A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling

This paper introduces a novel nonlocal partial difference equation (PDE) for labeling metric data on graphs. The 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 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 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.