  We consider the reconstruction of a two-dimensional discrete image from a set
of tomographic measurements corresponding to the Radon projection. Assuming
that the image has a structure where neighbouring pixels have a larger
probability to take the same value, we follow a Bayesian approach and introduce
a fast message-passing reconstruction algorithm based on belief propagation.
For numerical results, we specialize to the case of binary tomography. We test
the algorithm on binary synthetic images with different length scales and
compare our results against a more usual convex optimization approach. We
investigate the reconstruction error as a function of the number of tomographic
measurements, corresponding to the number of projection angles. The belief
propagation algorithm turns out to be more efficient than the
convex-optimization algorithm, both in terms of recovery bounds for noise-free
projections, and in terms of reconstruction quality when moderate Gaussian
noise is added to the projections.
