To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem >Pcurve of minimizing for a planar curve having fixed initial and final positions and directions. Here κ(s) is the curvature of the curve with free total length ℓ. This problem comes from a model of geometry of vision due to Petitot (in J. Physiol. Paris 97:265–309, ; Math. Inf. Sci. Humaines 145:5–101, ), and Citti & Sarti (in J. Math. Imaging Vis. 24(3):307–326, ). In previous work we proved that the range of the exponential map of the underlying geometric problem formulated on SE(2) consists of precisely those end-conditions (xfin,yfin,θfin) that can be connected by a globally minimizing geodesic starting at the origin (xin,yin,θin)=(0,0,0). From the applied imaging point of view it is relevant to analyze the sub-Riemannian geodesics and in detail. In this article we class="unordered" style="list-style-type:disc">show that is contained in half space x≥0 and (0,yfin)≠(0,0) is reached with angle π,show that the boundary consists of endpoints of minimizers either starting or ending in a cusp,analyze and plot the cones of reachable angles θfin per spatial endpoint (xfin,yfin),relate the endings of association fields to and compute the length towards a cusp,analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold and with spatial arc-length parametrization s in the plane . Surprisingly, s-parametrization simplifies the exponential map, the curvature formulas, the cusp-surface, and the boundary value problem,present a novel efficient algorithm solving the boundary value problem,show that sub-Riemannian geodesics solve Petitot’s circle bundle model (cf. Petitot in J. Physiol. Paris 97:265–309, []),show a clear similarity with association field lines and sub-Riemannian geodesics.
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机译：为了对心理物理学中感知组织（格式塔）底层的关联字段进行建模，我们考虑将具有固定初始位置和最终位置和方向的平面曲线最小化的问题> P 曲线。此处的κ（s）是自由总长度为ℓ的曲线的曲率。这个问题来自于Petitot（J. Physiol。Paris 97：265–309，; Math。Inf。Sci。Humaines 145：5–101，）和Citti＆Sarti（J. Phys。 《数学影像与视觉》 24（3）：307–326，）。在先前的工作中，我们证明了在SE（2）上公式化的基础几何问题的指数图的范围恰好包含可以通过从原点开始的全局最小化测地线连接的那些最终条件（xfin，yfin，θfin） （xin，yin，θin）=（0,0,0）。从应用的成像角度来看，分析亚黎曼测地线是很重要的。在本文中，我们 class =“ unordered” style =“ list-style-type：disc”> <！-list-behavior = unordered prefix-word = mark-type = disc max-label-size = 0- > 显示包含在半个空间x≥0中，并且以角度π到达（0，yfin）≠（0,0）， 显示边界由最小化器的端点组成 分析并绘制每个空间端点（xfin，yfin）的可到达角度θfin的圆锥， 将关联字段的结尾与关联并计算长度尖点 使用次黎曼流形中的公共弧长参数化t和平面中的空间弧长参数化 s 来分析指数图。令人惊讶的是， s 参数化简化了指数图，曲率公式，尖点表面和边值问题， 提出了一种新颖有效的算法来解决边值问题， 表明子黎曼测地线解决了Petitot的圆束模型（参见Petitot in J. Physiol。Paris 97：265–309，[]）， 与关联场线和次黎曼测地线。
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