Davide Barbieri

An uncertainty principle underlying the functional architecture of V1

We present a model of the morphology of orientation maps in V1 based on the uncertainty principle of the SE(2) group. Starting from the symmetries of the cortex, suitable harmonic analysis instruments are used to obtain coherent states in the Fourier domain as minimizers of the uncertainty. Cortical activities related to orientation maps are then obtained by projection on a suitable cortical Fourier basis.

A Cortical-Inspired Geometry for Contour Perception and Motion Integration

In this paper we develop a geometrical model of functional architecture for the processing of spatio-temporal visual stimuli. The model arises from the properties of the receptive field linear dynamics of orientation and speed-selective cells in the visual cortex, that can be embedded in the definition of a geometry where the connectivity between points is driven by the contact structure of a 5D manifold. Then, we compute the stochastic kernels that are the approximations of two Fokker Planck operators associated to the geometry, and implement them as facilitation patterns within a neural population activity model, in order to reproduce some psychophysiological findings about the perception of contours in motion and trajectories of points found in the literature.

Cortical spatiotemporal dimensionality reduction for visual grouping

The visual systems of many mammals, including humans, are able to integrate the geometric information of visual stimuli and perform cognitive tasks 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 the single cell level and geometric processing by means of cell connectivity. We present a geometric model of such connectivities in the space of detected features associated with spatiotemporal visual stimuli and show how they can be used to obtain low-level object segmentation. The main idea is to define a spectral clustering procedure with anisotropic affinities over data sets consisting of embeddings of the visual stimuli into higher-dimensional spaces. Neural plausibility of the proposed arguments will be discussed.

Reproducing kernel Hilbert spaces of CR functions for the Euclidean Motion group

We study the geometric structure of the reproducing kernel Hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the Euclidean Motion

Spatiotemporal receptive fields of cells in V1 are optimally shaped for stimulus velocity estimation

SE(2)

Regularity of minimal intrinsic graphs in 3-dimensional sub-Riemannian structures of step 2

. A natural Hilbert norm for functions on the group is constructed that makes the wavelet transform an isometry, but since the considered representations are not square integrable the resulting Hilbert space will not coincide with

How Uncertainty Bounds the Shape Index of Simple Cells

L^2(SE(2))

Neuromorphology of Meaning

. The reproducing kernel Hilbert subspace generated by the wavelet transform, for the case of a minimal uncertainty mother wavelet, can be characterized in terms of the complex regularity defined by the natural

Group Riesz and frame sequences: the Bracket and the Gramian

CR

Riesz and frame systems generated by unitary actions of discrete groups

structure of the group. Relations with the Bargmann transform are presented.