Analysis of Planar Ornament Patterns via Motif Asymmetry Assumption and Local Connections

Adanova, Venera
Tarı, Zehra Sibel
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 p6, p6m, 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, 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.


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Adanova, V.; Tarı, Zehra Sibel (2016-01-01)
© 2015 Elsevier Inc.From art to science, ornaments constructed by repeating a base motif (tiling) have been a part of human culture. These ornaments exhibit various kinds of symmetries depending on the construction process as well as the symmetries of the base motif. The scientific study of the ornaments is the study of symmetry, i.e., the repetition structure. There is, however, an artistic side of the problem too: intriguing color permutations, clever choices of asymmetric interlocking forms, several symm...
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Ornaments are created by repeating a base motif via combination of four primitive geometric repetition operations: translation, rotation, reflection, and glide reflection. The way the operations are combined defines symmetry groups. Thus, the classical study of ornaments is based on group theory. However, the discrete and inflexible nature of symmetry groups fail to capture relations among ornaments when artistic freedom is used to break symmetry via intriguing choices of base motifs and color permutations....
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Citation Formats
V. Adanova and Z. S. Tarı, “Analysis of Planar Ornament Patterns via Motif Asymmetry Assumption and Local Connections,” JOURNAL OF MATHEMATICAL IMAGING AND VISION, pp. 269–291, 2019, Accessed: 00, 2020. [Online]. Available: