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Constraining by a family of strictly nonexpansive idempotent functions with applications in image reconstruction

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Abstract

When solving an image reconstruction problem a previous knowledge concerning the original image may lead to various constraining strategies. A convergence result has been previously proved for a constrained version of the Kaczmarz projection algorithm with a single strictly nonexpansive idempotent function with a closed image. In this paper we consider a more general projection based iterative method and a family of such constraining functions with some additional hypotheses in order to better use the a priori information for every approximation calculated. We present a particular family of box-constraining functions which satisfies our assumptions and we design an adaptive algorithm that uses an iteration-dependent family of constraining functions for some numerical experiments of image reconstruction on Tomographic Particle Image Velocimetry.

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References

  1. Censor, Y., Stavros, A.Z.: Parallel Optimization: Theory, Algorithms and Applications. Numer. Math. and Sci. Comp.. Oxford University Press, New York (1997)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Herman, G.T.: Image Reconstruction from Projections. The Fundamentals of Computerized Tomography. Academic Press, New York (1980)

    MATH  Google Scholar 

  4. Koltracht, I., Lancaster, P.: Constraining strategies for linear iterative processes. IMA J. Numer. Anal. 10, 555–567 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Nicola, A., Petra, S., Popa, C., Schnörr, C.: On a general extending and constraining procedure for linear iterative methods. Int. J. Comput. Math. 89, 231–253 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Petra, S., Popa, C., Schnörr, C.: Accelerating constrained SIRT with applications in tomographic particle image reconstruction (2009). Preprint. Available at http://www.ub.uni-heidelberg.de/archiv/9477

  7. Petra, S., Schnörr, C.: TomoPIV meets Compressed Sensing. Pure Math. Appl. 20, 49–76 (2009)

    MATH  Google Scholar 

  8. Popa, C.: Projection Algorithms—Classical Results and Developments. Applications to Image Reconstruction. Lambert Academic, Saarbrücken (2012)

    Google Scholar 

  9. Popa, C.: Extended and constrained diagonal weighting algorithm with application to inverse problems in image reconstruction. Inverse Probl. 26 (2010). doi:10.1088/0266-5611/26/6/065004

  10. Popa, C., Zdunek, R.: Kaczmarz extended algorithm for tomographic image reconstruction from limited-data. Math. Comput. Simul. 65, 579–598 (2004)

    Article  MathSciNet  Google Scholar 

  11. Simmons, G.F.: Introduction to Topology and Modern Analysis. McGraw-Hill, New York (1963)

    MATH  Google Scholar 

  12. Tanabe, K.: Projection method for solving a singular system of linear equations and its applications. Numer. Math. 17, 203–214 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We owe our deepest gratitude to Dr. Ştefania Petra and Prof. Cristoph Schnörr for the numerical data they provided. We wish to express our thanks to professor Yair Censor for his valuable comments and advices. Last but not least, we would like to thank our anonymous referees for their helpful sugguestions which much improved the first versions of the manuscript.

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Authors and Affiliations

  1. Ovidius University of Constanta, Blvd. Mamaia 124, Constanta, 900527, Romania

    Ioana Pantelimon & Constantin Popa

  2. “Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Mathematics, Calea 13 Septembrie, Nr. 13, Bucharest, 050711, Romania

    Constantin Popa

Authors
  1. Ioana Pantelimon
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  2. Constantin Popa
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Correspondence to Constantin Popa.

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Communicated by Per Christian Hansen.

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Pantelimon, I., Popa, C. Constraining by a family of strictly nonexpansive idempotent functions with applications in image reconstruction. Bit Numer Math 53, 527–544 (2013). https://doi.org/10.1007/s10543-012-0414-0

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  • DOI: https://doi.org/10.1007/s10543-012-0414-0

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