<title>Iterative image reconstruction algorithms based on cross-entropy minimization</title>

Abstract

The multiplicative algebraic reconstruction technique (MART) is an iterative procedure used in reconstruction of images from projections. The problem can be viewed as one of finding a nonnegative approximate solution of a certain linear system of equations y equals Px. In the consistent case, in which there are nonnegative solutions of y equals Px, the MART sequence converges to the unique nonnegative solution for which the Kullback-Leibler distance KL(x,x0) equals (summation) xjlog(xj/x0j + x0j-xj is minimized, where x0 > 0 is the starting vector for the iteration. When x0 is constant the sequence converges to the maximum Shannon entropy solution, at Lent has shown. The behavior of MART in the inconsistent case is an open problem. When y equals Px has no nonnegative solution the full MART sequence [zk, k equals 0,1,...] does not converge, while the 'simultaneously updated' version, SMART, converges to the nonnegative minimizer of KL(Px,y). In every example we have considered, the subsequences [znI+i, i fixed, n equals 0,1,...,] consisting of those iterates associated with completed cycles (I is the number of entries in y) do converge, but to distinct limits, which we denote z(infinity ,i.). Unlike most other reconstruction algorithms, if the new limiting projection data [Pz(infinity ,i)i] is used in place of the original data y and the algorithm repeated, we do not recapture [Pz(infinity i.)i]; this suggests that the MART algorithm as usually presented may be but part of a complete algorithm involving feeding back the new projection values until convergence. In all our simulations this expanded version of MART has converged, and the limit is the same as SMART; that is, the nonnegative minimizer of KL(Px,y). Both the MART and relaxed MART algorithms can be obtained through the alternating minimization of certain weighted Kullback- Leibler distances between convex sets. Orthogonality conditions in the form of Pythagorean- like identities play a useful role in the proofs concerning convergence of these algorithms.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.