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Symmetries of 1-D Images

  • Lewis D. Griffin
Article

Abstract

The possible automorphism groups of scalar functions of a one-dimensional Euclidean domain are presented. The groups are determined relative to a class of transformations that allow an isometry of the function domain, simultaneous with a separate isometry of the function co-domain. Ten non-trivial automorphism groups are found. Seven of these are related to five of the seven Frieze groups.

Keywords

Image structure Group theory Visual perception Image features 

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References

  1. 1.
    Rosen, J.: Resource letter SP-2: Symmetry and group theory in physics. Am. J. Phys. 49(4), 304–319 (1981) CrossRefGoogle Scholar
  2. 2.
    Cantwell, B.J.: Introduction to Symmetry Analysis. Cambridge University Press, Cambridge (2002) zbMATHGoogle Scholar
  3. 3.
    Hahn, T. (ed.): Space-Group Symmetry. International Tables for Crystallography, vol. A. International Union of Crystallography, Chester (2006) Google Scholar
  4. 4.
    Jones, O.: The Grammar of Ornament. Day, London (1856) Google Scholar
  5. 5.
    Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Raume I. Math. Ann. 70, 297–336 (1911) CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Conway, J.H., et al.: On three-dimensional space groups. Contrib. Algebra Geom. 42(2), 475–507 (2001) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Grünbaum, B., Shephard, G.C.: Tilings and Patterns. Freeman, New York (1987) zbMATHGoogle Scholar
  8. 8.
    Schattschneider, D.: MC Escher. Visions of Symmetry. Plenum Press, New York (1990) Google Scholar
  9. 9.
    Holser, W.T.: Classification of symmetry groups. Acta Crystallograph. 14, 1236–1242 (1961) CrossRefGoogle Scholar
  10. 10.
    Loeb, A.A.: Color and Symmetry. Krieger, Melbourne (1978) Google Scholar
  11. 11.
    Koenderink, J.J., van Doorn, A.J.: Image processing done right. In: ECCV 2002, Copenhagen. Springer, Berlin (2002) Google Scholar
  12. 12.
    Alexander, D.C., et al.: Spatial transformations of diffusion tensor magnetic resonance images. IEEE Trans. Med. Imaging 20(11), 1131–1139 (2001) CrossRefGoogle Scholar
  13. 13.
    Vovk, U., Pernus, F., Likar, B.: A review of methods for correction of intensity inhomogeneity in MRI. IEEE Trans. Med. Imaging 26(3), 405–421 (2007) CrossRefGoogle Scholar
  14. 14.
    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) CrossRefGoogle Scholar
  15. 15.
    Baylis, G.C., Driver, J.: Perception of symmetry and repetition within and across visual shapes: Part-descriptions and object-based attention. Vis. Cognit. 8(2), 163–196 (2001) CrossRefGoogle Scholar
  16. 16.
    Dakin, S.C., Watt, R.J.: Detection of bilateral symmetry using spatial filters. Spat. Vis. 8(4), 393–413 (1994) CrossRefGoogle Scholar
  17. 17.
    Levi, D.M., Saarinen, J.: Perception of mirror symmetry in amblyopic vision. Vis. Res. 44(21), 2475–2482 (2004) CrossRefGoogle Scholar
  18. 18.
    Mancini, S., Sally, S.L., Gurnsey, R.: Detection of symmetry and anti-symmetry. Vis. Res. 45(16), 2145–2160 (2005) CrossRefGoogle Scholar
  19. 19.
    Oka, S., et al.: VEPs elicited by local correlations and global symmetry: Characteristics and interactions. Vis. Res. 47(16), 2212–2222 (2007) CrossRefGoogle Scholar
  20. 20.
    Rainville, S.J.M., Kingdom, F.A.A.: The functional role of oriented spatial filters in the perception of mirror symmetry—psychophysics and modeling. Vis. Res. 40(19), 2621–2644 (2000) CrossRefGoogle Scholar
  21. 21.
    Sally, S., Gurnsey, R.: Symmetry detection across the visual field. Spat. Vis. 14(2), 217–234 (2001) CrossRefGoogle Scholar
  22. 22.
    Scognamillo, R., et al.: A feature-based model of symmetry detection. Proc. R. Soc. Lond. Ser. B—Biol. Sci. 270(1525), 1727–1733 (2003) CrossRefGoogle Scholar
  23. 23.
    Wilson, H.R., Wilkinson, F.: Symmetry perception: a novel approach for biological shapes. Vis. Res. 42(5), 589–597 (2002) CrossRefGoogle Scholar
  24. 24.
    Griffin, L.D.: The 2nd order local-image-structure solid. IEEE Trans. Pattern Anal. Mach. Intell. 29(8), 1355–1366 (2007) CrossRefGoogle Scholar
  25. 25.
    Griffin, L.D., Lillholm, M.: Hypotheses for image features, icons and textons. Int. J. Comput. Vis. 71(3), 213–230 (2006) CrossRefGoogle Scholar
  26. 26.
    Yale, P.B.: Geometry and Symmetry. Dover Press, New York (1968) zbMATHGoogle Scholar
  27. 27.
    Armstrong, M.A.: Groups and Symmetry. Springer, Berlin (1988) zbMATHGoogle Scholar
  28. 28.
    Glassner, A.: Frieze groups. IEEE Comput. Graph. Appl. 16(3), 78–83 (1996) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Computer ScienceUniversity College LondonLondonUK

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