<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,x<SUP>0</SUP>) equals (summation) x<SUB>j</SUB>log(x<SUB>j</SUB>/x<SUP>0</SUP><SUB>j</SUB> + x<SUP>0</SUP><SUB>j</SUB>-x<SUB>j</SUB> is minimized, where x<SUP>0</SUP> &gt; 0 is the starting vector for the iteration. When x<SUP>0</SUP> 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 [z<SUP>k</SUP>, 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 [z<SUP>nI+i</SUP>, 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<SUP>(infinity</SUP> ,i.). Unlike most other reconstruction algorithms, if the new limiting projection data [Pz<SUP>(infinity</SUP> ,i)<SUB>i</SUB>] is used in place of the original data y and the algorithm repeated, we do not recapture [Pz<SUP>(infinity</SUP> i.)<SUB>i</SUB>]; 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.