Advertisement

Framelet analysis of some geometrical illusions

  • Hitoshi AraiEmail author
  • Shinobu Arai
Open Access
Original Paper Area 1

Abstract

In this paper we study a spiral illusion generated by fractal islands. Furthermore, by a neuro-scientific consideration we present a new class of geometrical illusions. In order to analyse these illusions, we propose a new mathematical method.

Keywords

Geometrical illusion Wavelet frame Framelet Extrastriate visual cortex 

Mathematics Subject Classification (2000)

92C99 98A08 68U10 

References

  1. 1.
    Arai H.: A nonlinear model of visual information processing based on discrete maximal overlap wavelet. Interdiscip. Inf. Sci. 11, 177–190 (2005)MathSciNetGoogle Scholar
  2. 2.
    Arai H.: Wavelets. Kyoritsu Publ. Co., Tokyo (2010) (in Japanese)Google Scholar
  3. 3.
    Arai H., Arai S.: Common factor of a certain kind of tilt illusions clarified by a wavelet, VISION. J. Vis. Soc. Japan 17, 259–265 (2005) (in Japanese)MathSciNetGoogle Scholar
  4. 4.
    Arai H., Arai S.: Finite discrete, shift-invariant, directional filterbanks for visual information processing, I: Construction. Interdiscip. Inf. Sci. 13, 255–273 (2007)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Arai H., Arai S.: 2D tight framelets with orientation selectivity suggested by vision science. Invited paper. JSIAM Lett. 1, 9–12 (2009)Google Scholar
  6. 6.
    Cohen A., Daubechies I., Feauveau J.-C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coifman R.R., Donoho D.L.: Translation invariant de-noising. Lect. Notes Stat. 103, 125–150 (1995)Google Scholar
  8. 8.
    Daubechies I., Han B., Ron A., Shen Z.: Framelets: MRA-based construction of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fraser J.: A new visual illusion of direction. Br. J. Psychol. 2, 307–320 (1908)Google Scholar
  10. 10.
    Gallant J.L., Braun J., Van Essen D.C.: Selectivity for polar, hyperbolic, and Cartesian gratings in macaque visual cortex. Science 259, 100–103 (1993)CrossRefGoogle Scholar
  11. 11.
    Gallant J.L., Conner C.E., Rakshit S., Lewis J.W., Van Essen D.C.: Neural responses to polar, hyperbolic, and Cartesian gratings in area V4 of the macaque monkey. J. Neurophysiol. 76, 2718–2739 (1996)Google Scholar
  12. 12.
    Gregory R.L., Heard P.: Border locking and the Cafe Wall illusion. Perception 8, 365–380 (1979)CrossRefGoogle Scholar
  13. 13.
    Kitaoka A.: Tilt illusions after Oyama (1960): a review. Jpn. Psychol. Res. 49, 7–19 (2007)CrossRefGoogle Scholar
  14. 14.
    Kitaoka A.: Geometrical illusions. In: Goto, T., Tanaka, H. (eds) Handbook of the Science of Illusion., pp. 56–77. University of Tokyo Press, Tokyo (2005) (in Japanese)Google Scholar
  15. 15.
    Kitaoka A., Pinna B., Brelstaff G.: New variants of the spiral illusion. Perception 30, 637–646 (2001)CrossRefGoogle Scholar
  16. 16.
    Lauwerier H.A.: Fractals: Endlessly Repeated Geometrical Figures. Princeton University Press, Princeton (1991)zbMATHGoogle Scholar
  17. 17.
    Morgan M.J., Moulden B.: The Münsterberg figure and twisted cords. Vis. Res. 26, 1793–1800 (1986)CrossRefGoogle Scholar
  18. 18.
    Nason G.P., Silverman B.W.: The stationary wavelet transform and some statistical applications. Lect. Notes Stat. 103, 281–299 (1995)Google Scholar
  19. 19.
    Percival D.B., Walden A.T.: Wavelet Methods for Time Series Analysis. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2010

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan
  2. 2.TokyoJapan

Personalised recommendations