Abstract
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 subgroups within the group lead to a small set of candidate motifs that exhibit local symmetry consistent with the global symmetry of the entire pattern. We further consider affine distorted periodic patterns and show that each such pattern can be classified into a small set of symmetry groups that describe the patterns’ potential symmetries under affine transformation.
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References
Fedorov, E.S.: The elements of the study of figures. [Russian] (2) 21. In: Zapiski Imperatorskogo S. Peterburgskogo Mineralogichesgo Obshchestva [Proc. S. Peterb. Mineral. Soc.], pp. 1–289 (1885)
Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W.H. Freeman and Company, New York (1987)
Kali. Programs that can automatically generate 2d planar crystallographic patterns, http://www.geom.umn.edu/apps/kali/
Lin, H.-C., Wang, L.-L., Yang, S.-N.: Extracting periodicity of a regular texture based on autocorrelation functions. Pattern Recognition Letters 18, 433–443 (1997)
Liu, Y., Collins, R.T.: A Computational Model for Repeated Pattern Perception using Frieze and Wallpaper Groups. In: Computer Vision and Pattern Recognition Conference (June 2000)
Liu, Y., Collins, R.T.: Frieze and wallpaper symmetry groups classification under affine and perspective distortion. Technical Report CMU-RI-TR-98-37, The Robotics Institute, Carnegie Mellon University, Pittsburgh, PA (1998)
Schattschneider, D.: The plane symmetry groups: their recognition and notation. American Mathematical Monthly 85, 439–450 (1978)
Washburn, D.K., Crowe, D.W.: Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. University of Washington Press (1991)
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Liu, Y., Collins, R.T. (2000). Periodic Pattern Analysis under Affine Distortions Using Wallpaper Groups. In: Sommer, G., Zeevi, Y.Y. (eds) Algebraic Frames for the Perception-Action Cycle. AFPAC 2000. Lecture Notes in Computer Science, vol 1888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722492_18
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DOI: https://doi.org/10.1007/10722492_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41013-3
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