Document 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 BigotLaboratoire 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