Phd in MATHEMATICAL MODELS FOR ENGINEERING, ELECTROMAGNETICS AND
NANOSCIENCES

Thesis title: Differential Entropy Based Methods for Thresholding in Wavelet Bases and Other Applications

In this thesis the problem of automatically selecting expansion coefficients of a function f(t) in a wavelet basis expansion is considered. The problem is approached through an information-theoretic point of view and three different differential entropy based measures are proposed to compare expansion coefficients in order to find the optimal subset. Several theoretical results about differential entropy of a function are developed in order to provide a solid theoretical framework to the proposed measures. A numerical scheme to compute the differential entropy of a function is presented and its stability and convergence properties are proved. Numerical experiments concerning different wavelet bases and frames are presented, and the behaviour of the proposed measures is compared to state of the art methods. Moreover, numerical experiments highlight a connection between the proposed measures and the well-known information measure Normalized Compression Distance (NCD). In addition, Fourier basis is considered, and an application to Fourier shape descriptors is developed. As a last contribute, the problem of locating time-frequency interferences in multicomponent signals is considered. A method for time-domain location of modes interferences is presented relying on a filtered energy signal. The optimal amount of filtering is automatically detected in a rate/distortion-like curve by means of the proposed information measures.