  Phase contrast tomography is an alternative to classic absorption contrast
tomography that leads to higher contrast reconstructions in many applications.
We review how phase contrast data can be acquired by using a combination of
phase and absorption gratings. Using algebraic reconstruction techniques the
object can be reconstructed from the measured data. In order to solve the
resulting linear system we propose the Generalized Bidiagonal Tikhonov (GBiT)
method, an adaptation of the generalized Arnoldi-Tikhonov method that uses the
bidiagonal decomposition of the matrix instead of the Arnoldi decomposition. We
also study the effect of the finite difference operator in the model by
examining the reconstructions with either a forward difference or a central
difference approximation. We validate our conclusions with simulated and
experimental data.
