Enhanced spatial resolution through DFT rederivations of X-ray phase retrieval algorithms

Propagation-based phase-contrast imaging, used in conjunction with the phase retrieval algorithm based on the Transport-of-Intensity Equation (TIE) (Paganin et al., 2002), is commonly used to improve the sensitivity of X-ray imaging. Recently, a `Generalised Paganin Method' algorithm was published to correct the tendency of the TIE algorithm to over-blur images. The article, Paganin et al. 2020, provided a derivation of the new method and demonstrated a difference in the level of blurring applied by each algorithm. In this manuscript, we quantify the spatial resolution improvement and describe the optimal experimental conditions to observe this improvement. We link the effectiveness of the spatial resolution improvement to the imaging point spread function (PSF), incorporating the PSF to compare the blurring applied by each algorithm. We then validate this model through measurements of spatial resolution in experimental data imaging plastic phantoms and biological tissue, using detectors with different PSFs. By analysing edge-spread functions in CT data captured with indirect detectors with PSFs of several pixels in extent, we show negligible spatial resolution improvement when using the generalised Paganin method. However, a clear improvement in spatial resolution, up to 17%, was observed with direct detectors having PSFs of approximately one pixel in extent. Additionally, we demonstrate clear visual improvement in resolution in CT slices of rat lungs. Finally, we demonstrate the versatility of this improvement by generalising other phase retrieval algorithms, namely for multi-material samples and for spectral decomposition using propagation-based phase contrast, and experimentally verify improvements in spatial resolution.