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

Quentin Mérigot 1 Jocelyn Meyron 2, 3, * Boris Thibert 3
* Corresponding author
2 GIPSA-AGPIG - AGPIG
GIPSA-DIS - Département Images et Signal
3 CVGI - Calcul des Variations, Géométrie, Image
LJK - Laboratoire Jean Kuntzmann
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.
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Submitted on : Wednesday, July 5, 2017 - 11:34:31 AM
Last modification on : Thursday, July 11, 2019 - 11:44:01 AM
Long-term archiving on : Tuesday, January 23, 2018 - 8:33:02 PM

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Quentin Mérigot, Jocelyn Meyron, Boris Thibert. An algorithm for optimal transport between a simplex soup and a point cloud. SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2018, 11 (2), pp.1363-1389. ⟨10.1137/17M1137486⟩. ⟨hal-01556544⟩

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