Astronomical observations are often photon starved. Consequently many astronomical spectra have a poor signal-to-noise ratio (SNR) and are significantly corrupted by (white Gaussian) photon noise. This is even true for solar observations, since high resolution measurements of polarized light soon run out of photons (e.g., Stix 1991). The reduction of this noise is highly desirable for a number of reasons (cf. the papers in the volume edited by Cayrel De Strobel & Spite 1988, for examples of the merits of high SNR spectra).
Fourier smoothing has long been the method of choice to suppress noise (Brault & White 1971), but recently methods based on the wavelet transformation have become increasingly popular (Starck & Bijaoui 1994; Starck & Murtagh 1994; Murtagh et al. 1995). In principle they offer much greater flexibility for analyzing and processing data. The main advantage of wavelets lies in the additional "spatial'' resolution of the transformed signal. In contrast to the Fourier transformation, the signal is decomposed into waves of finite length, i.e. into waves which are spatially localized - hence the name wavelets. The wavelet transform of a one-dimensional signal has two independent variables - a frequency and a spatial location variable. It leads to a decomposition of, say, a spectrum into a series of spectra at finer and coarser resolutions. Indeed, there is a close mathematical relationship between the wavelet transformation and the multi-resolution analysis of a signal (Mallat 1989). Hence the wavelet transform furnishes us with the frequency spectrum of a signal at every spatial location. This feature, besides others, opens new and fruitful ways of processing and analyzing data of various kinds.
In particular, it allows the smoothing of astronomical spectra, typically composed of a continuum with interspersed spectral lines, to be optimized. Since high-frequency signals are present only at the wavelengths of the spectral lines, only at these wavelengths need they be kept in the de-noised spectrum. At the remaining positions, i.e. in the continuum, only the lowest frequencies are due to the source itself. Whereas Fourier filtering affects all data points in the same manner, wavelets allow different parts of spectra to be filtered individually, in principle promising a considerably refined and improved treatment.
In the present paper, we compare different wavelet smoothing methods to each other and to Fourier smoothing for the specific case of astronomical spectra. We also present a wavelet-packets based smoothing scheme which we find to be superior in recovering the true signal from a combination of signal and noise, at least for the cases we have considered.