Computing

, Volume 26, Issue 3, pp 189–195

Reconstructing pictures from projections: On the convergence of the ART algorithm with relaxation

  • M. R. Trummer
Article

Abstract

The convergence of the additive and linear ART algorithm with relaxation is proved in a new way and under weaker assumptions on the sequence of the relaation parameters than in earlier works. These algorithms are iterative methods for the reconstruction of digitized pictures from one-dimesional views. A second proof using elementary matrix algebra shows the geometric convergence of the linear ART algorithm with relaxation.

Über die Konvergenz des ART-Algorithmus für die Rekonstruktion von Bildern

Zusammenfassung

ART-Algorithmen sind iterative Methoden zur Rekonstruktion von digitalen Bildern aus ihren Projektionen. Die Konvergenz des additiven und linearen (nicht restringierten) ART-Algorithmus mit Relaxation wird unter weit schwächeren Voraussetzungen über die Relaxationsparameter als bei bisher bekannten Resultaten bewiesen. Ein anderer Beweis zeigt die geometrisch schnelle Konvergenz des linearen relaxierten ART-Algorithmus.

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References

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    Gordon, R., Bender, R., Herman, G. T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography. J. theor. Biol.29, 471–481 (1970).Google Scholar
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    Groetsch, C. W.: Generalized inverses of linear operators. Representation and approximation. (Monographs and textbooks in pure and applied mathematics, Vol. 37), p. 41. New York-Basel: M. Dekker 1977.Google Scholar
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    Herman, G. T., Lent, A., Lutz, P. H.: Iterative relaxation methods for image reconstruction. Proc. ACM '75 Annual Conf., Minneapolis, Minn., 169–174 (Oct. 1975).Google Scholar
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    Marti, J. T.: On the convergence of the discrete ART algorithm for the reconstruction of digital pictures from their projections. Computing21, 105–111 (1979).Google Scholar
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    Tanabe, K.: Projection method for solving a singular system of linear equations and its applications. Num. Math.17, 203–214 (1971).Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • M. R. Trummer
    • 1
  1. 1.Seminar für Angewandte MathematikEidgenössische Technische HochschuleZürichSwitzerland

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