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A scale-space approach with wavelets to singularity estimation

Published online by Cambridge University Press:  15 November 2005

Jérémie Bigot
Jérémie Bigot*
Affiliation:
Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse, France; jbigot@cict.fr
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Abstract

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, “the structural intensity”, that computes the “density” of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed.

Keywords

Lipschitz singularitycontinuous wavelet transformscale-space representationzero-crossingswavelet maximafeature extractionnon parametric estimationbagginglandmark-based matching.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 143 - 164
Copyright
© EDP Sciences, SMAI, 2005

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