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

Duits R, Boscain U, Rossi F, Sachkov Y - J Math Imaging Vis (2014)

Bottom Line: Math.Physiol.Paris 97:265-309, [2003]),show a clear similarity with association field lines and sub-Riemannian geodesics.

View Article: PubMed Central - PubMed

Affiliation: IST/e, Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, The Netherlands.

ABSTRACT

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem P curve 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 (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 [Formula: see text] in detail. In this article we show that [Formula: see text] is contained in half space x≥0 and (0,y fin)≠(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 θ fin per spatial endpoint (x fin,y fin),relate the endings of association fields to [Formula: see text] and compute the length towards a cusp,analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold [Formula: see text] and with spatial arc-length parametrization s in the plane [Formula: see text]. 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.

No MeSH data available.


Receptive fields in the visual cortex of many mammalians are tuned to various locations and orientations. Assemblies of oriented receptive fields are grouped together on the surface of the primary visual cortex in a pinwheel like structure. Orientation sensitivity in the primary visual cortex of a tree shrew, replicated from [17], © 1997 Society of Neuroscience. Black dots indicate horizontal connections to aligned neurons with an 80∘ orientation preference shown by the white dots. The figure on the right indicates horizontal connections at 160∘
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Fig5: Receptive fields in the visual cortex of many mammalians are tuned to various locations and orientations. Assemblies of oriented receptive fields are grouped together on the surface of the primary visual cortex in a pinwheel like structure. Orientation sensitivity in the primary visual cortex of a tree shrew, replicated from [17], © 1997 Society of Neuroscience. Black dots indicate horizontal connections to aligned neurons with an 80∘ orientation preference shown by the white dots. The figure on the right indicates horizontal connections at 160∘

Mentions: Orientation columns are connected between them in two different ways. The first kind is given by vertical connections, which connect orientation columns belonging to the same hypercolumn and sensible to similar directions. The second is given by the horizontal connections across the orientation columns which checks for alignment of local orientations. See Figs. 5 and 6. Fig. 5


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

Duits R, Boscain U, Rossi F, Sachkov Y - J Math Imaging Vis (2014)

Receptive fields in the visual cortex of many mammalians are tuned to various locations and orientations. Assemblies of oriented receptive fields are grouped together on the surface of the primary visual cortex in a pinwheel like structure. Orientation sensitivity in the primary visual cortex of a tree shrew, replicated from [17], © 1997 Society of Neuroscience. Black dots indicate horizontal connections to aligned neurons with an 80∘ orientation preference shown by the white dots. The figure on the right indicates horizontal connections at 160∘
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4551278&req=5

Fig5: Receptive fields in the visual cortex of many mammalians are tuned to various locations and orientations. Assemblies of oriented receptive fields are grouped together on the surface of the primary visual cortex in a pinwheel like structure. Orientation sensitivity in the primary visual cortex of a tree shrew, replicated from [17], © 1997 Society of Neuroscience. Black dots indicate horizontal connections to aligned neurons with an 80∘ orientation preference shown by the white dots. The figure on the right indicates horizontal connections at 160∘
Mentions: Orientation columns are connected between them in two different ways. The first kind is given by vertical connections, which connect orientation columns belonging to the same hypercolumn and sensible to similar directions. The second is given by the horizontal connections across the orientation columns which checks for alignment of local orientations. See Figs. 5 and 6. Fig. 5

Bottom Line: Math.Physiol.Paris 97:265-309, [2003]),show a clear similarity with association field lines and sub-Riemannian geodesics.

View Article: PubMed Central - PubMed

Affiliation: IST/e, Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, The Netherlands.

ABSTRACT

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem P curve 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 (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 [Formula: see text] in detail. In this article we show that [Formula: see text] is contained in half space x≥0 and (0,y fin)≠(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 θ fin per spatial endpoint (x fin,y fin),relate the endings of association fields to [Formula: see text] and compute the length towards a cusp,analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold [Formula: see text] and with spatial arc-length parametrization s in the plane [Formula: see text]. 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.

No MeSH data available.