Notions of optimal transport theory and how to implement them on a computer

Bruno Lévy 1 Erica Schwindt 1
1 ALICE - Geometry and Lighting
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, with active research communities, both on its theoretical aspects and on more applicative considerations, such as geometry processing and machine learning. This article aims at explaining the main principles behind the theory of optimal transport, introduce the different involved notions, and more importantly, how they relate, to let the reader grasp an intuition of the elegant theory that structures them. Then we will consider a specific setting, called semi-discrete, where a continuous function is transported to a discrete sum of Dirac masses. Studying this specific setting naturally leads to an efficient computational algorithm, that uses classical notions of computational geometry, such as a generalization of Voronoi diagrams called Laguerre diagrams.
Complete list of metadatas

Cited literature [30 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01717967
Contributor : Erica Schwindt <>
Submitted on : Thursday, March 8, 2018 - 11:09:32 AM
Last modification on : Tuesday, December 18, 2018 - 4:18:26 PM
Long-term archiving on : Saturday, June 9, 2018 - 2:09:10 PM

File

OT-preprint.pdf
Files produced by the author(s)

Identifiers

Citation

Bruno Lévy, Erica Schwindt. Notions of optimal transport theory and how to implement them on a computer. Computers and Graphics, Elsevier, 2018, pp.1-22. ⟨10.1016/j.cag.2018.01.009⟩. ⟨hal-01717967⟩

Share

Metrics

Record views

385

Files downloads

1069