The concept of pointwise regularity allows to measure how the smoothnesse of
a function changes from point to point (one estimates how close it is to its Taylor expansion). For example Weierstrass functions, or Brownian motion have the same Hölder
regularity exponent at every point (which is 1/2 in the case of Brownian motion); they are referred to as ``monohölder functions''. In the opposite, some functions have very intricate changes of regularity. Hardy in 1916 gave evidence that it is probably
the case of ``Riemann's nondifferentiable function'', and this was partially confirmed by Duistermaat in 1991: These results allowed to conjecture that the Hölder regularity of this function at a point depends on the Diophantine approximation properties of this point. I performed the complete determination of the pointwise Hölder exponent of Riemann's function in 1996 and confirmed this conjecture. This result required a wavelet characterization of pointwise regularity which I had obtained previously, and which has been applied since then in a wide variety of situations.
It turns out that the Hölder exponent of such functions is extremely erratic and, in particular, is discontinuous everywhere. One might expect such complicated behaviors to be exceptional; on the opposite, the same phenomenon occurs for large classes of random processes: For example, I showed
that it is the case for most Lévy processes. Note that this had already been shown for random mesures that were used in turbulence models (multiplicative cascades), following the pioneering works of Mandelbrot, Kahane, Peyrière, Barral, ... Such properties also turn out to be generic in a wide variety of settings: e.g. in the sense of Baire categories, or of prevalence, ``most functions'' in a given (smooth enough) Sobolev space also display an everywhere discontinuous Hölder exponent (the prevalent case is a joint work with Aurelia Fraysse).
The use of the Hölder exponent requires to make the assumption that the function analyzed is locally bounded. This is a strong limitation, both in theory (e.g. one may wish to understand in which sense the genericity results mentioned above extend to the setting of Sobolev spaces which are not imbedded in the space of continuous functions) and in applications, since large collections of signals cannot be modeled by locally bounded functions (for example, it is the case for internet data, heart-beat intervals, or many natural images). In such cases, a good substitute for Hölder regularity is the p-exponent, which was introduced by Calderón and Zygmund in 1961 in order ot obtain optimal pointwise regularity results for solutions of PDEs. Clothilde Melot and myself investigated the mathematical properties of this exponent and provided a wavelet characterization on which is based the corresponding formulation of the multifractal formalism (see below). In a recent joint work with Bruno Martin, I studied the Brjuno function (a nowhere locally bounded function introduced by Jean-Christophe Yoccoz, and which plays a key-role in the theory of holomorphic dynamical systems) and we determined its p-exponent at every point. In this case too, the p-exponent is an extremely erratic function which depends on the Diophantine approximation properties of the point considered.
The origin of multifractal analysis can be traced back to the seminal works of Kolmogorov on turbulence in the 1940s: He introduced a new quantity for classification purposes: In short, one computes the expectation of increments of the data raised to a power p, and performs a log-log plot regression in the limit of small scales. The limit exponent thus obtained (which is a function of p) is called the Kolmogorov scaling function of the signal. Kolmogorov introduced it for model validation purposes, and, indeed, it first allowed to dismiss Fractional Brownian Motion as a model of turbulence at small scale; later, it allowed to validate some cascade models. One can note that models with a constant Hölder exponent yield a linear scaling function (fractional Brownian motions, Weierstrass functions, etc.).
However, the birth of the subject really took place in 1985: In a fundamental short paper, Uriel Frisch and Giorgio Parisi interpreted the nonlinearity of the scaling function as a signature of the presence of different Hölder exponents in the signal. And, more precisely, they proposed a formula, the multifractal formalism, which relates the scaling function and the fractional dimensions of the sets of points where the Hölder exponent takes a given value (this function of the exponent H is refered to as the ``multifractal spectrum'' of the data). The relationship they proposed is simply given by a Legendre transform. This seminal work had huge consequences.
On the mathematical side, the idea of estimating the Haudorff dimension of the sets of points with a given regularity exponent supplies the right framework to analyze the mathematical functions mentioned above (and many more): Though their Hölder exponents are extremely complicated, I showed that their multifractal spectrums are simple and remarkable functions in many cases: linear or affine functions of H for the Riemann function, Lévy processes, the Brjuno function (joint work with Bruno Martin), selfsimilar functions, or generic functions in Sobolev spaces, either from the Baire or prevalence settings (the prevalent case is a joint work with Aurelia Fraysse),... On other hand, many mathematicians worked on measures given by multiplicative cascades and showed that they lead to bell-shaped multifractal spectrums (see the articles of Kahane, Peyrière, Barral, Seuret, Bacry, Muzy and their collaborators and students).
Understanding the validity of the multifractal formalism has been the source of a large amount of research, both on the theoretical and applied sides. A first question is that, in order to obtain the whole multifractal spectrum from the Frisch-Parisi formula, one has to extend the range of ps for which the scaling function is defined to negative values. If this is done without precaution, this leads to absurd results (models as simple as Brownian motion would be classified as multifractal). One therefore requires a ``renormalization''
of the (usually divergent) integral that defines the scaling function for p <0. At the end of the 1980s, a first key step in this direction was taken by Alain Arneodo, Emmanuel Bacry and Jean-François Muzy; they introduced wavelet methods, and, more precisely, they used the concept of ``Wavelet Transform Maxima Method'', introduced by Stéphane Mallat: One replaces increments of the function by local maxima of the continuous wavelet transform and one restricts the computation of the scaling function to these local maxima; this allows to get rid of divergences due to very small values of increments or wavelet transform. This approach led to remarkable numerical results in many applications. However the heuristic on which this procedure is constructed (log-log plots of the decay rate along lines of maxima of the continuous wavelet transform yields the Hölder exponent) fails for some types of singularities (such as chirps). This limitation prevents the obtention of general mathematical results backing this procedure.
This drawback led me to introduce a variant which is based on local suprema of discrete wavelet coefficients computed on an orthonormal wavelet basis. This method retains the key idea of renormalizing wavelet coefficients for negative ps by only retaining the locally largest values (called ``wavelet leaders'')
but it also offers the additional advantage of being backed by several mathematical results: First log-log plots of the decay rate along lines of maxima of wavelet leaders do yield the Hölder exponent; this property is a key requirement in the derivation of the multifractal formalism. An important consequence is that the Legendre transform of the scaling function (built on wavelet leaders) always yields an upper bound for the multifractal spectrum, so that the multifractal formalism is at least ``half-true''. Additionnally, equality holds for many models (random wavelet series, cascade based models, etc). Furthermore, the wavelet leader scaling function is invariant by smooth perturbations and changes of variable, which is a key-requirement for its use in image processing.
The wavelet leader technique and beyond
From 2005 on, I developed a long range collaboration with Patrice Abry, Bruno Lashermes and Herwig Wendt with the purpose of systematically investigating the possibilities offered by the wavelet leaders based multifractal formalism. On the applied side, we have shown that the wavelet leader scaling function provides a new classification tool which proved crucial in many settings; e.g. it allows to determine which cascade models can model fully developed turbulence (e.g. the scaling function of log-Poisson models is not compatible with experimental data, whereas for log-normal models, it is); it also allowed to draw differences between the textures of different periods of Van Gogh's paintings, or between genuine Van Goghs and fakes; the upper bound supplied by the multifractal formalism (when based on wavelet leaders) allow to conclude that, if the scaling function is linear for all (positive and negative) values of p, then the data are monoHölder (i.e. the Hölder exponent is constant), which yields an extremely precise information concerning possible models; we showed that it is the case for massively aggregated internet data.
Multifractal analysis presents two different sides:
In signal and image processing, its purpose is to construct, from experimental data, global quantities that can be computed at each scale; regressions on log-log plots allow to deduce scaling exponents, which are then used for signal classification, model selection, or parameter fitting. In 1941, A. Kolmogorov introduced the ``scaling function'' of a signal, which was the first example of such a family of exponents, in the context of fully developed turbulence. In the mid-eighties, A. Arneodo and his collaborators showed that wavelets (families of functions that deduce from each other by translations and dilations, see [15,34]) supply a priviledged tool to derive such exponents and build alternative ones. In [7,16,17] it is shown that a natural quantity to use in the construction of scaling functions is supplied by wavelet leaders (the ``building blocks'' of the scaling functions are local suprema of wavelet coefficients computed on an orthonormal wavelet basis): the reason is that the wavelet criterium of pointwise regularity of [5,6] can be naturally expressed in terms of wavelet leaders, so that such scaling functions would naturally yield information on the Hölder singularities of the signal. Since the beginning of the 2000s, I developed a collaboration with P. Abry (ENS Lyon) and his group, the purpose of which is to study and use the method of wavelet leaders. This method offers several advantages: simplicity (especially in dimensions larger than 1), efficiency, and it is backed by a collection of mathematical results which yield either theorems valid in full generality, but also exact results for a many natural models in signal and image processing [16,17]. Such examples are particularly useful in order to check the accuracy of a numerical method; a spectacular application is the refutation of turbulence models . Recently, in collaboration with H. Wendt (Purdue University), we showed that the use of statistical nonparametric methods (bootstrap)  yields sharp tools in image processing [36,38], and in particular for automatic classification of large collections of images , or help for fake detection among paintings (``Van Gogh Challenge''). With P. Abry and S. Roux, we are now studying more general ``grand-canonical'' scaling functions, which supply a richer classification tool, and yield informations on the presence of very strong local oscillations in the data (``oscillating singularities'') ; such detections are an important issue in several fields of applications; for instance, some turbulence models predict the existence of such behaviors, so that their validation depends on settling the issue of the presence of oscillating singularities in turbulence data.
In 1985, U. Frisch and G. Parisi proposed a relationship between the scaling function and the fractal dimensions of the pointwise singularities of the signal. It was the first avatar of the ``multifractal formalism'', which, since then, has found many variants, and whose domain of validity is the subject of intensive studies. In mathematics, the determination of the spectrum of singularties of functions or measures (i.e. the Hausdorff dimensions of the sets of points where the Hölder exponent takes a given value), is a topic which, since the mid-eighties, has known a growing interest, both in deterministic and random settings. I have been particularly interested in this quest, and I showed the multifractal nature of self-similar processes , of the ``nowhere differentiable Riemann function'' , of large clases of Davenport series [22,36], of most Lévy processes , and of several models of random wavelet series [23,24]. A special topic of interest now is to understand specific features (such as directional regularity) that show up for functions of several variables (and, in the probabilistic setting, for random fields). Note that multifractality has been put into light in many other settings, such as measures obtained through multiplicative constructions (turbulence models), fragmentation processes (J. Bertoin and his group),... One of my top interests has been to show that, far from being exceptional, such behaviors are ``generic''; a first result was that, in a given Sobolev space ``quasi-every'' function (in the sense of Baire categories) is multifractal . Since then several additional results (in particular generic results in the sense supplied by prevalence) obtained in collaboration with A. Fraysse, confirmed these first ones [4,25]. Recently the study of some classes of Markov processes (in collaboration with J. Barral, N. Fournier and S. Seuret ) led to the introduction of ``local multifractal analysis'' which is fitted to non-stationary cases, and more generally to situations where the spectrum can evolve with time.
With Clothilde Melot, we obtained wavelet characterizations of the p-exponent
This explains why, in signal processing, one usually does not try to determine pointwise regularity exponents, but more ``global'' quantities, such as the spectrum of singularities of the data, which measures the sizes of the sets where the regularity exponent of a function (or a measure) takes a given value (see below for a more precise definition).
The numerical determination of these spectra requires a priori information: wavelet characterizations of the pointwise exponent. In the case of the Hölder exponent, a first characterization of this exponent was obtained in [5,6]. However, it turned out that a reformulation in terms of local quantities based on wavelet coefficents (``wavelet leaders'') was actually the required ingredient, as shown in . Such a reformulation was obtained in  for the Hölder exponent, and in [8,9,10] for the p-exponent (introduced by Calderon and Zygmund) and their extensions (note that such extensison have to be used as soon as singularités of negative exponent are present in the data). Adding an ``oscillation exponents'' allows to characterize in a much more precise way the local behavior of a function, see [11,12,13,14].
I am now particularly interested in the following topics:
-study of directional regularity (see [27,28] for a first directional wavelet type criterium)
-in collaboration with A. Durand, the multifractal analysis of multivariate random fields (Lévy fields ) and deterministic functions.
-in collaboration with P. Abry et S. Roux, we study the extensions of multifractal analysis when an additional oscillation exponent is taken into account. First encouraging results tend to confirm the absence of oscillating singularities in experimental turbulence data, see [14,32].