Minimal Lipschitz and∞-harmonic extensions of vector-valued functions on finite graphs

摘要

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→∞. Furthermore, we examine the relation betweenminimal Lipschitz extensions and iterated weighted midrange filters and address their connection to∞-Laplacians for scalarvalued functions. A convergence proof for an iterative algorithm proposed by Elmoataz et al. (2014) for finding the zero of the∞-Laplacian is given. Finally, we present applications in image inpainting.