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

Abstract

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 V1 is a functionnal architecture which can be mathematically idealized by the fibration having the retinian plane M as base and the projective line P1 as fiber F. The total space E of Pi p is isomorphic to the direct product M x 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. 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 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.