Multi-resolution internal template cleaning: An application to the
  Wilkinson Microwave Anisotropy Probe 7-yr polarization data
  Cosmic microwave background (CMB) radiation data obtained by different
experiments contain, besides the desired signal, a superposition of microwave
sky contributions. We present a fast and robust method, using a wavelet
decomposition on the sphere, to recover the CMB signal from microwave maps. An
application to \textit{WMAP} polarization data is presented, showing its good
performance particularly in very polluted regions of the sky. The applied
wavelet has the advantages of requiring little computational time in its
calculations, being adapted to the \textit{HEALPix} pixelization scheme, and
offering the possibility of multi-resolution analysis. The decomposition is
implemented as part of a fully internal template fitting method, minimizing the
variance of the resulting map at each scale. Using a $\chi^2$ characterization
of the noise, we find that the residuals of the cleaned maps are compatible
with those expected from the instrumental noise. The maps are also comparable
to those obtained from the \textit{WMAP} team, but in our case we do not make
use of external data sets. In addition, at low resolution, our cleaned maps
present a lower level of noise. The E-mode power spectrum $C_{\ell}^{EE}$ is
computed at high and low resolution; and a cross power spectrum $C_{\ell}^{TE}$
is also calculated from the foreground reduced maps of temperature given by
\textit{WMAP} and our cleaned maps of polarization at high resolution. These
spectra are consistent with the power spectra supplied by the \textit{WMAP}
team. We detect the E-mode acoustic peak at $\ell \sim 400$, as predicted by
the standard $\Lambda CDM$ model. The B-mode power spectrum $C_{\ell}^{BB}$ is
compatible with zero.
