Erratum to: Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI

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.

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

Duits

et al. 2013

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Erratum to: Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI

Abstract: In the type-setting of this article, an unfortunate error has occurred. Within this article the captions of Fig. 22 (experiment on human dataset) and Fig. 23 (experiment on artificial dataset) were accidently interchanged.

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

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

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