Image denoising by statistical area thresholding

Area openings and closings are morphological filters which efficiently suppress impulse noise from an image, by removing small connected components of level sets. The problem of an objective choice of threshold for the area remains open. Here, a mathematical model for random images will be considered. Under this model, a Poisson approximation for the probability of appearance of any local pattern can be computed. In particular, the probability of observing a component with size larger than $k$ in pure impulse noise has an explicit form. This permits the definition of a statistical test on the significance of connected components, thus providing an explicit formula for the area threshold of the denoising filter, as a function of the impulse noise probability parameter. Finally, using threshold decomposition, a denoising algorithm for grey level images is proposed

Potentially semi-stable deformation rings for representations with values in PGLnPGL_n

We study the potentially semi-stable deformation rings for Galois representations taking their values in $PGL_n$, by comparing them to the deformation rings for $GL_n$. As an application, we state an analogue of the Breuil-M\'ezard conjecture, and we show that the case of $PGL_n$ follows from the case of $GL_n$.Comment: minor update ; 28 page

Un calcul d'anneaux de déformations potentiellement Barsotti--Tate

57 pages, en françaisInternational audienceLet F be an unramified extension of Qp. The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformations rings with tame Galois type of irreducible two-dimensional representations of the absolute Galois group of F. We then apply our method in the particular case where F has degree 2 over Q_p and determine this way almost all these deformations rings. In this particular case, we observe a close relationship between the structure of these deformations rings and the geometry of the associated Kisin variety. As a corollary and still assuming that F has degree 2 over Q_p, we prove, except in two very particular cases, a conjecture of Kisin which predicts that intrinsic Galois multiplicities are all equal to 0 or 1