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 advancing 2D and 3D point cloud analysis for machine learning and computational geometry applications.

Can we compute a “mean shape” from a set of objects, such as handwritten digits or 3D scans of bones, in such a way to preserve their finest geometric details, while ignoring differences solely due to rotations, reflections or rigid transformations in general? And can such a “mean” be used for wider purposes such as tracking a shape's evolution in time?Traditional machine learning methods often struggle with shapes appearing in different poses or orientations, such as rotated or mirrored. To address this problem, we formalize the Procrustes-Wasserstein (PW) distance as an optimal transport metric for measuring the similarity between discrete distributions, such as shapes represented as point clouds. Unlike previous approaches, PW is both mathematically rigorous (we prove that it is a true distance) and fits naturally to recover rigid transformations in the Euclidean space. We also introduce PW barycenters as a novel technique for computing interpolations from a set of point clouds in an isometric invariant way. Our method provides faithful and interpretable representations and highlights better geometric details than existing optimal transport tools. We demonstrate its effectiveness in real-world tasks, including shape clustering and tracking evolutionary changes in ancient animal bones. By making point cloud comparison more accurate and robust, our research has the potential to open new perspectives in fundamental research in zoo-archaeology and, more generally, in the the analysis of complex 2D and 3D data.