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

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