An algorithm for optimal transport between a simplex soup and a point cloud

TL;DR

This paper introduces a numerical method for computing optimal transport maps between measures supported on complex lower-dimensional structures and point clouds, with proven convergence and practical applications in 3D geometry processing.

Contribution

It presents a convergent damped Newton algorithm for solving a nonlinear system related to optimal transport on simplex soups, extending previous methods to more general supports.

Findings

Proven linear convergence of the Newton algorithm under genericity and connectedness conditions.

Application of the method to compute optimal transport plans in 3D for complex geometries.

Demonstrated usefulness in tasks like surface quantization, remeshing, and point set registration.

Abstract

We propose a numerical method to find the optimal transport map between a measure supported on a lower-dimensional subset of R^d and a finitely supported measure. More precisely, the source measure is assumed to be supported on a simplex soup, i.e. on a union of simplices of arbitrary dimension between 2 and d. As in [Aurenhammer, Hoffman, Aronov, Algorithmica 20 (1), 1998, 61--76] we recast this optimal transport problem as the resolution of a non-linear system where one wants to prescribe the quantity of mass in each cell of the so-called Laguerre diagram. We prove the convergence with linear speed of a damped Newton's algorithm to solve this non-linear system. The convergence relies on two conditions: (i) a genericity condition on the point cloud with respect to the simplex soup and (ii) a (strong) connectedness condition on the support of the source measure defined on the simplex soup. Finally, we apply our algorithm in R^3 to compute optimal transport plans between a measure supported on a triangulation and a discrete measure. We also detail some applications such as optimal quantization of a probability density over a surface, remeshing or rigid point set registration on a mesh.