Wasserstein Barycentric Coordinates: Histogram Regression Using Optimal Transport

Nicolas Bonneel (1, 2) , Gabriel Peyré (3, 4) , Marco Cuturi (5)

1 M2DisCo - Geometry Processing and Constrained Optimization
2 GeoMod - Modélisation Géométrique, Géométrie Algorithmique, Fractales
3 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
5 Kyoto University

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.

Mots clés: Sinkhorn algorithm; barycentric coordinates; Wasserstein distance; optimal transport; fitting

Domaines: Traitement du signal et de l'image [eess.SP]

Nicolas Bonneel, Gabriel Peyré, Marco Cuturi. Wasserstein Barycentric Coordinates: Histogram Regression Using Optimal Transport. ACM Transactions on Graphics, 2016, 35 (4), pp.71:1--71:10. ⟨10.1145/2897824.2925918⟩. ⟨hal-01303148⟩