Notiziario Scientifico
Settimana dal 3 al 9 febbraio 2014
Lunedì 3 febbraio 2014
Martedì 4 febbraio 2014
Martedì 4 febbraio 2014
Martedì 4 febbraio 2014
Mercoledì 5 febbraio 2014
Tutte le informazioni relative a questo notiziario devono pervenire
all'indirizzo di posta elettronica
seminari@mat.uniroma1.it
entro le ore 9 del venerdì precedente la settimana di pubblicazione.
Ore 14:30, Aula di Consiglio
Seminario di Analisi Matematica
We present some recent results concerning the homogenization of uniformly elliptic equations in
nondivergence form. The equations are assumed to have coefficients which are independent at unit
distance. We give optimal results on the order of the error in homogenization in every dimension,
measured in L^infty and Holder spaces up to C^[1,alpha], alpha < 1. As a corollary, we obtain the
existence of stationary correctors exist in dimensions five and higher (and their nonexistence, in
general, in dimensions four and smaller). Finally, we give regularity results which state that a
generic equation has essentially the same regularity as Laplace's equation, up to C^[1,1].
Ore 14:30, Aula 211, Università di Roma III
Seminario di Probabilità
We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G
are either open or closed and refresh their status at rate mu while at the same time a random walker
moves on G at rate 1 but only along edges which are open. On the d-dimensional torus with side
length n, we prove that in the subcritical regime, the mixing times is of order n^2/mu. We also
obtain results concerning mean squared displacement and hitting times.This is a joint work with
Yuval Peres and Jeff Steif.
Ore 15:00, Aula di Consiglio
Seminario di Modellistica differenziale numerica
Shape from Shading and Photometric Stereo are two fundamental problems in Computer Vision aimed at
reconstructing surface depth given either a single image taken under a known light source or
multiple images taken under different illuminations, respectively. Whereas the former utilizes
partial differential equation (PDE) techniques to solve the image irradiance equation, the latter
can be expressed as a linear system of equations in surface derivatives when 3 or more images are
given. It therefore seems that current Photometric Stereo techniques do not extract all possible
depth information from each image by itself. We use PDE techniques for the solution of the Shape
from Photometric Stereo problem when only 2 images are available. Extending our previous results on
this problem, we consider the more realistic perspective projection of surfaces during the
photographic process. Under these assumptions, there is a unique weak (Lipschitz continuous)
solution to the problem at hand, solving the well known convex/concave ambiguity of the
Shape-from-Shading problem. We propose two approximation schemes for the numerical solution of this
problem, an Up-Wind finite difference scheme and a semi-Lagrangian scheme, and analyze their
properties. We show that both schemes converge linearly and accurately reconstruct the original
surfaces. Our results thus show that using methodologies common in the field of Shape-from-Shading
it is possible to recover more depth information for the Photometric Stereo problem under the more
realistic perspective projection assumption. Even an extension to more than 2 images will be
presented. Starting from the 2 images basic model we generalise the linearisation process when
several pictures are taken into account with the advantages to have a fast and direct method of
surface reconstruction in height resolution even in presence of shadows.
Ore 16:00, Aula di Consiglio
Seminario di Modellistica differenziale numerica
The Shape from Shading problem is a well known ill-posed problem. Several contributions have
addressed the case of Lambertian surfaces improving the model with the introduction of perspective
deformations or studying the corresponding photometric stereo problem. In our study we focus the
attention on a different improvement which is intended to reduce the assumptions on the properties
of the surface dealing with more general (and real) non-Lambertian surfaces. Our goal is to find a
unique model which should be flexible enough to handle many different kinds of real images. As a
starting point for this rather big project, we consider the basic model of a single nonlinear
partial differential equation (PDE) where we need to introduce new terms to tackle the general
non-Lambertian case. In particular, in this talk we will consider the non-Lambertian diffusive
Oren-Nayar reflectance model and the specular Phong model, we will construct semi-Lagrangian
approximation schemes for the corresponding nonlinear PDEs and we will compare their performances
with the classical Lambertian model in terms of some error indicators on a series of benchmarks
images.
Ore 16:00, Aula di Consiglio
Seminario di Fisica Matematica
We study the Maxwell's equations with dissipative boundary conditions. The solutions of the mixed
problems are given by a contraction semigroup V(t)f = e^[tG_b]f, t > = 0. If f is an eigenfunction
of the generator G_b with eigenvalues lambda, Re lambda < 0, the corresponding solution is called
asymptotically disappearing (ADS). If we have (ADS), the wave operators are not complete and the
inverse back-scattering problems become very complicated. Thus the existence of (ADS) is important
for scattering. First we prove that the spectrum of the generator G_b in the open half plane Re
lambda < 0 is formed only by isolated eigenvalues with finite multiplicity. Second we establish the
existence of (ADS) for the Maxwell system in the exterior of a sphere. Finally, we show that the
(ADS) are stable under small perturbations of the boundary conditions and the boundary. The above
results are generalized for symmetric first order systems with constant coefficients whose principal
symbol has constant rank. This is a joint work with F. Colombini and J. Rauch.
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