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 Pcurve of minimizing 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 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 in detail. In this article we show that is contained in half space x≥0 and (0,yfin)≠(0,0) is reached with angle π,show that the boundary consists of endpoints of minimizers either starting or ending in a cusp,analyze and plot the cones of reachable angles θfin per spatial endpoint (xfin,yfin),relate the endings of association fields to and compute the length towards a cusp,analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold and with spatial arc-length parametrization s in the plane . 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.

On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

“…Fig. 8) seem to reveal the same homeo/diffeomorphic properties of the exponential map integrating the canonical ODE's in Pontryagin maximum principle, that were formally shown on the SE(2) case [11,Thm.6&9]. In future work, we plan to analyze the sub-Riemannian spheres via viscosity solutions of sub-Riemannian HJBequations, i.e.…”

“…There, the presence of cusps in the spatial projection of a geodesic is typically undesirable, as it yields a non-smooth path in 3D images. Likewise, to the 2D case [11], it may even be used as a criterium for not connecting two given boundary conditions in an image.…”

“…[6,11]. To state the problem P mec on this Lie group quotient, we first resort to the corresponding left-invariant sub-Riemannian problem P MEC in the Lie group SE(3).…”

“…Now by the global optimality results in [6,11], we have the following corollary. (2), which coincides with the set of admissible end conditions of P curve in SE (2).…”

“…Based on our numerical experiments, we pose the following conjecture which is analogous to a result in the 2D case of finding cuspless sub-Riemannian geodesics in (SE (2),˜ ,G 1 ) [6,11]. Note that S B ∈ R and S R ∈ R but S L ∈ R. This conjecture would imply that no conjugate points arise within R and the problem P curve (1.1) is well posed for all end conditions in R. The proof of this conjecture would be on similar lines as in Appendix F of [11].…”