Speech Title: Wavelet Techniques in Multifractal Analysis: Recent Advances and New Challenges
Abstract:
Multifractal analysis yields insightful parameters based on scaling invariance properties which are used for classification and model selection. Among the various methods available, those based on the expansion of the data on orthonormal wavelet bases now represent the state of the art in image and signal processing for the following reasons:
- they yield a large panel of variants that can be tuned to the global regularity of the data, depending if they are modelled by continuous functions, Lp functions, measures or Schwartz distributions;
- they are backed by mathematical results which yield access to the fractional dimensions of the singularity sets of the data.
We will give an overview of these techniques, and show applications in several domains. We will also expose the remaining deadlocks, which are met for sparse data or when analyzing a collections of correlated signals.
Biography:
Stéphane Jaffard is a professor of mathematics at LAMA (Laboratory of Analysis and Applied Mathematics) at Université Paris Est Créteil (UPEC). He graduated from École Polytechnique, where he obtained a PhD under the supervision of Yves Meyer. He spent a postdoc at the Institute for Advanced Study. In 1995, he became a professor at Université Paris Est Créteil. From 2000 to 2005, he was a junior member of the Institut de France. He was the President of the French Mathematical Society from 2007 to 2010. In 2022, he coordinated the project "Assises des Mathématiques", which aimed at promoting the importance of mathematics in science, industry, and society.
His main works concern wavelet and multifractal analysis, with applications in signal and image processing. In 1991, together with Ingrid Daubechies and Jean-Lin Journé, he built Wilson bases, the existence of which had been conjectured by Nobel Prize-winning physicist Kenneth Wilson. Their exceptional properties for the temporal and frequency analysis of signals play a key role in the signal processing algorithm that led to the discovery of gravitational waves in 2015.
He determined the wavelet characterization of pointwise Hölder regularity and applied it to the analysis of large classes of functions and stochastic processes. Recently, with Bruno Martin, he determined the pointwise regularity of Jean-Christophe Yoccoz's Brjuno function, which plays a central role in the theory of holomorphic dynamical systems.
He introduced the wavelet leader method in multifractal analysis, whose purpose is to estimate the size of the sets of points where a function or a signal displays a given pointwise regularity exponent. Together with Patrice Abry, Herwig Wendt, and their collaborators, he applied this technique to the analysis and classification of e.g. turbulence data, finance, physiological signals or the analysis of Vincent van Gogh paintings. In 2021, he was awarded the Jacques-Louis-Lions prize by the French Academy of Sciences. In 2023, he held the Aisentstadt Chair (CRM, Université de Montréal). In 2024, he held the Francqui Chair at VUB (Vrije Universiteit Brussel).