Rigid structures such as cars or any other solid objects are often represented by finite clouds of unlabeled points. The most natural equivalence on these point clouds is rigid motion or isometry maintaining all inter-point distances. Rigid patterns of point clouds can be reliably compared only by complete isometry invariants that can also be called equivariant descriptors without false negatives (isometric clouds having different descriptions) and without false positives (non-isometric clouds with the same description). Noise and motion in data motivate a search for invariants that are continuous under perturbations of points in a suitable metric. We propose the first continuous and complete invariant of unlabeled clouds in any Euclidean space. For a fixed dimension, the new metric for this invariant is computable in a polynomial time in the number of points.

翻译：摘要：汽车或其他固体物体等刚性结构通常由有限的无标签点云表示。这些点云最自然的等价关系是保持所有点间距离的刚体运动或等距变换。只有通过完全等距不变量（也可称为等变描述符），才能可靠地比较点云的刚性模式——该不变量既无假阴性（等距点云有不同的描述），也无假阳性（非等距点云具有相同描述）。数据中的噪声和运动促使我们寻找在适当度量下对点的扰动具有连续性的不变量。本文提出了首个适用于任意欧氏空间中无标签点云的连续且完全的不变量。对于固定维度，该不变量的新度量可在与点数呈多项式关系的时间内计算得出。