DISCRETE RADON-TRANSFORM AND ITS APPROXIMATE INVERSION VIA THE EM ALGORITHM

The problem of reconstructing a binary function, f, defined on a finit e subset of a lattice Z, from an arbitrary collection of its partial-s ums is considered. The approach we present is based on (a) relaxing th e binary constraints f(z) = 0 or 1 to interval constraints 0 less than or equal to f(z) less than or equal to 1, z is an element of Z, and ( b) applying a minimum distance method (using Kullback-Leibler's inform ation divergence index as our distance function) to find such an f -sa y, (f) over cap-for which the distance between the observed and the th eoretical partial sums is as small as possible. (Turning this f into a binary function can be done as a separate postprocessing step: for in stance, through thresholding, or through some additional Bayesian mode ling.) To derive this minimum-distance solution, we develope a new EM algorithm. This algorithm is different from the often-studied EM/maxim um likelihood algorithm in emission tomography and other linear-invers e positively constrained problems because of the additional upper-boun d constraint (f less than or equal to 1) on the signal f. Properties o f the algorithm, as well as similarities with and differences from som e other methods, such as the linear-programming approach or the algebr aic reconstruction technique, are discussed. The methodology is demons trated on three recently studied phantoms, and the simulation results are very promising, suggesting that the method could also work well un der field conditions which may include a small or moderate revel of me asurement noise in the observed partial sums. The methodology has impo rtant applications in high-resolution electron microscopy for the reco nstruction of the atomic structure of crystals from their projections. (C) 1998 John Wiley & Sons, Inc.