Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux

Petitot, Jean; Tondut, Yannick

Mathématiques informatique et sciences humaines, Tome 145 (1999), pp. 5-101.

Ce travail propose certains modèles variationnels pour les processus corticaux d’intégration des contours subjectifs modaux (de type contours illusoires à la Kanizsa), modèles fondés sur les concepts géométriques de fibration et de structure de contact. La structure rétinotopique des hypercolonnes d’orientation de l’aire $V_1$ (telle qu’elle est décrite depuis les travaux pionniers de Hubel, Wiesel et Mountcastle) est une architecture fonctionnelle qui peut être mathématiquement idéalisée par la fibration $\pi :E\to M$ ayant pour base le plan $M$ de la rétine et pour fibre $F$ la droite projective ${\mathbb{P}}^1$ des directions du plan, l’espace total $E$ de $\pi$ étant isomorphe au produit direct $M\times F$. Au-dessus de chaque position rétinienne se trouve implémenté un exemplaire (discrétisé) de $F$. Les connexions horizontales cortico-corticales implémentent ce que l’on appelle la trivialité locale de cette fibration et sans doute également une connexion (au sens d’Elie Cartan) définissant un transport parallèle. Après avoir rappelé ces données, le papier se focalise sur l’interprétation géométrique des résultats de Field, Hayes et Hess sur le champ d’association. Ces travaux semblent montrer que ce que l’on appelle en géométrie symplectique la structure de contact de la fibration $\pi :E\to M$ se trouve neuralement implémenté. Le champ d’association correspond dans ce cadre à une condition d’intégrabilité des courbes dans $E$ : elles doivent être les relevées de leur projection sur le plan rétinien $M$. Ce modèle d’une fibration munie d’une structure de contact naturelle est ensuite appliqué à l’interprétation des contours subjectifs modaux et conduit à des variantes du modèle dit de l’elastica développé par B.K.P. Horn et D. Mumford. L’idée est que les contours subjectifs modaux ont des relevées qui sont «géodésiques» dans le fibré cortical $E$, c’est-à-dire de longueur minimale (pour une métrique appropriée) dans la classe des courbes satisfaisant la condition d’intégrabilité. Les modèles «géodésiques» sont ensuite reformulés, à la suite de R. Bryant et P. Griffiths, dans un cadre géométrique plus fondamental, celui des groupes de Lie et du repère mobile d’Elie Cartan. Quelques possibilités de test expérimentaux sont enfin considérées.

This work presents some variational models for the cortical algorithms processing Kanizsa modal subjective contours. These models are based on the geometric concepts of fibration and contact structure. The retinoptic structure of the orientation hypercolumns in the visual area $V_1$ is a functional architecture which can be mathematically idealized by the fibration $\pi :E\to M$ having the retinian plane $M$ as base and the projective line ${\mathbb{P}}^1$ as fiber $F$. The total space $E$ of $\pi$ is isomorphic to the direct product $M\times F$. The cortico-cortical horizontal connections implement what is called the local triviality of this fibration, and also a Cartan connection defining a parallel transport between neighboring fibers. Then, the paper focuses on the geometrical interpretation of the results of Field, Hayes and Hess concerning the association field. It shows that the latter implements what is called the contact structure of the fibration $\pi :E\to M$. The association field expresses an integrability condition for the skew curves in $E$ : they have to be a lifting of their projection on the retinian plane $M$. This model of a fibration endowed with a contact structure is then applied to the modal subjective contours and provides a variant of the elastica model developped by B.K.P. Horn and D. Mumford. The key idea is that the lifting of subjective contours satisfy a «geodesic» condition in the cortical fibration $E$ : they have to be of minimal lenght (for an appropriate metrics) among the class of curves satisfying the integrability condition. These «geodesic» models are then reformulated, according to R. Bryant and P. Griffiths, in the more fondamental geometric framework of Lie groups and Cartan’s “repère mobile” (Vielbein). Finally, some experimental possibilities are suggested.

Abraham R., Marsden J. (1978), Foundations of Mechanics, Benjamin/Cummings, Reading. | MR | Zbl

Arnold V. (1976), Méthodes mathématiques de la Mécanique classique, Ed. Mir, Moscou. | MR | Zbl

Berest P. (1995), Calcul des Variations, Application à la Mécanique et à la Physique, Cours de l'Ecole Polytechnique, Paris.

Berthoz A. (1997), Le Sens du Mouvement, Odile Jacob, Paris.

Bonhoeffer T., Grinvald A. (1991), Iso-orientation domains in cat visual cortex are arranged in pinwheel-like patterns, Nature, vol. 353, pp. 429-431.

Born R.T., Tootell R.B. (1990), Spatial frequency tuning of single units in macaque supragranular striate cortex, Proc. Natl. Acad. Sci. USA, vol. 88, pp. 7066-7070.

Bourguignon J.P. (1993), Calcul variationnel, Cours de l'Ecole Polytechnique, Paris.

Brady M., Grimson W.E.L., Langridge D. (1980), Shape encoding and subjective contours, Proceedings of the AAAI, Stanford University, pp. 15-17.

Bryant R., Griffiths P. (1986), Reduction for constrained variational problems and ∫k2/2 ds, American Journal of Mathematics, vol. 108, pp. 525-570. | Zbl

Buser P., Imbert M. (1987), Vision, Hermann.

Caselles V., Coll T., Morel J-M. (forthcoming), A Kanizsa Programme.

Das A., Gilbert C.D. (1995), Long range horizontal connections and their role in cortical reorganization revealed by optical recording of cat primary visual cortex, Nature, vol. 375, pp. 780-784.

Das A., Gilbert C.D. (1997), Distorsions of visuotopic map match orientation singularities in primary visual cortex, Nature, vol. 387, pp. 594-598.

Davis G., Driver J. (1994), Parallel detection of Kanizsa subjective figures in the human visual system, Nature, vol. 371, pp. 791-793.

Dresp B., Bonnet C. (1991), Psychophysical evidence for low level processing of illusory contours and surfaces in the Kanizsa square, Vision Research, vol. 31, pp. 1813-1817.

Dresp B., Bonnet C. (1993), Psychophysical measures of illusory form perception : further evidence for local mechanisms, Vision Research, vol. 33, n° 5/6, pp. 759-766.

Dresp B., Bonnet C. (1995), Subthreshold summation with illusory contours, Vision Research, vol. 35, pp. 1071-1078.

Dresp B., Grossberg S. (1997), Contour integration across polarities and spatial gaps : from local contrast filtering to global grouping, Vision Research, vol. 37, n° 7, pp. 913-924.

Field D.J. (1987), Relations between the statistics of natural images and the response properties of cortical cells, Journal of the Optical Society of America, A, vol. 4, n° 12, pp. 2379-2394.

Field D.J., Hayes A., Hess R.F. (1993), Contour integration by the human visual system : evidence for a local "association field" , Vision Research, vol. 33, n° 2, pp. 173-193.

Finkel L.H., Edelman G.M. (1989), Integration of distributed cortical systems by reentry : a computer simulation of interactive functionnally segregated visual areas, Journal of Neuroscience, vol 9, n° 9, pp 3188-3208.

Florack L.M.J., TER Haar Romeny B.M., Koenderink J.J., Vierge-Ver M.A. (1992), Scale and the differential structure of images, Image and Vision Computing, vol. 10, n° 6, pp. 376-388.

Fregnac Y., Bringuier V., Chavane F.,. Glaeser L., Lorenceau J. (1996), An intracellular study of space and time representation in primary visual cortical receptive fields, Journal of Physiology, vol. 90, pp. 189-197.

GAMKRELIDZE R.V. (Ed.) (1991), Geometry 1, Encyclopoedia of Mathematical Sciences, Springer Verlag, Berlin. | MR | Zbl

Gilbert C.D. (1992), Horizontal integration and cortical dynamics, Neuron, vol. 9, pp. 1-13.

Gilbert C.D. (1994), Circuitry, architecture and functional dynamics of visual cortex, in Higher order processing in the visual system, Ciba Foundation Symposium 184, pp. 35-62.

Gilbert C.D., Das A., Ito M., Kapadia M., Westheimer G. (1996), Spatial integration and cortical dynamics, Proc. Nat. Acad. Sci. USA, vol. 93, pp. 615-622.

Gilbert C.D., Wiesel T.N. (1989), Columnar specificity of intrinsic horizontal and corticocortical connections in cat visual cortex, Journal of Neuroscience, vol. 9, n° 7, pp. 2432-2442.

Gray C.M., Singer W. (1989), Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex, Proc. Natl. Acad. Sci. USA, vol. 86, pp. 1698-1702.

Gray C.M., Kônig P., Engel A.K., Singer W. (1989), Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties, Nature, vol. 338, pp. 334-337.

Grinvald A., Lieke E.E., Frostig R.D., Hildesheim R. (1994), Cortical point-spread function and long-range lateral interactions revealed by real-time optical imaging of macaque monkey primary visual cortex, Journal of Neuroscience, vol. 14, n° 5, pp. 2545-2568.

Grosof D.H., Shapley R.M., Hawken M.J. (1993), Macaque V1 neurons can signal "illusory" contours, Nature, vol. 365, pp. 550-552.

Grossberg S., Mingolla E. (1985a), Neural dynamics of form perception : boundary completion, illusory figures and neon color spreading, Psychological Review, vol. 92, pp. 173-211.

Grossberg S., Mingolla E. (1985b), Neural dynamics of perceptual grouping : textures, boundaries and emergent segmentation, Perception and Psychophysics, vol. 38, pp. 141-171.

Grossberg S., Mingolla E. (1987), The role of illusory contours in visual segmentation, in The Perception of Illusory Contours, in Petry S. et Meyer G.E. eds, Springer, pp. 116-125.

Heitger F., Rosenthaler L., Von Der Heydt R., Peterhans E., Kübler O. (1992), Simulation of neural contour mechanisms : from simple to endstopped cells, Vision Research, vol. 32, n° 5, pp. 963-981.

Heitger F., VON Der Heydt R. (1993), A computational model of neural contour processing : figure-ground segregation and illusory contours, Proc. 4th Int. Conf. Comp. Vis., IEEE, pp. 32-40.

Hirsch J., Delapaz R.L., Relkin N.R., Victor J., Kim K., Li T., Borden P., Rubin N., Shapley R. (1995), Illusory contours activate specific regions in human visual cortex : Evidence from functional magnetic resonance imaging, Proc. Natl. Acad. Sci. USA, vol. 92, pp. 6469-6473.

Hoffman W.C. (1985), Some reasons why algebraic topology is important in neuropsychology : perceptual and cognitive systems as fibrations, International Journal of Man-Machine Studies, vol. 22, pp. 613-650. | Zbl

Hoffman W.C. (1989), The visual cortex is a contact bundle, Applied Mathematics and Computation, vol. 32, pp. 137-167. | MR | Zbl

Hoffman W.C. (1994a), Conformal structures in perceptual psychology, Spatial Vision, vol. 8, n° 1, pp. 19-31.

Hoffman W.C. (1994b), Equivariant dynamical systems : a formal model for the generation of arbitrary shapes, Proceedings of the NATO Conference on "Shape in Picture", Springer Verlag, Berlin.

Horn B.K.P. (1983), The curves of least energy, ACM Transactions on Mathematical Software, vol. 9, n° 4, pp. 441-460. | MR | Zbl

Hubel D.H. (1988), Eye, Brain and Vision, Scientific American Library.

Hummel H.E., Biederman I. (1992), Dynamic binding in a neural network for shape recognition, Psychological Review, vol. 99, n° 3, pp. 480-517.

Kanizsa G. (1979), Organization in Vision : Essays on Visual Perception, Praeger. Trad. française La . Grammaire du Voir, Diderot Editeur, Paris.

Kapadia M.K., Ito M., Gilbert C.D., Westheimer G. (1995), 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.

Kellman P.J., Shipley T.F. (1991), A theory of visual interpolation in object perception, Cognitive Psychology, vol. 23, pp. 141-221.

Koenderink J.J. (1984a), Simultaneous order in nervous nets from a functional standpoint, Biological Cybernetics, vol. 50, pp. 35-41. | Zbl

Koenderink J.J. (1984b), The structure of images, Biological Cybernetics, vol. 50, pp. 363-370. | MR | Zbl

Koenderink J.J. (1988), Operational significance of receptive field assemblies, Biological Cybernetics, vol. 58, pp. 163-171. | MR | Zbl

Koenderink J.J. (1990), The brain as a geometry engine, Psychological Research, vol. 52, pp. 122-127.

Koenderink J.J., Richards W.A. (1988), Two-dimensional curvature operators, Journal of the Optical Society of America, vol. 5, n° 7, pp. 1136-1141. | MR

Koenderink J.J., Van Doorn A.J. (1987), Representation of local geometry in the visual system, Biological Cybernetics, vol. 55, pp. 367-375. | MR | Zbl

Kovacs I., Julesz B. (1993), A closed curve is much more than an incomplete one : effect of closure in figure-ground segmentation, Proc. Nat. Acad. Sci. USA, vol. 90, pp. 7495-7497.

Lamme A.F., Van Dijk B.W., Spekreijse H. (1993), Contour from motion occurs in primary visual cortex, Nature, vol. 363, pp. 541-543.

Lesher G.W., Mingolla E. (1993), The role of edges and line-ends in illusory contour formation, Vision Research, vol. 33, n° 16, pp. 2253-2270.

Marr D. (1982), Vision, Freeman.

Mumford D. (1992), Elastica and computer vision, in Algebraic Geometry and Applications, C. Bajaj ed., Springer Verlag, Berlin. | MR | Zbl

Ninio J. (1996), L'Empreinte des Sens, Odile Jacob.

Nitzberg M., Mumford D., Shiota T. (1993), Filtering, segmentation and depth, Lecture Notes in Computer Science, n° 662, Springer Verlag, Berlin. | MR | Zbl

Peterhans E., Von Der Heydt R. (1989), Mechanisms of contour perception in monkey visual cortex. II. Contours bridging gaps, Journal of Neuroscience, vol. 9, n° 5, pp. 1749-1763.

Peterhans E., VON Der Heydt R. (1991), Subjective contours. Bridging the gap between psychophysics and psychology, Trends in Neuroscience, vol. 14, n° 3, pp. 112-119.

Petitot J. (1990), Le physique, le morphologique, le symbolique : remarques sur la vision, Revue de Synthèse, n° 1-2, pp. 139-183.

Petitot J. (1992), Physique du Sens, Editions du CNRS, Paris.

Petitot J. (1995), Sheaf mereology and Husserl's morphological ontology, Int. J. Human-Computer Studies, vol. 43, pp. 741-763.

Petitot J. (1999), Morphological Eidetics for a Phenomenology of Perception, Naturalizing Phenomenology: Issues in Contemporary Phenomenology and Cognitive Science, (J. Petitot, F. J. Varela, J.-M. Roy, B. Pachoud, eds.), Stanford University Press.

Petry S., Gannon R. (1987), Time motion and objectedness in illusory contours, in The Perception of Illusory Contours, Petry S. et Meyer G.E. eds, Springer.

PETRY S. et MEYER G.E. eds. (1987), The Perception of Illusory Contours, Springer.

Polat U., Sagi D. (1993), Lateral interactions between spatial channels : suppression and facilitation revealed by lateral masking experiment, Vision Research, vol. 33, n° 7, pp. 993-999.

Rock I. (1987), A problem solving approach of illusory contours, in The Perception of Illusory Contours, Petry S. et Meyer G.E. eds, Springer, pp. 62-70.

Shapley R., Gordon J. (1987), The existence of interpolated illusory contours depends on contrast and spatial separation, in The Perception of Illusory Contours, Petry S. et Meyer G.E. eds, Springer, pp. 109-115.

Sharpe R. (1997), Differential Geometry, Springer, New-York. | MR | Zbl

Sheth B.R., Sharma J., Rao C., Sur M. (1996), Orientation maps of subjective contours in visual cortex, Nature, vol. 274, pp. 2110-2115.

Shipley T.F., Kellman P.J. (1992a), Perception of partly occluded objects and illusory figures : evidence for an identity hypothsesis, Journal of Experimental Psychology, vol. 18, n° 1, pp. 106-120.

Shipley T.F., Kellman P.J. (1992b), Strength of visual interpolation depends on the ratio of physically specified to total edge length, Perception and Psychophysics, vol. 52, pp. 97-106.

Spillman L., Dresp B. (1995), Phenomena of illusory form : can we bridge the gap between levels of explanation ?, Perception, vol. 24, pp. 1333-1364.

Spivak M. (1974), A Comprehensive Treatise on Differential Geometry, Publish or Perish, Boston, Mass.

Stemmler M., Usher M., Niebur E. (1995) Lateral interactions in primary visual cortex : a model bridging physiology and psychophysics, Science, vol. 269, p. 1877-1880.

Tondut Y., Petitot J. (1997), Géométrie de contact et Champ d'association dans le cortex visuel, Rapport n° 9725, CREA, Ecole Polytechnique, Paris.

Ts'O D., Gilbert C.D., Wiesel T.N. (1986), Relationships between horizontal interactions and functional architecture in cat striate cortex as revealed by crosscorrelation analysis, Journal of Neuroscience, vol. 6, n° 4, pp. 1160-1170.

Ullman S. (1976), Filling in the gaps : the shape of subjective contours and a model for their generation, Biological Cybernetics, vol. 25, pp. 1-6.

Von Der Heydt R., Peterhans E. (1989a), Mechanisms of contour perception in monkey visual cortex. I. Lines of pattern discontinuity, Journal of Neuroscience, vol. 9, n° 5, pp. 1731-1748.

Von Der Heydt R., Peterhans E. (1989b), Cortical contour mechanisms and geometrical illusions, in Neural mechanisms of visual perception, Lam D.M. et Gilbert C.D. eds, Portfolio Publishing Company, pp. 158-170.

Von Der Heydt R., Peterhans E., Baumgartner G. (1984), Illusory contours and cortical neuron responses, Science, vol. 224, pp. 1260-1262.

Webb J.A., Pervin E. (1984), The shape of subjective contours, Proceedings of the IAAA, pp. 340-343.

Williams L.R., Jacobs D.W. (1995), Stochastic completion fields : a neural model of illusory contour shape and salience, Proc. Intern. Conf. Comp. Vis., n° 5, pp. 408-415.

Zeki S. (1993), A Vision of the Brain, Blackwell Scientific Publications.

Zucker S.W., David C., Dobbins A., Iverson L. (1988), The organization of curve detection : coarse tangent fields and fine spline covering, Proc. 2nd Intern. Conf. Comp. Vis., IEEE.