610 research outputs found
A sub-Riemannian model of the visual cortex with frequency and phase
In this paper we present a novel model of the primary visual cortex (V1) based on orientation, frequency and phase selective behavior of the V1 simple cells. We start from the first level mechanisms of visual perception: receptive profiles. The model interprets V1 as a fiber bundle over the 2-dimensional retinal plane by introducing orientation, frequency and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as a rotated, frequency modulated and phase shifted Gabor function. We start from the Gabor function and show that it induces in a natural way the model geometry and the associated horizontal connectivity modeling the neural connectivity patterns in V1. We provide an image enhancement algorithm employing the model framework. The algorithm is capable of exploiting not only orientation but also frequency and phase information existing intrinsically in a 2-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm
Minimal Surfaces in Sub-Riemannian Structures and Functional Geometry of the Visual Cortex
We develop geometrical models of vision consistent with the characteristics of the visual cortex and study geometric flows in the relevant model geometries. We provide a novel sub-Riemannian model of the primary visual cortex, which models orientation-frequency selective phase shifted cortex cell behavior and the associated horizontal connectivity. We develop an image enhancement algorithm using sub-Riemannian diffusion
and Laplace-Beltrami flow in the model framework. We provide two geometric models for multi-scale orientation map and orientation-frequency preference map construction which employ Bargmann transform in high dimensional cortical spaces. We prove the uniqueness of the solution to sub-Riemannian mean curvature flow equation in the Heisenberg group geometry. An iterative diffusion process followed by a maximum selection mechanism was proposed by Citti and Sarti in the sub-Riemannian setting of the roto-translation group. They conjectured that this two-fold procedure is equivalent to a mean curvature flow. However a complete proof was missing, even in the Euclidean setting. We prove in the Euclidean
setting that this two fold procedure is equivalent to mean curvature flow
A sub-Riemannian model of the visual cortex with frequency and phase
International audienceIn this paper we present a novel model of the primary visual cortex (V1) based on orientation, frequency and phase selective behavior of the V1 simple cells. We start from the first level mechanisms of visual perception: receptive profiles. The model interprets V1 as a fiber bundle over the 2-dimensional retinal plane by introducing orientation, frequency and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as a rotated, frequency modulated and phase shifted Gabor function. We start from the Gabor function and show that it induces in a natural way the model geometry and the associated horizontal connectivity modeling the neural connectivity patterns in V1. We provide an image enhancement algorithm employing the model framework. The algorithm is capable of exploiting not only orientation but also frequency and phase information existing intrinsically in a 2-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm
Highly corrupted image inpainting through hypoelliptic diffusion
We present a new image inpainting algorithm, the Averaging and Hypoelliptic
Evolution (AHE) algorithm, inspired by the one presented in [SIAM J. Imaging
Sci., vol. 7, no. 2, pp. 669--695, 2014] and based upon a semi-discrete
variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The
AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic
diffusion and ad-hoc local averaging techniques. In particular, we focus on
reconstructing highly corrupted images (i.e. where more than the 80% of the
image is missing), for which we obtain reconstructions comparable with the
state-of-the-art.Comment: 15 pages, 10 figure
Cortical Functional architectures as contact and sub-Riemannian geometry
In a joint paper, Jean Petitot together with the authors of the present paper
described the functional geometry of the visual cortex as the symplectization
of a contact form to describe the family of cells sensitive to position,
orientation and scale. In the present paper, as a "homage" to the enormous
contribution of Jean Petitot to neurogeometry, we will extend this approach to
more complex functional architectures built as a sequence of contactization or
a symplectization process, able to extend the dimension of the space. We will
also outline a few examples where a sub-Riemannian lifting is needed
Hyperbolic planforms in relation to visual edges and textures perception
We propose to use bifurcation theory and pattern formation as theoretical
probes for various hypotheses about the neural organization of the brain. This
allows us to make predictions about the kinds of patterns that should be
observed in the activity of real brains through, e.g. optical imaging, and
opens the door to the design of experiments to test these hypotheses. We study
the specific problem of visual edges and textures perception and suggest that
these features may be represented at the population level in the visual cortex
as a specific second-order tensor, the structure tensor, perhaps within a
hypercolumn. We then extend the classical ring model to this case and show that
its natural framework is the non-Euclidean hyperbolic geometry. This brings in
the beautiful structure of its group of isometries and certain of its subgroups
which have a direct interpretation in terms of the organization of the neural
populations that are assumed to encode the structure tensor. By studying the
bifurcations of the solutions of the structure tensor equations, the analog of
the classical Wilson and Cowan equations, under the assumption of invariance
with respect to the action of these subgroups, we predict the appearance of
characteristic patterns. These patterns can be described by what we call
hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of
the planforms that were used in [1, 2] to account for some visual
hallucinations. If these patterns could be observed through brain imaging
techniques they would reveal the built-in or acquired invariance of the neural
organization to the action of the corresponding subgroups.Comment: 34 pages, 11 figures, 2 table
Sub-Riemannian geometry and its applications to Image Processing
Master's Thesis in MathematicsMAT399MAMN-MA
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