Talbot-Lau X-ray phase-contrast imaging is a novel imaging modality, which provides not only an X-ray absorption image, but also additionally a differential phase image and a dark-field image. The dark-field image is related to small angle scattering and has an interesting property when canning oriented structures: the recorded signal depends on the relative orientation of the structure in the imaging system. Exactly this property allows to draw conclusions about the orientation and to reconstruct the structure. However, the reconstruction is a complex, non-trivial challenge. A lot of research was conducted towards this goal in the last years and several reconstruction algorithms were proposed. A key step of the reconstruction algorithm is the inversion of a forward projection model. Up until now, only 2-D projection models are available, with effectively limit the scanning trajectory to a 2-D plane. To obtain true 3-D information, this limitation requires to combine several 2-D scans, which leads to quite complex, impractical acquisitions schemes. Furthermore, it is not possible with these models to use 3-D trajectories that might allow simpler protocols, like for example a helical trajectory. To address these limitations, we propose in this work a very general 3-D projection model. Our projection model defines the dark-field signal dependent on an arbitrarily chosen ray and sensitivity direction. We derive the projection model under the assumption that the observed scatter distribution has a Gaussian shape. We theoretically show the consistency of our model with more constrained existing 2-D models. Furthermore, we experimentally show the compatibility of our model with dark-field measurements of two matchsticks. We believe that this 3-D projection model is an important step towards more flexible trajectories and imaging protocols that are much better applicable in practice.
Dark-field imaging is a scattering-based X-ray imaging method that can be performed with laboratory X-ray tubes. The possibility to obtain information about unresolvable structures has already seen a lot of interest for both medical and material science applications. Unlike conventional X-ray attenuation, orientation dependent changes of the dark-field signal can be used to reveal microscopic structural orientation. To date, reconstruction of the three-dimensional dark-field signal requires dedicated, highly complex algorithms and specialized acquisition hardware. This severely hinders the possible application of orientation-dependent dark-field tomography. In this paper, we show that it is possible to perform this kind of dark-field tomography with common Talbot-Lau interferometer setups by reducing the reconstruction to several smaller independent problems. This allows for the reconstruction to be performed with commercially available software and our findings will therefore help pave the way for a straightforward implementation of orientation-dependent dark-field tomography.