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.

A plane-probing algorithm computes the normal vector of a digital plane from a starting point and a predicate "Is a point x in the digital plane?". This predicate is used to probe the digital plane as locally as possible and decide on-the-fly the next points to consider. However, several existing planeprobing algorithms return the correct normal vector only for some specific starting points and an approximation otherwise, e.g. the Hand R-algorithm proposed in Lachaud et al.

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