Analysis of planar ornament patterns via motif asymmetry assumption and local connections

TL;DR

This paper introduces an automatic method to classify planar ornament patterns into 13 symmetry groups and extract fundamental domains, even from small image fragments, focusing on asymmetric motifs similar to Escher's art.

Contribution

The method uniquely classifies ornaments and extracts fundamental domains without requiring global translational repetition, handling asymmetric motifs and small image fragments.

Findings

Successfully classifies ornaments into 13 symmetry groups

Extracts fundamental domains from minimal ornament fragments

Works with asymmetric motifs like Escher's art

Abstract

Planar ornaments, a.k.a. wallpapers, are regular repetitive patterns which exhibit translational symmetry in two independent directions. There are exactly $17$ distinct planar symmetry groups. We present a fully automatic method for complete analysis of planar ornaments in $13$ of these groups, specifically, the groups called $p6m, \, p6, \, p4g, \,p4m, \,p4, \, p31m, \,p3m, \, p3, \, cmm, \, pgg, \, pg, \, p2$ and $p1$. Given the image of an ornament fragment, we present a method to simultaneously classify the input into one of the $13$ groups and extract the so called fundamental domain (FD), the minimum region that is sufficient to reconstruct the entire ornament. A nice feature of our method is that even when the given ornament image is a small portion such that it does not contain multiple translational units, the symmetry group as well as the fundamental domain can still be defined. This is because, in contrast to common approach, we do not attempt to first identify a global translational repetition lattice. Though the presented constructions work for quite a wide range of ornament patterns, a key assumption we make is that the perceivable motifs (shapes that repeat) alone do not provide clues for the underlying symmetries of the ornament. In this sense, our main target is the planar arrangements of asymmetric interlocking shapes, as in the symmetry art of Escher.