Phase retrieval in propagation-based X-ray imaging beyond the limits of transport of intensity and contrast transfer function approaches.

We derive a phase retrieval formula for propagation-based phase contrast X-ray imaging that does not require weakly attenuating objects or short propagation distances. It is directly applicable to both single- and multiple-distance scenarios. We show the validity conditions and study the error of the underlying mutual intensity approximation, which uses the common assumptions of weak phase shift variations and phase-attenuation duality. The approximation generalizes those behind the transport of intensity (TIE) and contrast transfer function (CTF) models, and it approaches them when their respective additional assumptions are satisfied. When they are not, it clearly outperforms them, which we show both theoretically and practically on synthetic and measured data.