Iterative image reconstruction algorithms based on cross-entropy minimization

The cross-entropy (or Kullback-Leibler) distance between two nonnegative vectors a and b is KL(a,b) = Sigma a(n) log(a(n)/b(n)) + b(n) -a(n). Several well-known iterative algorithms for reconstructing tomographic images lead to solutions that minimize certain combinations of KL distances, and can be derived from alternating minimization of related KL distances between convex sets; these include the expectation maximization (EM) algorithm for likelihood maximization (ML), and the Bayesian maximum a posteriori (MAP) method with gamma-distributed priors, as well as the multiplicative algebraic reconstruction technique (MART). Each of these algorithms can be viewed as providing approximate nonnegative solutions to a (possibly inconsistent) linear system of equations, y = Px. In almost all cases, the ML problem has a unique solution (and so the EM iteration has a limit that is independent of the starting point) unless the system of equations y = Px has a nonnegative solution, regardless of the dimensions of y and x. We introduce the "simultaneous" MART (SMART) algorithm and prove convergence: for 0 < alpha, < 1, SMART converges to the x >= 0 for which alpha KL(Px, y) + (1 - alpha)KL(x, p) is minimized, where p denotes a prior estimate of the desired x; for alpha = 1, the SMART algorithm converges in the consistent case (as does MART) to the unique solution of y = Px minimizing KL(x,x(0)), where x(0) is the starting point for the iteration, and in the inconsistent case, to the unique nonnegative minimizer of KL(Px,y).