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- Giovanna Citti, Alessandro Sarti
- Journal of Mathematical Imaging and Vision
- 2005

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)

We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance: ∂xxu+ u∂yu− ∂tu = f. Using a harmonic analysis technique on Lie groups, we prove that, if the solution u satisfies condition ∂xu 6= 0 in an open set Ω ⊂ R and f ∈ C∞(Ω), then u ∈ C∞(Ω).

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 study the notion of regular surface in the Carnot–Carathéodory spaces and we prove the implicit function theorem in this setting. We fix at every point of R a subset of the tangent space, called horizontal tangent plane, and we assume that it has a basis X1, . . . , Xm of C∞ vector fields satisfying the Hörmander condition of… (More)

- Giovanna Citti, Maria Manfredini, +4 authors Alessandro Sarti
- 2010

Abstract. The functionality of the visual cortex has been described in [63] and in [50] as a contact manifold of dimension three and in [62] 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,… (More)

– 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)

- Davide Barbieri, Giovanna Citti, Gonzalo Sanguinetti, Alessandro Sarti
- Journal of physiology, Paris
- 2012

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)

- Davide Barbieri, Giovanna Citti, Giacomo Cocci, Alessandro Sarti
- Journal of Mathematical Imaging and Vision
- 2013

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… (More)

- Alessandro Sarti, Giovanna Citti, Jean Petitot
- Biological Cybernetics
- 2007

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)

- Gonzalo Sanguinetti, Giovanna Citti, Alessandro Sarti
- VISAPP
- 2008