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

Mérigot, Quentin; Meyron, Jocelyn; Thibert, Boris

doi:10.1137/17m1137486

Cited by 35 publications

(26 citation statements)

References 24 publications

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“…This diagram can be efficiently computed by using libraries, such as for instance Cgal or Geogram. When c(x, y) = x − y 2 is the quadratic cost and X is a triangulated surface in R 3 , the Laguerre cells can be obtained by intersecting power cells with the triangulated surface X [72]. This approach is also used in several inverse problems arising in nonimaging optics that correspond to optimal transport problems.…”

Section: 68)mentioning

confidence: 99%

Optimal transport: discretization and algorithms

Mérigot

Thibert

2021

Handbook of Numerical Analysis

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Section: 68)mentioning

confidence: 99%

Optimal transport: discretization and algorithms

Mérigot

Thibert

2021

Handbook of Numerical Analysis

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“…Applications include optimal location of resources [13], signal and image compression [34,42], numerical integration [62, Sect. 2.3], mesh generation [32,58], finance [62], materials science [19, Sect. 3.2], and particle methods for PDEs (sampling the initial distribution) [15,Example 7.1].…”

Section: Literature On Crystallization Optimal Partitions and Quantizationmentioning

confidence: 99%

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

Bourne

Cristoferi

2021

Commun. Math. Phys.

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We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure $$f \mathrm {d}x$$ f d x by a discrete probability measure $$\sum _i m_i \delta _{z_i}$$ ∑ i m i δ z i , subject to a constraint on the particle sizes $$m_i$$ m i . The locations $$z_i$$ z i of the particles, their sizes $$m_i$$ m i , and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

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“…Sampling a mesh using a point cloud has recently been proposed with elegant computational geometry algorithms [30,47,46], generalizing the euclidean Centroidal Voronoi Tessellation (CVT) for surfaces. However, the best approaches have a time complexity in O(n 2 log n) which remain computationally non-affordable for computer vision applications.…”

Section: Related Workmentioning

confidence: 99%