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Analyse 2-microlocale et application au débruitage

Abstract : In order to analyse signals with variable regularity, it is necessary to use tools that characterize their regularity at one given point.

The 2-microlocal frontier supplies a very general viewpoint on these tools. To each point of the definition domain of a function is associated a curve that characterizes in a very complete way its regularity at this point, and that contains in particular the information given by the regularity exponents. We recall in a first part the original definition and the properties of the 2-microlocal frontier.

In the second part, we give a "time-domain" form of the 2-microlocal frontier that allows to consider it as a natural generalization of Hölder exponents. We show its properties directly from that expression. We also introduce a "time-domain 2-microlocal spectrum" that makes its calculation easier and treat a few examples.

In the third part, we clarify the parallelism between 2-microlocal spectrum and multifractal spectrum and prove in the same general frame the 2-microlocal formalism and the weak multifractal formalism. Moreover, we show in this common frame new results that concern their prescription.

In the fourth part, we apply 2-microlocal analysis to signal denoising. We suggest a denoising method that allows to recover the initial global regularity of the signal and a denoising method allows to recover the 2-microlocal spectrum at one point. The methods are illustrated by numerical applications on classical functions.

Finally, we show in the last part an applied work that was done in collaboration with the ENILIA Surgères: fractal tools characterizing global regularity are applied to the classification of microscopical butter images.
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Contributor : Antoine Echelard <>
Submitted on : Wednesday, May 28, 2008 - 9:24:37 PM
Last modification on : Wednesday, December 19, 2018 - 3:02:03 PM
Long-term archiving on: : Friday, May 28, 2010 - 6:36:49 PM


  • HAL Id : tel-00283008, version 1


Antoine Echelard. Analyse 2-microlocale et application au débruitage. Traitement du signal et de l'image [eess.SP]. Ecole Centrale de Nantes (ECN); Université de Nantes, 2007. Français. ⟨tel-00283008⟩



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