标题： 深入探讨Procrustes-Wasserstein距离：属性和重心 显示英文标题

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

摘要： 由于其对刚体变换（如旋转和反射）的不变性，Procrustes-Wasserstein（PW）被引入文献中作为一种最优传输（OT）距离，作为Wasserstein的替代方案，更适合点云对齐和比较等任务。 考虑到这一应用，我们精心构建了一个离散概率测度空间，并证明在该空间上PW实际上是一个距离。 已经存在解决PW问题的算法，然而我们通过讨论和测试多种初始化策略来扩展PW框架。 然后我们引入了PW质心的概念，并详细描述了一个从数据中估计它的算法。 结果是一种从点云集合中计算代表性形状的新方法。 我们将该方法与现有的OT方法进行基准测试，证明在需要精确对齐和形状保持的场景中表现更优。 最后，我们在考古学背景下展示了PW质心的实用性。 我们的结果突显了PW在提升机器学习和计算几何应用中的2D和3D点云分析方面的潜力。 显示英文摘要

摘要： 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.