Inversion of the attenuated Radon transform for non-parallel geometries

In this work, we first introduced a reorganized form of the Novikov´s inversion formula for the attenuated Radon transform with parallel-beam geometry which utilizes the conventional filters (such as the Shepp-Logan, Hanning, etc.) to simplify image reconstruction procedures. Then we developed an accurate reconstruction method for non-parallel beam geometries. The method contains three major steps: (1) one-dimensional rebinning using phase-shifting technique, (2) performing non-uniform Hilbert transform, and (3) applying the reorganized Novikov´s formula. Numerical evaluations demonstrated its computational efficiency and sound stability to different levels of Poisson noise. It also seemed to be very robust to different settings of fan-beam geometry from very long to very short focal lengths without affecting the reconstruction accuracy.