This paper is a contribution to the analysis of the statistical correlation of the wavelet packet coefficients resulting from the decomposition of a random process, stationary in the wide-sense, whose power spectral density is bounded with support in [−π, π].Consider two quadrature mirror filters (QMF) that depend on a parameter r, such that these filters tend almost everywhere to the Shannon QMF when r increases. The parameter r is called the order of the QMF under consideration. The order of the Daubechies filters (resp. the Battle-Lemarié filters) is the number of vanishing moments of the wavelet function (resp. the spline order of the scaling function).Given any decomposition path in the wavelet packet tree, the wavelet packet coefficients are proved to decorrelate for every packet associated with a large enough resolution level, provided that the QMF order is large enough and above a value that depends on this wavelet packet.Another consequence of our derivation is that, when the coefficients associated with a given wavelet packet are approximately decorrelated, the value of the autocorrelation function of these coefficients at lag 0 is close to the value taken by the power spectral density of the decomposed process at a specific point. This specific point depends on the path followed in the wavelet packet tree to attain the wavelet packet under consideration.Some simulations highlight the good quality of the "whitening" effect that can be obtained in practical cases.

This paper presents a new sigmoid-based WaveShrink function. The shrinkage obtained via this function is particularly suitable to reduce noise without impacting significantly the statistical properties of the signal to be recovered. The proposed WaveShrink function depends on a parameter that makes it possible to control the attenuation degree imposed to the data, and thus, allows for a flexible shrinkage.

International audienceA new threshold is presented for better estimating a signal by sparse transform and soft thresholding. This threshold derives from a non-parametric statistical approach dedicated to the detection of a signal with unknown distribution and unknown probability of presence in independent and additive white Gaussian noise. This threshold is called the detection threshold and is particularly appropriate for selecting the few observations, provided by the sparse transform, whose amplitudes are sufficiently large to consider that they contain information about the signal. An upper bound for the risk of the soft thresholding estimation is computed when the detection threshold is used. For a wide class of signals, it is shown that, when the number of observations is large, this upper bound is from about twice to four times smaller than the standard upper bounds given for the universal and the minimax thresholds. Many real-world signals belong to this class, as illustrated by several experimental results

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