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

Remco Duits, Ugo V. Boscain, Francesco Rossi, Yuri Sachkov

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 \(\mathcal{R} \subset\mathrm{SE}(2)\) 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