Galyaev Ivan, Mashtakov Alexey
V. A. Trapeznikov Institute of Control Sciences of RAS, 117997 Moscow, Russia.
Ailamazyan Program Systems Institute of RAS, 152021 Veskovo, Russia.
J Imaging. 2021 Dec 17;7(12):277. doi: 10.3390/jimaging7120277.
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 taken into account. The occluded contours are completed via sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and curvatures. Here, M=R2×SO(2)×R models the configuration space of neurons of the visual cortex. We study the problem of sub-Riemannian geodesics on via methods of geometric control theory. We prove complete controllability of the system and the existence of optimal controls. By application of the Pontryagin maximum principle, we derive a Hamiltonian system that describes the geodesics. We obtain the 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.
我们考虑了初级视觉皮层的Petitot-Citti-Sarti模型的自然扩展。在扩展模型中，考虑了轮廓的曲率。通过在位置、方向和曲率的四维空间M中的次黎曼测地线来完成被遮挡的轮廓。这里，M = R2×SO(2)×R对视觉皮层神经元的配置空间进行建模。我们通过几何控制理论的方法研究M上的次黎曼测地线问题。我们证明了系统的完全可控性和最优控制的存在性。通过应用庞特里亚金极大值原理，我们推导了一个描述测地线的哈密顿系统。我们得到了异常极值的显式参数化。在正常情况下，我们提供了三个功能独立的第一积分。数值模拟表明存在另一个第一积分，这导致了系统的刘维尔可积性。
