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

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 Ehl 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 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.