A Multiscale Approach to Optimal Transport

• Published in: Computer graphics forum Vol. 30; no. 5; pp. 1583 - 1592
• Main Author: Merigot, Quentin
• Format: Journal Article
• Language: English
• Published: Oxford, UK Blackwell Publishing Ltd 01.08.2011; Wiley
• Subjects: Algorithms; and systems; Computational Geometry; Computer Science; Density; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems; languages; Least squares method; Mathematical analysis; Mathematical models; Numerical Analysis; Optimization; Planes; Studies; Transport; Transportation problem (Operations research); convex programming; optimal transport; Power diagram; wasserstein distance
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/j.1467-8659.2011.02032.x

In this paper, we propose an improvement of an algorithm of Aurenhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Euclidean plane. Our algorithm exploits the multiscale nature of this optimal transport problem. We iteratively simplify the target using Lloyd's algorithm, and use the solution of the simplified problem as a rough initial solution to the more complex one. This approach allows for fast estimation of distances between measures related to optimal transport (known as Earth‐mover or Wasserstein distances). We also discuss the implementation of these algorithms, and compare the original one to its multiscale counterpart.