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.
Publié le : 2015-07-05
Classification:
Barycenter of measures,
Wasserstein distance,
Radon transform,
Optimal transport,
[INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing,
[SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing,
[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
@article{hal-00881872,
author = {Bonneel, Nicolas and Rabin, Julien and Peyr\'e, Gabriel and Pfister, Hanspeter},
title = {Sliced and Radon Wasserstein Barycenters of Measures},
journal = {HAL},
volume = {2015},
number = {0},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00881872}
}
Bonneel, Nicolas; Rabin, Julien; Peyré, Gabriel; Pfister, Hanspeter. Sliced and Radon Wasserstein Barycenters of Measures. HAL, Tome 2015 (2015) no. 0, . http://gdmltest.u-ga.fr/item/hal-00881872/