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 omitted} for a planar curve having fixed initial and final positions and directions. Here ¿(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 R¿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,¿ fin) that can be connected by a globally minimizing geodesic starting at the origin (x in,y in,¿ in)=(0,0,0). From the applied imaging point of view it is relevant to analyze the sub-Riemannian geodesics and R in detail. In this article we - show that R is contained in half space x=0 and (0,y fin)¿(0,0) is reached with angle p, - show that the boundary ¿R consists of endpoints of minimizers either starting or ending in a cusp, - analyze and plot the cones of reachable angles ¿ fin per spatial endpoint (x fin,y fin), - relate the endings of association fields to ¿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 (SE(2),Ker(-sin¿dx+cos¿dy),G¿:=¿2(cos¿dx+sin¿dy)¿(cos¿dx+sin¿dy)+d¿¿d¿) and with spatial arc-length parametrization s in the plane R2 . 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. Keywords: Sub-Riemannian geometric control; Association fields; Pontryagin’s maximum principle; Boundary value problem; Geodesics in roto-translation space