Signal extraction and power spectrum estimation using wavelet transform scale space filtering and Bayes shrinkage

This paper introduces two novel methods for signal extraction and denoising through the use of the wavelet transform scale space filtering and Bayes shrinkage. In the first method, the noisy signal is decomposed into multiple scales by the dyadic wavelet transform. The scale space filtering algorithm then extracts the original signal modulus maxima by using the properties of the signal and noise modulus maxima across scales. Finally, a 'denoised' signal is reconstructed by the alternate projection algorithm. This denoising method can reduce noise to a high degree while preserving most of the important features of the signal such as edges and other singularities. In the second method, we employ a hierarchical Gaussian mixture model for the wavelet coefficients at different scales and we obtain the best signal estimate through the Bayesian posterior analysis technique. In an application example, we implement the algorithm to obtain a smooth and high-resolution power spectral density estimate from the signal's periodogram.