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Convolutional wasserstein distances: efficient optimal transportation on geometric domains

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Date
2015
Link to item file
https://hal.archives-ouvertes.fr/hal-01188953
Dewey
Probabilités et mathématiques appliquées
Sujet
displacement interpolation; Optimal transportation; Wasserstein distances; entropy
Journal issue
ACM Transactions on Graphics
Volume
34
Number
4
Publication date
2015
Article pages
art. 66
Publisher
Association for Computing Machinery
DOI
http://dx.doi.org/10.1145/2766963
URI
https://basepub.dauphine.fr/handle/123456789/15598
Collections
  • CEREMADE : Publications
Metadata
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Author
Solomon, Justin
De Goes, Fernando
Peyré, Gabriel
Cuturi, Marco
Butscher, Adrian
Nguyen, Andy
Du, Tao
Guibas, Leonidas
Type
Article accepté pour publication ou publié
Abstract (EN)
This paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The result- ing objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing.

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