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Wasserstein barycentric coordinates: histogram regression using optimal transport

Bonneel, Nicolas; Peyré, Gabriel; Cuturi, Marco (2016), Wasserstein barycentric coordinates: histogram regression using optimal transport, 43rd international conference and exhibition on Computer Graphics & Interactive Techniques (SIGGRAPH 2016), 2016-07, Anaheim, Etats-Unis

Type
Communication / Conférence
External document link
https://hal.archives-ouvertes.fr/hal-01303148
Date
2016
Conference title
43rd international conference and exhibition on Computer Graphics & Interactive Techniques (SIGGRAPH 2016)
Conference date
2016-07
Conference city
Anaheim
Conference country
Etats-Unis
Journal name
ACM Transactions on Graphics
Volume
35
Number
4
Publisher
Association for Computing Machinery
Pages
n°71
Publication identifier
10.1145/2897824.2925918
Metadata
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Author(s)
Bonneel, Nicolas cc

Peyré, Gabriel

Cuturi, Marco
Abstract (EN)
This article defines a new way to perform intuitive and geometrically faithful regressions on histogram-valued data. It leverages the theory of optimal transport, and in particular the definition of Wasserstein barycenters, to introduce for the first time the notion of barycentric coordinates for histograms. These coordinates take into account the underlying geometry of the ground space on which the histograms are defined, and are thus particularly meaningful for applications in graphics to shapes, color or material modification. Beside this abstract construction, we propose a fast numerical optimization scheme to solve this backward problem (finding the barycentric coordinates of a given histogram) with a low computational overhead with respect to the forward problem (computing the barycenter). This scheme relies on a backward algorithmic differentiation of the Sinkhorn algorithm which is used to optimize the entropic regularization of Wasserstein barycenters. We showcase an illustrative set of applications of these Wasserstein coordinates to various problems in computer graphics: shape approximation, BRDF acquisition and color editing.
Subjects / Keywords
Sinkhorn algorithm; barycentric coordinates; Wasserstein distance; optimal transport; fitting

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