  The aim of this research is to reconstruct the 3D X-ray refractive index
gradient maps by the proposed vector Radon transform and its inverse, assuming
that the small-angle deviation condition is met. Theoretical analyses show that
the X-ray beam can be modeled as a streamline with continuous change of
direction in a row when measured in one grating period, which allows the
extraction of the refraction angle signals. Experimental results show that all
the 2D refraction signals of different directions can be acquired by a standard
circular scanning procedure, which is typically used in the X-ray differential
phase-contrast computed tomography. Furthermore, the 3D refractive index
gradient maps that contain the directional density changes, can also be
accurately reconstructed.
