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arXiv · 1903.04873

Minimal Lipschitz and $\infty$-Harmonic Extensions of Vector-Valued Functions on Finite Graphs

Miroslav Bačák·Johannes Hertrich·Sebastian Neumayer·Gabriele Steidl
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Abstract

This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called lex and L-lex minimal extensions are actually the same and call them minimal Lipschitz extensions. Then we prove that the solution of the graph $p$-Laplacians converge to these extensions as $p\to \infty$. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to $\infty$-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed by Elmoataz et al.~(2014) for finding the zero of the $\infty$-Laplacian is given. Finally, we present applications in image inpainting.

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Miroslav Bačák, Johannes Hertrich, Sebastian Neumayer, Gabriele Steidl. 2019-03-12. Minimal Lipschitz and $\infty$-Harmonic Extensions of Vector-Valued Functions on Finite Graphs. https://arxiv.org/abs/1903.04873

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