New families of exact fan-beam and cone-beam image reconstruction formulae via filtering the backprojection image of differentiated projection data along singly measured lines

In this paper, new families of mathematically exact image reconstruction formulae for fan-beam and cone-beam computed tomography (CT) are developed. Using these new formulae, images may be reconstructed via filtering the backprojection image of differentiated projection data. We refer to them as FBPD image reconstruction algorithms. The novel features of these formulae include: (1) the filtering line may be along a line that passes through the image point and intersects the source trajectory at only one point, i.e., singly measured line (SM-line); and (2) for all the points along the filtering line, projection data acquired from a segment of source trajectory with the same angular range will be utilized to calculate the backprojection image. Although the filtering operations are not guaranteed to be shift-invariant along the filtering lines, these formulae are mathematically exact, and are novel solutions to the image reconstruction problem using fan-beam and cone-beam projections for a large class of complete source trajectories in which the existence of the doubly measured lines (DM-lines) may be established (A DM-line is a straight line that passes through an image point and intersects source trajectory at two different points). In each family of reconstruction formulae, we have demonstrated that we may have infinitely many mathematically exact image reconstruction formulae for a given scanning configuration.