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Local regularity analysis, some applications to multifractal analysis

Analyse de régularité locale, quelques applications à l'analyse multifractale

Abstract : In many scientific domains (signal and image processing, fully developed turbulence), it is important both from a theoretical and a practical viewpoint to be able to detect and to characterize the
singularities of an object. This object can either be a function, a
distribution, a measure or a process.

The first part of my work is focused on the characterization of the
local regularity of a function. One often measures the local regularity of a function $f$ around a point $x_0$ by computing the
(\em pointwise \ho exponent) of $f$ at $x_0$, denoted by $\alp(x_0)$. Unfortunately this exponent does not describe completely the behavior of $f$ around $x_0$.

The local \ho exponent of $f$ at $x_0$, denoted by $\all(x_0)$,
brings new informations on this behavior. I studied the functions
$x\mapsto \all(x)$ and $x\mapsto \alp(x)$, and their mutual
relationships: they must coincide on an uncountable dense set, but
they can differ everywhere except on a set of Hausdorff dimension
0. This shows that these exponents are somehow complementary. The
proof of these results involves wavelet coefficients.


The 2-microlocal spaces introduced by J.M. Bony, denoted by $\css'$, allow us to generalize the notion of regularity exponents for a distribution.I first found a new spatial characterization of these spaces, valid for the functions $f\in C^\ep$ ($\ep>0$). This characterization appears to be useful in signal processing, since it is very accessible from a numerical point of view.

Using 2-microlocal spaces $\css'$, one can associate with every point $x_0$ not only a single exponent, but a whole curve in $\R^2$. This curve $\Gamma$, called 2-microlocal frontier, possesses remarkable properties: it is convex, with a derivative always smaller than -1, and all the usual regularity exponents can be recovered from the knowledge of $\Gamma$. It thus yields a geometrical description of the local regularity of a distribution at a point. With J. Lévy Véhel, I proved that the 2-microlocal frontier of a distribution $f$ at $x$ is the Legendre transform of a function called 2-microlocal spectrum (denoted by $\chi_(x_0)$): this relationship is called 2-microlocal formalism, by analogy with the multifractal formalism. $\chi_(x_0)$ is related to the value of the wavelet coefficients of $f$ around $x_0$. The study of this function $\chi_(x_0)$ and of the 2-microlocal formalism brings in fact a new and unifying point of view on the local regularity analysis of continuous functions, and is fruitful: for instance, we investigated the relations with the usual exponents, we discovered new properties and new compatibility conditions between 2-microlocal frontiers. The explicit computation of $\chi_(x_0)$ is performed for several famous functions: a ``chirp'', a ``cusp'', the Weierstrass Function, the ``non-differentiable'' Riemann function.

Another part of my work is devoted to multifractal analysis of
measures and functions.
With J. Barral, I constructed multifractal functions and processes as follows. Let $\mu$ be a positive Borel measure, and let $s_0$ and $p_0$ be two positive real numbers. We define the functions $F_\mu$ by $$F_\mu(x)=\sum_(j\geq 0) \sum_(k\in \mathbb(Z)) \pm
2^(-j(s_0-\frac(1)(p_0))) |\mu\big ([k2^(-j),(k+1)2^(-j))\big
)|^(\frac(1)(p_0)) \psijk(x).$$ We proved that if $\mu$ satisfies some multifractal formalism (for measures), then $F_\mu$ also
satisfies a multifractal formalism (for functions). This result
applies to classical families of measures: Quasi-Bernoulli, Random
Cascades, Mandelbrot Cascades, ... This theorem also brings an answer to a conjecture of Arnéodo, Bacry, Muzy on the value of the multifractal spectrum of ``Random Wavelet Cascades'', which serve as a model for a turbulent fluid.

Finally I investigated, for a function $f$, in the relationships between verification of the multifractal formalism and presence of fast oscillations. This work has an unexpected consequence: I showed that a threshold (comparable to the hard threshold) applied on the wavelet coefficients can create oscillations and can make the multifractal formalism fail.