Framelet analysis of some geometrical illusions

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.

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)

    MathSciNet  Google 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)

    MathSciNet  Google 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)

    MATH  MathSciNet  Google 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)

    MATH  Article  MathSciNet  Google 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)

    MATH  Article  MathSciNet  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  13. 13

    Kitaoka A.: Tilt illusions after Oyama (1960): a review. Jpn. Psychol. Res. 49, 7–19 (2007)

    Article  Google 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)

    Article  Google Scholar 

  16. 16

    Lauwerier H.A.: Fractals: Endlessly Repeated Geometrical Figures. Princeton University Press, Princeton (1991)

    Google Scholar 

  17. 17

    Morgan M.J., Moulden B.: The Münsterberg figure and twisted cords. Vis. Res. 26, 1793–1800 (1986)

    Article  Google 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)

    Google Scholar 

Download references

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Correspondence to Hitoshi Arai.

Additional information

H. Arai was supported partly by Precursory Research for Embryonic Science and Technology, Japan Science and Technology Agency, and by Grant-in-Aid for Scientific Research (B), Japan Society for the Promotion of Science.

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Arai, H., Arai, S. Framelet analysis of some geometrical illusions. Japan J. Indust. Appl. Math. 27, 23–46 (2010). https://doi.org/10.1007/s13160-010-0009-6

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Keywords

  • Geometrical illusion
  • Wavelet frame
  • Framelet
  • Extrastriate visual cortex

Mathematics Subject Classification (2000)

  • 92C99
  • 98A08
  • 68U10