IEEE Trans Med Imaging. 1982;1(2):113-22. doi: 10.1109/TMI.1982.4307558.
Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density lambda = lambda(x, y, z) generates, and is to be reconstructed from, the number of counts n()(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate lambdainsertion mark of lambda which maximizes the probability p(n()|lambda) of observing the actual detector count data n() over all possible densities lambda. Let independent Poisson variables n(b) with unknown means lambda(b), b = 1, ..., B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ..., D with p(b,d) a one-step transition matrix, assumed known. We observe the total number n() = n()(d) of emissions in each detector unit d and want to estimate the unknown lambda = lambda(b), b = 1, ..., B. For each lambda, the observed data n() has probability or likelihood p(n()|lambda). The EM algorithm of mathematical statistics starts with an initial estimate lambda(0) and gives the following simple iterative procedure for obtaining a new estimate lambdainsertion mark(new), from an old estimate lambdainsertion mark(old), to obtain lambdainsertion mark(k), k = 1, 2, ..., lambdainsertion mark(new)(b)= lambdainsertion mark(old)(b)Sum of (n()p(b,d) from d=1 to D/Sum of lambdainsertion mark()old(b('))p(b('),d) from b(')=1 to B), b=1,...B.
先前的发射断层成像 (ET) 模型无法区分发射断层成像和透射断层成像的物理学原理。我们给出了一个更准确的通用数学模型，其中一个未知的发射密度 $\lambda = \lambda(x,y,z)$ 产生并从每个探测器单元 $d$ 的计数 $n()(d)$ 中重建。在模型中，我们给出了一种确定估计值 $\lambda$ 的算法，该算法可以最大化观察到的实际探测器计数数据 $n()$ 的概率 $p(n()|\lambda)$，对于所有可能的密度 $\lambda$。假设独立的泊松变量 $n(b)$ 具有未知的均值 $\lambda(b)$，$b=1,\ldots,B$，代表在一个包含发射器的物体的每个 $B$ 个盒子（像素）中的未观察到的发射数量。假设盒子 $b$ 中的每个发射都以概率 $p(b,d)$ 在探测器单元 $d$ 中被检测到，$d=1,\ldots,D$，其中 $p(b,d)$ 是一步转移矩阵，假设已知。我们观察每个探测器单元 $d$ 中发射的总数 $n() = n()(d)$，并希望估计未知的 $\lambda = \lambda(b)$，$b=1,\ldots,B$。对于每个 $\lambda$，观察到的数据 $n()$ 具有概率或似然 $p(n()|\lambda)$。数理统计的 EM 算法从初始估计值 $\lambda(0)$ 开始，并为从旧估计值 $\lambda(0)$ 获得新的估计值 $\lambda(new)$ 提供了以下简单的迭代过程，以获得估计值 $\lambda(k)$，$k=1,2,\ldots$，$\lambda(new)(b)=\lambda(old)(b)\times\frac{\sum\limits_{d=1}^Dn()\times p(b,d)}{\sum\limits_{b'=1}^B\lambda(old)(b')\times p(b',d)}$，$b=1,\ldots,B$。
