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

Journal of Mathematical Imaging and Vision
R DuitsY Sachkov

Abstract

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem Pcurve of minimizing [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] in detail. In this article we show that [Formula: see text] is contained in half space x≥0 and (0,yfin)≠(0,0) is reached with angle π,show that the boundary [Formula: see text] consists of endpoints of minimizers either starting or ending in a cusp,analyze and plot the cones of reachable angles θfin per sp...Continue Reading

References

Nov 27, 2002·Vision Research·Sébastien GeorgesJean Lorenceau
Feb 10, 2004·Journal of Physiology, Paris·Jean Petitot
Jul 15, 2009·Journal de gynécologie, obstétrique et biologie de la reproduction·J-P LucotD Vinatier
Jan 5, 2011·Journal of Vision·Gonzalo SanguinettiAlessandro Sarti
Dec 28, 2011·IEEE Transactions on Pattern Analysis and Machine Intelligence·Guy Ben-Yosef, Ohad Ben-Shahar
Apr 7, 2012·Journal of Physiology, Paris·Davide BarbieriAlessandro Sarti

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Citations

Apr 14, 2019·Journal of Computational Neuroscience·Noemi MontobbioAlessandro Sarti
Jan 1, 2018·Journal of Mathematical Imaging and Vision·R DuitsJ M Portegies
Jan 1, 2017·Journal of Mathematical Imaging and Vision·A MashtakovI Beschastnyi
Jul 31, 2020·Journal of Mathematical Neuroscience·E BaspinarG Citti
Jan 1, 2018·Journal of Mathematical Imaging and Vision·Erik J BekkersJorg M Portegies

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Software Mentioned

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