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

作者：Duits R^*; Boscain U; Rossi F; Sachkov Y

来源：Journal of Mathematical Imaging and Vision, 2014, 49(2): 384-417.

DOI：10.1007/s10851-013-0475-y

• Sub-Riemannian geometric control
• Association fields
• Pontryagin%26apos
• s maximum principle
• Boundary value problem
• Geodesics in roto-translation space

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem P (curve) of minimizing for a planar curve having fixed initial and final positions and directions. Here kappa(s) is the curvature of the curve with free total length a%26quot;%26quot;. 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 %26 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 (x (fin),y (fin),theta (fin)) that can be connected by a globally minimizing geodesic starting at the origin (x (in),y (in),theta (in))=(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 %26lt;UnorderedList Mark=%26quot;Bullet%26quot;%26gt; %26lt;ItemContent%26gt; %26lt;Para%26gt;show that is contained in half space xa parts per thousand yen0 and (0,y (fin))not equal(0,0) is reached with angle pi, %26lt;br%26gt;show that the boundary consists of endpoints of minimizers either starting or ending in a cusp, %26lt;br%26gt;analyze and plot the cones of reachable angles theta (fin) per spatial endpoint (x (fin),y (fin)), %26lt;br%26gt;relate the endings of association fields to and compute the length towards a cusp, %26lt;br%26gt;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, %26lt;br%26gt;present a novel efficient algorithm solving the boundary value problem, %26lt;br%26gt;show that sub-Riemannian geodesics solve Petitot%26apos;s circle bundle model (cf. Petitot in J. Physiol. Paris 97:265-309, [2003]), %26lt;br%26gt;show a clear similarity with association field lines and sub-Riemannian geodesics.

• 出版日期2014-6