CT gradient image reconstruction directly from projections

An algorithm is proposed to directly reconstruct a CT gradient image in a region of interest(ROI). First, the central slice theorem is generalized and a differential constraint condition (DCC) is introduced in parallel-beam geometry. Then, an algorithm is developed to reconstruct the gradient images in both Cartesian and polar coordinate systems based on a two-step Hilbert transform method. Finally, the reconstruction algorithm is extended into the equi-distant fan-beam geometry. Meanwhile, a conditional truncation for projection data acquisition is permitted by using a one-dimensional(1-D) finite Hilbert transform in image domain. Because the reconstructed gradient image is in terms of local operator, it have a better performance in CT image analysis and other CT applications compared to the global Calderon operator in Lambda Tomography.