A nonlocal graph-PDE and higher-order geometric integration for image labeling - Sitenko, Dmitrij; Boll, Bastian; Schnörr, Christoph (Arxiv, 2022)
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.
Version 1 vom 9 Mai 2022, Version 2 vom 4 Oktober 2022
Gesehen am 14.10.2022