To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem P _curve of minimizing \(\int _{0}^{\ell} \sqrt{\xi^{2} +\kappa^{2}(s)} {\rm d}s \) 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, 2003; Math. Inf. Sci. Humaines 145:5–101, 1999), and Citti & Sarti (in J. Math. Imaging Vis. 24(3):307–326, 2006). In previous work we proved that the range \(\mathcal{R} \subset\mathrm{SE}(2)\) of the exponential map of the underlying geometric problem formulated on SE(2) consists of precisely those end-conditions (x _fin,y _fin,θ _fin) that can be connected by a globally minimizing geodesic starting at the origin (x _in,y _in,θ _in)=(0,0,0). From the applied imaging point of view it is relevant to analyze the sub-Riemannian geodesics and \(\mathcal{R}\) in detail. In this article we

• show that \(\mathcal{R}\) is contained in half space x≥0 and (0,y _fin)≠(0,0) is reached with angle π,
• show that the boundary \(\partial\mathcal{R}\) consists of endpoints of minimizers either starting or ending in a cusp,
• analyze and plot the cones of reachable angles θ _fin per spatial endpoint (x _fin,y _fin),
• relate the endings of association fields to \(\partial\mathcal {R}\) and compute the length towards a cusp,
• analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold \((\mathrm{SE}(2),\mathrm{Ker}(-\sin\theta{\rm d}x +\cos\theta {\rm d}y), \mathcal{G}_{\xi}:=\xi^{2}(\cos\theta{\rm d}x+ \sin\theta {\rm d}y) \otimes(\cos\theta{\rm d}x+ \sin\theta{\rm d}y) + {\rm d}\theta \otimes{\rm d}\theta)\) and with spatial arc-length parametrization s in the plane \(\mathbb{R}^{2}\). 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, [2003]),
• show a clear similarity with association field lines and sub-Riemannian geodesics.

中文翻译：

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通过SE（2）中的无尖子黎曼大地测量学建立的关联字段。

为了模拟在心理物理学中潜在的感知组织（格式塔）不足的关联字段，我们考虑最小化\（\ int _ {0} ^ {\ ell} \ sqrt {\ xi ^ {2} + \ kappa ^ {2}的问题P _曲线（s）} {\ rm d} s \），用于具有固定的初始和最终位置和方向的平面曲线。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（在J. Physiol。Paris 97：265–309，2003; Math。Inf。Sci。Humaines 145：5-101，1999）和Citti＆Sarti（in J. Math。Imaging Vis。24（3）：307-326，2006）。在先前的工作中，我们证明了范围\（\ mathcal {R} \ subset \ mathrm {SE}（2）\）制定了SE底层几何问题的指数映射的（2）由正是那些最终条件（X _鳍，ÿ _翅片，θ _翅片），其可以由一个全局最小测地起始在原点连接（X _中，ÿ _在，θ _在）=（0,0,0）。从应用的成像角度来看，详细分析亚黎曼测地线和\（\ mathcal {R} \）很重要。在本文中，我们

• 表明，\（\ mathcal {R} \）被包含在半空间X ≥0和（0，ÿ _翅片）≠（0,0）达到与角π，
• 证明边界\（\ partial \ mathcal {R} \）由最小化器的端点组成，这些端点的起点或终点均为尖峰，
• 分析并画出的可到达角锥体θ _鳍每空间端点（X _鳍，ÿ _翅片），
• 将关联字段的结尾与\（\ partial \ mathcal {R} \）相关联，并计算到顶点的长度，
• 分析指数映射都与公共弧长参数化吨在副黎曼流形\（（\ mathrm {SE}（2），\ mathrm {ker的}（ - \罪\ THETA {\ RM d} X + \ cos \ theta {\ rm d} y），\ mathcal {G} _ {\ xi}：= \ xi ^ {2}（\ cos \ theta {\ rm d} x + \ sin \ theta {\ rm d} y ）\ otimes（\ cos \ theta {\ rm d} x + \ sin \ theta {\ rm d} y）+ {\ rm d} \ theta \ otimes {\ rm d} \ theta）\）且具有空间弧平面\（\ mathbb {R} ^ {2} \）中的长度参数化s。令人惊讶的是，s参数化简化了指数图，曲率公式，尖端表面和边值问题，
• 提出了一种解决边值问题的新颖有效算法，
• 证明了黎曼次大地测量学可以解决Petitot的圆束模型（参见Petitot in J. Physiol。Paris 97：265–309，[2003]），
• 与关联场线和次黎曼测地线显示出明显的相似性。