Minimal Lipschitz and $\infty$-Harmonic Extensions of Vector-Valued Functions on Finite Graphs
For researchers in graph-based signal processing and image processing, this work provides theoretical foundations and algorithmic guarantees for minimal Lipschitz and ∞-harmonic extensions, though the results are incremental.
This paper unifies two notions of minimal Lipschitz extensions for vector-valued functions on finite graphs, proves convergence of graph p-Laplacian solutions to these extensions as p→∞, and provides a convergence proof for an iterative algorithm for the ∞-Laplacian. Applications in image inpainting are shown.
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.