Image recovery by convex combinations of projections.

*(English)*Zbl 0752.65045The functional analytic question discussed in this paper is: For which \(T\) one has weak convergence of the sequences \(\{T^ nx\}^ \infty_{n=0}\) 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. It is shown via more abstract results that one may choose \(T=\alpha_ 0id+\sum^ r_{i=1}\alpha_ iT_ i\) with \(T_ i=id+\lambda_ i(P_ i-id)\), \(0<\lambda_ i<2\), \(\alpha_ j>0\), \(\sum^ r_ 0\alpha_ j=1\). It is argued that this choice is more suitable for parallel computer implementation than the classical \(T=T_ r\dots T_ 1\).

Reviewer: H.von Weizsäcker (Kaiserslautern)

##### MSC:

65J10 | Numerical solutions to equations with linear operators |

65Y05 | Parallel numerical computation |

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

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

##### Keywords:

parallel computing; projections; weak approximations; image analysis; weak convergence; common fixed point
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\textit{G. Crombez}, J. Math. Anal. Appl. 155, No. 2, 413--419 (1991; Zbl 0752.65045)

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