Talk:Wavelet-based multifractal analysis

I am happy to be given the opportunity of expressing my deep admiration for the seminal work achieved by Alain Arenodo and his group. Arneodo wrote a new chapter in signal or image processing which is now named « multifractal signal processing ». Empirical evidence of multifractal behaviour in signal processing in velocity fields of fully developed turbulent lays its roots in the founding papers of Kolmogorov and Obukov in 1962. In 1985 Uriel Frisch and Giorgio Parisi defined the « spectrum of singularities » of an arbitrary signal. This « spectrum of singularities » is a function of one real variable which has a meaning for any signal. In fluid dynamics the « spectrum of singularities » measures the intermittent behavior of turbulent flows. Frisch and Parisi speculated that the « spectrum of singularities » is always the Legendre transform of a « structure function ». This detour is needed since the « spectrum of singularities » which is based on Hölder exponents is unstable and cannot be computed in real time. On the contrary, the « structure function » is a stable functional.

Alain Arneodo and his group reshaped this program and built better algorithms for computing the « spectrum of singularities ». Ordinary increments were used by Frisch and Parisi to define the « structure function ». Arneodo used wavelet coefficients instead. This permits to compute the « spectrum of singularities » when the signal under investigation is corrupted by a white noise, a situation where the increments do not make any sense (a reference is Arnaud Gloter and Marc Hoffmann).

With their remarkable algorithm Arneodo and his group treated a variety of applications and obtained spectacular results. They unveiled some deep structural properties of the signals under study. One of the main issues raised by their method has been solved by S. Jaffard. Jaffard proved that in some instances the WTMM algorithm yields a function which is completely unrelated to the spectrum of singularites. But Jaffard also proved that the WTMM algorithm gives the right answer for a generic class of signals. Being optimistic, there are no reasons to believe that mother Nature is deceiving and I am backing the WTMM algorithm.

Yves Meyer CMLA ENS-Cachan 94235 Cachan Cedex ymeyer@cmla.ens-cachan.fr

This article gives a very nice overview of the work done on continuous wavelet analysis of multifractal functions and measures, with a short account of some selected (and spectacular) applications. This is of course not surprising since the authors are the main experts in the field and which was, so to say, invented by A. Arneodo and his collaborators. The WTMM method, described in this contribution, has very significantly evolved since its first version, and applies now to a variety of situations, and had yielded deep results in various domains of physics and biology. As such, it qualifies as a very generic approach, to be used for exploratory signal analysis whenever complex multiscale behavior is expected. This contribution provides a very nice introduction to this domain.

Bruno Torrésani