4 Conclusions

We have considered the 2D wavelet transform with two scales to analyse CMB maps. First of all we present the continuous approach to the 2D wavelet. A discrete analysis is performed for a finite image. In this case, a scaling function is introduced in order to define a 2D wavelet basis.

The 2D wavelet technique has been applied to denoise CMB maps for S/N ratios ranging between 0.7 and 3. We have also considered the case of non-uniform Gaussian noise. A factor between 5 and 3 is gained for the reconstructed images/original signal map in relation to the signal plus noise maps/original signal map. Regarding the $C_{\ell}'s$, the relative errors are below a $20\%$ up to $\ell = 1700$ for all the cases with S/N $\geq 1$. A comparison with Wiener Filter and Maximum Entropy Method has also been performed. The later gives comparable reconstructions to the wavelet method, being however slower and more complicated to apply. Wiener Filter provides reconstructed maps with errors comparable to the wavelet technique we propose. However, unlike the wavelet method, Wiener filter requires previous knowledge of the signal power spectrum. Moreover, the $C_{\ell}'s$ of the denoised map obtaining by applying Wiener filter are clearly underestimated.

Finally, we would like to remark that linear wavelet denoising, which preserves Gaussianity, gives reconstruction

errors similar to those obtained with the non-linear soft thresholding techniques. Moreover, linear denoising is better achieved by 2D wavelets than by the multiresolution ones discussed in Sanz et al. ([1999]).

Acknowledgements

This work has been supported by the DGESIC Project No. PB95-1132-C02-02, CICYT Acción Especial No. ESP98-1545-E and Comisión Conjunta Hispano-Norteamericana de Cooperación Científica y Tecnológica with Ref. 98138. RBB has been supported by a Spanish MEC fellowship. JLS acknowledges partial financial support from a Sabbatical Grant by the Spanish DGESIC. JLS, RBB and LC thank the hospitality of the Center for Particle Astrophysics and the Astronomy Department of the University of California at Berkeley. LT acknowledges partial financial support from the "Agenzia Spaziale Italiana'' (ASI) and Consiglio Nazionale delle Ricerche (CNR).