Reconstruction Algorithms and Scanning Geometries in Tomographic Imaging

Chapter

Abstract

In general tomographic imaging consists of two steps: the acquisition of data and the reconstruction. Thereby the following triple problem has to be solved: choose an efficient reconstruction algorithm, identify the optimal sampling conditions imposed on measured data as required by this reconstruction algorithm, and find an efficient way of collecting such data in the practice.

Keywords

Filter Back Projection Inversion Formula Singular Line Algebraic Reconstruction Technique Radon Data 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Matej S, Lewitt RM (1996) Practical consideration for 3-D image reconstruction using spherically symmetric volume elements. IEEE Trans Med Imaging 15:68–78PubMedCrossRefGoogle Scholar
  2. 2.
    Higgins JR (1996) Sampling theory in Fourier and signal analysis: foundations. Oxford University, New York, NYGoogle Scholar
  3. 3.
    Petersen DP, Middleton D (1962) Sampling and reconstruction of wave-number limited functions in N-dimensional Euclidean spaces. Inf Control 5:279–323CrossRefGoogle Scholar
  4. 4.
    Schwartz L (1950–1951) Théorie des distributions, 1–2. HermannGoogle Scholar
  5. 5.
    Gelfand IM, Shilov GE (1964) Generalized functions, vol I. Academic, New York, NYGoogle Scholar
  6. 6.
    Kak A, Slaney M (2001) Principles of computerized tomographic imaging. SIAM, Philadelphia, PACrossRefGoogle Scholar
  7. 7.
    Rattey PA, Lindgren AG (1981) Sampling the 2-D Radon transform. IEEE Trans Acoust Speech Signal Process ASSP-29:994–1002CrossRefGoogle Scholar
  8. 8.
    Natterer F (1986) The mathematics of computerized tomography. Wiley, ChichesterGoogle Scholar
  9. 9.
    Faridani A, Ritman EL (2000) High-resolution computed tomography from efficient sampling. Inverse Probl 16:635–650CrossRefGoogle Scholar
  10. 10.
    Lewitt RM, Matej S (2003) Overview of methods for image reconstruction from projections in emission computed tomography. Proc IEEE 91:1588–1611CrossRefGoogle Scholar
  11. 11.
    Herman GT, Lakshminarayanan AV, Naparstek A (1976) Convolution reconstruction techniques for divergent beams. Comput Biol Med 6:259–274PubMedCrossRefGoogle Scholar
  12. 12.
    Lakshminarayanan AV (1975) Reconstruction from divergent ray data. Department of Computer Science technical report TR-92, State University of New York, Buffalo, NYGoogle Scholar
  13. 13.
    Natterer F (1993) Sampling in fan beam tomography. SIAM J Appl Math 53(2):358–380CrossRefGoogle Scholar
  14. 14.
    Marr RB (1974) On the reconstruction of a function on a circular domain from a sampling of its line integrals. J Math Anal Appl 45:357–374CrossRefGoogle Scholar
  15. 15.
    Davison ME (1981) A singular value decomposition for the Radon transform in n-dimensional Euclidean space. Numer Funct Anal Optim 3(3):321–340CrossRefGoogle Scholar
  16. 16.
    Lisin FS (1976) Conditions for the completeness of the system of polynomials. Math Notes18(4):891–894CrossRefGoogle Scholar
  17. 17.
    Xu Y (2006) A new approach to the reconstruction of images from Radon projections. Adv Appl Math 36:388–420CrossRefGoogle Scholar
  18. 18.
    Xu Y, Tischenko O, Hoeschen C. A new reconstruction algorithm for Radon data. Proceedings of SPIE, vol 6142. Medical imaging 2006: physics of medical imaging, pp 791–798Google Scholar
  19. 19.
    Hakopian H (1982) Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type. J Approx Theory 34:286–305CrossRefGoogle Scholar
  20. 20.
    Tischenko O, Xu Y, Hoeschen C (2010) Main features of the tomographic reconstruction algorithm OPED. Radiat Prot Dosimetry 139(1–3):204–207, Advance Access publication 12 Feb 2010PubMedCrossRefGoogle Scholar
  21. 21.
    Noo F, Clackdoyle R, Pack JD (2004) A two-step Hilbert transform method for 2D image reconstruction. Phys Med Biol 49:3903–3023PubMedCrossRefGoogle Scholar
  22. 22.
    Mikhlin SG (1957) Integral equations and their applications to certain problems in mechanics, mathematical physics and technology. Pergamon, New YorkGoogle Scholar
  23. 23.
    Tuy HK (1983) An inversion formula for cone-beam reconstruction. SIAM J Appl Math 43:546–552CrossRefGoogle Scholar
  24. 24.
    Feldkamp LA, Davis LC, Kress JW (1984) Practical cone-beam algorithm. J Opt Soc Am A1(6):612–619CrossRefGoogle Scholar
  25. 25.
    Wang G, Lin T-H, Cheng PC (1993) A general cone-beam reconstruction algorithm. IEEE Trans Med Imaging 12:486–496PubMedCrossRefGoogle Scholar
  26. 26.
    Turbell H (2001) Ph.D. ThesisGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Research Unit Medical Radiation Physics and DiagnosticsHelmholtz Zentrum München - German Research Center for Environmental HealthNeuherbergGermany

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