Iterative image reconstruction algorithms based on cross-entropy minimization

作     者：Byrne, Charles L. 

作者机构：Univ Massachusetts Dept Math Lowell MA 01854 USA 

出 版 物：《IEEE TRANSACTIONS ON IMAGE PROCESSING》 (IEEE Trans Image Process)

年 卷 期：1993年第2卷第1期

页      面：96-103页

核心收录：

学科分类：0808[工学-电气工程] 08[工学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：National Cancer Institute, NCI, (CA42165, CA51071) National Cancer Institute, NCI 

主　　题：Image reconstruction Iterative algorithms Minimization methods Maximum likelihood estimation Detectors Signal to noise ratio Tomography Bayesian methods Equations Cancer 

摘      要：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 = 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).