A Graph Framework for Manifold-Valued Data

Ronny Bergmann; Daniel Tenbrinck

doi:10.1137/17m1118567

Graph-based methods have been proposed as a unified framework for discrete calculus of local and nonlocal image processing methods in the recent years. In order to translate variational models and partial differential equations to a graph, certain operators have been investigated and successfully applied to real-world applications involving graph models. So far the graph framework has been limited to real- and vector-valued functions on Euclidean domains. In this paper we generalize this model

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... o the case of manifold-valued data. We introduce the basic calculus needed to formulate variational models and partial differential equations for manifold-valued functions and discuss the proposed graph framework for two particular families of operators, namely, the isotropic and anisotropic graph p-Laplacian operators, p≥1. Based on the choice of p we are in particular able to solve optimization problems on manifold-valued functions involving total variation (p=1) and Tikhonov (p=2) regularization. Finally, we present numerical results from processing both synthetic as well as real-world manifold-valued data, e.g., from diffusion tensor imaging (DTI) and light detection and ranging (LiDAR) data.

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A Graph Framework for Manifold-Valued Data

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