Pitfall in Multifractal Analysis of Negative Regularity

Roberto Leonarduzzi (1) , Herwig Wendt (2, 3, 4) , Stéphane Jaffard (5) , Patrice Abry (1)

1 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 IRIT-SC - Signal et Communications
4 CNRS - Centre National de la Recherche Scientifique
5 IRIT - Institut de recherche en informatique de Toulouse

The recent introduction of p-exponents and p-leaders extends the application of wavelet leader multifractal analysis to functions or signals with negative regularity. These new quantities are defined only for functions that are locally in Lp. However, in practice, estimations from discrete data can always be computed, even if the underlying function that models the data is not in Lp. In this case, the analysis is meaningless but indistinguishable from a valid one. In this contribution, we use a very simple function model provided by deterministic wavelet cascades to study the behavior of multifractal estimates when they are computed from discrete data that is not modeled by a function in Lp, and show that the result is a spectrum with correct shape but shifted so as to exactly be in the Lp limit. We also use numerical simulations on various multifractal random processes to show that the validity of our results extends beyond the simple model that we used.

Mots clés: Multifractal analysis

Domaines: Synthèse d'image et réalité virtuelle [cs.GR]; Traitement du signal et de l'image [eess.SP]; Traitement des images [eess.IV]; Vision par ordinateur et reconnaissance de formes [cs.CV]; Intelligence artificielle [cs.AI]

Roberto Leonarduzzi, Herwig Wendt, Stéphane Jaffard, Patrice Abry. Pitfall in Multifractal Analysis of Negative Regularity. 25eme Colloque Groupe de Recherche et d'Etudes du Traitement du Signal et des Images (GRETSI 2015), Sep 2015, Lyon, France. pp. 1-4. ⟨hal-01511908⟩