Geodesic
A scale-space approach with wavelets to singularity estimation
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 143-164

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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.

DOI : 10.1051/ps:2005007
Classification : 62G05, 62G08, 65Dxx
Keywords: Lipschitz singularity, continuous wavelet transform, scale-space representation, zero-crossings, wavelet maxima, feature extraction, non parametric estimation, bagging, landmark-based matching 
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     author = {Bigot, J\'er\'emie},
     title = {A scale-space approach with wavelets to singularity estimation},
     journal = {ESAIM: Probability and Statistics},
     pages = {143--164},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2005},
     doi = {10.1051/ps:2005007},
     mrnumber = {2148964},
     zbl = {1136.62030},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2005007/}
}
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Bigot, Jérémie. A scale-space approach with wavelets to singularity estimation. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 143-164. doi: 10.1051/ps:2005007

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