Maximum likelihood reconstruction for emission tomography.
Maximum likelihood reconstruction for emission tomography.
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DOI：
10.1109/tmi.1982.4307558
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发表时间：
1982-01-01
影响因子：
10.6
通讯作者：
Vardi, Y
中科院分区：
文献类型：
--
作者：
Shepp, L A;Vardi, Y
以前的模型发射断层扫描（ET）不区分从透射断层扫描的ET的物理。我们给出了ET的更精确的通用数学模型，其中未知的发射密度lambda = lambda（x，y，z）生成D个检测器单元d中的每个检测器单元d中的计数n（*）（d）的数量，并且将从D个检测器单元d中的每个检测器单元d中的计数n（*）（d）的数量重构。在该模型中，我们给出了一个算法，用于确定λ的估计值，使概率p（n（*））最大化|在所有可能的密度λ上观察实际的探测器计数数据n（*）。设独立泊松变量n（B）的均值λ（B）未知，B = 1，...，B表示划分包含发射器的对象的每个B框（像素）中未观测到的发射的数量。假设在检测器单元d中以概率p（B，d）检测到框B中的每个发射，d = 1，.，其中p（B，d）是一步转移矩阵，假设已知。我们观察每个检测器单元d中的发射的总数n（*）= n（*）（d），并且想要估计未知的lambda = lambda（B），B = 1，.，B。对于每个lambda，观测数据n（*）具有概率或似然p（n（*））|lambda）。数理统计的EM算法从初始估计λ（0）开始，并给出以下简单的迭代过程，用于从旧估计λ_dainsertion_mark（old）获得新估计λ_dainsertion_mark（new），以获得λ_dainsertion_mark（k），k = 1，2，.，重定位标记（新）（B）=重定位标记（旧）（B）（n（*）p（B，d）从d=1到D的总和/重定位标记（）旧（B（'））p（B（'），d）从B（'）=1到B）的总和，B=1，... B。
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.