Wasserstein barycenters and applications in image processing

In optimal transport, a Wasserstein barycenter 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 $\mathbb {R} ^{n}$ 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 distance of two probability measures, $W_{2}(\rho ,\mu )$.

Definition

Let $\Omega$ be a domain and ${\mathcal {P}}(\Omega )$ be the set of probability measures on $\Omega$. Given a collection of probability measures $\{\mu _{i}\}_{i\in I}$ and nonnegative weights $\{\lambda _{i}\}_{i\in I}$, we define a weighted barycenter of $\{\mu \}$ as any probability measure $\mu$ that minimizes  $\sum _{i\in I}\lambda _{i}W_{2}^{2}(\mu _{i},\mu )^{2}$ over the space $\mu \in {\mathcal {P}}(\Omega )$. Here $W_{2}$ denotes the 2-Wasserstein distance.

Generalizations

Wasserstein barycenters are examples of Karcher and Fréchet means where the distance function used is the Wasserstein distance.