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Methods for Improving the Quality of Image Reconstruction in Computerized Tomography

  • Doina Carp
  • Constantin Popa
  • Cristina Şerban
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 179)

Abstract

Appeared in early 50s in medical applications, but essentially developed since early 70s Computerized Tomography (CT) has became nowadays one of the most powerful investigation tool in science and technology. In this chapter we present several classes of methods which can rise the efficiency and/or robustness of classical projection-based algorithms (as Kaczmarz and Cimmino type algorithms) for algebraic reconstruction of images in Computerized Tomography.

References

  1. Bautu, A., Bautu, E., Popa, C.: Evolutionary algorithms in image reconstruction from limited data. In: Proceedings of the 4th Workshop on Mathematical Modelling on Environmental and Life Sciences Problems, 7–10 Sept 2005. Constanta, Romania, Publishing House of the Romanian Academy, Bucharest, Romania, pp. 15–26 (2006)Google Scholar
  2. Butz F., Fügenschuh, Wood J.N., Breuer M.: Particle-image velocimetry and the assignement problem. In: Fink, A., Fgenschuh, A., Geiger, M. (eds.), Operations Research Proceedings (GOR (Gesellschaft fr Operations Research e.V.)). Springer, Cham, pp. 243–249 (2016)Google Scholar
  3. Censor, Y., Stavros, A.Z.: Parallel optimization: theory, algorithms and applications. In: Numerical Mathematics and Scientific Computation Series. Oxford Univ. Press, New York (1997)Google Scholar
  4. Censor, Y., Gordon, D., Gordon, R.: Component averaging: an efficient iterative parallel algorithm for large and sparse unstructured problems. Parallel Comput. 27, 777–808 (2001)MathSciNetCrossRefGoogle Scholar
  5. Censor, Y., Pantelimon, I., Popa, C.: Family constraining of iterative algorithms. Numer. Alg. 66(2), 323–338 (2014)MathSciNetCrossRefGoogle Scholar
  6. Cimmino, G.: Calcolo approssiomatto per le soluzioni dei sistemi di equazioni lineari. Ric. Sci. progr. tecn. econom. naz. 1, 326–333 (1938)zbMATHGoogle Scholar
  7. Herman, G.T.: Image reconstruction from projections. In: The Fundamentals of Computerized Tomography. Academic Press, New York (1980)Google Scholar
  8. Kaczmarz, S.: Angenaherte Auflosung von Systemen linearer Gleichungen. Bull. Acad. Polonaise Sci. et Lettres A , 355–357 (1937)Google Scholar
  9. Kaczmarz, S.: Approximate solution of systems of linear equations. Int. J. Control. 57(6), 1269–1271 (1993)MathSciNetCrossRefGoogle Scholar
  10. Koltracht, I., Lancaster, P.: Constraining strategies for linear iterative processes. IMA J. Numer. Anal. 10, 555–567 (1990)MathSciNetCrossRefGoogle Scholar
  11. Köstler, H., Popa, C., Preclik, T., Rüde, U.: On Kaczmarz’s projection iteration as a direct solver for linear least squares problems. Linear Alg. Appl. 436(2), 389–404 (2012)MathSciNetCrossRefGoogle Scholar
  12. Natterer, F.: The Mathematics of Computerized Tomography. Wiley, New York (1986)Google Scholar
  13. Nicola, A., Petra, S., Popa, C., Schnörr, C.: A general extending and constraining procedure for linear iterative methods. Int. J. Comput. Math. 89(2), 231–253 (2012)MathSciNetCrossRefGoogle Scholar
  14. 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)MathSciNetzbMATHGoogle Scholar
  15. Petra, S., Popa, C., Schnörr, C.: Accelerating Constrained SIRT with Applications in Tomographic Particle Image Reconstruction (2009). http://www.ub.uni-heidelberg.de/archiv/9477
  16. Popa, C., Zdunek, R.: Kaczmarz extended algorithm for tomographic image reconstruction from limited-data. Math. Comput. Simul. 65, 579–598 (2004)MathSciNetCrossRefGoogle Scholar
  17. Popa, C.: Constrained Kaczmarz extended algorithm for image reconstruction. Linear Alg. Appl. 429(8–9), 2247–2267 (2008)MathSciNetCrossRefGoogle Scholar
  18. Popa, C.: A hybrid Kaczmarz—conjugate gradient algorithm for image reconstruction. Math. Comput. Simul. 80(12), 2272–2285 (2010)MathSciNetCrossRefGoogle Scholar
  19. Popa, C.: Extended and constrained diagonal weighting algorithm with application to inverse problems in image reconstruction. Inv. Probl. 26(6), 17p (2010)MathSciNetCrossRefGoogle Scholar
  20. Tanabe, K.: Projection method for solving a singular system of linear equations and its applications. Numer. Math. 17, 203–214 (1971)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceOvidius UniversityConstantaRomania

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