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In this paper, the mathematical theory of wallpaper groups is used to construct a computational tool for symmetry analysis
of periodic patterns. Starting with a novel peak detection algorithm based on “regions of dominance”, an input periodic pattern
can be automatically classified into one of the 17 wallpaper groups. The orbits of stabilizer subgr...
Contexts in source publication
Context 1
... symmetries of S form the symmetry group of S under composition. It has been proven that there are seventeen wallpaper groups Figure 2 describing patterns extended by t w o linearly independent translational generators 7,22. Mathematically, w allpaper groups are deened only for innnite patterns that cover the whole plane. ...
Context 2
... practice, we analyze a periodic pattern P of a nite area, and use the phrase symmetry group G of P" to mean that G is the symmetry group of the innnite periodic pattern that has P as a nite patch. Figure 2 depicts unit lattices for the 17 distinct wallpaper groups from 77. Each unit is characterized in terms of its translation generators, rotation, reeec- tion and glide-reeection symmetries. ...
Citations
... Some works move further beyond computing a translational lattice to address classifying repeated patterns according to the 17 plane symmetry groups (wallpapers) [24][25][26][27]29]. As a first step, lattice detection is performed. ...
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.
