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Decomposition Algorithms for Reconstructing Discrete Sets with Disjoint Components

  • P. Balázs
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

The reconstruction of discrete sets from their projections is a frequently studied field in discrete tomography with applications in electron microscopy, image processing, radiology, and so on. Several efficient reconstruction algorithms have been developed for certain classes of discrete sets having some good geometrical properties. On the other hand, it has been shown that the reconstruction under certain circumstances can be very time-consuming, even NP-hard. In this chapter we show how prior information that the set to be reconstructed consists of several components can be exploited in order to facilitate the reconstruction. We present some general techniques to decompose a discrete set into components knowing only its projections and thus reduce the reconstruction of a general discrete set to the reconstruction of single components, which is usually a simpler task.

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References

  1. 1.
    Balazs, P.: A decomposition technique for reconstructing discrete sets from four projections. Image and Vision Computing, submitted.Google Scholar
  2. 2.
    Balázs, P.: Reconstruction of discrete sets from four projections: Strong decomposability. Electr. Notes Discr. Math., 20, 329–345 (2005).CrossRefGoogle Scholar
  3. 3.
    Balazs, P., Balogh, E., Kuba, A.: Reconstruction of 8-connected but not 4-connected hv-convex discrete sets. Discr. Appl. Math., 147, 149–168 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Balogh, E., Kuba, A., Dévényi, C., Del Lungo, A.: Comparison of algorithms for reconstructing hv-convex discrete sets. Lin. Algebra Appl. 339, 23–35 (2001).zbMATHCrossRefGoogle 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.
    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Medians of polyominoes: A property for the reconstruction. Int. J. Imaging Systems Techn., 9, 69–77 (1998).CrossRefGoogle Scholar
  7. 7.
    Brunetti, S., Daurat, A.: An algorithm reconstructing convex lattice sets. Theor. Comput. Sci., 304, 35–57 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chrobak, M., Dürr, C.: Reconstructing hv-convex polyominoes from orthogonal projections. Inform. Process. Lett., 69, 283–289 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Del Lungo, A.: Polyominoes defined by two vectors. Theor. Comput. Sci., 127, 187–198 (1994).zbMATHCrossRefGoogle Scholar
  10. 10.
    Del Lungo, A., Nivat, M.: Reconstruction of connected sets from two projections. In [14], pp. 163–188 (1999).Google Scholar
  11. 11.
    Del Lungo, A., Nivat, M., Pinzani, R.: The number of convex polyominoes reconstructible from their orthogonal projections. Discr. Math., 157, 65–78 (1996).zbMATHCrossRefGoogle Scholar
  12. 12.
    Gardner, R.J., Gritzmann, P.: Uniqueness and complexity in discrete tomography. In [14], pp. 85–113 (1999).Google Scholar
  13. 13.
    Golomb, S.W.: Polyominoes. Scribner, New York, NY (1965).Google Scholar
  14. 14.
    Herman, G.T., Kuba, A. (eds.), Discrete Tomography: Foundations, Algorithms, and Applications. Birkhauser, Boston, MA (1999).zbMATHGoogle Scholar
  15. 15.
    Kuba, A.: The reconstruction of two-directionally connected binary patterns from their two orthogonal projections. Comp. Vision, Graphics, Image Proc. 27, 249–265 (1984).CrossRefGoogle Scholar
  16. 16.
    Kuba, A., Balogh, E.: Reconstruction of convex 2D discrete sets in polynomial time. Theor. Comput. Sci., 283, 223–242 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9, 371–377 (1957).zbMATHMathSciNetGoogle Scholar
  18. 18.
    Woeginger, G.W.: The reconstruction of polyominoes from their orthogonal projections. Inform. Process. Lett., 77, 225–229 (2001).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • P. Balázs
    • 1
  1. 1.Dept.of Computer Algorithms and Arti ?cial IntelligenceUniversity of SzegedSzegedHungary

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