Multifractal analysis for signal and image classification
Stéphane Jaffard, UPEC

Seminar • 23 March 2017

Abstract
Multifractal analysis was introduced at the end of the 1980s by physicists whose purpose was to relate global regularity indices associated to a signal (velocity of turbulent fluids) with the distribution of pointwise singularities present in the data. Several variants were later proposed, including methods based on local suprema of the continuous wavelet transform, or Detrended Fluctuation Analysis (DFA). We will focus on methods based on wavelet coefficients, using an orthonormal wavelet basis. We will see how the tools supplied by multifractal analysis can be adapted to particular types of data: e.g. the use of p-leaders (local L^p norms of wavelet coefficients) vs. leaders (local suprema of wavelet coefficients) depending on the global regularity of the data, or anisotropic wavelet transforms for the analysis of anisotropic textures. We will also see and how to adapt the analysis when the data do not present a global self-similarity. These ideas will be illustrated by a wide range of examples, such as : (in 1D) turbulence, internet traffic, heart-beat intervals, literary texts, and (in 2D) natural images, paintings and photographic papers. As regards literary texts and paintings, we will see that parameters originating from multifractal tools can give rise to new methods in textometry and stylometry.