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An investigation of convergence rates in expectation maximization (EM) iterative reconstruction

机译：期望最大化收敛率的调查（EM）迭代重建

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The non-uniform convergence property, i.e. the low-frequency components of the image will converge earlier than the high ones, is an important property to the expectation maximization (EM) iterative process, by which some researchers [1] [2] tried to improve the convergence rates of EM algorithm. In our previous work [3][4], we proposed a method based on this property for scatter compensation in SPECT imaging called fast estimation of scatter components (FESC), by which we can estimate the scatter components in projections with good accuracy in high speed. However, there are still many problems remaining unclear about the non-uniform convergence properties of EM iteration. And it is not convenient to analyze the properties of EM algorithm directly by general linear methods because EM iteration belongs to a nonlinear process. In this paper, we completed an investigation by which we can comprehend the non-uniform convergence properties of EM iteration more clearly. A more significant result is that, with the same analysis method in our investigation, we can prove theoretically that the ordered subsets expectation maximization (OS-EM) algorithm possesses a more uniform convergence property than the maximum likelihood expectation maximization (ML-EM) algorithm, which contributes to OS-EM algorithm having much higher convergence rates than ML-EM algorithm. We divided the EM iteration into two processes, a back-projection process to acquire the information for updating image and an image-update process to modify the image. The former belongs to a linear process that can be analyzed directly by singular value decomposition (SVD) and the late belongs to a nonlinear process that can be considered to be a modulation process and analyzed by Fourier transform analysis. The results showed that the non-uniform convergence property of EM algorithm is determined by its back-projection process, and the responses of frequency components proportionate to the square of the singular values of system transform matrix which always appears higher values to the low-frequency components than the high-frequency ones.
机译：非均匀收敛性，即，图像的低频分量会更早收敛比高的，是期望最大化（EM）迭代过程，通过该一些研究者[1] [2]尝试的重要性质提高EM算法的收敛速率。在我们以前的工作[3] [4]中，我们提出了一种基于该属性的方法，用于SPACT成像中的散射补偿，称为散射组件（FEC）的快速估计，我们可以通过高精度来估计投影中的散点分量速度。然而，仍存在许多问题仍然不清楚EM迭代的非均匀收敛性能。通过通用线性方法直接分析EM算法的性质是不方便的，因为EM迭代属于非线性过程。在本文中，我们完成了调查，我们可以更清楚地理解EM迭代的非均匀收敛性质。更重要的是，通过对我们调查中的相同的分析方法，可以从理论上证明有序子集期望最大化（OS-EM）算法具有比最大似然预期最大化（ML-EM）算法更均匀的会聚属性，这有助于OS-EM算法比ML-EM算法更高的收敛速率。我们将EM迭代划分为两个进程，是回投影过程，以获取更新图像的信息和图像更新过程来修改图像。前者属于线性过程，可以通过奇异值分解（SVD）直接分析，并且已经被视为非线性过程，其可以被认为是由傅里叶变换分析分析的调制过程。结果表明，EM算法的非均匀收敛性由其背投过程决定，频率分量的响应与系统变换矩阵的奇异值的平方成比例，始终看起来更高的低频值组件比高频。

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