Wasserstein barycenters and applications in image processing

In optimal transport, a Wasserstein barycenter (Insert reference of Sant) is a 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

Barycenters in physics and geometry are points that represent a notion of a mean of a number of objects. 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 of all the points in an object. Given countably many points $\{x_{i}\}_{i\in I}$ in $R^{n}$ with nonnegative weights $\{\lambda _{i}\}_{i\in I}$, the weighted $L^{2}$ barycenter of the points is the unique point $y$ minimizing $\sum _{i\in I}\lambda _{i}||y-x_{i}||_{2}^{2}$. Wasserstein barycenters attempt to capture this concept for probability measures by replacing the Euclidean distance with the Wasserstein metric.

Definition

To generalize this to Wasserstein spaces, let $\Omega$ be a domain and $P(\Omega )$ be the set of probability measures on $\Omega$. Given a collection of probability measures $\{\mu _{i}/\rho _{i}\}_{i\in I}$ and nonnegative weights $\{\lambda _{i}\}_{i\in I}$, we define a weighted barycenter of $\{\mu \}$ as any probability measure $\mu /\rho$ that minimizes  $\sum _{i\in I}\lambda _{i}W_{2}^{2}(\rho _{i},\rho )^{2}$ over the space $\rho \in P(\Omega )$. 

Generalizations

Wasserstein barycenters are examples of Karcher and Fr%C3%A9chet means where the distance function used is the Wasserstein distance.