Wavelet thresholding techniques for power spectrum estimation

Abstract

Estimation of the power spectrum S( f ) of a stationary random process can be viewed as a nonparametric statistical estimation problem. We introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown S( f ). This approach offers the ability to capture statistically significant components of lnS( f ) at different resolution levels and guarantees nonnegativity of the spectrum estimator. The spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non-Gaussian noise. We propose a wavelet thresholding technique to solve this problem under specified noise/resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables. The thresholds are computed using a saddle-point approximation to the distribution of the noise coefficients.

Wavelet thresholding via MDL for natural images

“…To provide one further benchmark for our MDL results, we also implemented the simple twostage MDL thresholds suggested by Moulin 22,23 and Saito 32 and tested them on the Lena and Baboon images. Saito's procedure produced spectacular decreases in the numbers of kept coe cients sometimes throwing away up to 80 more than lMDL , with a considerable rise in MSE and a corresponding degradation in the visual quality of the reconstruction.…”

“…Several applications of two-stage MDL to wavelet thresholding for images have appeared previously in the literature. Moulin 22,23 and Saito 32 , for example, derive thresholds similar in form to that employed by VisuShrink; while Moulin and Liu 24 employ Rissanen's universal prior on integers to construct an MDL criterion that has met with success in denoising natural images.…”

“…In general, wavelet thresholding can be thought of as a special case of statistical model selection where we h a ve a s m a n y orthogonal predictor variables corresponding to wavelet basis elements as there are data points 22,23,32,7,8 . Viewing the problem in this way is attractive because it allows one to separate the kill" action setting coe cients to zero from the keep" action estimating the remaining coe cients via shrinkage or some other procedure .…”

Compound Explosives Detection and Component Analysis via Terahertz Time-Domain Spectroscopy

“…is sufficiently smooth and N large, the periodogram and the power spectrum satisfy the following [26]:…”

“…The wavelet power spectrum estimation technique (WPSET) is a wavelet smoothing method of the log-scale periodogram of the obtained signal [18,26], enabling effective spectral oscillation maintenance in the THz range.…”