In this paper, we consider the standard form of two kinds of Kaczmarz-Tanabe type methods, one derived from the Kaczmarz method and the other derived from the symmetric Kaczmarz method. As a famous image reconstruction method in computed tomography, the Kaczmarz method has both advantage and disadvantage. The advantage are simple and easy to implement, while the disadvantages are slow convergence speed, and the symmetric Kaczmarz method is the same. For the standard form of this method, once the iterative matrix is generated, it can be used continuously in the subsequent iterations. Moreover, the iterative matrix can be stored in the image reconstructive devices, which makes the Kaczmarz method and the symmetric Kaczmarz method can be used like the simultaneous iterative reconstructive techniques (SIRT). Meanwhile, theoretical analysis shows that the convergence rate of symmetric Kaczmarz method is better than the Kaczmarz method but is slightly worse than that of two iterations Kaczmarz method, which is verified numerically. Numerical experiments also show that the convergence rates of the Kaczmarz method and the symmetric Kaczmarz method are better than the SIRT methods and slightly worse than CGMN method in some cases. However, the Kaczmarz Tanabe type methods have better problem adaptability.
翻译：在本文中, 我们考虑两种Kaczmarz- Tanabe类型方法的标准形式, 一种来自Kaczmarz 方法, 另一种来自对称 Kaczmarz 方法。 Kaczmarz 方法在计算断层法中是一种著名的图像重建方法, 作为一种著名的图像重建方法, Kaczmarz 方法具有优势和劣势。 其优点既简单又容易实施, 而劣势是缓慢的趋同速度, 而对称的Kaczmarz 方法也是一样的。 对于这种方法的标准形式, 一旦生成迭代矩阵, 它就可以在随后的迭代中持续使用。 此外, 迭代矩阵可以存储在图像重建设备中, 使Kaczmarz 方法和对称卡茨 方法的对称重塑集成比 。 然而, 理论分析表明, 辛兹 方法的趋同比卡茨 的平流方法要好得多。 然而, 卡茨 的对调制方法也比卡茨 方法的对称方法更差。
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