Giovanna Citti

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 rototranslation group and the completion process is modeled 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.

The symplectic structure of the primary visual cortex

We propose to model the functional architecture of the primary visual cortex V1 as a principal fiber bundle where the two-dimensional retinal plane is the base manifold and the secondary variables of orientation and scale constitute the vertical fibers over each point as a rotation–dilation group. The total space is endowed with a natural symplectic structure neurally implemented by long range horizontal connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures, such as the medial axis of a shape.

IMPLICIT FUNCTION THEOREM IN CARNOT–CARATHÉODORY SPACES

In this paper, we prove an implicit function theorem and we study the regularity of the function implicitly defined. The implicit function theorem had already been proved in homogeneous Lie groups by Franchi, Serapioni and Serra Cassano, while the regularity problem of the function implicitly defined was still open even in the simplest Lie group.

Smoothness of Lipschitz-continuous graphs with nonvanishing Levi curvature

In (1), (2), ~=(x , y, t) denotes the point of R 3, ut is the first derivative of u with respect to t, and analogous notations are used for the other firstand second-order derivatives of u. The notion of Levi curvature for a real manifold was introduced by E.E. Levi in 1909 in order to characterize the holomorphy domains of C 2. Since then, it has played a crucial role in the geometric theory of several complex variables. In looking for the polynomial hull of a graph, Slodkowski and Tomassini implicitly introduced in 1991 the following definition of Levi curvature for Lipschitz-continuous graphs [16].

Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian

AbstractThe paper is concerned with the Dirichlet problem(P)

An uncertainty principle underlying the functional architecture of V1

- \Delta _H u + au = u^{{{\left( {q + 2} \right)} \mathord{\left/ {\vphantom {{\left( {q + 2} \right)} {\left( {q - 2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {q - 2} \right)}}} in \Omega , u = 0 on \partial \Omega

A model of natural image edge co-occurrence in the rototranslation group

where Ω is a smooth, bounded domain in R2n+1,ΔH is the Kohn-Laplacian on the Heisenberg group Hn, and q=2n+2 is the homogeneous dimension of Hn. We first prove a representation formula for the Palais Smale sequences of the functional naturally associated to (P). Then we use this expression to prove that, if 0⩾a> - λ1 (λ1 is the smallest eigenvalue of ΔH), then (P) has at least a nonnegative solution. This theorem extends to this setting a previous result of Brezis and Niremberg for the classical Laplacian.

Generalized Mean Curvature Flow in Carnot Groups

Aim of this study is to provide a formal link between connectionist neural models and variational psycophysical ones. We show that the solution of phase difference equation of weakly connected neural oscillators gamma-converges as the dimension of the grid tends to 0, to the gradient flow relative to the Mumford-Shah functional in a Riemannian space. The Riemannian metric is directly induced by the pattern of neural connections. Next, we embed the energy functional in the specific geometry of the functional space of the primary visual cortex, that is described in terms of a subRiemannian Heisenberg space. Namely, we introduce the Mumford-Shah functional with the Heisenberg metric and discuss the applicability of our main gamma-convergence result to subRiemannian spaces.