New Methods for Signal Analysis: Multifractal Formalisms based on Profiles. From Theory to Practice.

The multifractal formalisms allow to numerically approximate the Hölder spectrum of a real-life signal f. In this thesis, we study some multifractal formalisms based on profiles: these are functions allowing to study the histograms of coefficients obtained by the wavelet transform of f. The profile-based formalisms studied here are the Wavelet Profile Method (WPM) and the Leaders Profile Method (LPM). An advantage of these methods, compared to those that use a Legendre transform on a structure function, as for example the Wavelet Leaders Method (WLM), lies in the fact that they are able to approximate non-concave spectra.

The contributions on a practical level are directly related to the profile-based methods. An efficient algorithm for these methods is proposed. It is applied and compared with the WLM on classical examples, as the fractional Brownian motion, the Lévy process and the Mandelbrot cascades, as well as on processes with prescribed pointwise regularity. A new method to distinguish the mono- and multifractality of a signal is also proposed. We apply these methods on a practical example: the Mars' topography. We show that the simultaneous use of the WLM and the LPM allows to obtain additional information on the nature of signals. We also show that it is possible to detect major surface features of Mars in the spatial distribution of the Hölder exponents.

The contributions on a theoretical level can be divided into two parts. For the first one, let us recall that the amplitudes of the wavelet leaders of a signal have not necessarily the same asymptotic pointwise Hölderian behaviour: a logarithmic correction can appear. We prove that the three well-known Hölderian behaviours of the Brownian motion (known as ordinary, rapid and slow) are also present in the behaviour of the amplitudes of its wavelet leaders. The proof provides a new way to study fine properties of stochastic processes. We also construct a multifractal process that has a local regularity similar to that of the Brownian motion. The second part of these contributions is a generalisation of the Snu spaces with the help of the admissible sequences. The topological properties holding for the usual Snu spaces are preserved. The robustness of these new spaces is also presented, which implies the independence of the chosen wavelet basis. Finally a link with the generalised Besov spaces is given.