Parallel methods in image recovery by projections onto convex sets.

*(English)*Zbl 0789.65039This paper is a continuation of the author’s work [J. Math. Anal. Appl. 155, No. 2, 413-419 (1991; Zbl 0752.65045)]. He studies weak convergence of sequences \(\{S^ n x\}^ \infty_{n=0}\) to a common fixed point of a finite number of projections \(P_ 1,\dots,P_ r\) (onto convex closed subsets \(C_ 1,\dots,C_ r\)) in a Hilbert space where \(S-1\) is a suitable linear combination of the \(P_ i - 1\). This paper proves this if the operator \(S\) does not have a multiple of the identity as a convex component. The author refers to experimental studies, but without details.

Reviewer: H.von Weizsäcker (Kaiserslautern)

##### MSC:

65J05 | General theory of numerical analysis in abstract spaces |

65Y05 | Parallel numerical computation |

47A50 | Equations and inequalities involving linear operators, with vector unknowns |

##### Keywords:

parallel computing; weak approximations; image analysis; weak convergence; common fixed point; projections; Hilbert space
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##### References:

[1] | L. M. Bregman: Finding the common point of convex sets by the method of successive projection. Dokl. Akad. Nauk SSSR 162 (1965), 487-490. |

[2] | G. Crombez: Image recovery by convex combinations of projections. J. Math. Anal. Appl. 155 (1991), 413-419. · Zbl 0752.65045 |

[3] | M. I. Sezan, H. Stark: Image restoration by the method of convex projections: part 2 - applications and numerical results. IEEE Trans. Medical Imaging 1 (1982), 95-102. |

[4] | M. I. Sezan, H. Stark: Applications of convex projection theory to image recovery in tomography and related areas. Image recovery: theory and application, Academic Press Inc. New York, 1987. · Zbl 0627.65133 |

[5] | D. C. Youla, H. Webb: Image restoration by the method of convex projections: part 1 - theory. IEEE Trans. Medical Imaging 1 (1982), 81-94. |

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