Three dimensional image reconstruction on an extended field—a fast, stable algorithm

Abstract

WE describe an improved alogrithm (EFIRT) for solving iteratively the linear equations relating a three-dimensional density to a given set of its projections, when the unknown density is expressed as samples on a grid. The need to solve such equations arises in electron microscopy, medical radiography, radio and X-ray astronomy and other fields. The projection equations comprise a very large, sparse and often under-determined set. If the number of unknowns is not too large, the equations may be solved by a least squares procedure, using filtering to combat the ill-conditioning¹. The number of unknowns is often too great for such a procedure to be computationally feasible and iterative techniques^2,3 must be used. Alternatively the Fourier transform provides a stable and computationally efficient means of solution¹. Sometimes use of the Fourier transform is not convenient; for example, if point rather than plane projections are given or if the projection data are weighted by a non-uniform but linear response of the measuring device. Such problems are much more readily expressed in terms of projection equations and the development of stable methods for their solution is therefore important.