergmann (TU Kaiserslautern).