Density reconstruction using arbitrary ray-sampling schemes

Methods for calculating the distribution of absorption densities in a cross section through an object from density intergrals along rays in the plane of the cross section are well-known, but are restricted to particular geometries of data collection. So-called convolutional-back projection-summation methods, used now for parallelray data, have recently been extended to special cases of the fan-beam reconstruction problem by the addition of pre- and post-multiplication steps. In this paper, a technique for deriving reconstruction algorithms for arbitrary ray-sampling schemes is presented; the resulting algorithms entail the use of a general linear operator, but require little more computation than the convolutional methods, which represent special cases. The key to the derivation is the observation that the contribution of a particular ray sum to a particular point in the reconstruction essentially depends on the negative inverse square of the perpendicular distance from the point to the ray, and that this contribution has to be weighted by the ray-sampling density. The remaining task is the efficient arrangement of this computation, so that the contribution of each ray sum to each point in the reconstruction does not have to be calculated explicitly. The exposition of the new method is informal in order to facilitate the application of this technique to various scanning geometries. The frequency domain is not used, since it is inappropriate for the space-variant operators encountered in the general case. The technique is illustrated by the derivation of an algorithm for parallel-ray sampling with uneven spacing between rays and uneven spacing between projection angles. A reconstruction is shown which attains high spatial resolution in the central region of an object by sampling central rays more finely than those passing through outer portions of the object.