Mathematical Tools for Multifractal Signal Processing

Large classes of signals exhibit a very irregular behavior. In the most complicated situations this irregular behavior may follow different regimes, and can switch from one regime to another almost instantaneously. This is obviously the case for recordings of speech signals; precise recordings of turbulence data (which became available at the beginning of the 80’s) show that turbulence also falls in this category. Such signals cannot be modeled by standard stationary increments processes, such as Fractional Brownian Motion (or related Gaussian processes) for instance. The techniques of multifractal signal analysis have been specifically designed to analyze such behavior. Initially developed in the mid 80’s in the context of turbulence analysis, they were applied successfully to a large range of signals, including traffic data (cars and internet), stock market prices, speech signals, texture analysis, etc. We give an overview of the mathematical tools that were developed for that purpose, and we present some of the most successful applications. We start by introducing some simple mathematical tools that will be useful to model the above notions. First, what is meant by pointwise regularity? It is a way to quantify, by using a positive real number , the fact that the graph of a function has a certain smoothness at a point x0. The lowest possible level of regularity is continuity: A function F is continuous at x0 if jF (x) F (x0)j ! 0 as x! x0; continuity corresponds to a regularity index = 0. Similarly, F is differentiable if there exists a linear function P such that jF (x) P (x x0)j ! 0 faster than jx x0j as x! x0; this corresponds to a regularity index = 1. The following definition is a direct generalization of these two particular cases. Let be a positive real number and x0 2 IR; a function F : IR ! IR is C (x0) if there exists a polynomial P of degree less than such that jF (x) P (x x0)j Cjx x0j : (1)