The neurogeometry of pinwheels as a sub-Riemannian contact structure

Petitot, Jean

doi:10.1016/j.jphysparis.2003.10.010

Cited by 111 publications

(154 citation statements)

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“…This problem has important applications in robotics [8] and vision [13]. On the other hand, this is the simplest sub-Riemannian problem where the conjugate and cut loci differ one from another in the neighborhood of the initial point.…”

Section: Introductionmentioning

confidence: 99%

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Моисеев¹,

Sachkov²

2009

ESAIM: COCV

114

149

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Abstract.The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained. Mathematics Subject IntroductionProblems of sub-Riemannian geometry have been actively studied by geometric control methods, see books [2,3,7,10]. One of the central and hard questions in this domain is a description of cut and conjugate loci. Detailed results on the local structure of conjugate and cut loci were obtained in the 3-dimensional contact case [1,6]. Global results are restricted to symmetric low-dimensional cases, primarily for left-invariant problems on Lie groups (the Heisenberg group [5,22], the growth vector (n, n(n + 1)/2) [9,11,12], the groups SO(3), SU(2), SL(2) and the Lens Spaces [4]).The paper continues this direction of research: we start to study the left-invariant sub-Riemannian problem on the group of motions of a plane SE(2). This problem has important applications in robotics [8] and vision [13]. On the other hand, this is the simplest sub-Riemannian problem where the conjugate and cut loci differ one from another in the neighborhood of the initial point.The main result of the work is an upper bound on the cut time t cut given in Theorem 5.4: we show that for any sub-Riemannian geodesic on SE(2) there holds the estimate t cut ≤ t, where t is a certain function defined on the cotangent space at the identity. In a forthcoming paper [21] we prove that in fact t cut = t. The bound on the cut time is obtained via the study of discrete symmetries of the problem and the corresponding Maxwell points -points where two distinct sub-Riemannian geodesics of the same length intersect one another.This work has the following structure. In Section 1 we state the problem and discuss existence of solutions. In Section 2 we apply Pontryagin Maximum Principle to the problem. The Hamiltonian system for normal extremals is triangular, and the vertical subsystem is the equation of mathematical pendulum. In Section 3Keywords and phrases. Optimal control, sub-Riemannian geometry, differential-geometric methods, left-invariant problem, Lie group, Pontryagin Maximum Principle, symmetries, exponential mapping, Maxwell stratum. * The second author is partially supported by Russian Foundation for Basic Research, and we endow the cotangent space at the identity with special elliptic coordinates induced by the flow of the pendulum, and integrate the normal Hamiltonian system in these coordinates. Sub-Riemannian geodesics are parameterized by Jacobi's functions. In Section 4 we construct a discrete group of symmetries of the exponential mapping by continuation of reflections in the phase cylinder of the pendulum. In the main Section 5 we obtain an explicit description of Maxwell strata corresponding to the group of discrete symmetries, and prove the upper ...

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Section: Introductionmentioning

confidence: 99%

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Моисеев¹,

Sachkov²

2009

ESAIM: COCV

114

149

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“…(Right) A useful view is to understand this topological arrangement as a feature image, that is as the representation of the selectivity to the different features (a vector) as a function of space. We may understand the previous arrangement as an optimal projection of this feature space on the two-dimensional map of the cortex [Petitot, 2003]. particular retinotopic in low-level visual areas (that is that they represent the space as it is on the image formed on the retina).…”

Section: Feature Maps: Distributed Visual Representationsmentioning

confidence: 99%

“…In fact, the structure of V1 revealed for instance by optical imaging [Grinvald et al, 2001] shows that the information of orientation is distributed in macro-column, forming "pin-wheels" of orientation selectivity 15 . This representation may be viewed as an optimal projection of the multidimensional feature image on the cortical surface [Petitot, 2003] (see Fig. 3), thus forming a distributed representation of the different features 16 .…”

Section: Feature Maps: Distributed Visual Representationsmentioning

confidence: 99%

Dynamical neural networks: Modeling low-level vision at short latencies

Perrinet

2007

Eur. Phys. J. Spec. Top.

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Our goal is to understand the dynamics of neural computations in low-level vision. We study how the substrate of this system, that is local biochemical neural processes, could combine to give rise to an efficient and global perception. We will study these neural computations at different scales from the single-cell to the whole visual system to infer generic aspects of the underlying neural code which may help to understand this cognitive ability. In fact, the architecture of cortical areas, such as the Primary Visual Cortex (V1), is massively parallel and we will focus on cortical columns as generic adaptive micro-circuits. To stress on the dynamical aspect of the processing, we will also focus on the transient response, that is during the first milliseconds after the presentation of a stimulus. In a generic model of a visual area, we propose to study the neural code as implementing visual pattern matching, that is as efficiently inverting a known model of image synthesis. A possible solution offered by the architecture of the visual pathways could be to represent at first on the surface of the cortical area how well the prototypical visual features are matched by a combination of inferential mechanisms as ideal observers. We studied the efficiency of this representation by rating the statistics of the output using natural scenes, that is scenes occurring frequently. We show that this may be finally used to provide a behavioral output such as an eye movement. However, constraints specific to the visual system imply that the set of prototypical features is not independent and that the cortical columns should communicate to produce an efficient, sparse solution. We will present efficient algorithms and representations based on the event-based nature of neural computations. By explicitely defining this efficiency, we propose then a simple implementation of Sparse Spike Coding using greedy inference mechanisms but also how the system may adapt in a unsupervised fashion. These computations may be implemented in simple models of neural networks by explicitly setting the lateral connectivity between populations of columns. Using natural scenes, this algorithm provides a model of V1 which exhibit prototypical properties of neural activities in that area. We show simple applications in the field of image processing as a quantitative method to evaluate these different cortical models.

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“…It must be emphasized that 2D images contain important structures (e.g., luminance saddle points) that have no analog in the 1D signals to which the energy model is ideally suited. A realistic framework for spatial vision must be capable of representing the full variety of 2D structures (Ben-Shahar & Zucker, 2004;Dobbins, Zucker, & Cynader, 1987;Petitot, 2003).…”

Section: Introductionmentioning

confidence: 99%

A Differential Model of the Complex Cell

Hansard

Horaud

2011

Neural Computation

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The receptive fields of simple cells in the visual cortex can be understood as linear filters. These filters can be modeled by Gabor functions or gaussian derivatives. Gabor functions can also be combined in an energy model of the complex cell response. This letter proposes an alternative model of the complex cell, based on gaussian derivatives. It is most important to account for the insensitivity of the complex response to small shifts of the image. The new model uses a linear combination of the first few derivative filters, at a single position, to approximate the first derivative filter, at a series of adjacent positions. The maximum response, over all positions, gives a signal that is insensitive to small shifts of the image. This model, unlike previous approaches, is based on the scale space theory of visual processing. In particular, the complex cell is built from filters that respond to the 2D differential structure of the image. The computational aspects of the new model are studied in one and two dimensions, using the steerability of the gaussian derivatives. The response of the model to basic images, such as edges and gratings, is derived formally. The response to natural images is also evaluated, using statistical measures of shift insensitivity. The neural implementation and predictions of the model are discussed.

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The neurogeometry of pinwheels as a sub-Riemannian contact structure

Cited by 111 publications

References 54 publications

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Dynamical neural networks: Modeling low-level vision at short latencies

A Differential Model of the Complex Cell

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