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

2025-07-01

AI Summary

Balancing rigid transformation invariance and shape fidelity remains challenging in point cloud alignment and comparison. Method: This paper systematically establishes the theoretical framework of the Procrustes–Wasserstein (PW) distance, rigorously proving its metric properties over discrete probability measures. It introduces the PW barycenter and proposes an iterative estimation algorithm that jointly leverages optimal transport and Procrustes analysis, augmented with a multi-initialization strategy for enhanced robustness. Contribution/Results: Unlike conventional optimal transport (OT) methods, the PW distance inherently satisfies rigid invariance, while the barycenter formulation significantly improves shape representativeness and alignment accuracy. Experiments demonstrate superior performance on both 2D and 3D point cloud tasks, and real-world archaeological data further validate its effectiveness. The proposed framework establishes a novel paradigm for geometric data analysis.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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.

Problem

• Develops Procrustes-Wasserstein distance for point cloud alignment
• Introduces algorithms for computing PW barycenters from data
• Benchmarks PW against OT methods for shape preservation

Innovation

• Procrustes-Wasserstein distance for point cloud alignment
• Novel initialization strategies for PW framework
• PW barycenter algorithm for representative shape computation