Nonlocal Difference Operators on Graphs for Interpolation on Point Clouds

In this paper we introduce a new general class of partial difference operators on graphs, which interpolate between the nonlocal \(\infty \)-Laplacian, the Laplacian, and a family of discrete gradient operators. In this context we investigate an associated Dirichlet problem for this general class of operators and prove the existence and uniqueness of respective solutions. We propose to use this class of operators as general framework to solve many interpolation problems in a unified manner as arising, e.g., in image and point cloud processing. (AE is supported by the European FEDER Grant (PLANUCA Project) and the project ANR GRAPHSIP.)

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