Yavuz M, Fessler J A
Department of EECS, University of Michigan, Ann Arbor 48109-2122, USA.
Med Image Anal. 1998 Dec;2(4):369-78. doi: 10.1016/s1361-8415(98)80017-0.
Positron emission tomography (PET) measurements are usually precorrected for accidental coincidence events by real-time subtraction of the delayed-window coincidences. Randoms subtraction compensates on average for accidental coincidences but destroys the Poisson statistics. We propose and analyze two new approximations to the exact log-likelihood of the precorrected measurements, one based on a 'shifted Poisson' model, the other based on saddle-point approximations to the measurement of probability mass function (PMF). The methods apply to both emission and transmission tomography; however, in this paper we focus on transmission tomography. We compare the new models to conventional data-weighted least-squares (WLS) and conventional maximum-likelihood methods [based on the ordinary Poisson (OP) model] using simulations and analytic approximations. The results demonstrate that the proposed methods avoid the systematic bias of the WLS method, and lead to significantly lower variance than the conventional OP method. The saddle-point method provides a more accurate approximation to the exact log-likelihood than the WLS, OP and shifted Poisson alternatives. However, the simpler shifted Poisson method yielded comparable bias-variance performance to the saddle-point method in the simulations. The new methods offer improved image reconstruction in PET through more realistic statistical modeling, yet with negligible increase in computation time over the conventional OP method.
正电子发射断层扫描（PET）测量通常通过实时减去延迟窗符合事件来对偶然符合事件进行预校正。随机符合减法平均补偿偶然符合，但会破坏泊松统计。我们提出并分析了两种对预校正测量的精确对数似然的新近似方法，一种基于“移位泊松”模型，另一种基于对概率质量函数（PMF）测量的鞍点近似。这些方法适用于发射断层扫描和透射断层扫描；然而，在本文中我们专注于透射断层扫描。我们使用模拟和解析近似将新模型与传统的数据加权最小二乘法（WLS）和传统的最大似然方法[基于普通泊松（OP）模型]进行比较。结果表明，所提出的方法避免了WLS方法的系统偏差，并且比传统的OP方法具有显著更低的方差。鞍点方法比WLS、OP和移位泊松替代方法更精确地近似精确对数似然。然而，在模拟中，更简单的移位泊松方法产生了与鞍点方法相当的偏差 - 方差性能。新方法通过更现实的统计建模在PET中提供了改进的图像重建，并且与传统的OP方法相比，计算时间增加可忽略不计。
