  Spectral and grating-based differential phase-contrast X-ray imaging are two
emerging technologies that offer additional information compared with
conventional attenuation-based X-ray imaging. In the case of spectral imaging,
energy-resolved measurements allow the generation of material-specific images
by exploiting differences in the energy-dependent attenuation. Differential
phase-contrast imaging uses the phase shift that an X-ray wave exhibits when
traversing an object as contrast generation mechanism. Recently, we have
investigated the combination of these two imaging techniques (spectral
differential phase-contrast imaging) and demonstrated potential advantages
compared with spectral imaging. In this work, we present a noise analysis
framework that allows the prediction of (co-) variances and noise power spectra
for all three imaging methods. Moreover, the optimum acquisition parameters for
a particular imaging task can be determined. We use this framework for a
performance comparison of all three imaging methods. The comparison is focused
on (projected) electron density images since they can be calculated with all
three imaging methods. Our study shows that spectral differential
phase-contrast imaging enables the calculation of electron density images with
strongly reduced noise levels compared with the other two imaging methods for a
large range of clinically relevant pixel sizes. In contrast to conventional
differential phase-contrast imaging, there are no long-range noise correlations
for spectral differential phase-contrast imaging. This means that excessive low
frequency noise can be avoided. We confirm the analytical predictions by
numerical simulations.
