In this paper, we consider the standard forms of two kinds of Kaczmarz-Tanabe type methods, one is derived from the Kaczmarz method and the other is derived from the symmetric Kaczmarz method. As a famous image reconstruction method in computerized tomography, the Kaczmarz method is simple and easy to implement, but its convergence speed is slow, so is the symmetric Kaczmarz method. When the standard forms of the Kaczmarz-Tanabe type methods are obtained, their iteration matrices can be used continuously in the subsequent iterations. Moreover, the iteration matrices can be stored in the image reconstruction devices, which enables the Kaczmarz method and the symmetric Kaczmarz method to be used like the simultaneous iterative reconstructive techniques (SIRT). Meanwhile, theoretical analysis shows that the convergence rate of the symmetric Kaczmarz-Tanabe method is better than that of the Kaczmarz-Tanabe method but is slightly worse than that of two-step Kaczmarz-Tanabe method, which is verified numerically. Numerical experiments also show that the convergence rates of the Kaczmarz-Tanabe method and the symmetric Kaczmarz-Tanabe method are better than those of the SIRT methods.

翻译：本文考虑两类Kaczmarz-Tanabe型方法的标准形式，一类源自Kaczmarz方法，另一类源自对称Kaczmarz方法。作为计算机断层扫描中著名的图像重建方法，Kaczmarz方法具有简单易实现的优点，但其收敛速度较慢，对称Kaczmarz方法同样如此。获得Kaczmarz-Tanabe型方法的标准形式后，其迭代矩阵可在后续迭代中持续使用。此外，该迭代矩阵可存储于图像重建设备中，使Kaczmarz方法和对称Kaczmarz方法能够像同步迭代重建技术（SIRT）一样使用。同时，理论分析表明，对称Kaczmarz-Tanabe方法的收敛速度优于Kaczmarz-Tanabe方法，但略逊于两步Kaczmarz-Tanabe方法，这一结果得到了数值验证。数值实验还表明，Kaczmarz-Tanabe方法和对称Kaczmarz-Tanabe方法的收敛速度均优于SIRT方法。