Wasserstein barycenters and applications in image processing

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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

Barycenters in physics and geometry are points that represent the center of mass 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 position of all the points in an object. Wasserstein barycenters attempt to

Definition

Given points in with nonnegative weights , the weighted barycenter of the points is the unique point minimizing . To generalize this to Wasserstein spaces, let be a domain and  be the set of probability measures on . Given a collection of probability measures and nonnegative weights , we define a weighted barycenter of as any probability measure  that minimizes  over the space


Existence and Uniqueness

Other spaces

Applications

Barycenters in Image processing

References