Fan-beam reconstruction methods

In a previous paper a technique was developed for finding reconstruction algorithms for arbitrary ray-sampling schemes. The resulting algorithms use a general linear operator, the kernel of which depends on the details of the scanning geometry. Here this method is applied to the problem of reconstructing density distributions from arbitrary fan-beam data. The general fan-beam method is then specialized to a number of scanning geometries of practical importance. Included are two cases where the kernel of the general linear operator can be factored and rewritten as a function of the difference of coordinates only and the superposition integral consequently simplifies into a convolution integral. Algorithms for these special cases of the fan-beam problem have been developed previously by others. In the general case, however, Fourier transforms and convolutions do not apply, and linear space-variant operators must be used. As a demonstration, details of a fan-beam method for data obtained with uniform ray-sampling density are developed.