  Gravitational lensing distorts the cosmic microwave background (CMB)
anisotropies and imprints a characteristic pattern onto it. The distortions
depend on the projected matter density between today and redshift $z \sim
1100$. In this paper we develop a method for a direct reconstruction of the
projected matter density from the CMB anisotropies. This reconstruction is
obtained by averaging over quadratic combinations of the derivatives of CMB
field. We test the method using simulations and show that it can successfully
recover projected density profile of a cluster of galaxies if there are
measurable anisotropies on scales smaller than the characteristic cluster size.
In the absence of sufficient small scale power the reconstructed maps have low
signal to noise on individual structures, but can give a positive detection of
the power spectrum or when cross correlated with other maps of large scale
structure. We develop an analytic method to reconstruct the power spectrum
including the effects of noise and beam smoothing. Tests with Monte Carlo
simulations show that we can recover the input power spectrum both on large and
small scales, provided that we use maps with sufficiently low noise and high
angular resolution.
