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The statistics of the temperature anisotropies in the primordial Cosmic Microwave Background radiation field
provide a wealth of information for cosmology and the estimation of cosmological parameters. An even more
acute inference should stem from the study of maps of the polarization state of the CMB radiation. Measuring
the latter extremely weak CMB polarization signal requires very sensitive instruments. The full-sky maps of
both temperature and polarization anisotropies of the CMB to be delivered by the upcoming Planck Surveyor
satellite experiment are hence awaited with excitement. Still, analyzing CMB data requires tackling a number
of practical difficulties, notably that several other astrophysical sources emit radiation in the frequency range
of CMB observations. Separating the different astrophysical foreground components and the CMB proper from
available multichannel data is a problem that has drawn much attention in the community. Nevertheless, some
level of residual contributions, most significantly in the galactic region and at the locations of strong radio
point sources will unavoidably contaminate the estimated spherical CMB map. Masking out these regions is
common practice but the gaps in the data need proper handling. In order to restore the stationarity of a partly
incomplete CMB map and thus lower the impact of the gaps on non-local statistical tests, we developed an
inpainting algorithm on the sphere to fill in the gaps, based on an iterative thresholding scheme in a sparse
representation of the data. This algorithm relies on the variety of recently developed transforms on the sphere
among which several multiscale transforms which we will review. We also contribute to enlarging the set of
available transforms for polarized data on the sphere. We describe new multiscale decompositions namely the
isotropic undecimated wavelet and curvelet transforms for polarized data on the sphere. The proposed transforms
are invertible and so allow for applications in image restoration and denoising.
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