Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs. (28th February 2020)

• Record Type: Journal Article
• Authors: Bačák, Miroslav; Hertrich, Johannes; Neumayer, Sebastian; Steidl, Gabriele
• Is Part Of: Information and inference. Volume 9:Number 4(2020)
• Page Start: 935
• Page End: 959
• Publication Date: 2020-02-28
• Subjects: p-Laplacian -- ∞-Laplacian -- graph Laplacian -- ∞-harmonic extension -- absolutely minimal Lipschitz extension -- midrange filter -- image inpainting -- non-local techniques; Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaz033
• Languages: English
• ISSNs: 2049-8764

Abstract

Abstract: This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called $\mathrm{lex}$ and $L\mbox{-}\mathrm{lex}$ 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.