Wavelet Based Multifractal Formalism: Applications to DNA Sequences, Satellite Images of the Cloud Structure, and Stock Market Data

We elaborate on a unified thermodynamic description of multifractal distributions including measures and functions. This new approach relies on the computation of partition functions from the wavelet transform skeleton defined by the wavelet transform modulus maxima (WTMM). This skeleton provides an adaptive space-scale partition of the fractal distribution under study, from which one can extract the D(h) singularity spectrum as the equivalent of a thermodynamic function. With some appropriate choice of the analyzing wavelet, we show that the WTMM method provides a natural generalization of the classical box-counting and structure function techniques. We then extend this method to multifractal image analysis, with the specific goal to characterize statistically the roughness fluctuations of fractal surfaces. As a very promising perspective, we demonstrate that one can go even deeper in the multifractal analysis by studying correlation functions in both space and scales. Actually, in the arborescent structure of the WT skeleton is somehow uncoded the multiphcative cascade process that underhes the multifractal properties of the considered deterministic or random function. To illustrate our purpose, we report on the most significant results obtained when applying our concepts and methodology to three experimental situations, namely the statistical analysis of DNA sequences, of high resolution satellite images of the cloud structure, and of stock market data.