Issue ESAIM: PS Volume 9, 2005 Page(s) 143 - 164 DOI 10.1051/ps:2005007

ESAIM: P&S, April 2005, Vol. 9, pp. 143-164
DOI: 10.1051/ps:2005007

A scale-space approach with wavelets to singularity estimation

Jérémie Bigot

Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse, France; jbigot@cict.fr

(Received January 9, 2003. Revised July 5, 2004.)

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.

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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