Computable complete invariants for finite clouds of unlabeled points under Euclidean isometry

Abstract

A finite cloud of unlabeled points is the simplest representation of many real objects such as rigid shapes considered modulo rigid motion or isometry preserving inter-point distances. The distance matrix uniquely determines any finite cloud of labeled (ordered) points under Euclidean isometry but is intractable for comparing clouds of unlabeled points due to a huge number of permutations.The past work developed approximate algorithms for Hausdorff-like distances minimized over translations, rotations, and general isometries. One big success is the exact algorithm by Paul Chew et al for matching sets in the plane under Euclidean motion with a polynomial complexity of degree five in the number of points. We introduce continuous and complete isometry invariants on the spaces of finite clouds of unlabeled points considered under isometry in any Euclidean space. The continuity under perturbations of points in the bottleneck distance is proved in terms of new metrics that are exactly computable in polynomial time in the number of points for a fixed dimension.