A Primal-Dual Level Set Method for Computing Geodesic Distances

TL;DR

This paper presents a robust and efficient primal-dual level set method for computing geodesic distances on surfaces, leveraging implicit surface representation and convergence analysis to improve accuracy and implementation ease.

Contribution

It introduces a novel primal-dual level set approach for geodesic computation that is robust, efficient, and theoretically convergent, with practical numerical validation.

Findings

Method converges to geodesic with refinement

Algorithm is robust and easy to implement

Numerical results confirm convergence and efficiency

Abstract

The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high-resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.