Wasserstein barycenters and applications in image processing

In optimal transport, a Wasserstein barycenter (Insert reference of Sant) is a probability measure probability measure that acts as a center of mass between two or more probability measures. It generalizes the notions of physical barycenters and geometric centroids.

Introduction

Motivation

In astronomy, the barycenter is the center of mass of two or more objects that orbit each other, and in geometry, a centroid is the arithmetic mean position of all the points in an object. Wasserstein barycenters attempt to

Definition

Given points <math>\{x_i\}_{i \in I}<\math> in <math>R^n<\math> with nonnegative weights <math>\{\lambda_i \}_{i \in I}<\math>, the weighted <math>L^2<\math> barycenter of the points is the unique point <math>\{y\}<\math> minimizing <math>\sum_{i \in I} \lambda _i ||y - x_i||^2<\math>. To generalize this to Wasserstein spaces, let <math>\Omega<\math> be a domain and <math>P(\Omega)<\math> be the set of probability measures on <math>\Omega<\math>. Given a collection of probability measures <math>\{\mu_i/\rho_i\}_{i \in I}<\math> and nonnegative weights <math>\{\lambda_i\}_{i \in I}<\math>, we define a weighted barycenter of <math>\{\mu\}<\math> as any probability measure <math>\mu/\rho<\math> that minimizes  <math>\sum_{i \in I} \lambda_i W_2^2( \rho_i, \rho)^2<\math> over the space <math>\rho \in P(\Omega)<\math>.