Shape spaces in terms of Wasserstein geometry

TL;DR

This paper introduces a new concept of Shape space derived from Wasserstein space by factoring out isometric transformations, establishing its metric properties and relating geodesics to those in Wasserstein space.

Contribution

It defines Shape spaces as quotients of Wasserstein spaces by isometry group actions, proves their Polish space structure, and relates their geodesics to Wasserstein geodesics.

Findings

Shape space $ ext{S}_p(X)$ is Polish under proper actions.

The Wasserstein distance induces a natural shape distance.

Geodesics in shape space relate to Wasserstein geodesics.

Abstract

For a Polish space $X$, we define the Shape space $\mathcal{S}_p(X)$ to be the Wasserstein space $W_p(X)$ modulo the action of a subgroup $G$ of the isometry group $ISO(X)$ of $X$, where the action is given by the pushforward of measures. The Wasserstein distance can then naturally be transformed into a \emph{Shape distance} on Shape space if $X$ and the action of $G$ are proper. This is shown for example to be the case for complete connected Riemannian manifolds with $G$ being equipped with the compact-open topology. Before finally proposing a notion for tangent spaces on the Shape space $\mathcal{S}_2(\mathbb{R}^n)$, it is shown that $\mathcal{S}_p(X)$ is Polish as well in case $X$ and the action of $G$ are indeed proper. Also, the metric geodesics in $\mathcal{S}_p(X)$ are put in relation to the ones in $W_p(X)$.