An imaging set-up for reverse projection to obtain quantitative (hard) x-ray images from a sample and to quantitatively extract both absorption and phase information from the sample, comprising: a. an X-ray beam generated by an X-ray source; b. a beam splitter grating and an analyzer grating, spaced from each other by a distance D and having their respective lines parallel to each other, wherein the beam splitter grating is a line grating and the analyzer grating is a line absorption grating with high X-ray absorption wherein at least one of the gratings (G1, G2) is positioned relative to the probe in a direction ( x g) substantially perpendicular to both the incoming beam and the orientation of the lines of the grating to make the imaging set-up on the center of the linear region of the shifting curve s x g D; wherein a mechanism is comprised to place the sample to be investigated either between the X-ray source and the beam splitter grating or between the beam splitter grating and the analyzer grating; c. a position-sensitive detector with spatially modulated detection sensitivity having a number of individual pixels; d. means for recording the images of the detector, wherein a series of M images is collected by continuously or stepwise rotating from 0 to pi or 2pi either sample or the gratings (G1, G2) and the X-ray source relative to the sample, wherein each image taken at an angle 0 ≤ Φ ≤ pi comprises a corresponding reverse projection image taken at an angle pi ≤ Φ+pi ≤ 2pi, yielding in total a number of M/2 pairs of specular images; e. means for calculating pixel-wise an absorption image M and an refraction angle θ r image out of the pairs of specular images - without the need of phase stepping - according to: ln 2 S x g D I 0 I x r φ z + I − x r, φ + π, z = M x r φ z = ∫ − ∞ ∞ μ x y z dy r; 1 C I x r φ z − I − x r, φ + π, z I x r φ z + I − x r, φ + π, z = θ r x r φ z = − ∫ − ∞ ∞ ∂ δ x y z ∂ x r d y r a. an X-ray beam generated by an X-ray source; b. a beam splitter grating and an analyzer grating, spaced from each other by a distance D and having their respective lines parallel to each other, wherein the beam splitter grating is a line grating and the analyzer grating is a line absorption grating with high X-ray absorption wherein at least one of the gratings (G1, G2) is positioned relative to the probe in a direction ( x g) substantially perpendicular to both the incoming beam and the orientation of the lines of the grating to make the imaging set-up on the center of the linear region of the shifting curve s x g D; wherein a mechanism is comprised to place the sample to be investigated either between the X-ray source and the beam splitter grating or between the beam splitter grating and the analyzer grating; c. a position-sensitive detector with spatially modulated detection sensitivity having a number of individual pixels; d. means for recording the images of the detector, wherein a series of M images is collected by continuously or stepwise rotating from 0 to pi or 2pi either sample or the gratings (G1, G2) and the X-ray source relative to the sample, wherein each image taken at an angle 0 ≤ Φ ≤ pi comprises a corresponding reverse projection image taken at an angle pi ≤ Φ+pi ≤ 2pi, yielding in total a number of M/2 pairs of specular images; e. means for calculating pixel-wise an absorption image M and an refraction angle θ r image out of the pairs of specular images - without the need of phase stepping - according to: ln 2 S x g D I 0 I x r φ z + I − x r, φ + π, z = M x r φ z = ∫ − ∞ ∞ μ x y z dy r; 1 C I x r φ z − I − x r, φ + π, z I x r φ z + I − x r, φ + π, z = θ r x r φ z = − ∫ − ∞ ∞ ∂ δ x y z ∂ x r d y r