We consider a four-dimensional extension of the classical Petitot-Citti-Sarti model of contour completion by the visual cortex. The configuration space of the neurons is interpreted as the similarity transformations group M = SIM(2). According to the model, the damaged image contours are restored via sub-Riemannian geodesics in the space M of positions, orientations and thicknesses (scales). The left-invariant distribution of the tangent subspaces models the possible directions for establishing a neural connection. The sub-Riemannian distance is proportional to the energy expended in the activation of interneurons between two excited border neurons. Using the methods of geometric control theory, we study the geodesic problem in M. We prove the complete controllability and the existence of optimal controls. Via the Pontryagin maximum principle, we derive a Hamiltonian system that determines the geodesics. We provide a qualitative analysis of the Hamiltonian system. In special cases, we derive an explicit expression for the geodesics.
Международная конференция по дифференциальным уравнениям и динамическим системам, г. Суздаль