  Reconstructing the matter density field from galaxy counts is a problem
frequently addressed in current literature. Two main sources of error are shot
noise from galaxy counts and insufficient knowledge of the correct galaxy
position caused by peculiar velocities and redshift measurement uncertainty.
Here we address the reconstruction problem of a Poissonian sampled log-normal
density field with velocity distortions in a Bayesian way via a maximum a
posteriory method. We test our algorithm on a 1D toy case and find significant
improvement compared to simple data inversion. In particular, we address the
following problems: photometric redshifts, mapping of extended sources in coded
mask systems, real space reconstruction from redshift space galaxy distribution
and combined analysis of data with different point spread functions.
