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**Convergence and performance of parallel multiplicative algorithm for image reconstruction from total image.**
*(English)*
Zbl 1031.65146

The problem of reconstruction of the internal structure of an object using some input data called projections is one of the most important problems in image processing. Usually, it reduces to the problem of solving an integral equation and when the number of projections is large enough, then it can be solved analytically using methods based on the Radon inversion [V. N. Troyan and G. A. Ryzhikov, J. Math. Sci., New York 91, 2873-2882 (1998; Zbl 0902.65082)]. However, when the number of projections is limited, then it is better to apply iterative algorithms [Y. Censor and S. A. Zenios, Parallel optimization. Theory, algorithms, and applications. (Oxford University Press, Oxford) (1997; Zbl 0945.90064)].

In this paper two parallel iterative algorithms, namely PMSUM and ASUM, which is an asynchronous variation of PMSUM are introduced and discussed. Both algorithms are suitable for reconstructing an image when the input data are given as a function of projections. The author gives sufficient conditions for their convergence and discusses the results of several experiments. The main conclusion is that for image reconstruction with the very limited number of projections, the use of ASUM gives better results.

In this paper two parallel iterative algorithms, namely PMSUM and ASUM, which is an asynchronous variation of PMSUM are introduced and discussed. Both algorithms are suitable for reconstructing an image when the input data are given as a function of projections. The author gives sufficient conditions for their convergence and discusses the results of several experiments. The main conclusion is that for image reconstruction with the very limited number of projections, the use of ASUM gives better results.

Reviewer: Przemyslaw Stpiczynski (Lublin)

### MSC:

65R10 | Numerical methods for integral transforms |

65Y05 | Parallel numerical computation |

65R30 | Numerical methods for ill-posed problems for integral equations |

44A12 | Radon transform |

92C55 | Biomedical imaging and signal processing |

65Y20 | Complexity and performance of numerical algorithms |