Computing the cut locus of a Riemannian manifoldviaoptimal transport

Abstract

In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver, based on the so-called dynamical Monge-Kantorovich approach, we propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. We show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in $R^{3}$, and discuss advantages and limitations.

Numerical solution of the $ L^1 $-optimal transport problem on surfaces

“…It turns out that the proposed OTPbased approach on surfaces is a general method for the computation of the distance function. Indeed, the method has been applied in [27] for the identification of the cut locus of a triangulated surface with respect of a point, i.e., the set of points where the distance function becomes non differentiable, or, equivalently, the minimizing geodesics are not unique. Moreover, our approach can be extended to the approximation of medial axes and Voronoi diagrams of general surfaces embedded in R 3 , quantities that are of great interest in computational geometry [34].…”