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.
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 study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hörmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity solutions are indeed classical solutions. 2001… (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)
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)
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)
In this paper we consider long time behavior of a mean curvature flow of non parametric surface in R n , with respect to a conformal Riemannian metric. We impose zero boundary value, and we prove that the solution tends to 0 exponentially fast as t → ∞. Its normalization u sup u tends to the first eigenfunction of the associated linearized problem.