A Fast Algorithm for Reconstructing hv-Convex 8-Connected but Not 4-Connected Discrete Sets

  • Péter Balázs
  • Emese Balogh
  • Attila Kuba
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2886)


One important class of discrete sets where the reconstruction from two given projections can be solved in polynomial time is the class of hv-convex 8-connected sets. The worst case complexity of the fastest algorithm known so far for solving the problem is of O(mn· min { m 2,n 2} ) [2]. However, as it is shown, in the case of 8-connected but not 4-connected sets we can give an algorithm with worst case complexity of O(mn· min { m,n} ) by identifying the so-called \({\cal S}_4\)-components of the discrete set. Experimental results are also presented in order to investigate the average execution time of our algorithm.


Discrete tomography reconstruction convex and connected discrete set 


  1. 1.
    Balázs, P., Balogh, E., Kuba, A.: Reconstruction of 8-connected but not 4-connected discrete sets, Technical Report at the University of Szeged (2002),
  2. 2.
    Balogh, E., Kuba, A., Dévényi, C., Del Lungo, A.: Comparison of algorithms for reconstructing hv-convex discrete sets. Lin. Alg. and Its Appl. 339, 23–35 (2001)zbMATHCrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    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
  5. 5.
    Brunetti, S., Del Lungo, A., Del Ristoro, F., Kuba, A., Nivat, M.: Reconstruction of 8- and 4-connected convex discrete sets from row and column projections. Lin. Alg. and Its Appl. 339, 37–57 (2001)zbMATHCrossRefGoogle Scholar
  6. 6.
    Chrobak, M., Dürr, C.: Reconstructing hv-convex polyominoes from orthogonal projections. Information Processing Letters 69(6), 283–289 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography, Special Issue. Int. J. Imaging Systems and Techn. 9(2/3) (1998)Google Scholar
  9. 9.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations. Algorithms and Applications. Birkhäuser, Boston (1999)zbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Kuba, A.: Reconstruction in different classes of 2D discrete sets. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 153–163. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Kuba, A., Balogh, E.: Reconstruction of convex 2D discrete sets in polynomial time. Theor. Comput. Sci. 283, 223–242 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Latecki, L., Eckhardt, U., Rosenfeld, A.: Well-Composed Sets. Computer Vision and Image Understanding 61(1), 70–83 (1995)CrossRefGoogle Scholar
  14. 14.
    Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9, 371–377 (1957)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Woeginger, G.W.: The reconstruction of polyominoes from their orthogonal projections. Information Processing Letters 77, 225–229 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Péter Balázs
    • 1
  • Emese Balogh
    • 1
  • Attila Kuba
    • 1
  1. 1.Department of InformaticsUniversity of Szeged Árpád tér 2SzegedHungary

Personalised recommendations