Reconstruction of Connected Sets from Two Projections

  • Alberto Del Lungo
  • Maurice Nivat
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The problem of reconstructing a two-dimensional discrete set from its projections has been studied in discrete mathematics and applied in several areas. It has some interesting applications in image processing, electron microscopy, statistical data security, biplane angiography and graph theory. This chapter presents the computational complexity results of the problem of reconstructing a set from its horizontal and vertical projections with respect to some classes of sets on which some connectivity constrzints are imposed. We show that this reconstruction problem can be solved in polynomial time in a class of discrete sets, and is NP-complete otherwise.


Reconstruction Problem Vertical Projection Disjoint Cycle Median Column Filling Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    A. R. Shliferstein and Y. T. Chien, “Switching components and the ambiguity problem in the reconstruction of pictures from their projections,” Pattern Recognition,10, 327–340 (1978).CrossRefGoogle Scholar
  2. [2]
    R. W. Irving and M. R. Jerrum, “Three-dimensional statistical data security problems,” SIAM Journal of Computing, 23, 170–184 (1994).CrossRefGoogle Scholar
  3. [3]
    G. P. M. Prause and D. G. W. Onnasch, “Binary reconstruction of the heart chambers from biplane angiographic image sequence,” IEEE Trans. Medical Imaging, 15, 532–559 (1996).CrossRefGoogle Scholar
  4. [4]
    R. P. Anstee, “Invariant sets of arcs in network flow problems,” Discrete Applied Mathematics, 13, 1–7 (1986).CrossRefGoogle Scholar
  5. [5]
    P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, “The discrete Radon transform and its approximate inversion via linear programming,” Discrete Applied Mathematics, 75, 39–61 (1997).CrossRefGoogle Scholar
  6. [6]
    R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the computational complexity of reconstructing lattice sets from their X-rays,” Technical Report 970.05012, Techn. Univ. Miinchen, Fak. f. Math.,Miinchen, (1997).Google Scholar
  7. [7]
    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, 131–155 (1995).CrossRefGoogle Scholar
  8. [8]
    P. Schwander, C. Kisielowski, M. Seibt, F. H. Baumann, Y. Kim, and A. Ourmazd, “Mapping projected potential, interfacial roughness, and composition in general crystalline solids by quantitative transmission electron microscopy,” Physical Review Letters, 71, 4150–4153 (1993).CrossRefPubMedGoogle Scholar
  9. [9]
    H. J. Ryser, “Combinatorial properties of matrices of zeros and ones,” Canad. J. Math., 9, 371–377 (1957).CrossRefGoogle Scholar
  10. [10]
    S. K. Chang, “The reconstruction of binary patterns from their projections,” Communications ACM, 14, 21–24 (1971).CrossRefGoogle Scholar
  11. [11]
    R. A. Brualdi, “Matrices of zeros and ones with fixed row and column sum vectors,” Linear Algebra and Applications, 33, 159–231 (1980).CrossRefGoogle Scholar
  12. [12]
    A. Del Lungo, “Polyominoes defined by two vectors,” Theoretical Computer Science, 127, 187–198 (1994).CrossRefGoogle Scholar
  13. [13]
    A. Kuba, “The reconstruction of two-directionally connected binary patterns from their two orthogonal projections,” Computer Vision, Graphics, and Image Processing, 27, 249–265 (1984).CrossRefGoogle Scholar
  14. [14]
    S. K. Chang and C. K. Chow, “The reconstruction of three-dimensional objects from two orthogonal projections and its application to cardiac cineangiography,” IEEE Trans. on Computers, 22, 18–28 (1973).CrossRefGoogle Scholar
  15. [15]
    S. W. Golomb, Polyominoes, Revised and Expanded Edition, (Princeton University Press, Princeton, NJ), 1994.Google Scholar
  16. [16]
    E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “Reconstructing convex polyominoes from horizontal and vertical projections,” Theoretical Computer Science, 155, 321–347 (1996).CrossRefGoogle Scholar
  17. [17]
    M. R. Garey and D.S. Johnson, Computers and intractability: A guide to the theory of NP-completeness, (Freeman, New York), 1979.Google Scholar
  18. [18]
    G. J. Woeginger, “The reconstruction of polyominoes from their orthogonal projections,” Technical Report SFB-65, TU Graz,Graz, (1996).Google Scholar
  19. [19]
    E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “Medians of polyominoes: a property for the reconstruction,” International Journal of Imaging Systems and Technology, 8, 69–77 (1998).CrossRefGoogle Scholar
  20. [20]
    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, 121–123 (1979).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Alberto Del Lungo
    • 1
  • Maurice Nivat
    • 2
  1. 1.Dipartimento di Sistemi e Informatica (DSI)Università di FirenzeFirenzeItaly
  2. 2.Laboratoire d’Informatique, Algorithmique, Fondements et Applications (LIAFA)Université Denis DiderotParisFrance

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