A unified treatment of some iterative algorithms in signal processing and image reconstruction.

*(English)*Zbl 1051.65067The 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.

Reviewer: Karel Najzar (Praha)

##### MSC:

65J15 | Numerical solutions to equations with nonlinear operators (do not use 65Hxx) |

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 |