Analysis of a hyperbolic geometric model for visual texture perception

Grégory Faye 1, * Pascal Chossat 1, 2 Olivier Faugeras 1
* Corresponding author
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : We study the neural field equations introduced by Chossat and Faugeras to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincaré disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, that is, time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.
Mathematics Subject Classification: 30F45, 33C05, 34A12, 34D20, 34D23, 34G20, 37M05, 43A85, 44A35, 45G10, 51M10, 92B20, 92C20.
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Grégory Faye, Pascal Chossat, Olivier Faugeras. Analysis of a hyperbolic geometric model for visual texture perception. Journal of Mathematical Neuroscience, BioMed Central, 2011, 1 (1), pp.4. ⟨10.1186/2190-8567-1-4⟩. ⟨hal-00784424⟩

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