Bridging the gap between theory and applications : A long term program. Many of the key advances in multifractal analysis were obtained through interactions either of mathematicians working in different subjects (functional analysis, harmonic analysis, geometric measure theory, stochastic processes, statistics), or of scientists from different fields (mathematicians, physicists, specialists in signal and image processing). We strongly believe that cross-disciplinarity and dialectic interactions between theory and applications remain a key for substantial scientific progresses and steps forwards ; the AMATIS project is constructed in that spirit. Thus, it intends to cover the full range from theoretical advances to practical and implementation of statistical tools and their inclusion in user-friendly softwares, disseminated via a dedicated WEB platform.
Our researches focus on four main projects :
Understanding the range of validity of the different formulations of the multifractal formalism has been a central goal of mathematicians working in multifractal analysis, since the mid-eighties. This motivation has widened, since the range of variants has been considerably increased recently. The main focus used to be on constructing formalisms adapted to the Hölder exponent of functions or measures ; this scope was strongly widened with the irruption of p-exponents, general regularity exponents, and oscillation exponents.
Multifractal analysis of stochastic processes fieds, and random measures, has always been a key issue and a central theme in our domain (such as multiplicative cascades and their generalizations, Lévy processes, and Lévy fields, Markov processes,…). Although a number of questions have already been addressed, opening the way to several developments, many challenging issues remain to be studied.
Though the concepts, such as scaling, underlying it are commonly used in applications, mul- tifractal analysis remains often used only partially compared to the potential it offers and is also sometimes subject to misunderstanding and misuse in the transfer of its advanced based theore- tical results to their applied transpositions. This is often the case because, while the theoretical construction of multifractal analysis lie in a deterministic framework supplied by functional analy- sis, whereas its practical use for the characterization and/or simulation of real word data (signals or images stemming from actual applications) must naturally be considered within a ”statistical” or ”stochastic processes” perspective. These two different parts of multifractal analysis are often studied independently and only rarely can results obtained from functional analysis be directly translated in a stochastic framework and used practically. In this context, one aims at promoting the development of a statistical multifractal signal and image processing framework. It is organized around three classical issues in statistical analysis : Estimation, Test, Classification.
Multifractal analysis is largely used in applications. Yet, in most cases, practitioners resort to codes that they have redeveloped themselves or that they found on the web. Often, these codes are poorly documented or consist of ad hoc development that suit a specific task or set of data but do not aim at being applicable in a general or generic framework. Multifractal analysis theory per se is intricate. Therefore its practical implementation requires that a number of pre- and post-processing steps be validated for the practical outputs to yield valid theoretical conclusions. These theoretical and practical knowledges are rarely put together at work : Either the outputs of practical algo- rithms yield misleading conclusions with respect to the actual multifractal properties of the data (because practitioners do not have a rigorous enough knowledge of multifractal theory) or theore- tical developments are not considered in applications (because theorists do not develop codes that can actually and properly be used in applications). In this context, the central goal of this task is to create, develop and maintain a website/platform dedicated to multifractal analysis.