Iterative image reconstruction algorithms based on cross-entropy minimization

摘要

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 $EQ Px. In the consistent case, in which there are nonnegative solutions of y $EQ Px, the MART sequence converges to the unique nonnegative solution for which the Kullback-Leibler distance KL(x,x$+0$/) $EQ $SUM x$-j$/log(x$-j$//x$+0$/$-j$/ $PLU x$+0$/$-j$/-x$-j$/ is minimized, where x$+0$/ $GRT 0 is the starting vector for the iteration. When x$+0$/ 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 $EQ Px has no nonnegative solution the full MART sequence $LB@z$+k$/, k $EQ 0,1,...$RB 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 $LB@z$+nI$PLU@i$/, i fixed, n $EQ 0,1,...,$RB 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$+$INF@,i.$/. Unlike most other reconstruction algorithms, if the new limiting projection data $LB@Pz$+$INF@,i$/$-i$/$RB is used in place of the original data y and the algorithm repeated, we do not recapture $LB@Pz$+$INF@i.$/$-i$/$RB@; 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.!19