Abstract
The different ways in which images, defined as scalar functions of the Euclidean plane, can be symmetrical is considered. The symmetries analyzed are relative to the class of image isometries, each of which is a combined spatial and intensity isometry. All symmetry types, apart from those with discrete periodic translations, are derived. Fifteen such types are found, including one that has not previously been reported. The novel type occurs when an image has a continuous line of centres of symmetry each like the one found in the Taiji (Yin-Yang) symbol.
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References
Rosen, J.: Resource letter SP-2: Symmetry and group theory in physics. Am. J. Phys. 49(4), 304–319 (1981)
Cantwell, B.J.: Introduction to Symmetry Analysis. Cambridge University Press, Cambridge (2002)
Hahn, T.: Space-Group Symmetry. International Tables for Crystallography, vol. A. International Union of Crystallography, Chester (2006)
Klein, F.: A comparative review of recent researches in geometry (trans. by MW Haskell). Bull. NY Math. Soc. 2, 215–249 (1892)
Baylis, G.C., Driver, J.: Perception of symmetry and repetition within and across visual shapes: Part-descriptions and object-based attention. Vis. Cogn. 8(2), 163–196 (2001)
Levi, D.M., Saarinen, J.: Perception of mirror symmetry in amblyopic vision. Vis. Res. 44(21), 2475–2482 (2004)
Oka, S., : VEPs elicited by local correlations and global symmetry: Characteristics and interactions. Vis. Res. 47(16), 2212–2222 (2007)
Sally, S., Gurnsey, R.: Symmetry detection across the visual field. Spat. Vis. 14(2), 217–234 (2001)
Braitenberg, V.: Vehicles. Experiments in Synthetic Psychology. MIT Press, Cambridge (1984)
Perrett, D.: Symmetry and human facial attractiveness. Evol. Hum. Behav. 20(5), 295–307 (1999)
Cairns, P., Thimbleby, H.: Affordance and symmetry in user interfaces. Comput. J. 51, 650–661 (2008)
Bonneh, Y., Reisfeld, D., Yeshurun, Y.: Quantification of local symmetry—application to texture-discrimination. Spat. Vis. 8(4), 515–530 (1994)
Mellor, M., Brady, M.: A new technique for local symmetry estimation. In: Scale Space and Pde Methods in Computer Vision, Proceedings, vol. 3459, pp. 38–49 (2005)
Scognamillo, R., : A feature-based model of symmetry detection. Proc. R. Soc. Lond. Ser. B-Biol. Sci. 270(1525), 1727–1733 (2003)
Liu, Y.X., Collins, R.T., Tsin, Y.H.: A computational model for periodic pattern perception based on frieze and wallpaper groups. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 354–371 (2004)
Griffin, L.D.: Symmetries of 1-D images. J. Math. Imaging Vis. 31(2–3), 157–164 (2008)
Yale, P.B.: Geometry and Symmetry. Dover, New York (1968)
Koenderink, J.J., van Doorn, A.J.: Image processing done right. In: Computer Vision—Eccv 2002, Pt 1, pp. 158–172 (2002)
Conway, J.H., Burgiel, H., Goodman-Strauss, C.: The Symmetries of Things. AK Peters, Wellesley (2008)
Browne, C.: Taiji variations: yin and yang in multiple dimensions. Comput. Graph. 31(1), 142–146 (2007)
Schattschneider, D.: MC Escher. Visions of Symmetry. Plenum, New York (1990)
Loeb, A.A.: Color and Symmetry. Krieger, Melbourne (1978)
Shubnikov, A.V., Kopstik, V.A.: Symmetry in Science and Art. Plenum, New York (1974)
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Griffin, L.D. Symmetries of 2-D Images: Cases without Periodic Translations. J Math Imaging Vis 34, 259–269 (2009). https://doi.org/10.1007/s10851-009-0148-z
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DOI: https://doi.org/10.1007/s10851-009-0148-z