General fan-beam and cone-beam reconstruction algorithms formula 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. [1]. 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 condition that the source-to-detectors distance changes slowly enough relative to the length of itself, we obtain a very good approximate reconstruction formula. And we also develop an reconstruction algorithm for nonstandard spirals with different pitches, based on some developments in the recent two years. In the end the fan-beam formula is tested using the Shepp-Logan phantom the formula can get perfect numerical results.