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A Neuromathematical Model for Geometrical Optical Illusions

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Abstract

Geometrical optical illusions have been object of many studies due to the possibility they offer to understand the behavior of low-level visual processing. They consist in situations in which the perceived geometrical properties of an object differ from those of the object in the visual stimulus. Starting from the geometrical model introduced by Citti and Sarti (J Math Imaging Vis 24(3):307–326, 2006), we provide a mathematical model and a computational algorithm which allows to interpret these phenomena and to qualitatively reproduce the perceived misperception.

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References

  1. Angelucci, A., Levitt, J.B., Walton, E.J., Hupe, J.M., Bullier, J., Lund, J.S.: Circuits for local and global signal integration in primary visual cortex. J. Neurosci. 22(19), 8633–8646 (2002)

    Google Scholar 

  2. Bekkers, E., Duits, R., Berendschot, T., ter Haar Romeny, B.: A multi-orientation analysis approach to retinal vessel tracking. J. Math. Imaging Vis. 49(3), 583–610 (2014)

    Article  MATH  Google Scholar 

  3. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 417–424. ACM Press/Addison-Wesley Publishing Co. (2000)

  4. Bigun, J.: Optimal orientation detection of linear symmetry. In: Proc. of the IEEE-First International Conference on Computer Vision, pp. 433–438, London, June 8–11 (1987)

  5. Bosking, W.H., Zhang, Y., Schofield, B., Fitzpatrick, D.: Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. J. Neurosci. 17(6), 2112–2127 (1997)

    Google Scholar 

  6. Brox, T., Weickert, J., Burgeth, B., Mrázek, P.: Nonlinear structure tensors. Image Vis. Comput. 24(1), 41–55 (2006)

    Article  Google Scholar 

  7. Citti, G., Sarti, A.: A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24(3), 307–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Colman, A.M.: A Dictionary of Psychology. Oxford University Press, Oxford (2015)

    Google Scholar 

  9. Coren, S., Girgus, J.S.: Seeing is Deceiving: The Psychology of Visual Illusions. Lawrence Erlbaum, Hillsdale, NJ (1978)

    Google Scholar 

  10. Daugman, J.G.: Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. JOSA A 2(7), 1160–1169 (1985)

    Article  Google Scholar 

  11. DeAngelis, G.C., Ohzawa, I., Freeman, R.D.: Receptive-field dynamics in the central visual pathways. Trends Neurosci. 18(10), 451–458 (1995)

    Article  Google Scholar 

  12. Duits, R., Felsberg, M., Granlund, G., ter Haar Romeny, B.: Image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the euclidean motion group. Int. J. Comput. Vis. 72(1), 79–102 (2007)

    Article  Google Scholar 

  13. Eagleman, D.M.: Visual illusions and neurobiology. Nat. Rev. Neurosci. 2(12), 920–926 (2001)

    Article  Google Scholar 

  14. Ehm, W., Wackermann, J.: Modeling geometric-optical illusions: a variational approach. J. Math. Psychol. 56(6), 404–416 (2012)

    Article  MathSciNet  Google Scholar 

  15. Favali, M., Citti, G., Sarti, A.: Local and global gestalt laws: a neurally based spectral approach. Neural Comput. 29(2), 394–422 (2017)

    Article  Google Scholar 

  16. Felsberg, M.: Adaptive filtering using channel representations. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, M.C., Davies, L. (eds.) Mathematical Methods for Signal and Image Analysis and Representation, pp. 31–48. Springer-Verlag London (2012)

  17. Fermüller, C., Malm, H.: Uncertainty in visual processes predicts geometrical optical illusions. Vis. Res. 44(7), 727–749 (2004)

    Article  Google Scholar 

  18. Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: Proceedings of ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, pp. 281–305 (1987)

  19. Franken, E., van Almsick, M., Rongen, P., Florack, L., ter Haar Romeny, B.: An efficient method for tensor voting using steerable filters. In: European Conference on Computer Vision, pp. 228–240. Springer (2006)

  20. Freeman, W.T., Adelson, E.H., et al.: The design and use of steerable filters. IEEE Trans. Pattern Anal. Mach. Intell. 13(9), 891–906 (1991)

    Article  Google Scholar 

  21. Geisler, W.S., Kersten, D.: Illusions, perception and bayes. Nat. Neurosci. 5(6), 508–510 (2002)

    Article  Google Scholar 

  22. Gibson, J.J.: The concept of the stimulus in psychology. Am. Psychol. 15(11), 694 (1960)

    Article  Google Scholar 

  23. Hering, H.E.: Beiträge zur physiologie, pp. 1–5 (1861)

  24. Hoffman, W.C.: Visual illusions of angle as an application of lie transformation groups. SIAM Rev. 13(2), 169–184 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hubel, D.H., Wiesel, T.N.: Ferrier lecture: functional architecture of macaque monkey visual cortex. Proc. R. Soc. Lond. B Biol. Sci. 198(1130), 1–59 (1977)

    Article  Google Scholar 

  26. Jones, J.P., Palmer, L.A.: An evaluation of the two-dimensional gabor filter model of simple receptive fields in cat striate cortex. J. Neurophysiol. 58(6), 1233–1258 (1987)

    Article  Google Scholar 

  27. Jost, J.: Riemannian Geometry and Geometric Analysis. Springer Science & Business Media, Springer-Verlag Berlin Heidelberg (2008)

    MATH  Google Scholar 

  28. Kennedy, H., Martin, K., Orban, G., Whitteridge, D.: Receptive field properties of neurones in visual area 1 and visual area 2 in the baboon. Neuroscience 14(2), 405–415 (1985)

    Article  Google Scholar 

  29. Knill, D.C., Richards, W.: Perception as Bayesian Inference. Cambridge University Press, New York (1996)

    Book  MATH  Google Scholar 

  30. Koenderink, J.J., Van Doorn, A.: Receptive field families. Biol. Cybern. 63(4), 291–297 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Koffka, K.: Principles of Gestalt Psychology, vol. 44. Routledge, London (2013)

    Google Scholar 

  32. Lee, T.S.: Image representation using 2d gabor wavelets. IEEE Trans. Pattern Anal. Mach. Intell. 18(10), 959–971 (1996)

    Article  Google Scholar 

  33. Levitt, J.B., Kiper, D.C., Movshon, J.A.: Receptive fields and functional architecture of macaque v2. J. Neurophysiol. 71(6), 2517–2542 (1994)

    Article  Google Scholar 

  34. Lubliner, J.: Plasticity Theory. Courier Corporation, North Chelmsford, MA (2008)

    MATH  Google Scholar 

  35. Marsden, J.E., Hughes, T.J.: Mathematical Foundations of Elasticity. Courier Corporation, North Chelmsford (1994)

    MATH  Google Scholar 

  36. Masnou, S., Morel, J.M.: Level lines based disocclusion. In: Proceedings. 1998 International Conference on Image Processing, 1998. ICIP 98, pp. 259–263. IEEE (1998)

  37. Medioni, G., Tang, C.K., Lee, M.S.: Tensor voting: theory and applications. In: Proceedings of RFIA, vol. 2000 (2000)

  38. Mordohai, P., Medioni, G.: Tensor voting: a perceptual organization approach to computer vision and machine learning. Synth. Lect. Image Video Multimed. Process. 2(1), 1–136 (2006)

    Article  Google Scholar 

  39. Murray, M.M., Herrmann, C.S.: Illusory contours: a window onto the neurophysiology of constructing perception. Trends Cogn. Sci. 17(9), 471–481 (2013)

    Article  Google Scholar 

  40. Murray, M.M., Wylie, G.R., Higgins, B.A., Javitt, D.C., Schroeder, C.E., Foxe, J.J.: The spatiotemporal dynamics of illusory contour processing: combined high-density electrical mapping, source analysis, and functional magnetic resonance imaging. J. Neurosci. 22(12), 5055–5073 (2002)

    Google Scholar 

  41. Oppel, J.J.: Uber geometrisch-optische tauschungen. Jahresbericht des physikalischen Vereins zu Frankfurt am Main (1855)

  42. Perona, P.: Deformable kernels for early vision. IEEE Trans. Pattern Anal. Mach. Intell. 17(5), 488–499 (1995)

    Article  Google Scholar 

  43. Petitot, J.: Neurogéométrie de la vision (2008)

  44. Robinson, J.O.: The Psychology of Visual Illusion. Courier Corporation, Dover, Mineola, NY (2013)

    Google Scholar 

  45. Romeny, B.M.H.: Front-End Vision and Multi-Scale Image Analysis: Multi-Scale Computer Vision Theory and Applications, Written in Mathematica, vol. 27. Springer Netherlands (2008)

  46. Sanguinetti, G., Citti, G., Sarti, A.: A model of natural image edge co-occurrence in the rototranslation group. J. Vis. 10(14), 37–37 (2010)

    Article  Google Scholar 

  47. Sarti, A., Citti, G., Petitot, J.: The symplectic structure of the primary visual cortex. Biol. Cybern. 98(1), 33–48 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sarti, A., Citti, G., Petitot, J.: Functional geometry of the horizontal connectivity in the primary visual cortex. J. Physiol. Paris 103(1), 37–45 (2009)

    Article  Google Scholar 

  49. Smith, D.A.: A descriptive model for perception of optical illusions. J. Math. Psychol. 17(1), 64–85 (1978)

    Article  MathSciNet  Google Scholar 

  50. Song, C., Schwarzkopf, D.S., Lutti, A., Li, B., Kanai, R., Rees, G.: Effective connectivity within human primary visual cortex predicts interindividual diversity in illusory perception. J. Neurosci. 33(48), 18781–18791 (2013)

    Article  Google Scholar 

  51. Van Almsick, M., Duits, R., Franken, E., ter Haar Romeny, B.: From stochastic completion fields to tensor voting. In: Fogh Olsen, O., Florack, L., Kuijper, A (eds.) Deep Structure, Singularities, and Computer Vision, pp. 124–134. Springer, Berlin, Heidelberg (2005)

  52. Von Der Heyclt, R., Peterhans, E., Baurngartner, G.: Illusory contours and cortical neuron responses. Science 224, 1260–1262 (1984)

    Article  Google Scholar 

  53. von Helmholtz, H., Southall, J.P.C.: Treatise on physiological optics. In: Southall, J.P.C (ed.) Helmholtz’s Treatise on Physiological Optics, vol. 3. Dover Publications (1962) (Anybook Ltd., Lincoln, UK, translated from 3rd German edition)

  54. Wade, N.: The Art and Science of Visual Illusions. Routledge Kegan & Paul, London (1982)

    Google Scholar 

  55. Walker, E.H.: A mathematical theory of optical illusions and figural aftereffects. Percept. Psychophys. 13(3), 467–486 (1973)

    Article  Google Scholar 

  56. Weickert, J.: Anisotropic Diffusion in Image Processing, vol. 1. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

  57. Weiss, Y., Simoncelli, E.P., Adelson, E.H.: Motion illusions as optimal percepts. Nat. Neurosci. 5(6), 598–604 (2002)

    Article  Google Scholar 

  58. Westheimer, G.: Illusions in the spatial sense of the eye: geometrical-optical illusions and the neural representation of space. Vis. Res. 48(20), 2128–2142 (2008)

    Article  Google Scholar 

  59. Wundt, W.M.: Die geometrisch-optischen Täuschungen, vol. 24. BG Teubner, Leipzig (1898)

    Google Scholar 

  60. Young, R.A.: The gaussian derivative model for spatial vision: I. retinal mechanisms. Spat. Vis. 2(4), 273–293 (1987)

    Article  Google Scholar 

  61. Zhang, J., Duits, R., Sanguinetti, G., ter Haar Romeny, B.M.: Numerical approaches for linear left-invariant diffusions on se (2), their comparison to exact solutions, and their applications in retinal imaging. Numer. Math. Theory Methods Appl. 9(01), 1–50 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This project has received funding from the European Unions Seventh Framework Programme, Marie Curie Actions- Initial Training Network, under Grant Agreement No. 607643, “Metric Analysis For Emergent Technologies (MAnET).” We would like to thank B. ter Haar Romeny, University of Technology Eindhoven, and M. Ursino, University of Bologna, for their important comments and remarks to the present work.

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  1. Centre d’analyse et mathématiques sociales, CNRS - EHESS, Paris, France

    B. Franceschiello & A. Sarti

  2. Dipartimento di Matematica, Università di Bologna, Bologna, Italy

    G. Citti

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  1. B. Franceschiello
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Correspondence to B. Franceschiello.

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Franceschiello, B., Sarti, A. & Citti, G. A Neuromathematical Model for Geometrical Optical Illusions. J Math Imaging Vis 60, 94–108 (2018). https://doi.org/10.1007/s10851-017-0740-6

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