A scale-space approach with wavelets to singularity estimation

Jérémie Bigot (1)

1 LSP - Laboratoire de Statistique et Probabilités

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.

Mots clés: Lipschitz singularity; Continuous wavelet transform; Scale-space representation; Zero-crossings; Wavelet maxima; Feature extraction; Non parametric estimation; Bagging; Landmark-based matching

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

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