# Multifractal Analysis: From Theory to Applications and Back (14w5045)

Arriving in Banff, Alberta Sunday, February 23 and departing Friday February 28, 2014

## Organizers

(CNRS, ENS de Lyon)

(Université Paris est Créteil)

(University of North Carolina)

## Objectives

Aim and timeliness. \quad In view of the context described above, the proposed workshop thus intends to gather a selected audience of experts in fractals and multifractals from three different fields of expertise:
• functional analysis and geometric measure theory,
• probabilistic modeling, statistics and stochastic processes,
• signal and image processing, and applications at large.
This workshop aims at:
• fostering cross-fertilization between communities, easing transfers
• both from theoretical results to real applications and from relevant applied questions to theoretical formalization,
• elaborating the formalization and precise statements of
• open and crucial issues in multifractal analysis,
• being the starting point of new collaborations between
• researchers with different cultures and backgrounds in order to go towards the resolution of these issues.
Workshops gathering researchers from these three communities are rare. Too often, researchers meet within their own community. In contrast, the relatively small size of the meeting and the unique environment will favor the break of cultural and vocabulary walls between different communities and will therefore provide a great opportunity to make these new collaborations possible. The workshop will take place at a tipping point in the developments of multifractal analysis. New applications of multifractals are emerging, ranging from the detection of fake paintings to the analysis of brain signals. New fundamental results are discovered in probability theory, for example, in connection with harmonic measure, quasiconformal mappings and branching processes. New analysis tools (e.g. variants and extensions of wavelet leaders) are introduced on the functional analysis side. Enough critical mass has accumulated across the three communities for multifractal analysis to grow into a mature theory.Though data exhibiting multifractal features are abundant and have been analyzed for years, the understanding of probabilistic models for such data, statistical inference techniques and relations to fundamental mathematical properties through functional analysis, however, have been lacking at a number of levels. We intend to address exactly this weakness. Therefore, this workshop should impact several areas at both theoretical and practical levels.\\Specific objectives. \quad From a theoretical standpoint, this workshop will address questions of interest in probability theory and statistics, functional analysis and geometric measure theory. Regarding applications, the workshop should be of interest and use to practitioners working with multifractals in numerous and diverse applied areas such as physics and chemistry, earth and environmental sciences, image processing, medicine and physiology, economics and finance. To initiate scientific interactions and discussions, the organizers have identified a series of questions that can serve as a list of focused goals and issues that can be addressed during the workshop. \\At the analysis level, the so-called multifractal formalism provides a potentially powerful tool to measure the multifractal spectrum from real-world data. Yet, a significant number of theoretical questions about its practical use remain open. What appropriate multiresolution quantities, function increments, wavelet coefficients, or more complicated quantities derived from wavelet coefficients, should be used? What function spaces do they correspond to? The multifractal formalism relies on a Legendre transform and thus on a concavity assumption. What can be done for data that do not fulfill such an assumption? As such, the multifractal formalism does not bring any additional information related to the nature of the singularities that exist in data. How should the formalism be modified to enable the detection of the local regularity of potentially oscillating nature (chirp-type), against the simpler case of non-oscillating (cusp-type) singularities? In several dimensions, how can anisotropy phenomena be analyzed? \\At the modeling level, practitioners are often facing situations where the outcome of the analysis does not resemble the output of standard theoretical models. It would thus certainly be of interest to design a wide collection of deterministic and stochastic models, whose multifractal properties would be well-understood and could serve as a frame and guideline in the analysis of real-world data. For instance, developing models which incorporate oscillating singularities, or anisotropy, or which display non-concave or non-smooth multifractal spectra would be of great interest. \\At the probabilistic and statistic inference levels, the gap between the deterministic definition of the multifractal spectrum and its application to each given sample path of stochastic processes remain to be formalized in a precise manner. Because real-world data are naturally envisaged as realizations of random models by practitioners, bridging that gap would pave the way towards crucial practical issues such as parameter estimation or hypothesis testing performance evaluation: What are the confidence intervals for given estimates? How can one test formally what model better fits data? Such issues could be addressed on a number of stochastic models designed to serve as reference and benchmark to compare applications to. Along the same lines, the (multi)fractal properties of data are often expected by practitioners to serve as a tool permitting to finely quantify or measure intricate statistical properties of data, such as strong dependencies or subtle departures of data from joint Gaussianity. Such questions can also serve as fruitful discussion launchers. \\At the application level, in many situations, because of the explosion of sensor designs and deployment, the data to analyze no longer consist of univariate signals (or functions). They could either take values in a $d$-dimensional space (a collection of time series is recorded jointly from one same system) or be indexed by a multiparameter (a field). Notably, there has been a growing interest to incorporate fractal and multifractal analysis into image processing toolboxes, with diverse applications ranging from medicine to satellite imagery. This change in the nature of data to be analyzed raised theoretical and practical issues that have so far been barely addressed: How should the multivariate multifractal spectrum be defined? How does the notion of anisotropy that naturally comes with image take its place in the multifractal analysis? How could one distinguish between an anisotropic but regular texture that has been superimposed to an isotropic fractal texture from a texture where the anisotropy is truly built-in its (multi-)fractal properties? Images can exhibit boundaries (that split them into homogeneous subregions) that may themselves have fractal properties. Can they be distinguished from the fractal properties of the textures? Images may consist of the juxtaposition of fractal and regular patches of textures. How can they be identified?\\Finally, we plan to invite about 10 advanced graduate or post-doctoral students from different communities, working on or interested in the themes of the workshop. \\