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Parallel methods in image recovery by projections onto convex sets. (English) Zbl 0789.65039
This 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.

##### MSC:
 65J05 General theory of numerical analysis in abstract spaces 65Y05 Parallel numerical computation 47A50 Equations and inequalities involving linear operators, with vector unknowns
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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. [6] D. C. Youla: Mathematical theory of image restoration by the method of convex projections. Image recovery: theory and application, Academic Press Inc. New York, 1987.
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