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 $c\left ( x, y \right ) = {\left | {x - y} \right |^2}$ 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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L. Zajíček, On the differentiability of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29 (104), 340–348 (1979)
G. Alberti, On the structure of singular sets of convex functions, Calc. Var. Partial Differential Equations 2, 17–27 (1994)
G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in ${R^{n}}$, Math. Z. 230, 259–316 (1999)
Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 305, 805–808 (1987)
Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44, 375–417 (1991)
L. A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc. 5, 99–104 (1992)
L. A. Caffarelli, Boundary regularity of maps with convex potentials. II, Ann. of Math. (2) 144, 453–496 (1996)
L. A. Caffarelli and R. J. McCann, Free boundary problems in optimal transport, in preparation
F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983
D. S. Fry, Shape recognition using metrics on the space of shapes, Ph.D. thesis, Harvard University, 1993
W. Gangbo, An elementary proof of the polar factorization of vector-valued functions, Arch. Rational Mech. Anal. 128, 381–399 (1994)
W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math. 177, 113–161 (1996)
C. R. Givens and R. M. Shortt, A class of Wasserstein metrics for probability distributions, Michigan Math. J. 31, 231–240 (1984)
L. Kantorovich, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37, 199–201 (1942)
L. V. Kantorovich and G. S. Rubinstein, On a functional space and certain extremum problems, Dokl. Akad. Nauk SSSR (N.S.) 115, 1058–1061 (1957)
H. G. Kellerer, Duality theorems for marginal problems, Z. Wahrsch. Verwandte Gebiete 67, 399–432 (1984)
R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80, 309–323 (1995)
R. J. McCann, Equilibrium shapes for planar crystals in an external field, Comm. Math. Phys. 195, 699–723 (1998)
D. Mumford, Mathematical theories of shape: Do they model perception, in Geometrical Methods of Computer Vision, vol. 1570 of the Proceedings of the Society for Photo-Optical Instrumentation Engineers, Bellingham, 1991, pp. 2 10
S. T. Rachev, The Monge-Kantorovich mass transference problem and its stochastic applications, Theory Probab. Appl. 29, 647 676 (1984)
R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1972
L. Rüschendorf and S. T. Rachev, A characterization of random variables with minimum ${L^{2}}$-distance, J. Multivariate Anal. 32, 48–54 (1990)
T. L. Seethoff and R. C. Shiflett, Doubly stochastic measures with prescribed support, Z. Wahrsch. Verwandte Gebiete 41, 283–288 (1978)
C. Smith and M. Knott, On the optimal transportation of distributions, J. Optim. Theory Appl. 52, 323–329 (1987)
L. N. Wasserstein, Markov processes over denumerable products of spaces describing large systems of automata, Problems of Information Transmission 5, 47–52 (1969)
L. Zajíček, On the differentiability of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29 (104), 340–348 (1979)
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