Reconstruction of Decomposable Discrete Sets from Four Projections

  • Péter Balázs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3429)


In this paper we introduce the class of decomposable discrete sets and give a polynomial algorithm for reconstructing discrete sets of this class from four projections. It is also shown that the class of decomposable discrete sets is more general than the class \(\mathcal {S}'_{8}\) of hv-convex 8- but not 4-connected discrete sets which was studied in [3]. As a consequence we also get that the reconstruction from four projections in \(\mathcal {S}'_{8}\) can be solved in O(mn) time.


discrete tomography reconstruction from projections decomposable discrete set 


  1. 1.
    Balázs, P.: Reconstruction of decomposable discrete sets from four projections, Technical Report at the University of Szeged (2004),
  2. 2.
    Balázs, P., Balogh, E., Kuba, A.: A fast algorithm for reconstructing hv-convex 8-connected but not 4-connected discrete sets. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 388–397. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Balázs, P., Balogh, E., Kuba, A.: Reconstruction of 8-connected but not 4-connected hv-convex discrete sets, Discrete Applied Mathematics, (accepted)Google Scholar
  4. 4.
    Balogh, E., Kuba, A.: Comparison of algorithms for reconstructing hv-convex discrete sets. Lin. Alg. and Its Appl. 339, 23–35 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Reconstructing convex polyominoes from horizontal and vertical projections. Theor. Comput. Sci. 155, 321–347 (1996)zbMATHCrossRefGoogle Scholar
  6. 6.
    Brualdi, R.A.: Matrices of zeros and ones with fixed row and column sum vectors. Lin. Algebra and Its Appl. 33, 159–231 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brunetti, S., Daurat, A.: Reconstruction of discrete sets from two or more X-rays in any direction. In: Proceedings of the seventh International Workshop on Combinatorial Image Analysis, pp. 241–258 (2000)Google Scholar
  8. 8.
    Chrobak, M.: Reconstructing hv-convex polyominoes from orthogonal projections. Information Processing Letters 69(6), 283–289 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Daurat, A.: Convexité dans le plan discret. Application à la tomographie, Thèse de doctorat de l‘Université Paris 7 (2000),
  10. 10.
    Del Lungo, A.: Polyominoes defined by two vectors. Theor. Comput. Sci. 127, 187–198 (1994)zbMATHCrossRefGoogle Scholar
  11. 11.
    Del Lungo, A., Nivat, M., Pinzani, R.: The number of convex polyominoes reconstructible from their orthogonal projections. Discrete Math. 157, 65–78 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gardner, R.J., Gritzmann, P.: Uniqueness and complexity in discrete tomography. In: [15], pp. 85–113 (1999)Google Scholar
  13. 13.
    Hajdu, L., Tijdeman, R.: Algebraic aspects of discrete tomography. Journal für die reine und angewandte Mathematik 534, 119–128 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Herman, G.T., Kuba, A.: Discrete Tomography, Special Issue. Int. J. Imaging Systems and Techn. 9(2/3) (1998)Google Scholar
  15. 15.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms and Applications. Birkhäuser, Basel (1999)zbMATHGoogle Scholar
  16. 16.
    Kuba, A.: The reconstruction of two-directionally connected binary patterns from their two orthogonal projections. Comp. Vision, Graphics, and Image Proc. 27, 249–265 (1984)CrossRefGoogle Scholar
  17. 17.
    Kuba, A.: Reconstruction in different classes of 2D discrete sets. Lecture Notes on Computer Sciences 1568, 153–163 (1999)CrossRefGoogle Scholar
  18. 18.
    Kuba, A., Balogh, E.: Reconstruction of convex 2D discrete sets in polynomial time. Theor. Comput. Sci. 283, 223–242 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9, 371–377 (1957)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Soille, P.: From binary to grey scale convex hulls. Fundamenta Informaticae 41, 131–146 (2000)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Woeginger, G.W.: The reconstruction of polyominoes from their orthogonal projections. Inform. Process. Lett. 77, 225–229 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Péter Balázs
    • 1
  1. 1.Department of InformaticsUniversity of SzegedSzegedHungary

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