  Phase-wrapping artifacts, statistical image noise and the need for a minimum
amount of phase steps per projection limit the practicability of x-ray grating
based phase-contrast tomography, when using filtered back projection
reconstruction. For conventional x-ray computed tomography, the use of
statistical iterative reconstruction algorithms has successfully reduced
artifacts and statistical issues. In this work, an iterative reconstruction
method for grating based phase-contrast tomography is presented. The method
avoids the intermediate retrieval of absorption, differential phase and dark
field projections. It directly reconstructs tomographic cross sections from
phase stepping projections by the use of a forward projecting imaging model and
an appropriate likelihood function. The likelihood function is then maximized
with an iterative algorithm. The presented method is tested with tomographic
data obtained through a wave field simulation of grating based phase-contrast
tomography. The reconstruction results are in agreement with the expectations
and proof the validity of the concept.
