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Wavelet thresholding for some classes of non–Gaussian noise
Author(s) -
Antoniadis A.,
Leporini D.,
Pesquet J.–C.
Publication year - 2002
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/1467-9574.00211
Subject(s) - wavelet , mathematics , thresholding , estimator , minimax , pattern recognition (psychology) , smoothness , stationary wavelet transform , bayesian probability , discrete wavelet transform , prior probability , algorithm , cascade algorithm , wavelet transform , artificial intelligence , computer science , mathematical optimization , statistics , mathematical analysis , image (mathematics)
Wavelet shrinkage and thresholding methods constitute a powerful way to carry out signal denoising, especially when the underlying signal has a sparse wavelet representation. They are computationally fast, and automatically adapt to the smoothness of the signal to be estimated. Nearly minimax properties for simple threshold estimators over a large class of function spaces and for a wide range of loss functions were established in a series of papers by Donoho and Johnstone. The notion behind these wavelet methods is that the unknown function is well approximated by a function with a relatively small proportion of nonzero wavelet coefficients. In this paper, we propose a framework in which this notion of sparseness can be naturally expressed by a Bayesian model for the wavelet coefficients of the underlying signal. Our Bayesian formulation is grounded on the empirical observation that the wavelet coefficients can be summarized adequately by exponential power prior distributions and allows us to establish close connections between wavelet thresholding techniques and Maximum A Posteriori estimation for two classes of noise distributions including heavy–tailed noises. We prove that a great variety of thresholding rules are derived from these MAP criteria. Simulation examples are presented to substantiate the proposed approach.