The Multifractal Formalism Revisited With Wavelets

The multifractal formalism originally introduced for singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that the f(α) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal functions (the wavelets actually play the role of “generalized boxes”). We report on a systematic comparison between this alternative method and the structure function approach which is commonly used in the context of fully developed turbulence. We comment on the intrinsic limitations of the structure functions which possess fundamental drawbacks and do not provide a full characterization of the singularities of a signal in many cases. We show that our method based on the wavelet transform modulus maxima decomposition works in most situations and is likely to be the ground of a unified multifractal description of singular distributions. Our theoretical considerations are both illustrated on pedagogical examples, e.g., generalized devil staircases and fractional Brownian motions, and supported by numerical simulations. Recent applications of the wavelet transform modulus maxima method to experimental turbulent velocity signals at inertial range scales are compared to previous measurements based on the structure function approach. A similar analysis is carried out for the locally averaged dissipation and the validity of the Kolmogorov’s refined similarity hypothesis is discussed. To conclude, we elaborate on a wavelet based technique which goes further than a simple statistical characterization of the scaling properties of fractal objects and provides a very promising tool for solving the inverse fractal problem, i.e., for uncovering their construction rule in terms of a discrete dynamical system.