Generation and Empirical Investigation of hv-Convex Discrete Sets

  • Péter Balázs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4522)


One of the basic problems in discrete tomography is the reconstruction of discrete sets from few projections. Assuming that the set to be reconstructed fulfils some geometrical properties is a commonly used technique to reduce the number of possibly many different solutions of the same reconstruction problem. Since the reconstruction from two projections in the class of so-called hv-convex sets is NP-hard this class is suitable to test the efficiency of newly developed reconstruction algorithms. However, until now no method was known to generate sets of this class from uniform random distribution and thus only ad hoc comparison of several reconstruction techniques was possible. In this paper we first describe a method to generate some special hv-convex discrete sets from uniform random distribution. Moreover, we show that the developed generation technique can easily be adapted to other classes of discrete sets, even for the whole class of hv-convexes. Several statistics are also presented which are of great importance in the analysis of algorithms for reconstructing hv-convex sets.


discrete tomography hv-convex discrete set decomposable configuration random generation analysis of algorithms 


  1. 1.
    Balázs, P.: A decomposition technique for reconstructing discrete sets from four projections. Image and Vision Computing, accepted.Google Scholar
  2. 2.
    Balázs, P.: On the ambiguity of reconstructing hv-convex binary matrices with decomposable configurations. Acta Cybernetica, submitted.Google Scholar
  3. 3.
    Balázs, P.: Reconstruction of discrete sets from their projections using geometrical priors. Doctoral Dissertation at the University of Szeged (in preparation)Google Scholar
  4. 4.
    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Reconstructing convex polyominoes from horizontal and vertical projections. Theor. Comput Sci. 155, 321–347 (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Batenburg, K.J.: An evolutionary approach for discrete tomography. Discrete Applied Mathematics 151, 36–54 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Brunetti, S., Del Lungo, A., Del Ristoro, F., Kuba, A., Nivat, M.: Reconstruction of 4- and 8-connected convex discrete sets from row and column projections. Lin. Algebra Appl. 339, 37–57 (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chrobak, M., Dürr, C.: Reconstructing hv-convex polyominoes from orthogonal projections. Inform. Process. Lett. 69, 283–289 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Dahl, G., Flatberg, T.: Optimization and reconstruction of hv-convex (0,1)-matrices. Discrete Appl. Math. 151, 93–105 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Delest, M.P., Viennot, G.: Algebraic languages and polyominoes enumeration. Theor. Comput. Sci. 34, 169–206 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Gessel, I.: On the number of convex polyominoes. Ann. Sci. Math. Québec 24, 63–66 (2000)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms and Applications. Birkhäuser, Boston (1999)zbMATHGoogle Scholar
  12. 12.
    Herman, G.T., Kuba, A. (eds.): Advances in Discrete Tomography and Its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel (2007)zbMATHGoogle Scholar
  13. 13.
    Hochstättler, W., Loebl, M., Moll, C.: Generating convex polyominoes at random. Discrete Math. 153, 165–176 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Kuba, A., Nagy, A., Balogh, E.: Reconstruction of hv-convex binary matrices from their absorbed projections. Discrete Applied Mathematics 139, 137–148 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Shoup, V.: NTL: A library for doing number theory,
  17. 17.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences,
  18. 18.
    Woeginger, G.W.: The reconstruction of polyominoes from their orthogonal projections. Inform. Process. Lett. 77, 225–229 (2001)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Péter Balázs
    • 1
  1. 1.Department of Computer Graphics and Image Processing, University of Szeged, Árpád tér 2, H-6720 SzegedHungary

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