The standard forms of the Kaczmarz-Tanabe type methods and their convergence theory
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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.
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