Linear partial derivative matrix for iterative algorithm to reconstruct refractive index from refraction angle data

X-ray phase contrast imaging (PCI) has great potential in the medical diagnosis, material science et al and has made great progress in the last ten years. Diffraction enhanced imaging and grating-based imaging are the two significant methods for PCI, currently. Two methods are named differential phase contrast imaging because the projection data is partial derivatives of refractive index. Analytical algorithms to reconstruct the refractive index have been proposed. However, there is lack of an iterative algorithm to reconstruct the refractive index directly from the refraction angle data. In this paper, an algebraic iterative algorithm is proposed to reconstruct the distribution of refractive index from refraction angle data. Our algorithm is a complementarity to analytical reconstruction algorithms and inherits the benefits of iterative algorithms that can work well with in-complete projection data. The proposed algorithm is validated by numerical simulation and the results show that it can reconstruct refractive index accurately and the iteration is convergent and steady.