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

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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 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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Acknowledgements

We thank the anonymous reviewers and the handling editor for providing constructive feedback which improved our paper. We thank Devin Giray Tari for creating several iOrnament artworks that contributed to our database. We thank all contributors to iOrnament Gallery especially its creator Jürgen Richter-Gebert. Finally, we thank TUBITAK who funded our research via Grant 114E204.

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  1. Department of Computer Engineering, Middle East Technical University, 06800, Ankara, Turkey

    Venera Adanova & Sibel Tari

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  1. Venera Adanova
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Correspondence to Venera Adanova.

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Adanova, V., Tari, S. Analysis of Planar Ornament Patterns via Motif Asymmetry Assumption and Local Connections. J Math Imaging Vis 61, 269–291 (2019). https://doi.org/10.1007/s10851-018-0835-8

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