title: 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 problemPcurveof minimizing [Formula: see text] for a planar curve having fixed initial and final positions and directions. Hereκ(s) is the curvature of the curve with free total lengthℓ. 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 (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 [Formula: see text] in detail. In this article we show that [Formula: see text] is contained in half spacex≥0 and (0,yfin)≠(0,0) is reached with angleπ,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 anglesθfinper spatial endpoint (xfin,yfin),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 parametrizationtin the sub-Riemannian manifold [Formula: see text] and with spatial arc-length parametrizationsin 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.
authors: IST/e, Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, The Netherlands.
authors: R
authors: R
authors: Duits
authors: École Polytechnique Paris, CMAP, Route de Saclay, 91128 Palaiseau Cedex, France.
authors: U
authors: U
authors: Boscain
authors: Aix-Marseille University, LSIS, 13013 Marseille, France.
authors: F
authors: F
authors: Rossi
authors: Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalessky, 152140 Russia.
authors: Y
authors: Y
authors: Sachkov
keywords: Association fields; Boundary value problem; Geodesics in roto-translation space; Pontryagin’s maximum principle; Sub-Riemannian geometric control