A Geometric Model of Multi-scale Orientation Preference Maps via Gabor Functions

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Abstract

In this paper we present a new model for the generation of orientation preference maps in the primary visual cortex (V1), considering both orientation and scale features. First we undertake to model the functional architecture of V1 by interpreting it as a principal fiber bundle over the 2-dimensional retinal plane by introducing intrinsic variables orientation and scale. The intrinsic variables constitute a fiber on each point of the retinal plane and the set of receptive profiles of simple cells is located on the fiber. Each receptive profile on the fiber is mathematically interpreted as a rotated Gabor function derived from an uncertainty principle. The visual stimulus is lifted in a 4-dimensional space, characterized by coordinate variables, position, orientation and scale, through a linear filtering of the stimulus with Gabor functions. Orientation preference maps are then obtained by mapping the orientation value found from the lifting of a noise stimulus onto the 2-dimensional retinal plane. This corresponds to a Bargmann transform in the reducible representation of the \(\text {SE}(2)=\mathbb {R}^2\times S^1\) group. A comparison will be provided with a previous model based on the Bargmann transform in the irreducible representation of the \(\text {SE}(2)\) group, outlining that the new model is more physiologically motivated. Then, we present simulation results related to the construction of the orientation preference map by using Gabor filters with different scales and compare those results to the relevant neurophysiological findings in the literature.

Keywords

Orientation maps Neurogeometry Differential geometry Gabor functions Bargmann transform 

References

  1. 1.
    Barbieri, D., Citti, G., Cocci, G., Sarti, A.: A cortical-inspired geometry for contour perception and motion integration. J. Math. Imaging Vis. 49(3), 511–529 (2014)CrossRefMATHGoogle Scholar
  2. 2.
    Barbieri, D., Citti, G., Sanguinetti, G., Sarti, A.: Coherent states of the euclidean group and activation regions of primary visual cortex. arXiv preprint arXiv:1111.0669 (2011)
  3. 3.
    Barbieri, D., Citti, G., Sanguinetti, G., Sarti, A.: An uncertainty principle underlying the functional architecture of v1. J. Physiol. Paris 106(5), 183–193 (2012)CrossRefGoogle Scholar
  4. 4.
    Bargmann, V.: On a hilbert space of analytie functions and an associated integral transform. part II. A family of related function spaces application to distribution theory. Commun. Pure Appl. Math. 20(1), 1–101 (1967)CrossRefMATHGoogle Scholar
  5. 5.
    Bednar, J.A., Miikkulainen, R.: Constructing visual function through prenatal and postnatal learning. Neuroconstructivism Perspect. Prospects 2, 13–37 (2004)Google Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Bressloff, P.C., Cowan, J.D.: The functional geometry of local and horizontal connections in a model of v1. J. Physiol. Paris 97(2), 221–236 (2003)CrossRefGoogle Scholar
  8. 8.
    Bressloff, P.C., Cowan, J.D.: A spherical model for orientation and spatial-frequency tuning in a cortical hypercolumn. Philos. Trans. R. Soc. Lond. B Biol. Sci. 358(1438), 1643–1667 (2003)CrossRefGoogle Scholar
  9. 9.
    Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J., Wiener, M.C.: Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex. Philos. Trans. R. Soc. B Biol. Sci. 356(1407), 299–330 (2001)CrossRefGoogle Scholar
  10. 10.
    Cang, J., Rentería, R.C., Kaneko, M., Liu, X., Copenhagen, D.R., Stryker, M.P.: Development of precise maps in visual cortex requires patterned spontaneous activity in the retina. Neuron 48(5), 797–809 (2005)CrossRefGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Citti, G., Sarti, A.: Neuromathematics of Vision. Springer, Berlin (2014)CrossRefMATHGoogle Scholar
  13. 13.
    Cocci, G., Barbieri, D., Citti, G., Sarti, A.: Cortical spatiotemporal dimensionality reduction for visual grouping. Neural Comput. 27(6), 1252–1293 (2015)CrossRefGoogle Scholar
  14. 14.
    Das, A., Gilbert, C.D.: Long-range horizontal connections and their role in cortical reorganization revealed by optical recording of cat primary visual cortex. Nature 375(6534), 780 (1995)CrossRefGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Field, D., Tolhurst, D.: The structure and symmetry of simple-cell receptive-field profiles in the cat’s visual cortex. Proc. R. Soc. Lond. B Biol. Sci. 228(1253), 379–400 (1986)CrossRefGoogle Scholar
  17. 17.
    Field, D.J., Hayes, A., Hess, R.F.: Contour integration by the human visual system: evidence for a local association field. Vis. Res. 33(2), 173–193 (1993)CrossRefGoogle Scholar
  18. 18.
    Folland, G.B.: Harmonic Analysis in Phase Space, (AM-122), vol. 122. Princeton University Press, Princeton (2016)Google Scholar
  19. 19.
    Gabor, D.: Theory of communication. part 1: the analysis of information. J. Inst. Electr. Eng. Part III Radio Commun. Eng. 93(26), 429–441 (1946)Google Scholar
  20. 20.
    Hörmander, L.: Hypoelliptic second order differential equations. Acta Mathematica 119(1), 147–171 (1967)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jegelka, S., Bednar, J.A., Miikkulainen, R.: Prenatal development of ocular dominance and orientation maps in a self-organizing model of v1. Neurocomputing 69(10), 1291–1296 (2006)CrossRefGoogle Scholar
  22. 22.
    Koenderink, J.J.: The structure of images. Biol. Cybern. 50(5), 363–370 (1984)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Koenderink, J.J., van Doorn, A.J.: Representation of local geometry in the visual system. Biol. Cybern. 55(6), 367–375 (1987)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lindeberg, T.: Feature detection with automatic scale selection. Int. J. Comput. Vis. 30(2), 79–116 (1998)CrossRefGoogle Scholar
  25. 25.
    Lindeberg, T.: Generalized gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space. J. Math. Imaging Vis. 40(1), 36–81 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Lindeberg, T.: A computational theory of visual receptive fields. Biol. Cybern. 107(6), 589–635 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Petitot, J.: The neurogeometry of pinwheels as a sub-riemannian contact structure. J. Physiol. Paris 97(2), 265–309 (2003)CrossRefGoogle Scholar
  28. 28.
    Petitot, J.: Neurogéométrie de la vision. Modeles mathématiques et physiques des architectures fonctionelles. Éd. École Polytech, Paris (2008)Google Scholar
  29. 29.
    Petitot, J., Tondut, Y.: Vers une neurogéométrie. fibrations corticales, structures de contact et contours subjectifs modaux. Mathématiques informatique et sciences humaines 145, 5–102 (1999)Google Scholar
  30. 30.
    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)CrossRefGoogle Scholar
  31. 31.
    Sarti, A., Citti, G.: The constitution of visual perceptual units in the functional architecture of v1. J. Comput. Neurosci. 38(2), 285–300 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sarti, A., Citti, G., Petitot, J.: The symplectic structure of the primary visual cortex. Biol. Cybern. 98(1), 33–48 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    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)CrossRefGoogle Scholar
  34. 34.
    Sarti, A., Piotrowski, D.: Individuation and semiogenesis: an interplay between geometric harmonics and structural morphodynamics. In: Sarti, A., Montanari, F. & Galofaro, F. (eds.) Morphogenesis and Individuation, pp. 49–73. Springer (2015)Google Scholar
  35. 35.
    Sharma, U., Duits, R.: Left-invariant evolutions of wavelet transforms on the similitude group. Appl. Comput. Harmon. Anal. 39(1), 110–137 (2015)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Stellwagen, D., Shatz, C.: An instructive role for retinal waves in the development of retinogeniculate connectivity. Neuron 33(3), 357–367 (2002)CrossRefGoogle Scholar
  37. 37.
    Sugiura, M.: Unitary Representations and Harmonic Analysis: An Introduction, vol. 44. Elsevier, Amsterdam (1990)MATHGoogle Scholar
  38. 38.
    Tanaka, S., Miyashita, M., Ribot, J.: Roles of visual experience and intrinsic mechanism in the activity-dependent self-organization of orientation maps: theory and experiment. Neural Netw. 17(8), 1363–1375 (2004)CrossRefGoogle Scholar
  39. 39.
    Wertheimer, M.: Laws of organization in perceptual forms. In: A Source Book of Gestalt Psychology. Harcourt Brace, New York (1923)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Emre Baspinar
    • 1
    • 2
  • Giovanna Citti
    • 1
    • 2
  • Alessandro Sarti
    • 1
    • 2
  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  2. 2.CAMS/CNRS-EHESSParisFrance

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