A Cortical-Inspired Model for Orientation-Dependent Contrast Perception: A Link with Wilson-Cowan Equations

  • Marcelo Bertalmío
  • Luca CalatroniEmail author
  • Valentina Franceschi
  • Benedetta Franceschiello
  • Dario Prandi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11603)


We consider a differential model describing neuro-physiologi-cal contrast perception phenomena induced by surrounding orientations. The mathematical formulation relies on a cortical-inspired modelling [11] largely used over the last years to describe neuron interactions in the primary visual cortex (V1) and applied to several image processing problems [14, 15, 21]. Our model connects to Wilson-Cowan-type equations [26] and it is analogous to the one used in [3, 4, 16] to describe assimilation and contrast phenomena, the main novelty being its explicit dependence on local image orientation. To confirm the validity of the model, we report some numerical tests showing its ability to explain orientation-dependent phenomena (such as grating induction) and geometric-optical illusions [18, 24] classically explained only by filtering-based techniques [7, 20].


Orientation-dependent modelling Wilson-Cowan equations Primary visual cortex Contrast perception Variational modelling 


  1. 1.
    Barbieri, D., Citti, G., Cocci, G., Sarti, A.: A cortical-inspired geometry for contour perception and motion integration. JMIV 49(3), 511–529 (2014)CrossRefGoogle Scholar
  2. 2.
    Bekkers, E., Duits, R., Berendschot, T., ter Haar Romeny, B.: A multi-orientation analysis approach to retinal vessel tracking. JMIV 49(3), 583–610 (2014)CrossRefGoogle Scholar
  3. 3.
    Bertalmío, M.: From image processing to computational neuroscience: a neural model based on histogram equalization. Front. Comput. Neurosci. 8, 71 (2014)Google Scholar
  4. 4.
    Bertalmío, M., Caselles, V., Provenzi, E., Rizzi, A.: Perceptual color correction through variational techniques. IEEE Trans. Image Process. 16(4), 1058–1072 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bertalmío, M., Cowan, J.D.: Implementing the retinex algorithm with Wilson-Cowan equations. J. Physiol. Paris 103(1), 69–72 (2009)CrossRefGoogle Scholar
  6. 6.
    Blakeslee, B., Cope, D., McCourt, M.E.: The oriented difference of Gaussians (ODOG) model of brightness perception: overview and executable mathematica notebooks. Behav. Res. Methods 48(1), 306–312 (2016)CrossRefGoogle Scholar
  7. 7.
    Blakeslee, B., McCourt, M.E.: A multiscale spatial filtering account of the White effect, simultaneous brightness contrast and grating induction. Vision Res. 39(26), 4361–4377 (1999)CrossRefGoogle Scholar
  8. 8.
    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. Roy. Soc. Lond. B Biol. Sci. 356, 299–330 (2001)CrossRefGoogle Scholar
  9. 9.
    Bressloff, P.C., Cowan, J.D.: An amplitude equation approach to contextual effects in visual cortex. Neural Comput. 14(3), 493–525 (2002)CrossRefGoogle Scholar
  10. 10.
    Carandini, M., et al.: Do we know what the early visual system does? J. Neurosci. 25(46), 10577–10597 (2005)CrossRefGoogle Scholar
  11. 11.
    Citti, G., Sarti, A.: A cortical based model of perceptual completion in the roto-translation space. JMIV 24(3), 307–326 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Daugman, J.G.: Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. J. Opt. Soc. Am. A 2(7), 1160–1169 (1985)CrossRefGoogle Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Duits, R., Franken, E.: Left-invariant parabolic evolutions on \(SE(2)\) and contour enhancement via invertible orientation scores. Part I: linear left-invariant diffusion equations on \(SE(2)\). Quart. Appl. Math. 68(2), 255–292 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Franceschiello, B., Sarti, A., Citti, G.: A neuromathematical model for geometrical optical illusions. JMIV 60(1), 94–108 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kim, J., Batard, T., Bertalmío, M.: Retinal processing optimizes contrast coding. J. Vis. 16(12), 1151–1151 (2016)CrossRefGoogle Scholar
  17. 17.
    Martinez-Garcia, M., Cyriac, P., Batard, T., Bertalmío, M., Malo, J.: Derivatives and inverse of cascaded linear+nonlinear neural models. PLOS One 13(10), 1–49 (2018)CrossRefGoogle Scholar
  18. 18.
    McCourt, M.E.: A spatial frequency dependent grating-induction effect. Vis. Res. 22(1), 119–134 (1982)CrossRefGoogle Scholar
  19. 19.
    Olshausen, B.A., Field, D.J.: Vision and the coding of natural images: the human brain may hold the secrets to the best image-compression algorithms. Am. Sci. 88(3), 238–245 (2000)CrossRefGoogle Scholar
  20. 20.
    Otazu, X., Vanrell, M., Parraga, C.A.: Multiresolution wavelet framework models brightness induction effects. Vis. Res. 48(5), 733–751 (2008)CrossRefGoogle Scholar
  21. 21.
    Prandi, D., Gauthier, J.P.: A Semidiscrete Version of the Petitot Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition. SpringerBriefs in Mathematics. Springer, Cham (2017). Scholar
  22. 22.
    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
  23. 23.
    Self, M.W., et al.: Orientation-tuned surround suppression in mouse visual cortex. J. Neurosci. 34(28), 9290–9304 (2014)CrossRefGoogle Scholar
  24. 24.
    Weintraub, D.J., Krantz, D.H.: The Poggendorff illusion: amputations, rotations, and other perturbations. Attent. Percept. Psychol. 10(4), 257–264 (1971)CrossRefGoogle Scholar
  25. 25.
    Westheimer, G.: Illusions in the spatial sense of the eye: geometrical-optical illusions and the neural representation of space. Vis. Res. 48(20), 212–2142 (2008)CrossRefGoogle Scholar
  26. 26.
    Wilson, H.R., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. BioPhys. J. 12(1), 1–24 (1972)CrossRefGoogle Scholar
  27. 27.
    Yeonan-Kim, J., Bertalmío, M.: Retinal lateral inhibition provides the biological basis of long-range spatial induction. PLOS One 11(12), 1–23 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Marcelo Bertalmío
    • 1
  • Luca Calatroni
    • 2
    Email author
  • Valentina Franceschi
    • 3
  • Benedetta Franceschiello
    • 4
  • Dario Prandi
    • 5
  1. 1.DTICUniversitat Pompeu FabraBarcelonaSpain
  2. 2.CMAPÉcole Polytechnique CNRSPalaiseauFrance
  3. 3.IMOUniversité Paris-SudOrsayFrance
  4. 4.Fondation Asile des Aveugles and Laboratory for Investigative NeurophysiologyLausanneSwitzerland
  5. 5.CNRS, L2SCentraleSupélecGif-sur-YvetteFrance

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