2021/07/24 by I. A. Galyaev, Ivan Galyaev, Galyaev, Ivan +2 · 1 citation
Computer Science · Mathematics · #Applied mathematics #Computer science #Controllability #Curvature #FOS: Mathematics #Geodesic #Geometry #Hamiltonian (control theory) #Hamiltonian system #Mathematical analysis #Mathematical optimization #Mathematics #Morphological variations and asymmetry #Optimization and Control (math.OC) #Parametrization (atmospheric modeling) #Physics #Pure mathematics #Quantum mechanics #Space (punctuation) #Topological and Geometric Data Analysis #Visual cortex #math.OC
paper · pdf · doi:10.48550/arxiv.2107.11602
11 pages, in English, 2 figure English version. Introduction was extended, typos were corrected
openalex publication_date 2021/07/24 · arxiv created 2021/08/05 · arxiv updated 2021/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We consider a natural extension of the Petitot-Citti-Sarti model of the primary visual cortex. In the extended model, the curvature of contours is taking into account such that occluded contours are completed using sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and curvatures. Here, M=R2 x SO(2) x R models the configuration space of neurons of the visual cortex. We study the problem of sub-Riemannian geodesics on M via methods of geometric control theory. We prove complete controllability of the system and the existence of optimal controls. By application of Pontryagin maximum principle, we derive a Hamiltonian system that describes the geodesics. We obtain explicit parametrization of abnormal extremals. In the normal case, we provide three functionally independent first integrals. Numerical simulations indicate the existence of one more first integral that results in Liouville integrability of the system.