  The ill-posed problem of phase retrieval in optics, using one or more
intensity measurements, has a multitude of applications using electromagnetic
or matter waves. Many phase retrieval algorithms are computed on pixel arrays
using discrete Fourier transforms due to their high computational efficiency.
However, the mathematics underpinning these algorithms is typically formulated
using continuous mathematics, which can result in a loss in spatial resolution
in the reconstructed images. Herein we investigate how phase retrieval
algorithms for propagation-based phase-contrast X-ray imaging can be rederived
using discrete mathematics and result in more precise retrieval for single- and
multi-material objects and for spectral image decomposition. We validate this
theory through experimental measurements of spatial resolution using computed
tomography (CT) reconstructions of plastic phantoms and biological tissue,
using detectors with a range of imaging system point spread functions (PSFs).
We demonstrate that if the PSF substantially suppresses high spatial
frequencies, the potential improvement from utilising the discrete derivation
is limited. However, with detectors characterised by a single pixel PSF (e.g.
direct, photon-counting X-ray detectors), a significant improvement in spatial
resolution can be obtained, demonstrated here at up to 17%.
