  Emerging methods of x-ray imaging that capture phase and dark-field effects
are equipping medicine with complementary sensitivity to conventional
radiography. These methods are being applied over a wide range of scales, from
virtual histology to clinical chest imaging, and typically require the
introduction of optics such as gratings. Here, we consider extracting x-ray
phase and dark-field signals from bright-field images collected using nothing
more than a coherent x-ray source and detector. Our approach is based on the
Fokker--Planck equation for paraxial imaging, which is the diffusive
generalization of the transport-of-intensity equation. Specifically, we utilize
the Fokker--Planck equation in the context of propagation-based phase-contrast
imaging, where we show that two intensity images are sufficient for successful
retrieval of the projected thickness and dark-field signals associated with the
sample. We show the results of our algorithm using both a simulated dataset and
an experimental dataset. These demonstrate that the x-ray dark-field signal can
be extracted from propagation-based images, and that x-ray phase can be
retrieved with better spatial resolution when dark-field effects are taken into
account. We anticipate the proposed algorithm will be of benefit in biomedical
imaging, industrial settings, and other non-invasive imaging applications.
