1,520 research outputs found
The constitution of visual perceptual units in the functional architecture of V1
Scope of this paper is to consider a mean field neural model which takes into
account the functional neurogeometry of the visual cortex modelled as a group
of rotations and translations. The model generalizes well known results of
Bressloff and Cowan which, in absence of input, accounts for hallucination
patterns. The main result of our study consists in showing that in presence of
a visual input, the eigenmodes of the linearized operator which become stable
represent perceptual units present in the image. The result is strictly related
to dimensionality reduction and clustering problems
Regularity for Subelliptic PDE Through Uniform Estimates in Multi-Scale Geometries
We aim at reviewing and extending a number of recent results addressing
stability of certain geometric and analytic estimates in the Riemannian
approximation of subRiemannian structures. In particular we extend the recent
work of the the authors with Rea [19] and Manfredini [17] concerning stability
of doubling properties, Poincar\'e inequalities, Gaussian estimates on heat
kernels and Schauder estimates from the Carnot group setting to the general
case of H\"ormander vector fields
A cortical based model of perceptual completion in the roto-translation space
We present a mathematical model of perceptual completion and formation
of subjective surfaces, which is at the same time inspired by the architecture
of the visual cortex, and is the lifting in the 3-dimensional rototranslation group of
the phenomenological variational models based on elastica functional. The initial
image is lifted by the simple cells to a surface in the rototraslation group and the
completion process is modelled via a diffusion driven motion by curvature. The
convergence of the motion to a minimal surface is proved. Results are presented
both for modal and amodal completion in classic Kanizsa images
Regularity of minimal intrinsic graphs in 3 dimensional sub-Riemannian structures of step 2
This work provides a characterization of the regularity of noncharacteristic
intrinsic minimal graphs for a class of vector fields that includes non
nilpotent Lie algebras as the one given by Euclidean motions of the plane. The
main result extends a previous one on the Heisenberg group, using similar
techniques to deal with nonlinearities. This wider setting provides a better
understanding of geometric constraints, together with an extension of the
potentialities of specific tools as the lifting-freezing procedure and
interpolation inequalities. As a consequence of the regularity, a foliation
result for minimal graphs is obtained
From receptive profiles to a metric model of V1
In this work we show how to construct connectivity kernels induced by the
receptive profiles of simple cells of the primary visual cortex (V1). These
kernels are directly defined by the shape of such profiles: this provides a
metric model for the functional architecture of V1, whose global geometry is
determined by the reciprocal interactions between local elements. Our
construction adapts to any bank of filters chosen to represent a set of
receptive profiles, since it does not require any structure on the
parameterization of the family. The connectivity kernel that we define carries
a geometrical structure consistent with the well-known properties of long-range
horizontal connections in V1, and it is compatible with the perceptual rules
synthesized by the concept of association field. These characteristics are
still present when the kernel is constructed from a bank of filters arising
from an unsupervised learning algorithm.Comment: 25 pages, 18 figures. Added acknowledgement
A geometric model of multi-scale orientation preference maps via Gabor functions
In this paper we present a new model for the generation of orientation
preference maps in the primary visual cortex (V1), considering both orientation
and scale features. First we undertake to model the functional architecture of
V1 by interpreting it as a principal fiber bundle over the 2-dimensional
retinal plane by introducing intrinsic variables orientation and scale. The
intrinsic variables constitute a fiber on each point of the retinal plane and
the set of receptive profiles of simple cells is located on the fiber. Each
receptive profile on the fiber is mathematically interpreted as a rotated Gabor
function derived from an uncertainty principle. The visual stimulus is lifted
in a 4-dimensional space, characterized by coordinate variables, position,
orientation and scale, through a linear filtering of the stimulus with Gabor
functions. Orientation preference maps are then obtained by mapping the
orientation value found from the lifting of a noise stimulus onto the
2-dimensional retinal plane. This corresponds to a Bargmann transform in the
reducible representation of the group. A
comparison will be provided with a previous model based on the Bargman
transform in the irreducible representation of the group,
outlining that the new model is more physiologically motivated. Then we present
simulation results related to the construction of the orientation preference
map by using Gabor filters with different scales and compare those results to
the relevant neurophysiological findings in the literature
Regularity of mean curvature flow of graphs on Lie groups free up to step 2
We consider (smooth) solutions of the mean curvature flow of graphs over
bounded domains in a Lie group free up to step two (and not necessarily
nilpotent), endowed with a one parameter family of Riemannian metrics
\sigma_\e collapsing to a subRiemannian metric as \e\to 0. We
establish estimates for this flow, that are uniform as \e\to 0
and as a consequence prove long time existence for the subRiemannian mean
curvature flow of the graph. Our proof extend to the setting of every step two
Carnot group (not necessarily free) and can be adapted following our previous
work in \cite{CCM3} to the total variation flow.Comment: arXiv admin note: text overlap with arXiv:1212.666
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