Application of a Discrete Tomography Approach to Computerized Tomography

  • Y. Gerard
  • F. Feschet
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Linear programming is used in discrete tomography to solve the relaxed problem of reconstructing a function f with values in the interval [0,1]. The linear program minimizes the uniform norm Ax - 6∞ or the 1-norm ∞∞Ax - b∞∞1 of the error on the projections. We can add to this objective function a linear penalty function p(x) for trying to obtain smooth solutions. The same approach can be used in computerized tomography. The question is if it can provide better images than the classical methods of computerized tomography. The aim of this chapter is to provide a tentative answer. We present a preliminary study on real acquisitions from a phantom and provide reconstructions from a few projections, with qualities similar to the traditional methods.


Single Photon Emission Compute Tomography Gray Level Interior Point Method Order Subset Expectation Maximization Algebraic Reconstruction Technique 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aharoni, R., Herman, G.T., Kuba, A.: Binary vectors partially determined by linear equation systems. Discr. Math., 171, 1–16 (1997).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Censor, Y., Matej, S.: Binary steering of nonbinary iterative algorithms. In: Herman, G.T., Kuba, A. (ed.), Discrete Tomography, Foundations, Algorithms, and Applications. Birkhaüser, Boston, MA, pp. 285–296 (1999).Google Scholar
  3. 3.
    Feschet, F., Gerard, Y.: Computerized tomography with digital lines and linear programming. In: Andres, E., Damiand, G., Lienhardt, P. (eds.), Discrete Geometry in Computed Imagery, pp. 126–135 (2005).Google Scholar
  4. 4.
    Fishburn, P., Schwander, P., Shepp, L., Vanderbei, R.: The discrete Radon transform and its approximate inversion via linear programming. Discr. Appl. Math., 75, 39–61 (1997).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gerard, Y., Feschet, F.: Application of a discrete tomography algorithm to computerized tomography. Electr. Notes Discr. Math., 20, 501–517 (2005).CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gordon, R.: A tutorial on ART (algebraic reconstruction techniques). IEEE Trans. Nucl. Sci., 21, 31–43 (1974).Google Scholar
  7. 7.
    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).CrossRefGoogle Scholar
  8. 8.
    Gritzmann, P., de Vries, S., Wiegelmann, M.: Approximating binary images from discrete X-rays. SI AM J. Optimization, 11, 522–546 (2000).MATHCrossRefGoogle Scholar
  9. 9.
    Herman, G.T.: Image Reconstruction from Projections. Academic Press, New York, NY (1980).MATHGoogle Scholar
  10. 10.
    Herman, G.T.: Reconstruction of binary patterns from a few projections. In: Gunther, A., Levrat, B., and Lipps, H. (eds.), International Computing Symposium, North-Holland, Amsterdam, The Netherlands, pp. 371–378 (1974).Google Scholar
  11. 11.
    Herman, G.T., Lent, A.: Iterative reconstruction algorithms. Comput. Biol. Med., 6, 273–274 (1976).CrossRefGoogle Scholar
  12. 12.
    Herman, G.T., Lent, A.: A computer implementation of a Bayesian analysis of image reconstruction. Information and Control, 31, 364–384 (1976).CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Herman, G.T., Meyer, L.B.: Algebraic reconstruction techniques can be made computationally efficient. IEEE Trans. Med. Imag., 12, 600–609 (1993).CrossRefGoogle Scholar
  14. 14.
    Kaufman, L.: Maximum likelihood, least squares and penalized least squares for PET. IEEE Trans. Med. Imag., 12, 200–214 (1993).CrossRefGoogle Scholar
  15. 15.
    Kaufman, L., Neumaier, A.: PET regularization by envelope guided conjugate gradients. IEEE Trans. Med. Imag., 15, 385–389 (1996).CrossRefGoogle Scholar
  16. 16.
    Matej, S., Vardi, A., Herman, G.T., Vardi, E: Binary tomography using Gibbs priors. In: Herman, G.T., Kuba, A. (ed.), Discrete Tomography, Foundations, Algorithms, and Applications. Birkhauser, Boston, MA, pp. 191–211 (1999).Google Scholar
  17. 17.
    Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten. Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math.-Nat. Kl., 69, 262–277 (1917).Google Scholar
  18. 18.
    Reveilles, J.P.: Geometrie discrete, calcul en nombres entiers et algorithmique. M.A. thesis, ULP University, Strasbourg, France (1991).MATHGoogle Scholar
  19. 19.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley and Sons, New York, NY (1986).MATHGoogle Scholar
  20. 20.
    Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imag., 1, 113–122 (1982).CrossRefGoogle Scholar
  21. 21.
    Vardi, Y., Zhang, C.H.: Reconstruction of binary images via the EM algorithm. In: Herman, G.T., Kuba, A. (ed.), Discrete Tomography, Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA, pp. 297–316 (1999).Google Scholar
  22. 22.
    Weber, S., Schnorr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. Electr. Notes Discr. Math., 12 (2003).Google Scholar
  23. 23.
    Weber, S., Schiile, T, Hornegger, J., Schnorr, C.: Binary tomography by iterating linear programs from noisy projections. In: Klette, R, Zunic, J.D. (eds.), Combinatorial Image Analysis, Springer, Berlin, Germany, pp. 38–51 (2004).Google Scholar
  24. 24.
    Wunderling, R., Paralleler und Objektorientierter Simplex-Algorithmus, Ph.D. thesis, ZIB TR 96-09, Berlin, Germany (1996).MATHGoogle Scholar
  25. 25.
    Ye, Y: Interior Points Algorithms: Theory and Analysis. John Wiley and Sons, New York, NY (1997).Google Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • Y. Gerard
    • 1
  • F. Feschet
    • 1
  1. 1.University of Auvergne 1 - IUT/LAIC LaboratoryAubiere CedexFrance

Personalised recommendations