  Dark-field X-ray imaging is a novel modality which visualises scattering from
unresolved microstructure. Most dark-field imaging techniques rely on crystals
or structured illumination, but recent work has shown that dark-field effects
are observable in straightforward propagation-based imaging (PBI). Based on the
single-material X-ray Fokker--Planck equation with an a priori dark-field
energy dependence, we propose an algorithm to extract phase and dark-field
effects from dual-energy PBI images. We successfully apply the dark-field
retrieval algorithm to simulated and experimental dual-energy data, and show
that by accounting for dark-field effects, projected thickness reconstruction
is improved compared to the classic Paganin algorithm. With the emergence of
spectral detectors, the method could enable single-exposure dark-field imaging
of dynamic and living samples.
