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Graph Methods for Manifold-valued Data

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VMV - Variational methods and effective algorithms for imaging and vision

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).

This talk is part of the Isaac Newton Institute Seminar Series series.

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