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

Abstract

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} subsetmathrm{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 partialmathcal{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 partialmathcal {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}(-sintheta{rm d}x +costheta {rm d}y), mathcal{G}_{xi}:=xi^{2}(costheta{rm d}x+ sintheta {rm d}y) otimes(costheta{rm d}x+ sintheta{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.