Investigation of the multiscale statistics of natural and biomedical images

Thomas Pumir de Louvigny

Master Project

ENS, Ecole normale supérieure de Cachan, France

July 2013

Abstract

The innovation models provide a detailed framework for the study of sparse stochastic processes. The main strength of this model is that the images are not seen as a Gaussian process. Some predictions can be made concerning the statistical behavior of the wavelet coefficients. In a first part, we study those predictions. First of all, a power law of the wavelet coefficients is a widely observed fact, that we are able to explain using the innovation model. We investigate this fact (concerning the choice of the wavelet mainly). Secondly, a theorem shows a Gaussian behavior of the wavelet coefficients when the scale is increasing to the infinity. We try to illustrate this fact numerically. In a second part we investigate the M-terms approximation of stochastic processes, with an emphasis on Brownian motion. We talk of M-terms approximation when we keep only M coefficients in a transform domain. Most of the methods in M-terms approximations are either linear or non linear. We try to keep the non linear point of view. We investigate the law of the retained coefficients and look for a bound on the average approximation error.