Geometria differenziale per il completamento percettivo

G. Citti; A. Sarti

La Matematica nella Società e nella Cultura. Rivista dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 1, page 107-130
  • ISSN: 1972-7356

Abstract

top
In this paper we recall some classical experiments of cognitivive psycology due to Kanizsa and to Heiss, Fields and Hess, which allow to undestand the role of the orientation and distance in perceptive phenomena. Then we model the visual cortex with instruments of differential geometry and Analysis in Lie groups. The model is based on some neural mechanism, and in particular on the ability of the simple cells of detect at every point of an image the direction of level lines. In this way we justify from a microscopical point of view the perceptive phenomena just described.

How to cite

top

Citti, G., and Sarti, A.. "Geometria differenziale per il completamento percettivo." La Matematica nella Società e nella Cultura. Rivista dell'Unione Matematica Italiana 1.1 (2008): 107-130. <http://eudml.org/doc/290500>.

@article{Citti2008,
abstract = {In questo lavoro richiamiamo alcuni esperimenti classici di psicologia della percezione dovuti a Kanizsa e a Heiss, Fields e Hess, che indicano l'importanza del concetto di direzione nei processi percettivi. Poi presentiamo un modello di corteccia, che descrive la corteccia visiva con strumenti di geometria differenziale e analisi in gruppi di Lie. È basato sulla modellazione di alcuni meccanismi neurali, ed in particolare sulla capacità delle cellule semplici di individuare in ogni punto la direzione dei bordi degli oggetti, e giustifica da un punto di vista microscopico, i fenomeni percettivi precedentemente descritti.},
author = {Citti, G., Sarti, A.},
journal = {La Matematica nella Società e nella Cultura. Rivista dell'Unione Matematica Italiana},
language = {ita},
month = {4},
number = {1},
pages = {107-130},
publisher = {Unione Matematica Italiana},
title = {Geometria differenziale per il completamento percettivo},
url = {http://eudml.org/doc/290500},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Citti, G.
AU - Sarti, A.
TI - Geometria differenziale per il completamento percettivo
JO - La Matematica nella Società e nella Cultura. Rivista dell'Unione Matematica Italiana
DA - 2008/4//
PB - Unione Matematica Italiana
VL - 1
IS - 1
SP - 107
EP - 130
AB - In questo lavoro richiamiamo alcuni esperimenti classici di psicologia della percezione dovuti a Kanizsa e a Heiss, Fields e Hess, che indicano l'importanza del concetto di direzione nei processi percettivi. Poi presentiamo un modello di corteccia, che descrive la corteccia visiva con strumenti di geometria differenziale e analisi in gruppi di Lie. È basato sulla modellazione di alcuni meccanismi neurali, ed in particolare sulla capacità delle cellule semplici di individuare in ogni punto la direzione dei bordi degli oggetti, e giustifica da un punto di vista microscopico, i fenomeni percettivi precedentemente descritti.
LA - ita
UR - http://eudml.org/doc/290500
ER -

References

top
  1. BELLAICHE, A., The tangent space in sub-Riemannian geometry, in Proceedings of the satellite meeting of the 1st European congress of mathematics `Journées nonholonomes: géométrie sous-riemannienne, théorie du contrôle, robotique', Paris, France, June 30- July 1, 1992. Basel: Birkhuser. Prog. Math., Vol. 144 (1996), 1-78. MR1421822DOI10.1007/978-3-0348-9210-0_1
  2. BALLESTER, C. - BERTALMIO, M. - CASELLES, V. - SAPIRO, G. - VERDERA, J., Filling-in by interpolation of vector fields and gray levels, IEEE Transactions on Image Processing, Vol. 10, no. 8 (2001), 1200-1211. Zbl1037.68771MR1851781DOI10.1109/83.935036
  3. BENCE, J. - MERRIMAN, B. - OSHER, S., Diffusion generated motion by mena curvature, in Computational Crystal Growers Workshop, J. Taylor Sel. Taylor Ed. 
  4. BOSKING, W. - ZHANG, Y. - SCHOFIELD, B. - FITZPATRICK, D., Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex, J. Neurosi.17 (6) (1997), 2112-2127. 
  5. CITTI, G. - MANFREDINI, M. - SARTI, A., Neuronal Oscillations in the Visual Cortex: Γ -convergence to the Riemannian Mumford-Shah Functional, SIAM Jornal of Mathematical Analysis, Vol. 35, no. 6 (2004), 1394-1419. Zbl1058.49010MR2083784DOI10.1137/S0036141002398673
  6. CITTI, G. - SARTI, A., A cortical based model of perceptual completion in the Roto-Translation space, Journal of Mathematical Imaging and Vision, Vol. 24, Issue 3 (2006), 307-326. MR2235475DOI10.1007/s10851-005-3630-2
  7. DAUGMAN, J.G., 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 (1985), 1160-1169. 
  8. ESEDOGLU, S. - MARCH, R., Segmentation with Deph but without detecting junctions, Journal of Mathematical Imaging and Vision, Vol. 18 (2003), 7-15. Zbl1033.68132MR1966172DOI10.1023/A:1021837026373
  9. EVANS, L., Convergence of an Algorithm for mean curvature motion, Indiana Univ. Math J., Vol. 42, n. 2 (1993), 553-557. Zbl0802.65098MR1237058DOI10.1512/iumj.1993.42.42024
  10. FOLLAND, G.B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., Vol. 13 (1975), 161-207. Zbl0312.35026MR494315DOI10.1007/BF02386204
  11. FOLLAND, G.B. - STEIN, E.M., Estimates for the ∂ ¯ b Complex and Analysis on the Heisenberg Group, Comm. Pure Appl. Math., Vol. 20 (1974), 429-522. Zbl0293.35012MR367477DOI10.1002/cpa.3160270403
  12. GILBERT, C.D. - DAS, A. - ITO, M. - KAPADIA, M. - WESTHEIMER, G., Spatial integration and cortical dynamics. Proceedings of the National Academy of Sciences USA, 93, 615-622. 
  13. GROSSBERG, S. - MINGOLLA, E., Neural dynamics of perceptual grouping: textures, boundaries and emergent segmentations, in ``Perception and Psychophysics, 1985. 
  14. FIELD, D. - HAYES, A. - HESS, R.F., Contour integration by the human visual system: evidence for a local Association Field, Vision Research, Vol. 33 (1993), 173-193. 
  15. HÖRMANDER, L., Hypoelliptic second-order differential equations, Acta Math., Vol. 119 (1967), 147-171. MR222474DOI10.1007/BF02392081
  16. HÖRMANDER, L. - MELIN, A., Free systems of vector fields, Ark. Mat., Vol. 16 (1978), 83-88. MR650825DOI10.1007/BF02385983
  17. HUBEL, D. - WIESEL, T., Receptive fields, binocular interaction and functional architecture in the cat's visual cortex, Journal of Physiology, Vol. 160 (1962), 106-154. 
  18. KANIZSA, G., Grammatica del vedere, Il Mulino, Bologna, 1980. 
  19. KANIZSA, G., Organization in Vision, Praeger, New York, 1979. 
  20. KAPADIA, M.K. - ITO, M. - GILBERT, C.D. - WESTHEIMER, G., Improvement in visual sensitivity by changes in local context: parallel studies in human observers and in V1 of alert monkeys, Neuron, Vol. 15 (1995), 843-856. 
  21. KOVACS, I. - JULESZ, B., Perceptual sensitivity maps within globally defined visual shapes, Nature, Vol. 370 (1994), 644-646. 
  22. MUMFORD, D. - NITZBERG, M. - SHIOTA, T., Filtering, Segmentation and Deph, Springer-Verlag, Berlin, 1993. MR1226232DOI10.1007/3-540-56484-5
  23. MASNOU, S. - NOREL, J.M., Level lines based disocclusion, Proc. 5th. IEEE International Conference on Image Processing, Chicago, Illinois, October 4-7, 1998. MR1888912DOI10.1109/83.982815
  24. MILLER, K.D. - KAYSER, A. - PRIEBE, N. J., Contrast-dependent nonlinearities arise locally in a model of contrast-invariant orientation tuning, J. Neurophysiol., 85 (2001), 2130-2149. 
  25. NAGEL, A. - STEIN, E.M. - WAINGER, S., Balls and metrics defined by vector fields I: Basic properties, Acta Math., Vol. 155 (1985), 103-147. Zbl0578.32044MR793239DOI10.1007/BF02392539
  26. NELSON, S.B. - SUR, M. - SOMERS, D.C., An emergent model of orientation selectivity in cat visual cortical simples cells, J. Neurosci., Vol. 15, (1995), 5448-5465. 
  27. PERONA, P., Deformable kernels for early vision, IEEE-PAMI, Vol. 17, no. 5 (1995), 488- 499. 
  28. PETITOT, J. - TONDUT, Y., Vers une Neuro-geometrie. Fibrations corticales, structures de contact et contours subjectifs modaux, Mathématiques, Informatique et Sciences Humaines, EHESS, Paris, Vol. 145 (1998), 5-101. MR1697185
  29. PRIEBE, N.J. - MILLER, K.D. - TROYER, T.W. - KRUKOWSKY, A.E., Contrast-invariant orientation tuning in cat visual cortex: thalamocortical input tuning and correlation-based intracortical connectivity, J. Neurosci., Vol. 18 (1998), 5908-5927. 
  30. ROTHSCHILD, L. - STEIN, E.M., Hypoelliptic differential operators and nihilpotent Lie groups, Acta Math., Vol. 137 (1977), 247-320. Zbl0346.35030MR436223DOI10.1007/BF02392419
  31. SARTI, A. - CITTI, G. - MANFREDINI, M., From neural oscillations to variational problems in the visual cortex, Journal of Physiology, Vol. 97, no 2-3 (2003), 87-385. 
  32. YEN, S.C. - FINKEL, L.H., Extraction of Perceptually Salient Contours by Striate Cortical Networks, Vision Res., Vol. 38, no. 5 (1998), 719-741. 
  33. TS'O, D. - GILBERT, C.D - DIESEL, T.N, Relationship between horizontal interactions and functional architecture in cat striate cortex as revealed by cross-correlation analysis, J. Neurosc.6(4) (1986), 1160-1170. 
  34. WARNER, F.W., Foundations of differentiable manifolds and Lie groups. Glenview, Illinois-London: Scott, Foresman & Comp.270, 1971. Zbl0241.58001MR295244

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.