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

TL;DR

This paper presents a new nonlocal graph-PDE for image labeling, connecting geometric flows with higher-order integration techniques, and introduces accelerated schemes with proven convergence.

Contribution

It introduces a nonlocal graph-PDE derived from the assignment flow, linking geometric integration with higher-order methods and accelerated DC programming.

Findings

Equivalent to computing Riemannian gradient flow

Developed an accelerated DC scheme for higher-order integration

Provided convergence analysis and numerical experiments

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.