Direct x-ray scattering signal measurements in edge-illumination/beam-tracking imaging and their interplay with the variance of the refraction signals

X-ray dark-field or ultra-small angle scatter imaging has become increasingly important since the introduction of phase-based x-ray imaging and is having transformative impact in fields such as in vivo lung imaging and explosives detection. Here we show that dark-field images acquired with the edge-illumination method (either in its traditional double mask or simplified single mask implementation) provide a direct measurement of the scattering function, which is unaffected by system-specific parameters such as the autocorrelation length. We show that this is a consequence both of the specific measurement setup and of the mathematical approach followed to retrieve the dark-field images. We show agreement with theoretical models for datasets acquired both with synchrotron and laboratory x-ray sources. We also introduce a new contrast mechanism, the variance of refraction, which is extracted from the same dataset and provides a direct link with the size of the scattering centres. We show that this can also be described by the same theoretical models. We study the behaviour of both signals vs. key parameters such as x-ray energy and scatterer radius. We find this allows quantitative, direct, multi-scale scattering measurements during imaging, with implications in all fields where dark-field imaging is used.