Sliced and Radon Wasserstein Barycenters of Measures

Sliced and Radon Wasserstein Barycenters of Measures

Nicolas Bonneel, Julien Rabin, Gabriel Peyre, and Hanspeter Pfister.

(Journal of Mathematical Imaging and Vision, 2014.)

This article details two approaches to compute barycenters of measures using 1-D Wasserstein distances along radial projections of the input measures. The first method 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.