Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data

In this paper, a new image reconstruction scheme is presented based on Tuy's cone-beam inversion scheme and its fan-beam counterpart. It is demonstrated that Tuy's inversion scheme may be used to derive a new framework for fan-beam and cone-beam image reconstruction. In this new framework, images are reconstructed via filtering the backprojection image of differentiated projection data. The new framework is mathematically exact and is applicable to a general source trajectory provided the Tuy data sufficiency condition is satisfied. By choosing a piece-wise constant function for one of the components in the factorized weighting function, the filtering kernel is one dimensional, viz. the filtering process is along a straight line. Thus, the derived image reconstruction algorithm is mathematically exact and efficient. In the cone-beam case, the derived reconstruction algorithm is applicable to a large class of source trajectories where the pi-lines or the generalized pi-lines exist. In addition, the new reconstruction scheme survives the super-short scan mode in both the fan-beam and cone-beam cases provided the data are not transversely truncated. Numerical simulations were conducted to validate the new reconstruction scheme for the fan-beam case.