Discrete Radon transform and its approximate inversion via the EM algorithm

Abstract

The problem of reconstructing a binary function f, defined on a finite subset of a lattice ℤ, from an arbitrary collection of its partial-sums is considered. The approach we present is based on (a) relaxing the binary constraints f(z) = 0 or 1 to interval constraints 0 ≤ f(z) ≤ 1, z ∈ ℤ, and (b) applying a minimum distance method (using Kullback-Leibler's information divergence index as our distance function) to find such an f—say, fˆ—for which the distance between the observed and the theoretical partial sums is as small as possible. (Turning this fˆ into a binary function can be done as a separate postprocessing step: for instance, through thresholding, or through some additional Bayesian modeling.) To derive this minimum-distance solution, we develope a new EM algorithm. This algorithm is different from the often-studied EM/maximum likelihood algorithm in emission tomography and other linear-inverse positively constrained problems because of the additional upper-bound constraint (f ≤ 1) on the signal f. Properties of the algorithm, as well as similarities with and differences from some other methods, such as the linear-programming approach or the algebraic reconstruction technique, are discussed. The methodology is demonstrated on three recently studied phantoms, and the simulation results are very promising, suggesting that the method could also work well under field conditions which may include a small or moderate level of measurement noise in the observed partial sums. The methodology has important applications in high-resolution electron microscopy for the reconstruction of the atomic structure of crystals from their projections. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 155–173, 1998