Computerized Tomography with Digital Lines and Linear Programming

  • Fabien Feschet
  • Yan Gérard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3429)


We present a new method of computerized tomography based on linear programming. The approach is based on three main ideas: covering the set of pixels by digital lines, introducing a variable of maximal error in the linear constraints and adding in the objective function an entropy term.


Grey Level Synthetic Image Grey Level Image Topological Constraint 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.


  1. 1.
    Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. IEEE Computer Society Press, Los Alamitos (1988)zbMATHGoogle Scholar
  2. 2.
    Herman, G., Lent, A.: Iterative reconstruction algorithms. Comput. Biol. Med. 6, 273–274 (1976)CrossRefGoogle Scholar
  3. 3.
    Gordon, R., Herman, G.: Reconstruction of pictures from their projections. Communication of the ACM 14, 759–768 (1971)zbMATHCrossRefGoogle Scholar
  4. 4.
    Gordon, R.: A tutorial on ART (Algebraic Reconstruction Techniques). IEEE Transactions on Nuclear Science NS-21, 31–43 (1974)Google Scholar
  5. 5.
    Ben-Tal, A., Margalit, T., Nemirovski, A.: The ordered subsets mirror descent optimization method with applications to tomography. SIAM J. Optimization 12, 79–108 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Aharoni, R., Herman, G., Kuba, A.: Binary vectors partially determined by linear equation systems. Discrete Mathematics 171, 1–16 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kuba, A., Herman, G.: Discrete Tomography: Foundations, Algorithms and Applications. Birkhaüser, Basel (1999)zbMATHGoogle Scholar
  8. 8.
    Fishburn, P., Schwander, P., Shepp, L., Vanderbei, R.: The discrete radon transform and its approximate inversion via linear programming. Discrete Applied Math. 75, 39–61 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gritzmann, P., de Vries, S., Wiegelmann, M.: Approximating binary images from discrete X-rays. SIAM J. Optimization 11, 522–546 (2000)zbMATHCrossRefGoogle Scholar
  10. 10.
    Weber, S., Schnörr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. In: Int. Workshop on Combinatorial Image Analysis IWCIA 2003. Electronic Notes in Discrete Math., vol. 12, Elsevier, Amsterdam (2003)Google Scholar
  11. 11.
    Brunetti, S., Daurat, A.: Stability in discrete tomography: Linear programming, additivity and convexity. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 398–408. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Hajdu, L., Tijdeman, R.: An algorithm for discrete tomography. Linear Algebra and Appl. 339, 147–169 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Reveillès, J.P.: Géométrie discrète, calcul en nombres entiers et algorithmique. In: Thèse d’état, Université ULP - Strasbourg (1991)Google Scholar
  14. 14.
    Weber, S., Schüle, T., Hornegger, C.S., Discrete, J.: tomography by convex-concave regularization and d.c. programming. Technical report, Mannheim University (2003)Google Scholar
  15. 15.
    Schrijver, A.: Theory of Linear and Integer Programming. J. Wiley and Sons, Chichester (1986)zbMATHGoogle Scholar
  16. 16.
    Wunderling, R.: Paralleler und Objektorientierter Simplex-Algorithmus. PhD thesis, ZIB TR 96-09, Berlin (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Fabien Feschet
    • 1
  • Yan Gérard
    • 1
  1. 1.LLAIC – IUT Clermont-FerrandAubièreFrance

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