×

Image recovery by convex combinations of projections. (English) Zbl 0752.65045

The 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 \(T=T_ r\dots T_ 1\).

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amemiya, I.; Ando, T., Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged), 26, 239-244 (1965) · Zbl 0143.16202
[2] Bregman, L. M., Finding the common point of convex sets by the method of successive projection, Dokl. Akad. Nauk SSSR, 162, 487-490 (1965) · Zbl 0142.16804
[3] G. CrombezGlas. Mat.; G. CrombezGlas. Mat. · Zbl 0754.46015
[4] Groetsch, C. W., A nonstationary iterative process for nonexpansive mappings, (Proc. Amer. Math. Soc., 43 (1974)), 155-158 · Zbl 0254.47078
[5] Halperin, I., The product of projection operators, Acta Sci. Math., 23, 96-99 (1962) · Zbl 0143.16102
[6] Kirk, W. A., On successive approximations for non-expansive mappings in Banach spaces, Glasgow Math. J., 12, 6-9 (1971) · Zbl 0223.47024
[7] Lent, A.; Tuy, H., An iterative method for the extrapolation of bandlimited functions, J. Math. Anal. Appl., 83, 554-565 (1981) · Zbl 0472.65006
[8] Opial, Z., Nonexpansive and monotone mappings in Banach spaces, (Lecture Notes in Math., Vol. 67 (1967), Division of Applied Mathematics, Brown University: Division of Applied Mathematics, Brown University Providence, RI), No. 1 · Zbl 0179.19902
[9] Sezan, M. I.; Stark, H., Image restoration by the method of convex projections, Part 2, Applications and Numerical Results, IEEE Trans. Med. Imaging, 1, 95-102 (1982)
[10] Sezan, M. I.; Stark, H., Applications of convex projection theory to image recovery in tomography and related areas, (Image Recovery: Theory and Application (1987), Academic Press: Academic Press New York)
[11] Youla, D. C.; Webb, H., Image restoration by the method of convex projections, Part 1, Theory, IEEE Trans. Med. Imaging, 1, 81-94 (1982)
[12] Youla, D. C., Mathematical theory of image restoration by the method of convex projections, (Image Recovery: Theory and Application (1987), Academic Press: Academic Press New York) · Zbl 0161.13202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.