A unified treatment of some iterative algorithms in signal processing and image reconstruction. (English) Zbl 1051.65067

The author gives a unified treatment of several well-known algorithms in signal processing and image reconstruction. These are special case of the Krasnoselskii-Mann iterative procedure [cf. W. R. Mann, Proc. Am. Math. Soc. 4, 506–510 (1953; Zbl 0050.11603)] to finding fixed points of nonexpansive continuous operators on Hilbert space. They include the Gerchberg-Papoulis method [cf. A. Papoulis, IEEE Trans. Circuits and Systems CAS-22, No. 9, 735–742 (1975)] for bandlimited extrapolation, the SART algorithm of A. Anderson and A. Kak [Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm. Ultrason. Imaging 6, 81–94 (1984)], the Landweber and projected Landweber algorithm [cf. L. Landweber, Am. J. Math. 73, 615–624 (1951; Zbl 0043.10602)], simultaneous and sequential methods for solvig the convex feasibility problem, the ART and Cimmino methods [cf. G. Cimmino, Ric. Sci. Progr. Tecn. Econom. Naz. 1, 326–333 (1938; Zbl 0018.41802)]for solving linear systems of equations, the CQ algorithm for solving the split feasibility problem and Z. O. Dolidze’s procedure [Ekonom. i Mat. Metody 18, No. 5, 925–929 (1982)] for the variational inequality problem for monotone operators.


65J15 Numerical solutions to equations with nonlinear operators
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
47J25 Iterative procedures involving nonlinear operators
65K05 Numerical mathematical programming methods
90C25 Convex programming
65F10 Iterative numerical methods for linear systems
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