Sliced and Radon Wasserstein Barycenters of Measures

Nicolas Bonneel 1, 2, 3 Julien Rabin 4 Gabriel Peyré 5, 6, * Hanspeter Pfister 3
* Corresponding author
1 GeoMod - Modélisation Géométrique, Géométrie Algorithmique, Fractales
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
2 M2DisCo - Geometry Processing and Constrained Optimization
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
4 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : This article details two approaches to compute barycenters of measures using 1-D Wasserstein distances along radial projections of the input measures. The first me- thod makes use of the Radon transform of the measures, and the second is the solution of a convex optimization problem over the space of measures. We show several properties of these barycenters and explain their relationship. We show numerical approximation schemes based on a discrete Radon transform and on the resolution of a non-convex optimization problem. We explore the respective merits and drawbacks of each approach on applications to two image processing problems: color transfer and texture mixing.
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Nicolas Bonneel, Julien Rabin, Gabriel Peyré, Hanspeter Pfister. Sliced and Radon Wasserstein Barycenters of Measures. Journal of Mathematical Imaging and Vision, Springer Verlag, 2015, 1 (51), pp.22-45. ⟨10.1007/s10851-014-0506-3⟩. ⟨hal-00881872⟩

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