Reconstruction of Binary Images via the EM Algorithm

Abstract

The problem of reconstructing a binary function x defined on a finite subset of a lattice ℤ, from an arbitrary collection of its partial sums is considered. The approach is based on (a) relaxing the binary constraints \(x\left( i \right) = 0\) or 1 to interval constraints \(0x\left( i \right)1,i \in {\Bbb Z}\),and (b) applying a minimum distance method (using Kullback-Leibler’s information divergence index as our distance function) to find such an x — say, \(\hat x\)— for which the distance between the observed and the theoretical partial sums is as small as possible (Turning this \(\hat x\) into a binary function can be done as a separate postprocessing step: for instance,through thresholding, or through some additional Bayes modeling.) This minimum-distance solution is derived via a new EM algorithm that extends the often-studied EM/maximum likelihood (EM/ML) algorithm in emission tomography and certain linear-inverse problems to include lower-and upper-bound constraints on the function x. Properties of the algorithm including convergence and uniqueness conditions on the solution (or parts of it) are described.