Reciprocal-FDK reconstruction for x-ray diffraction computed tomography

Objective. X-ray diffraction (XRD) technology uses x-ray small-angle scattering signal for material analysis, which is highly sensitive to material inter-molecular structure. To meet the high spatial resolution requirement in applications such as medical imaging, XRD computed tomography (XRDCT) has been proposed to provide XRD intensity with improved spatial resolution from point-wise XRD scan. In XRDCT, 2D spatial tomography corresponds to a 3D reconstruction problem with the third dimension being the XRD spectrum dimension, i.e. the momentum transfer dimension. Current works in the field have studied reconstruction methods for either angular-dispersive XRDCT or energy-dispersive XRDCT for small samples. The approximations used are only suitable for regions near the XRDCT iso-center. A new XRDCT reconstruction method is needed for more general imaging applications. Approach. We derive a new FDK-type reconstruction method (Reciprocal-FDK) for XRDCT without limitation on object size. By introducing a set of reciprocal variables, the XRDCT model is transformed into a classical cone-parallel CT model, which is an extension of a circular-trajectory cone-beam CT model, after which the FDK method is applied for XRDCT reconstruction. Main results. Both analytical simulation and Monte Carlo simulation experiments are conducted to validate the XRDCT reconstruction method. The results show that when compared to existing analytical reconstruction methods, there are improvements in the proposed Reciprocal-FDK method with regard to relative structure reconstruction and XRD pattern peak reconstruction. Since cone-parallel CT does not satisfy the data completeness condition, cone-angle effect affects the reconstruction accuracy of XRDCT. The property of cone-angle effect in XRDCT is also analyzed with ablation studies. Significance. We propose a general analytical reconstruction method for XRDCT without constraint on object size. Reciprocal-FDK provides a complete derivation and theoretical support for XRDCT reconstruction by analogy to the well-studied cone-parallel CT model. In addition, the intrinsic problem with the XRDCT data model and the corresponding reconstruction error are discussed for the first time.