Time-scale analyses and self-similar stochastic processes

Abstract

A number of different physical situations (e.g.,“1 /f noise,” turbulence, texture analysis,… ) give rise to fractal or fractal-like signals, modeled as samples of self-similar processes. This motivates the development of specific methods for characterizing self-similarity structures in signals and for efficiently estimating the corresponding scaling laws. The recently introduced techniques of time-scale analysis (wavelet transforms and bilinear generalizations) offer such a possibility, especially in the case of locally self-similar processes, i.e., those for which scaling laws are time-dependent. In this respect, basics of linear and bilinear time-scale theories will be reviewed in connection with more classical methods. Starting from idealizations such as white noise, Poisson process, or (fractional) Brownian motion and variations thereof, we will show what advantages can be gained from time-scale approaches, either for obtaining almost Karhunen-Loeve (doubly orthogonal) representations via orthogonal wavelet bases, or for defining general classes of estimators (aimed at scaling exponents) via bilinear time-scale representations which generalize the usual Wigner-Ville distribution.