A Primal-Dual Level Set Method for Computing Geodesic Distances

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.

💡 Research Summary

The paper introduces a novel algorithm for computing geodesic distances on three‑dimensional surfaces by representing the surface implicitly as the zero‑level set of a scalar function ϕ(x). Instead of discretizing the surface into a mesh or a graph, the authors formulate the shortest‑path problem as a constrained variational problem: minimize ½∫₀¹‖γ̇(t)‖² dt subject to the constraint ϕ(γ(t)) = 0 for all t∈