An Algorithm for Reconstructing Special Lattice Sets from Their Approximate X-Rays

  • Sara Brunetti
  • Alain Daurat
  • Alberto Del Lungo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)

Abstract

We study the problem of reconstructing finite subsets of the integer lattice Z2 from their approximate X-rays in a finite number of prescribed lattice directions. We provide a polynomial-time algorithm for reconstructing Q-convex sets from their “approximate” X-rays. A Qconvex set is a special subset of Z2 having some convexity properties. This algorithm can be used for reconstructing convex subsets of Z2 from their exact X-rays in some sets of four prescribed lattice directions, or in any set of seven prescribed mutually nonparallel lattice directions.

References

  1. 1.
    B. Aspvall, M. F. Plass and R. E. Tarjan, A linear-time algorithm for testing the truth of certain quantified Boolean formulas, Information Processing Letters, 8 3 (1979) 121–123. 114, 120MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    E. Barcucci, A. Del Lungo, M. Nivat and R. Pinzani, Reconstructing convex polyominoes from their horizontal and vertical projections, Theoretical Computer Science, 155 (1996) 321–347. 114, 120MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Y. Boufkhad, O. Dubois and M. Nivat, Reconstructing (h,v)-convex bidimensional patterns of objects from approximate horizontal and vertical projections, preprint. 114, 120Google Scholar
  4. 4.
    S. Brunetti, A. Daurat, Reconstruction of Discrete Sets From Two or More Projections in Any Direction, Proc. of the Seventh International Workshop on Combinatorial Image Analysis (IWCIA 2000), Caen, (2000) 241–258. 114, 121Google Scholar
  5. 5.
    M. Chrobak, C. Dürr, Reconstructing hv-Convex Polyominoes from Orthogonal Projections, Information Processing Letters, 69 6 (1999) 283–289. 114, 120MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Daurat, Uniqueness of the reconstruction of Q-convex from their projections, preprint. 114, 124Google Scholar
  7. 7.
    A. Daurat, A. Del Lungo and M. Nivat, The medians of discrete sets according to some linear metrics, Discrete & Computational Geometry, 23 (2000) 465–483. 116MATHMathSciNetGoogle Scholar
  8. 8.
    R. J. Gardner and P. Gritzmann, Discrete tomography: determination of finite sets by X-rays, Trans. Amer. Math. Soc., 349 (1997) 2271–2295. 114MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. J. Gardner, P. Gritzmann and D. Prangenberg, On the computational complexity of reconstructing lattice sets from their X-rays, Discrete Mathematics, 202 (1999) 45–71. 113MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    A. Kuba and G. T. Herman, Discrete Tomography: A Historical Overview, in Discrete Tomography: Foundations, Algorithms and Applications, editors G. T. Herman and A. Kuba, Birkhauser, Boston, MA, USA, (1999) 3–34. 113Google Scholar
  11. 11.
    L. Mirski, Combinatorial theorems and integral matrices, Journal of Combinatorial Theory, 5 (1968) 30–44. 121CrossRefGoogle Scholar
  12. 12.
    R. W. Irving and M. R. Jerrum, Three-dimensional statistical data security problems, SIAM Journal of Computing, 23 (1994) 170–184. 113MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C. Kisielowski, P. Schwander, F. H. Baumann, M. Seibt, Y. Kim and A. Ourmazd, An approach to quantitative high-resolution transmission electron microscopy of crystalline materials, Ultramicroscopy, 58 (1995) 131–155. 113CrossRefGoogle Scholar
  14. 14.
    G. P. M. Prause and D. G. W. Onnasch, Binary reconstruction of the heart chambers from biplane angiographic image sequence, IEEE Transactions Medical Imaging, 15 (1996) 532–559. 113CrossRefGoogle Scholar
  15. 15.
    A. R. Shliferstein and Y. T. Chien, Switching components and the ambiguity problem in the reconstruction of pictures from their projections, Pattern Recognition, 10 (1978) 327–340. 113MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Sara Brunetti
    • 1
  • Alain Daurat
    • 2
  • Alberto Del Lungo
    • 3
  1. 1.Dipartimento di Sistemi e InformaticaFirenzeItaly
  2. 2.Laboratoire de Logique et d’Informatique de Clermont-1 (LLAIC1)Ensemble Universitaire des CézeauxAubière CedexFrance
  3. 3.Dipartimento di MatematicaSienaItaly

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