Provided the correct de-noising procedure is chosen, wavelet based de-noising methods give better results than classical Fourier filtering. Due to the additional spatial resolution and the frequency-adapted width of the wavelets, they are able to preserve sharp features of a signal while at the same time strongly suppressing noise within the low frequency parts. Our tests demonstrate, however, that the choice of wavelet basis and de-noising technique is crucial. We find that, at least for the considered astronomical spectra, hard thresholding using well-known orthogonal wavelet bases, such as the various Daubechies wavelets, give results that are generally not better than the best Fourier filter as far as their ability to recover the higher SNR features (spectral lines) is concerned. Donoho's wavelet shrinkage technique (based on orthogonal wavelets) and thresholding or Wiener-filtering techniques using non-orthogonal wavelets (specifically linear or cubic spline wavelets combined with the so-called à trous algorithm) as proposed by Starck & Bijaoui (1994) work better and in general are more successful than Fourier techniques in recovering the original signal. If an efficient, quick and simple-to-use de-noising algorithm is required then we recommend the techniques developed by Starck & Bijaoui. The technique of Bury et al. (1996) gives excellent results as well, but is not suited to de-noise large sets of measurements at once since each spectrum has to be treated separately. Their method of modelling the noise requires that the signal is divided into different parts, each of which contains only features of a similar size.
The most exact results, however, are achieved by our wavelet-packets method, namely three-level decomposition coupled with simple hard thresholding. At least for solar spectra it separates noise from the uncorrupted signal better than any other technique we have tested. In particular, it turns out to be superior to other techniques in recovering weak signals hidden in the noise. The main disadvantage of our technique is that it is slow (although not significantly slower than iterative techniques such as the adaptive filtering methods proposed by Starck & Bijaoui 1994). We recommend it for cases in which the highest possible accuracy is desired.
In this initial investigation we have applied the multi-level decomposition technique in a rather crude manner. Other wavelet bases and truncation schemes may lead to better results. Corresponding extensions of our technique will be presented in a following paper.
Acknowledgements
We thank Benedikt Oswald for many enlightening discussions and for making his Daubechies wavelet filtering routines available to us, David Rees for drawing the work of Donoho and co-workers to our attention and the referee Dr. Bijaoui for strongly stressing the work of his group.