  We outline some basics of imaging using both fully-coherent and
partially-coherent X-ray beams, with an emphasis on phase-contrast imaging. We
open with some of the basic notions of X-ray imaging, including the vacuum wave
equations and the physical meaning of the intensity and phase of complex scalar
fields. The projection approximation is introduced, together with the concepts
of attenuation contrast and phase contrast. We also outline the multi-slice
approach to X-ray propagation through thick samples or optical elements,
together with the Fresnel scaling theorem. Having introduced the fundamentals,
we then consider several aspects of the forward problem, of modelling the
formation of phase-contrast X-ray images. Several topics related to this
forward problem are considered, including the transport-of-intensity equation,
arbitrary linear imaging systems, shift-invariant linear imaging systems, the
transfer-function formalism, blurring induced by finite source size, the
space-frequency model for partially-coherent fields, and the Fokker-Planck
equation for paraxial X-ray imaging. Having considered these means for
modelling the formation of X-ray phase-contrast images, we then consider
aspects of the associated inverse problem of phase retrieval. This concerns how
one may decode phase-contrast images to gain information regarding the
sample-induced attenuation and phase shift.
