Analysis of planar ornament patterns via motif asymmetry assumption and local connections
This work addresses the challenge of analyzing complex planar patterns, such as Escher-like art, for applications in computer vision and symmetry detection, though it is incremental as it builds on known symmetry group theory with a novel assumption.
The authors tackled the problem of automatically analyzing planar ornaments (wallpapers) by developing a method that simultaneously classifies an ornament fragment into one of 13 symmetry groups and extracts the fundamental domain, even from small portions lacking multiple translational units, without relying on global translational repetition lattices.
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.