  The following paper describes a method for three-dimensional (3D)
reconstruction of multi-material objects based on propagation-based X-ray
phase-contrast tomography (PB-CT) with phase retrieval using the homogenous
form of the Transport-of-Intensity equation (TIE-Hom). Unlike conventional
PB-CT algorithms that perform phase retrieval of individual projections, the
described post-reconstruction phase-retrieval method is applied in 3D to a
localised region of the CT-reconstructed volume. We demonstrate via numerical
simulations the accuracy and noise characteristics of the method under a
variety of experimental conditions, comparing it to both conventional
absorption tomography and two-dimensional (2D) TIE-Hom phase retrieval applied
to projection images. The results indicate that the 3D post-reconstruction
method generally achieves a modest improvement in noise suppression over
existing PB-CT methods. It is also shown that potentially large computational
gains over projection-based phase retrieval for multi-material samples are
possible. In particular, constraining phase retrieval to a localised 3D region
of interest reduces the overall computational cost and eliminates the need for
multiple CT reconstructions and global 2D phase retrieval operations for each
material within the sample.
