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Convergence analysis for column-action methods in image reconstruction

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An Erratum to this article was published on 18 November 2016

Abstract

Column-oriented versions of algebraic iterative methods are interesting alternatives to their row-version counterparts: they converge to a least squares solution, and they provide a basis for saving computational work by skipping small updates. In this paper we consider the case of noise-free data. We present a convergence analysis of the column algorithms, we discuss two techniques (loping and flagging) for reducing the work, and we establish some convergence results for methods that utilize these techniques. The performance of the algorithms is illustrated with numerical examples from computed tomography.

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References

  1. Aharoni, R., Censor, Y.: Block-iterative projection methods for parallel computation of solutions to convex feasibility problems. Lin. Alg. Appl. 120, 165–175 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andersen, A.H., Kak, A.C.: Simultaneous algebraic reconstruction technique (SART): a superior implementation of the art algorithm. Ultrasonic Imag. 76, 81–94 (1984)

    Article  Google Scholar 

  3. Bai, Z.-Z., Jin, C.-H.: Column-decomposed relaxation methods for the overdetermined systems of linear equations. Int. J. Appl. Math. 13(1), 71–82 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Björck, Å., Elfving, T.: Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations. BIT 19(2), 145–163 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cao, Z.H.: On the convergence of iterative methods for solving singular linear systems. J. Comp. Appl. Math. 145(1), 1–9 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM Rev. 23, 444–466 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Censor, Y., Eggermont, P.P.B., Gordon, D.: Strong underregularization in Kaczmarz’s method for inconsistent systems. Numer. Math. 41(1), 83–92 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Censor, Y., Elfving, T.: Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem. SIAM J. Matrix Anal. Appl. 24 (1), 40–58 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Censor, Y., Elfving, T., Herman, G.T., Nikazad, T.: On diagonally relaxed orthogonal projection methods. SIAM J. Sci. Comput. 30(1), 473–504 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Censor, Y., Gordon, D., Gordon, R.: BICAV: an inherently parallel algorithm for sparse systems with pixel-dependent weighting. IEEE Trans. Med. Imag. 20, 1050–1060 (2001)

    Article  Google Scholar 

  11. Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford University Press, New York (1997)

    MATH  Google Scholar 

  12. Eggermont, P.P.B., Herman, G.T., Lent, A.: Iterative algorithms for large partitioned linear systems, with applications to image reconstruction. Lin. Alg. Appl. 40, 37–67 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Elfving, T.: Block-iterative methods for consistent and inconsistent linear equations. Numer. Math. 35(1), 1–12 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Elfving, T., Nikazad, T.: Properties of a class of block-iterative methods. Inverse Probl. 25(11), 115011 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29(3), 47–81 (1970)

    Article  Google Scholar 

  16. Haltmeier, M.: Convergence analysis of a block iterative version of the loping Landweber-Kaczmarz iteration. Nonlin. Anal. 71, e2912–e2919 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hansen, P.C., Saxild-Hansen, M.: AIR tools—a MATLAB package of algebraic iterative reconstruction methods. J. Comp. Appl. Math. 236(8), 2167–2178 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jørgensen, J.S., Sidky, E.Y., Hansen, P.C., Pan, X.: Empirical average-case relation between undersampling and sparsity in X-ray CT. Inverse Probl. Imag. 9(2), 431–446 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiang, M., Wang, G.: Convergence studies on iterative algorthms for image reconstruction. IEEE Trans. Med. Imag. 22(5), 569–579 (2003)

    Article  Google Scholar 

  20. Keller, H.B.: On the solution of singular and semidefinite linear systems by iteration. J. Soc. Indus. Appl. Math. Ser. B 2, 281–290 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kindermann, S., Leitão, A.: Convergence rates for Kaczmarz-type regularization methods. Inverse Probl. Imag. 8(1), 149–172 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Needell, D., Tropp, J.O.: Paved with good intentions: analysis of a randomized Kaczmarz method. Lin. Alg. Appl. 441, 199–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Watt, D.W.: Column-relaxed algebraic reconstruction algorithm for tomography with noisy data. Appl. Opt. 33(20), 4420–4427 (1994)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Linköping University, 581 83, Linköping, Sweden

    Tommy Elfving

  2. Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    Per Christian Hansen

  3. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran

    Touraj Nikazad

Authors
  1. Tommy Elfving
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  2. Per Christian Hansen
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  3. Touraj Nikazad
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Corresponding author

Correspondence to Per Christian Hansen.

Additional information

This work is a part of the project HD-Tomo funded by Advanced Grant No. 291405 from the European Research Council.

An erratum to this article is available at http://dx.doi.org/10.1007/s11075-016-0232-6.

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Elfving, T., Hansen, P.C. & Nikazad, T. Convergence analysis for column-action methods in image reconstruction. Numer Algor 74, 905–924 (2017). https://doi.org/10.1007/s11075-016-0176-x

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