Volume 9, June 2005
|Page(s)||143 - 164|
|Published online||15 November 2005|
A scale-space approach with wavelets to singularity estimation
Laboratoire de Statistique et Probabilités,
Université Paul Sabatier,
Toulouse, France; firstname.lastname@example.org
Revised: 5 July 2004
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.
Mathematics Subject Classification: 62G05 / 62G08 / 65Dxx
Key words: Lipschitz singularity / continuous wavelet transform / scale-space representation / zero-crossings / wavelet maxima / feature extraction / non parametric estimation / bagging / landmark-based matching.
© EDP Sciences, SMAI, 2005
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