In the work of Donoho and collaborators at Stanford [.Donoho 1993, Donoho 1994.], denoising is simply signal extraction from data, but done via wavelets. This is quite different from traditional filtering approaches -- it is nonlinear, due to a thresholding step. Denoising by soft thresholding (given data = signal + noise), involves the following steps ([.Donoho 1994.]): Step 1: perform a suitable wavelet transform of the noisy data; (the wavelet basis may be chosen based on various factors including computational burden, and ability to compress the L2 energy of the signal into a very few, very large coefficients); Step 2: perform a soft thresholding of the wavelet coefficients where the threshold depends on the noise variance; (when the wavelet bases are chosen as in step 1, thresholding kills the effect of the noise without killing the effect of the signal); Step 3: the coefficients obtained from step 2 are then padded with zeros to produce a legitimate wavelet transform and this is inverted to obtain the signal estimate. Various examples show how the noise is largely suppressed while features in the original signal remain sharp after denoising by the above approach (in contrast with traditional linear methods of smoothing which trade-off noise suppression against a broadening of signal features). Donoho and Johnstone give rigorous justification of denoising via soft thresholding [.Donoho 1994.], for restricted types of signal + noise models.
We propose to extend wavelet-based denoising to signals with nontrivial statistical correlations over time, and where possible, determine the soft thresholds adaptively, in response to varying noise characteristics (see also [. Krim 1995 .]).