The asymptotic behaviour and a near linear time algorithm for isometry invariants of periodic sets

TL;DR

This paper introduces a new, faster isometry invariant for periodic sets that is continuous, distinguishes sets with identical previous invariants, and has an explicit asymptotic analysis, enabling efficient processing of large datasets.

Contribution

It presents a novel near linear time algorithm for computing isometry invariants, improving speed and distinguishing power over previous methods.

Findings

New invariants are faster to compute and continuous.

Explicit asymptotic behavior described for various sets.

Algorithm efficiently processed hundreds of thousands of structures.

Abstract

The fundamental model of a periodic structure is a periodic point set up to rigid motion or isometry. Our recent paper in SoCG 2021 defined isometry invariants (density functions), which are complete in general position and continuous under perturbations. This work introduces much faster isometry invariants (average minimum distances), which are also continuous and distinguish some sets that have identical density functions. We explicitly describe the asymptotic behaviour of the new invariants for a wide class of sets including non-periodic. The proposed near linear time algorithm processed a dataset of hundreds of thousands of real structures in a few hours on a modest desktop.