Inversion of the attenuated radon transform for non-parallel geometries

Abstract

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.