Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Shape recognition via Wasserstein distance

Authors: Wilfrid Gangbo and Robert J. McCann
Journal: Quart. Appl. Math. 58 (2000), 705-737
MSC: Primary 94A08; Secondary 28A35, 49Q20
DOI: https://doi.org/10.1090/qam/1788425
MathSciNet review: MR1788425
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Abstract: The Kantorovich-Rubinstein-Wasserstein metric defines the distance between two probability measures $ \mu $ and $ \nu $ on $ {R^{d + 1}}$ by computing the cheapest way to transport the mass of $ \mu $ onto $ \nu $, where the cost per unit mass transported is a given function $ c\left( x, y \right)$ on $ {R^{2d + 2}}$. Motivated by applications to shape recognition, we analyze this transportation problem with the cost $ c\left( x, y \right) = {\left\vert {x - y} \right\vert^2}$ and measures supported on two curves in the plane, or more generally on the boundaries of two domains $ \Omega , \Lambda \subset {R^{d + 1}}$. Unlike the theory for measures that are absolutely continuous with respect to Lebesgue, it turns out not to be the case that $ \mu - a.e.x \in \partial \Omega $ is transported to a single image $ y \in \partial \Lambda $; however, we show that the images of $ x$ are almost surely collinear and parallel the normal to $ \partial \Omega $ at $ x$. If either domain is strictly convex, we deduce that the solution to the optimization problem is unique. When both domains are uniformly convex, we prove a regularity result showing that the images of $ x \in \partial \Omega $ are always collinear, and both images depend on $ x$ in a continuous and (continuously) invertible way. This produces some unusual extremal doubly stochastic measures.

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DOI: https://doi.org/10.1090/qam/1788425
Article copyright: © Copyright 2000 American Mathematical Society

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