An in depth look at the Procrustes-Wasserstein distance: properties and barycenters

Due to its invariance to rigid transformations such as rotations and reflections, Procrustes-Wasserstein (PW) was introduced in the literature as an optimal transport (OT) distance, alternative to Wasserstein and more suited to tasks such as the alignment and comparison of point clouds. Having that application in mind, we carefully build a space of discrete probability measures and show that over that space PW actually is a distance. Algorithms to solve the PW problems already exist, however we extend the PW framework by discussing and testing several initialization strategies. We then introduce the notion of PW barycenter and detail an algorithm to estimate it from the data. The result is a new method to compute representative shapes from a collection of point clouds. We benchmark our method against existing OT approaches, demonstrating superior performance in scenarios requiring precise alignment and shape preservation. We finally show the usefulness of the PW barycenters in an archaeological context. Our results highlight the potential of PW in boosting 2D and 3D point cloud analysis for machine learning and computational geometry applications.