Bi-continuous and complete SE(2)-invariants parametrize all clouds of unordered points

Yury Elkin, Vitaliy Kurlin

Keywords: point cloud, rigid motion, isometry, complete invariant, continuous metric, molecule

TL;DR: This paper parametrizes the space of unordered point clouds under rigid motion by complete and realizable invariants with Lipschitz continuous distance metrics, illustrated on the QM9 database of 130K+ molecules.

Abstract: The most basic form of a rigid object is a cloud of unordered points, for example, a set of corners or other salient features. The rigid shape of a point cloud in the Euclidean plane is its SE(2)-equivalence class under rigid motion (a composition of translations and rotations). We introduce complete invariants (with no false negatives, no false positives) and a bi-Lipschitz continuous metric that satisfies all axioms, provides a 1-1 matching between points in clouds, and is computable in a quadratic time of the number $m$ of points. The realizability property implies that the space of all rigid clouds is efficiently parametrized by vectorial invariants like geographic coordinates. The new invariants justified that any of 130K+ molecules in the QM9 database is uniquely determined by the rigid shape of its atomic cloud.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Submission Number: 11715