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An Algorithm for Optimal Transport between a Simplex Soup and a Point Cloud

https://doi.org/10.1137/17M1137486

Abstract

We propose a numerical method to find the optimal transport map between a measure supported on a lower-dimensional subset of $\mathbb{R}^d$ 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 [F. Aurenhammer, F. Hoffman, and B. Aronov, Algorithmica, 20 (1998), pp. 61--76] we recast this optimal transport problem as the resolution of a nonlinear 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 nonlinear 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.

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