Journal of Mathematical Imaging and Vision

, Volume 24, Issue 3, pp 307–326 | Cite as

A Cortical Based Model of Perceptual Completion in the Roto-Translation Space

Article

Abstract

We present a mathematical model of perceptual completion and formation of subjective surfaces, which is at the same time inspired by the architecture of the visual cortex, and is the lifting in the 3-dimensional rototranslation group of the phenomenological variational models based on elastica functional. The initial image is lifted by the simple cells to a surface in the rototranslation group and the completion process is modeled via a diffusion driven motion by curvature. The convergence of the motion to a minimal surface is proved. Results are presented both for modal and amodal completion in classic Kanizsa images.

Keywords

visual cortex perceptual completion cognitive neuroscience Lie group partial differential equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Ambrosio and S. Masnou, “A direct variational approach to a problem arising in image recostruction,” Interfaces and Free Boundaries, Vol. 5, No. 1, pp. 63–81, 2003.MathSciNetMATHGoogle Scholar
  2. 2.
    G. Barles and C. Georgelin, “A simple proof for the convergence for an approximation scheme for computing motions by mean curvature,” SIAM J. Numerical Analysis, Vol. 32, pp. 484–500, 1995.CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    O. Bar, H. Sompolinsky, and R. Ben-Yishai, “ Theory of orientation tuning in visual cortex” Proc. Natl. Acad. Sci. U.S.A., Vol. 92, pp. 3844–3848, 1995.Google Scholar
  4. 4.
    A. Bellaiche, “The tangent space in sub-Riemannian geometry” in Proceedings of the satellite Meeting of the 1st European, Congress of Mathematics ‘Journees nonholonomes: Geometrie sous-riemannienne, theorie du controle, robotique,’ Paris, France, June 30–July 1, 1992. Basel: Birkhäuser. Prog. Math., Vol. 144, pp. 1–78, 1996.Google Scholar
  5. 5.
    G. Bellettini and R. March, “An image segmentation variational model with free discontinuities and contour curvature,” Math. Mod. Meth. Appl. Sci., Vol. 14, pp. 1–45, 2004.CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    C. Ballester, M. Bertalmio, V. Caselles, G. Sapiro, and J. Verdera, “Filling-in by interpolation of vector fields and gray levels,” IEEE Transactions on Image Processing, Vol. 10, No. 8, pp. 1200–1211, 2001.CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    J. Bence, B. Merriman, and S. Osher, “Diffusion generated motion by mena curvature,” in Computational Crystal Growers Workshop, J. Taylor Sel. Taylor (Ed).Google Scholar
  8. 8.
    A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, “Fundamental solutions for non-divergence form operators on stratified groups,” Trans. Amer. Math. Soc., Vol. 356, No. 7, pp. 2709–2737, 2004.CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    L. Capogna, D. Danielli, and N. Garofalo, “The geometric Sobolev embedding for vector fields and the isoperimetric inequality” Comm Anal. Geo., Vol. 12, pp. 203–215, 1994.MathSciNetGoogle Scholar
  10. 10.
    M. Carandini and D.L. Ringach, “Predictions of a recurrent model of orientation selectivity,” Vision Res., Vol. 37, pp. 3061–3071, 1997.CrossRefGoogle Scholar
  11. 11.
    G. Citti, M. Manfredini, and A. Sarti, “Neuronal oscillations in the visual cortex: Γ-convergence to the Riemannian Mumford-Shah functional” SIAM Journal of Mathematical Analysis, Vol. 35, No. 6, pp. 1394–1419, 2004.CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    J.G. Daugman, “Uncertainty—relation for resolution in space spatial frequency and orientation optimized by two dimensional visual cortical filters,” J. Opt. Soc. Amer., Vol. 2, No. 7, pp. 1160–1169, 1985.Google Scholar
  13. 13.
    E. De Giorgi, “Some remarks on Γ convergence and least square methods,” in Composite Media and Homogeniziation Theory G. Dal Masoand and G. F. Dell’Antonio (Eds.), Birkhauser Boston, 1991, pp. 153–142.Google Scholar
  14. 14.
    S. Esedoglu and R. March, “Segmentation with Deph but without detecting junctions,” Journal of Mathematical Imaging and Vision, Vol. 18, pp. 7–15, 2003.CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    A.K. Engel, P. Konig, C.M. Gray, and W. Singer, “Stimulus dependent neuronal oscillations, in cat visual cortex: Intercolumnar interaction as determined by cross-correlation analysis” European Journal of Neuroscience, Vol. 2, pp. 558–606, 1990.CrossRefGoogle Scholar
  16. 16.
    A.K. Engel, A.K. Kreiter, P. Konig, and W. Singer, “Syncronization of oscillatory neuronal responses between striate and extrastriate visual cortical areas of the cat,” PNAS, Vol. 88, pp. 6048–6052, 1991.Google Scholar
  17. 17.
    L. Evans, “Convergence of an Algorithm for mean curvature motion,” Indiana Univ. Math. J., Vol. 42, No. 2, pp. 553–557, 1993.CrossRefGoogle Scholar
  18. 18.
    B. Franchi, R. Serapioni, and F. Serra Cassano, “On the structure of finite perimeter sets in step 2 Carnot groups,” J. Geom. Anal., Vol. 13, No. 3, pp. 421–466, 2003.MathSciNetMATHGoogle Scholar
  19. 19.
    G.B. Folland, “Subelliptic estimates and function spaces on nilpotent Lie groups,” Ark. Mat., Vol. 13, pp. 161–207, 1975.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    G.B. Folland, “On the Rothschild-Stein lifting theorem,” Commun. Partial Differ. Equations Vol. 2, pp. 165–191, 1977.MATHMathSciNetGoogle Scholar
  21. 21.
    G.B. Folland and E.M. Stein, “Estimates for the \(\bar\partial_b\) complex and analysis on the Heisenberg group,” Comm. Pure Appl. Math., Vol. 20, pp. 429–522, 1974.MathSciNetGoogle Scholar
  22. 22.
    R. Goodman, “Lifting vector fields to nilpotent Lie groups,” J. Math. Pures Appl., Vol. 57, pp. 77–85, 1978.MATHMathSciNetGoogle Scholar
  23. 23.
    C.D. Gilbert, A. Das, M. Ito, M. Kapadia, and G. Westheimer, “Spatial integration and cortical dynamics,” Proceedings of the National Academy of Sciences USA, Vol. 93, pp. 615–622.Google Scholar
  24. 24.
    C.M. Gray, P. Konig, A.K. Engel, and W. Singer, “Oscillatory responses in cat visual cortex exhibit inter-columnar syncronization which reflects global stimulus properties,” Nature, Vol. 338, pp. 334–337, 1989.CrossRefGoogle Scholar
  25. 25.
    S. Grossberg and E. Mingolla, “Neural dynamics of perceptual grouping: Textures, boundaries and emergent segmentations,” in Perception and Psychophysics, 1985.Google Scholar
  26. 26.
    Field, A. Heyes, and R.F. Hess, “Contour integration by the human visual system: Evidence for a local Association Field,” Vision Research, Vol. 33, pp. 173–193, 1993.CrossRefGoogle Scholar
  27. 27.
    W.C. Hoffman, “The visual cortex is a contact bundle,” Applied Mathematics and Computation, Vol 32, pp. 137–167, 1989.CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    W.C. Hoffman and M. Ferraro, “Lie transformation groups, integral transforms, and invariant pattern recognition,” Spatial Vision, Vol. 8, pp. 33–44, 1994.Google Scholar
  29. 29.
    H. Hörmander, “Hypoelliptic second-order differential equations,” Acta Math., Vol. 119, pp. 147–171, 1967.MATHMathSciNetGoogle Scholar
  30. 30.
    H. Hörmander and A. Melin, “Free systems of vector fields,” Ark. Mat, Vol. 16, pp. 83–88, 1978.CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    D. Hubel and T. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex,” Journal of Physiology, Vol. 160, pp. 106–154, 1962.Google Scholar
  32. 32.
    D. Jerison and A. Sánchez-Calle, “Subelliptic, second order differential operators,” Complex analysis III, Proc. Spec. Year, College Park/Md. 1985–86, Lect. Notes Math. 1277, pp. 46–77, 1987.Google Scholar
  33. 33.
    J.P. Jones and L.A. Palmer “An evaluation of the two-dimensional gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiology, Vol. 58, pp. 1233–1258, 1987.Google Scholar
  34. 34.
    G. Kanizsa, Grammatica del vedere, Il Mulino, Bologna, 1980.Google Scholar
  35. 35.
    G. Kanizsa, Organization in Vision, Hardcover, 1979.Google Scholar
  36. 36.
    M.K. Kapadia, M. Ito, C.D. Gilbert, and G. Westheimer, “Improvement in visual sensitivity by changes in local context: Parallel studies in human observers and in V1 of alert monkeys,” Neuron, Vol. 15, pp. 843–856, 1995.CrossRefGoogle Scholar
  37. 37.
    S. Kusuoka and D. Stroock, “Applications of the Malliavin calculus III,” J. Fac. Sci. Univ. Tokio, Sect. IA, Math, Vol. 34, pp. 391–442, 1987.Google Scholar
  38. 38.
    S. Kusuoka and D. Stroock, “Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator,” Ann. of Math., Vol. 127, pp. 165–189, 1988.CrossRefMathSciNetGoogle Scholar
  39. 39.
    I. Kovacs and B. Julesz, “A closed curve is much more than an incomplete one: effect of closure in figure-ground segmentation,” PNAS, 90, pp. 7495–7497, 1993.Google Scholar
  40. 40.
    I. Kovacs and B. Julesz, “Perceptual sensitivity maps within globally defined visual shapes,” Nature, Vol. 370, pp. 644–646, 1994.CrossRefGoogle Scholar
  41. 41.
    LeVeque and J. Randall, Nonlinear conservation laws and finite volume methods. (English) Steiner, Oskar et al., Computational methods for astrophysical fluid flow. Saas-Fee advanced course 27. Lecture notes 1997. Swiss Society for Astrophysics and Astronomy. Berlin: Springer, pp. 1–159, 1998.Google Scholar
  42. 42.
    S. Marcelja, “Mathematical description of the response of simple cortical cells,” J. Opt. Soc. Amer., Vol. 70, pp. 1297–1300, 1980.MathSciNetCrossRefGoogle Scholar
  43. 43.
    V. Magnani, “Differentiability and area formula on stratified Lie groups,” Houston J. Math., Vol. 27, No. 2, pp. 297–323, 2001.MATHMathSciNetGoogle Scholar
  44. 44.
    D. Mumford, M. Nitzberg, and T. Shiota, Filtering, Segmentation and Deph, Springer-Verlag: Berlin, 1993.Google Scholar
  45. 45.
    S. Masnou and J.M. Norel, “Level lines based disocclusion,” Proc. 5th. IEEE International Conference on Image Processing, Chicago, Illinois, October 4–7, 1998.Google Scholar
  46. 46.
    K.D. Miller, A. Kayser, and N.J. Priebe, “Contrast-dependent nonlinearities arise locally in a model of contrast-invariant orientation tuning,” J. Neurophysiol., Vol. 85, pp. 2130–2149, 2001.Google Scholar
  47. 47.
    E. Mingolla, “Le unità della visione,” IX Kanitza lecture, Trieste symposium on perception and cognition, 26 October 2001.Google Scholar
  48. 48.
    A. Nagel, E.M. Stein, and S. Wainger, “Balls and metrics defined by vector fields I: Basic properties,” Acta Math., Vol. 155, pp. 103–147, 1985.MathSciNetMATHGoogle Scholar
  49. 49.
    S.B. Nelson, M. Sur, and D.C. Somers, “An emergent model of orientation selectivity in cat visual cortical simples cells,” J. Neurosci., Vol. 15, pp. 5448–5465, 1995.Google Scholar
  50. 50.
    S.D. Pauls, “A notion of rectifiability modeled on Carnot groups,” Indiana Univ. Math. J., Vol. 53, No. 1, pp. 49–81, 2004.CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    S.D. Pauls, “Minimal surfaces in the Heisenberg group,” Geom. Dedicata, Vol. 104, pp. 201–231, 2004.CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    P. Perona, “Deformable kernels for early vision,” IEEE-PAMI, Vol. 17, No. 5, pp. 488–499, 1995.Google Scholar
  53. 53.
    J. Petitot, “Phenomenology of Perception, Qualitative Physics and Sheaf Mereology,” Proceedings of the 16th International Wittgenstein Symposium, Vienna, Verlag Hölder-Pichler-Tempsky, 1994, pp. 387–408.Google Scholar
  54. 54.
    J. Petitot and Y. Tondut, “Vers une Neuro-geometrie. Fibrations corticales, structures de contact et contours subjectifs modaux, Mathématiques, Informatique et Sciences Humaines,” EHESS, Paris, Vol. 145, pp. 5–101, 1998.Google Scholar
  55. 55.
    J. Petitot, Morphological Eidetics for Phenomenology of Perception, in Naturalizing Phenomenology: Issues in Contemporary Phenomenology and Cognitive Science, J. Petitot, F.J. Varela, J.-M. Roy, B. Pachoud (Eds.), Stanford, Stanford University Press, 1998, pp. 330–371.Google Scholar
  56. 56.
    N.J. Priebe, K.D. Miller, T.W. Troyer, and A.E. Krukowsky, “Contrast-invariant orientation tuning in cat visual cortex: Thalamocortical input tuning and correlation-based intracortical connectivity.” J. Neurosci., Vol. 18, pp. 5908–5927, 1998.Google Scholar
  57. 57.
    L. Rothschild and E.M. Stein, “Hypoelliptic differential operators and nihilpotent Lie groups,” Acta Math., Vol. 137, pp. 247–320, 1977.MathSciNetMATHGoogle Scholar
  58. 58.
    A. Sarti, R. Malladi and J.A. Sethian, Subjective surfaces: A method for completion of missing boundaries, in Proceedings of the National Academy of Sciences of the United States of America, Vol. 12, No.97, pp. 6258–6263, 2000.CrossRefMathSciNetGoogle Scholar
  59. 59.
    A. Sarti, G. Citti, and M. Manfredini, “From neural oscillations to variational problems in the visual cortex,” Journal of Physiology, Vol. 97, No. 2–3, pp. 87–385, 2003.Google Scholar
  60. 60.
    M. Shelley, D.J. Wielaard, D. McLaughlin and R. Shapley, “A neuronal network model of macaque primary visual cortex (v1): Orientation selectivity and dynamics in the input layer 4calpha”. Proc. Natl. Acad. Sci. U.S.A., Vol. 97, pp. 8087–8092, 2000.CrossRefGoogle Scholar
  61. 61.
    S.C. Yen and L.H. Finkel, “Extraction of perceptually salient contours by striate cortical networks,” Vision Res., Vol. 38, No. 5, pp. 719–741, 1998.CrossRefGoogle Scholar
  62. 62.
    Y.Q. Song and X.P. Yang, “BV function in the Heisenberg group,” Chinese Ann. Math. Ser. A, Vol. 24, No. 5, pp. 541–554, 2003; translation in Chinese J. Contemp. Math., Vol. 24, No. 4, pp. 301–316, 2004.MathSciNetMATHGoogle Scholar
  63. 63.
    E.M. Stein, Harmonic Analysis, Princeton University Press, 1993.Google Scholar
  64. 64.
    S.K. Vodop’yanov and A.D. Ukhlov, “Approximately differentiable transformations and change of variables on nilpotent groups,” Sib. Math. J., Vol. 37, No.1, pp. 62–78, 1996, translation from Sib. Mat. Zh., Vol. 37, No. 1, pp. 70–89, 1996.CrossRefMathSciNetMATHGoogle Scholar
  65. 65.
    V.S. Varadarajan, “Lie groups, Lie algebras, and their representations,” Graduate Texts in Mathematics. 102, New York, Springer, 1984.MATHGoogle Scholar
  66. 66.
    N.T. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups Cambridge texts in Mathematics, 100, Cambridge University Press, Cambredge, 1992.Google Scholar
  67. 67.
    F.W. Warner, Foundations of differentiable manifolds and Lie groups. Glenview, Illinois-London: Scott, Foresman & Comp. 270, 1971.MATHGoogle Scholar
  68. 68.
    C. Wang, “The comparsion principle for viscosity solutions of fully nonlinear subelliptic equations in Carnot groups,” Preprint.Google Scholar
  69. 69.
    F. Worgotter and C. Koch, “A detailed model of the primary visual pathway in the cat: Comparison of afferent excitatory and intracortical inhibitory connection schemes for orientation selectivity,” J. Neurosci., Vol. 11, pp. 1959–1979, 1991.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BolognaBologna
  2. 2.Department of Electronics, Information and SystemsUniversity of BolognaBologna

Personalised recommendations