Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2)

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem P (curve) of minimizing [Formula: see text] 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 [Formula: see text] 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 [Formula: see text]: show that [Formula: see text] is contained in half space x≥0 and (0,y (fin))≠(0,0) is reached with angle π, show that the boundary [Formula: see text] 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 [Formula: see text] and compute the length towards a cusp, analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold [Formula: see text] and with spatial arc-length parametrization s in the plane [Formula: see text]. 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.