Stein Unbiased Estimate of Risk (Sure) to denoise signals

Question

I wish to use Stein Unbiased Estimate of Risk (Sure) for denoising signals.

I have a 1-Dimensional signal. I am using wavelets to decompose the signal into multiple levels of approximate and detail coefficients.

For denoising the original signal, do I need to do a thresholding for every level of detail coefficients or doing it on the last level of detail coeffcient will do the job ?

Answer 1

Thresholding is usually applied to all the frequencies of a signal because the procedure exploits the fact that the wavelet transform maps white noise (purely random, uncorrelated and constant power spectral density noise) in the signal domain to white noise in the transform domain and as such is spread across the different frequencies Thus, while signal energy becomes more concentrated into fewer coefficients in the transform domain, noise energy does not. Other noises given that have different spectrum properties will map differently and this is where the selection of the type of thresholding procedure becomes important.

In thresholding the highest decomposition level (lowest frequencies) while leaving the lower levels (higher frequencies) not denoised sounds a little extrange if you want to reconstruct the signal. However you could also extract a level and denoise its related range of frequencies (e.g. from level 1 to level 2) if you have a range of frequencies you may have interest for.

Speaking about the thresholding function be aware in any case that Sure has different results depending on the type of noises the signal has. For example it will reduce the distribution of white noise in horizontal components but will only decrease large amplitudes. For signals where togueter with white you may have other noise colors like random walk and flicker noise sure is not an efective procedure.