Dynamic off-the-grid untangling of curves by Riemannian metric

TL;DR

This paper introduces a novel off-the-grid method for untangling and tracking point source trajectories in a temporal sequence by leveraging Riemannian geometry and a lifted problem formulation, improving robustness and control.

Contribution

It presents a new regularisation based on the relaxed Reeds-Shepp metric and proves convergence properties, enabling effective reconstruction of tangled trajectories.

Findings

Successfully untangles complex trajectories in numerical experiments

Achieves comparable or improved localization accuracy against state-of-the-art methods

Provides theoretical guarantees through $ extGamma$-convergence analysis

Abstract

We propose an improved strategy for point sources tracking in a temporal stack through an off-the-grid fashion, inspired by the Benamou-Brenier regularisation in the literature. We define a lifting of the problem in the higher-dimensional space of the roto-translation group. This allows us to overcome the theoretical limitation of the off-the-grid method towards tangled point source trajectories, thus enabling the reconstruction and untangling even from the numerical standpoint. We define accordingly a new regularisation based on the relaxed Reeds-Shepp metric, an approximation of the sub-Riemannian Reeds-Shepp metric, further allowing control on the local curvature of the recovered trajectories. Then, we derive some properties of the discretisation and prove a $\Gamma$-convergence result, fostering interest for practical applications of polygonal, B\'ezier, and piecewise geodesic discretisation. We finally test our proposed method on a localisation problem example, and give a fair comparison with the state-of-the-art off-the-grid method.