Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2) |
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Authors: | R Duits U Boscain F Rossi Y Sachkov |
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Affiliation: | 1. IST/e, Eindhoven University of Technology, Den Dolech 2, 5600 MB, Eindhoven, The Netherlands 2. école Polytechnique Paris, CMAP, Route de Saclay, 91128, Palaiseau Cedex, France 3. Aix-Marseille University, LSIS, 13013, Marseille, France 4. Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalessky, 152140, Russia
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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 ?. 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 - 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 $\partial\mathcal{R}$ 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 $\partial\mathcal {R}$ and compute the length towards a cusp,
- analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold $(\mathrm{SE}(2),\mathrm{Ker}(-\sin\theta{\rm d}x +\cos\theta {\rm d}y), \mathcal{G}_{\xi}:=\xi^{2}(\cos\theta{\rm d}x+ \sin\theta {\rm d}y) \otimes(\cos\theta{\rm d}x+ \sin\theta{\rm d}y) + {\rm d}\theta \otimes{\rm d}\theta)$ 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.
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