NOISE PROPERTIES OF THE EM ALGORITHM .1. THEORY
NOISE PROPERTIES OF THE EM ALGORITHM .1. THEORY
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DOI：
10.1088/0031-9155/39/5/004
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发表时间：
1994-05-01
影响因子：
3.5
通讯作者：
TSUI, BMW
中科院分区：
文献类型：
--
作者：
BARRETT, HH;WILSON, DW;TSUI, BMW
期望最大化（EM）算法是最大似然（ML）估计和图像重建的重要工具，特别是在医学成像中。它是一种非线性迭代算法，尝试找到生成数据集的对象的 ML 估计。该算法的收敛性和其他确定性属性已得到很好的确立，但对于数据中的噪声如何影响最终重建图像中的噪声知之甚少。在本文中，我们详细介绍了这些统计特性。我们想到的具体应用是发射断层扫描中的图像重建，但结果对于 EM 算法的任何应用都有效，其中数据集可以通过泊松统计来描述。我们证明图像中像素灰度级的概率密度函数可以很好地通过对数正态定律来近似。推导出灰度级方差和像素间协方差的表达式。 方差起初随着迭代次数而快速增加，但最终随着接近 ML 估计而饱和。此外，任何迭代次数的方差都有一个与平均图像的平方成比例的因子（尽管其他因子也可能取决于平均图像），因此标准差的图类似于对象本身。因此，图像的低强度区域往往具有低噪声。相比之下，线性重建方法（例如断层扫描中的滤波反投影）显示出更加全局的噪声模式，物体的高强度区域在相当远的低强度区域产生噪声。本文的理论结果取决于两个近似值，但在本系列的第二篇论文中，我们通过蒙特卡罗模拟证明了这些近似值在发射断层扫描的各种条件下都是合理的。因此，该理论可以用作计算图像质量客观品质因数的基础。
The expectation-maximization (EM) algorithm is an important tool for maximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. It is a nonlinear iterative algorithm that attempts to find the ML estimate of the object that produced a data set. The convergence of the algorithm and other deterministic properties are well established, but relatively little is known about how noise in the data influences noise in the final reconstructed image. In this paper we present a detailed treatment of these statistical properties. The specific application we have in mind is image reconstruction in emission tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistics. We show that the probability density function for the grey level at a pixel in the image is well approximated by a log-normal law. An expression is derived for the variance of the grey level and for pixel-to-pixel covariance. The variance increases rapidly with iteration number at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportional to the square of the mean image (though other factors may also depend on the mean image), so a map of the standard deviation resembles the object itself. Thus low-intensity regions of die image tend to have low noise. By contrast, linear reconstruction methods, such as filtered back-projection in tomography, show a much more global noise pattern, with high-intensity regions of the object contributing to noise at rather distant low-intensity regions. The theoretical results of this paper depend on two approximations, but in the second paper in this series we demonstrate through Monte Carlo simulation that the approximations are justified over a wide range of conditions in emission tomography. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.