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… (More)
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.
The functionality of the visual cortex has been described in  and in  as a contact manifold of dimension three and in  the Mumford and Shah functional has been proposed to segment lifting of an image in the three dimensional cortical space. Hence, we study here this functional and we provide a constructive approach to the problem, extending to… (More)
We consider the following nonlinear degenerate para-bolic equation which arises in some recent problems of mathematical finance: ∂ xx u + u∂ y u − ∂ t u = f. Using a harmonic analysis technique on Lie groups, we prove that, if the solution u satisfies condition ∂ x u = 0 in an open set Ω ⊂ R 3 and f ∈ C ∞ (Ω), then u ∈ C ∞ (Ω).
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… (More)
A geometric model for segmentation of images with missing boundaries is presented. Some classical problems of boundary completion in cognitive images, like the pop up of subjective contours in the famous triangle of Kanizsa, are faced from a surface evolution point of view. The method is based on the mean curvature evolution of a graph with respect to the… (More)
In this paper, we propose to model the edge information contained in natural scenes as points in the 3D space of positions and orientations. This space is equipped with a strong geometrical structure and it is identified as the rototranslation group. In this space, we compute a histogram of co-occurrence of edges from a database of natural images and show… (More)
In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: L = m i=1 X 2 i + ∆, in R n where ∆ is the Laplace operator, m < n, and the limit operator L = m i=1 X 2 i is hypoelliptic. It is well known that L admits a fundamental solution Γ. Here we establish some a priori estimates uniform in of it, using a… (More)
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… (More)
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… (More)