A new framework of image reconstruction from fan beam projections

In this paper, we present a unified framework to reconstruct images for both fan beam and cone beam projections. The important feature of our theoretical framework is that it does not depend on the classical concept of the Radon transform at all. This property allows us to directly generalize the ideas and techniques developed in this paper to the cone beam reconstruction problem. In this paper, we extract such a framework from developing a new image reconstruction scheme from fan beam projections. Our new scheme also provides us new understanding of fan beam reconstruction problem. Our main results for the fan beam reconstruction are the following: First, we derive a general reconstruction scheme, in which the data sufficiency condition is transparently revealed by the reconstruction formula. Specifically, the data sufficiency condition for an accurate reconstruction of a region of interest (ROI) is that all the lines passing through the ROI must intersect the source trajectory at least once. Second, we further simplify the general reconstruction scheme by following three major steps of our new framework: (i) using symmetries of intermediate function; (ii) handling the data redundancy; (iii) changing discrete summation over the possible focal points into an integral along the source trajectory. After these steps, we obtain a new filtered backprojection algorithm. The key characteristic of this new algorithm is to take derivative of measured data with respect to the trajectory parameter. In practice, we can trade this derivative to some other continuous functions. In the configuration of a circular source trajectory with a third generation arc/collinear detector, we demonstrate how to remove the undesirable derivative of measured projection data. It results in a new algorithm for the sequential reconstruction of a ROI with a general normalized weighting function.