Iterative image reconstruction algorithms based on cross-entropy minimization
Iterative image reconstruction algorithms based on cross-entropy minimization
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DOI：
10.1109/83.210869
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发表时间：
1993-01-01
影响因子：
10.6
通讯作者：
Byrne, Charles L.
中科院分区：
文献类型：
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作者：
Byrne, Charles L.
两个非负向量a和b之间的交叉熵（或Kullback-Leibler）距离是KL(a,b) = Sigma a(n) log(a(n)/b(n)) + b(n) -a(n)。用于重建层析图像的几种著名的迭代算法导致最小化KL距离的某些组合的解决方案，并且可以从凸集之间相关KL距离的交替最小化中得到；其中包括用于似然最大化（ML）的期望最大化（EM）算法，以及具有伽马分布先验的贝叶斯最大后验（MAP）方法，以及乘法代数重建技术（MART）。每一种算法都可以被看作是为一个线性方程组y = Px（可能不一致）提供近似的非负解。在几乎所有情况下,毫升的问题有一个独特的解决方案(EM迭代有限制,是独立于起点),除非方程组y = Px非负解,无论x和y的尺寸。我们引入“同时”集市(智能)算法和证明收敛:0 <α< 1,智能收敛于x > = 0的αKL (Px, y) +(1 -α)吉隆坡(x, p)最小化,p表示所需的x的先验估计;对于alpha = 1， SMART算法在一致的情况下（MART也是如此）收敛到y = Px最小化KL（x,x(0)）的唯一解，其中x(0)是迭代的起点，在不一致的情况下，收敛到KL（Px,y）的唯一非负最小值。
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).