A general fan-beam reconstruction algorithm for free-form trajectories

In this paper we develop a general exact fan-beam reconstruction algorithm for free-form trajectories not only closed but also unclosed,based on the fan-beam reconstruction formula recently developed by Noo et al.. A mathematical proof is then provided with the geometrical explanation of equi-spatial detectors. With this algorithm we can obtain exact region of interest (ROI) reconstruction if and only if every projecting line passing through the ROI intersects the free-form source trajectory,when the projections are not truncated. Furthermore,under the condi- tion that the source-to-detectors distance changes slowly enough relative to the length itself,we obtain a very good approximate reconstruction algorithm,which is the same as the algorithm of the circular trajectory except that the source-to-detectors distance is a function of the rotation angle. Then the algorithms are tested using the Shepp-Logan phantom and the experiment shows that the algorithms can get perfect numerical results.