Digital Deblurring of CMB Maps:
Performance and Efficient Implementation

R. Vio, J. Nagy, L. Tenorio, P. Andreani, C. Baccigalupi and W. Wamsteker

Digital deblurring of images is an important problem that arises in multifrequency observations of the Cosmic Microwave Background (CMB) where, because of the width of the point spread functions (PSF), maps of different frequencies suffer a different loss of spatial resolution. Deblurring is useful for various reasons: first, it helps restore some of the signal's high frequencies lost through the smoothing effect of the instrument's PSF; second, emissions at various frequencies observed with different resolutions can be better studied on a comparable resolution; third, map-based component separation algorithms require input maps with similar level of degradation. Here we consider the performance of Tikhonov deblurring of noisy maps with applications to {\it PLANCK}. Compared to deblurring in frequency space, Tikhonov deblurring in real space allows the efficient implementation of algorithms in a wide range of applications that include deblurring with spatially dependent PSF, and deblurring with white or correlated noise. We discuss the use of the Tikhonov methodology in practical applications and provide details for an efficient implementation of the algorithms. Its performance is tested on simulated CMB maps; results indicate the possibility of a substantial gain in angular resolution. Matlab code is made available.