Wang W, Gindi G
Department of Electrical Engineering, SUNY at Stony Brook 11794, USA.
Phys Med Biol. 1997 Nov;42(11):2215-32. doi: 10.1088/0031-9155/42/11/015.
The ability to theoretically model the propagation of photon noise through PET and SPECT tomographic reconstruction algorithms is crucial in evaluating the reconstructed image quality as a function of parameters of the algorithm. In a previous approach for the important case of the iterative ML-EM (maximum-likelihood-expectation-maximization) algorithm, judicious linearizations were used to model theoretically the propagation of a mean image and a covariance matrix from one iteration to the next. Our analysis extends this approach to the case of MAP (maximum a posteriori)-EM algorithms, where the EM approach incorporates prior terms. We analyse in detail two cases: a MAP-EM algorithm incorporating an independent gamma prior, and a one-step-late (OSL) version of a MAP-EM algorithm incorporating a multivariate Gaussian prior, for which familiar smoothing priors are special cases. To validate our theoretical analyses, we use a Monte Carlo methodology to compare, at each iteration, theoretical estimates of mean and covariance with sample estimates, and show that the theory works well in practical situations where the noise and bias in the reconstructed images do not assume extreme values.
从理论上对光子噪声通过正电子发射断层扫描（PET）和单光子发射计算机断层扫描（SPECT）断层重建算法的传播进行建模的能力，对于评估作为算法参数函数的重建图像质量至关重要。在之前针对迭代最大似然期望最大化（ML-EM）算法这一重要情况的方法中，通过明智的线性化处理从理论上对平均图像和协方差矩阵从一次迭代到下一次迭代的传播进行建模。我们的分析将这种方法扩展到最大后验概率（MAP）-EM算法的情况，其中EM方法纳入了先验项。我们详细分析了两种情况：一种纳入独立伽马先验的MAP-EM算法，以及一种纳入多元高斯先验的MAP-EM算法的一步延迟（OSL）版本，常见的平滑先验是其特殊情况。为了验证我们的理论分析，我们使用蒙特卡罗方法在每次迭代时将均值和协方差的理论估计值与样本估计值进行比较，并表明在重建图像中的噪声和偏差不呈现极端值的实际情况下该理论效果良好。
