Direct Three-Dimensional Image Reconstruction from Divergent Rays

The solutions to the image reconstruction problem, in two and three dimensions, for both parallel and divergent ray geometries, are presented within a general linear reconstruction framework. It is shown that, with suitable parameterizations, each of these solutions reduces to a "convolve-and-backproject" algorithm. The exact solution to the three-dimensional divergent ray geometry problem is a new result and is treated in detail. This problem arises when the sensors are located around the object region in three-dimensional space, and the measurement rays diverge (i.e., fan out) from the individual sources. An approximation to the exact solution has been made in order to derive convenient and practical convolving functions for this geometry.