A nonlocal graph-PDE and higher-order geometric integration for image labeling

• Verfasst von: Sitenko, Dmitrij [VerfasserIn]
• Verfasst von: Boll, Bastian [VerfasserIn]
• Verfasst von: Schnörr, Christoph [VerfasserIn]
• Titel: A nonlocal graph-PDE and higher-order geometric integration for image labeling
• Verf.angabe: Dmitrij Sitenko, Bastian Boll, Christoph Schnörr
• Ausgabe: Version v2
• E-Jahr: 2022
• Jahr: 4 Oct 2022
• Umfang: 60 S.
• Fussnoten: Version 1 vom 9 Mai 2022, Version 2 vom 4 Oktober 2022 ; Gesehen am 14.10.2022
• Titel Quelle: Enthalten in: De.arxiv.org
• Ort Quelle: [S.l.] : Arxiv.org, 1991
• Jahr Quelle: 2022
• Band/Heft Quelle: (2022), Artikel-ID 2205.03991, Seite 1-60
• DOI: doi:10.48550/arXiv.2205.03991
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: Computer Science - Computer Vision and Pattern Recognition
• Sach-SW: Mathematics - Numerical Analysis
• Sach-SW: Mathematics - Optimization and Control
• K10plus-PPN: 1818950065

Abstract

This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was introduced in \textit{J.~Math.~Imaging \& Vision} 58(2), 2017. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions (DC) decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments.