Wavelet deconvolution in a periodic setting
| Autor: | Gerard Kerkyacharian, Iain M. Johnstone, Dominique Picard, Marc Raimondo |
|---|---|
| Přispěvatelé: | Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Benassù, Serena |
| Jazyk: | angličtina |
| Rok vydání: | 2004 |
| Předmět: |
Statistics and Probability
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] Fast Fourier transform 020206 networking & telecommunications 02 engineering and technology 01 natural sciences Convolution [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] 010104 statistics & probability symbols.namesake Fourier transform Wavelet Rate of convergence Fourier analysis 0202 electrical engineering electronic engineering information engineering symbols Calculus Deconvolution 0101 mathematics Statistics Probability and Uncertainty Algorithm Meyer wavelet Mathematics |
| Zdroj: | Journal of the Royal Statistical Society: Series B Journal of the Royal Statistical Society: Series B, 2004, 66 n.3, pp.547-573 Journal of the Royal Statistical Society: Series B, Royal Statistical Society, 2004, 66 n.3, pp.547-573 |
| ISSN: | 1369-7412 1467-9868 |
| Popis: | In this paper, we present an inverse estimation procedure which combines Fourier analysis with wavelet expansion. In the periodic setting, our method can recover a blurred function observed in white noise. The blurring process is achieved through a convolution operator which can either be smooth (polynomial decay of the Fourier transform) or irregular (such as the convolution with a box-car). The proposal is non-linear and does not require any prior knowledge of the smoothness class; it enjoys fast computation and is spatially adaptive. This contrasts with more traditional ltering methods which demand a certain amount of regularisation and often fail to recover non-homogeneous functions. A ne tuning of our method is derived via asymptotic minimax theory which reveals some key dierences with the direct case of Donoho et al. (1995): (a) band-limited wavelet families have nice theoretical and computing features; (b) the high frequency cut o depends on the spectral characteristics of the convolution kernel; (c) thresholds are level dependent in a geometric fashion. We tested our method using simulated lidar data for underwater remote sensing. Both visual and numerical results show an improvement over existing methods. Finally, the theory behind our estimation paradigm gives a complete characterisation of the ’Maxiset’ of the method i.e. the set of functions where the method attains a near-optimal rate of convergence for a variety of L p loss functions. |
| Databáze: | OpenAIRE |
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