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 for unlabeled points studied the binary problem of isometry detection, incomplete invariants, or approximations to Hausdorff-style distances, which require minimizations over infinitely many general isometries. This paper introduces the first continuous and complete isometry invariants for 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.

翻译：有限无标号点集是许多实际物体的最简表示，例如考虑模去刚体运动或保持点间距离的等距变换的刚性形状。距离矩阵在欧氏等距下唯一确定任意有限有标号（有序）点集，但由于存在大量置换，该方法无法用于比较无标号点集。针对无标号点集的先前工作主要研究等距检测的二元问题、不完备不变量或对豪斯多夫型距离的近似，这些方法需要对无穷多个一般等距变换进行最小化计算。本文首次提出针对任意欧氏空间中考虑等距变换的有限无标号点集的连续且完备的等距不变量。在瓶颈距离扰动下，本文证明了所提方法的连续性，并基于新度量实现在固定维度下关于点数的多项式时间内精确计算。