105 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
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
Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups
In this paper we study heat kernels associated to a Carnot group , endowed
with a family of collapsing left-invariant Riemannian metrics \sigma_\e which
converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on as
\e\to 0. The main new contribution are Gaussian-type bounds on the heat
kernel for the \sigma_\e metrics which are stable as \e\to 0 and extend the
previous time-independent estimates in \cite{CiMa-F}. As an application we
study well posedness of the total variation flow of graph surfaces over a
bounded domain in (G,\s_\e). We establish interior and boundary gradient
estimates, and develop a Schauder theory which are stable as \e\to 0. As a
consequence we obtain long time existence of smooth solutions of the
sub-Riemannian flow (\e=0), which in turn yield sub-Riemannian minimal
surfaces as .Comment: We have corrected a few typos and added a few more details to the
proof of the Gaussian estimate
Local and global gestalt laws: A neurally based spectral approach
A mathematical model of figure-ground articulation is presented, taking into
account both local and global gestalt laws. The model is compatible with the
functional architecture of the primary visual cortex (V1). Particularly the
local gestalt law of good continuity is described by means of suitable
connectivity kernels, that are derived from Lie group theory and are neurally
implemented in long range connectivity in V1. Different kernels are compatible
with the geometric structure of cortical connectivity and they are derived as
the fundamental solutions of the Fokker Planck, the Sub-Riemannian Laplacian
and the isotropic Laplacian equations. The kernels are used to construct
matrices of connectivity among the features present in a visual stimulus.
Global gestalt constraints are then introduced in terms of spectral analysis of
the connectivity matrix, showing that this processing can be cortically
implemented in V1 by mean field neural equations. This analysis performs
grouping of local features and individuates perceptual units with the highest
saliency. Numerical simulations are performed and results are obtained applying
the technique to a number of stimuli.Comment: submitted to Neural Computatio
Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian
We establish a Harnack inequality for a class of quasi-linear PDE modeled on
the prototype {equation*}
\partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where
, \ \X = (X_1,..., X_m) is a system of Lipschitz vector fields
defined on a smooth manifold \M endowed with a Borel measure , and
denotes the adjoint of with respect to . Our estimates are
derived assuming that (i) the control distance generated by \X induces
the same topology on \M; (ii) a doubling condition for the -measure of
metric balls and (iii) the validity of a Poincar\'e inequality involving
\X and . Our results extend the recent work in
\cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting
including the model cases of (1) metrics generated by H\"ormander vector fields
and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci
curvature and Riemannian volume forms; and (3) metrics generated by non-smooth
Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the
Harnack inequality continues to hold when the Lebesgue measure is substituted
by any smooth volume form or by measures with densities corresponding to
Muckenhoupt type weights
Cortical spatio-temporal dimensionality reduction for visual grouping
The visual systems of many mammals, including humans, is able to integrate
the geometric information of visual stimuli and to perform cognitive tasks
already at the first stages of the cortical processing. This is thought to be
the result of a combination of mechanisms, which include feature extraction at
single cell level and geometric processing by means of cells connectivity. We
present a geometric model of such connectivities in the space of detected
features associated to spatio-temporal visual stimuli, and show how they can be
used to obtain low-level object segmentation. The main idea is that of defining
a spectral clustering procedure with anisotropic affinities over datasets
consisting of embeddings of the visual stimuli into higher dimensional spaces.
Neural plausibility of the proposed arguments will be discussed
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