Galyaev Ivan, Mashtakov Alexey
V. A. Trapeznikov Institute of Control Sciences of RAS, Moscow 117997, Russia.
Ailamazyan Program Systems Institute of RAS, Pereslavl-Zalessky 152021, Russia.
J Imaging. 2024 Jul 31;10(8):185. doi: 10.3390/jimaging10080185.
An extended four-dimensional version of the traditional Petitot-Citti-Sarti model on contour completion in the visual cortex is examined. The neural configuration space is considered as the group of similarity transformations, denoted as M=SIM(2). The left-invariant subbundle of the tangent bundle models possible directions for establishing neural communication. The sub-Riemannian distance is proportional to the energy expended in interneuron activation between two excited border neurons. According to the model, the damaged image contours are restored via sub-Riemannian geodesics in the space of positions, orientations and thicknesses (scales). We study the geodesic problem in using geometric control theory techniques. We prove the existence of a minimal geodesic between arbitrary specified boundary conditions. We apply the Pontryagin maximum principle and derive the geodesic equations. In the special cases, we find explicit solutions. In the general case, we provide a qualitative analysis. Finally, we support our model with a simulation of the association field.
研究了传统的Petitot-Citti-Sarti模型在视觉皮层轮廓完成方面的扩展四维版本。神经配置空间被视为相似性变换群，记为M = SIM(2)。切丛的左不变子丛对建立神经通信的可能方向进行建模。次黎曼距离与两个兴奋的边界神经元之间中间神经元激活所消耗的能量成正比。根据该模型，受损图像轮廓通过位置、方向和厚度（尺度）空间中的次黎曼测地线得以恢复。我们使用几何控制理论技术研究测地线问题。我们证明了任意指定边界条件之间存在最小测地线。我们应用庞特里亚金极大值原理并推导测地线方程。在特殊情况下，我们找到显式解。在一般情况下，我们进行定性分析。最后，我们通过关联场的模拟来支持我们的模型。
