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A scale-space approach with wavelets to singularity estimation

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.
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Jérémie Bigot. A scale-space approach with wavelets to singularity estimation. ESAIM: Probability and Statistics, EDP Sciences, 2005, vol. 9, pp. 143-164. ⟨10.1051/ps:2005007⟩. ⟨hal-00837732⟩

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