X-ray in-line phase retrieval for tomography

Phase contrast in X-ray imaging offers imaging of fine features at lower doses than absorption. Of the phasecontrast methods in use in-line phase contrast is interesting due to its experimental simplicity, but to extract information on absorption and phase distributions from the resulting images, phase retrieval is needed. Many phase-retrieval methods suitable for different situations have been developed, but few comparisons of those methods done. We consider a sub-group of phase-retrieval methods that are suitable for tomography, i.e., that use only one exposure (for practical experimental reasons) and are non-iterative (for speed). In total we have found seven suitable methods in the literature. All, though derived in different ways under different assumptions, follow the same pattern and can be outlined as a single method where each specific version is marked by variations in particular steps. We summarize this unified approach, and give the variations of the individual methods. In addition, we outline approximations and assumptions of each method. Using this approach it is possible to conclude which specific algorithms are most suitable in specific situations and to test this based on simulated and experimental data. Ultimately, this leads to conclusions on which methods are the most suitable in different situations.