Uniqueness and Complexity in Discrete Tomography

  • Richard J. Gardner
  • Peter Gritzmann
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We study the discrete inverse problem of reconstructing finite subsets of the n-dimensional integer lattice ℤn that are only accessible via their line sums (discrete X-rays) in a finite set of lattice directions. Special emphasis is placed on the question of when such sets are uniquely determined by the data and on the difficulty of the related algorithmic problems. Such questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high-resolution transmission electron microscopy.


Lattice Line Finite Subset Truth Assignment Cross Ratio Discrete Tomography 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    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
  2. [2]
    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
  3. [3]
    R. J. Gardner, Geometric Tomography (Cambridge University Press, New York) 1995.Google Scholar
  4. [4]
    G. G. Lorentz, “A problem of plane measure,” Amer. J. Math. 71, 417–426 (1949).CrossRefGoogle Scholar
  5. [5]
    A. Rényi, “On projections of probability distributions,” Acta Math. Acad. Sci. Hung. 3, 131–142 (1952).CrossRefGoogle Scholar
  6. [6]
    A. Heppes, “On the determination of probability distributions of more dimensions by their projections,” Acta Math. Acad. Sci. Hung. 7, 403–410 (1956).CrossRefGoogle Scholar
  7. [7]
    G. Bianchi and M. Longinetti, “Reconstructing plane sets from projections,” Discrete Comp. Geom. 5, 223–242 (1990).CrossRefGoogle Scholar
  8. [8]
    R. J. Gardner and P. Gritzmann, “Discrete tomography: Determination of finite sets by X-rays,” Trans. Amer. Math. Soc. 349 2271–2295 (1997).CrossRefGoogle Scholar
  9. [9]
    R. J. Gardner and P. McMullen, “On Hammer’s X-ray problem,” J. London Math. Soc. (2) 21, 171–175 (1980).CrossRefGoogle Scholar
  10. [10]
    F. Q. Gouvàa, p-adic Numbers (Springer, New York), 1993.CrossRefGoogle Scholar
  11. [11]
    E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “X-rays characterizing some classes of digital pictures,” (Technical Report RT 4/96, Dipartimento di Sistemi e Informatica, Universita di Firenze, Firenze), 1996.Google Scholar
  12. [12]
    E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “Reconstructing convex polyominoes from their horizontal and vertical projections,” Theor. Comput. Sci. 155, 321–347 (1996).CrossRefGoogle Scholar
  13. [13]
    R. J. Gardner, “Geometric tomography,” Notices Amer. Math. Soc. 42, 422–429 (1995).Google Scholar
  14. [14]
    P. R. Scott, “Equiangular lattice polygons and semiregular lattice polyhedra,” College Math. J. 18, 300–306 (1987).CrossRefGoogle Scholar
  15. [15]
    A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York), 1987.Google Scholar
  16. [16]
    R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the computational complexity of reconstructing lattice sets from their X-rays,” Discrete Math. 202, 45–71 (1999).CrossRefGoogle Scholar
  17. [17]
    P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Intern. J. Imaging Syst. Techn. 9, 101–109 (1998).CrossRefGoogle Scholar
  18. [18]
    R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the reconstruction of binary images from their discrete Radon transforms,” Proc. Intern. Symp. Optical Science, Engineering, and Instrumentation, SPIE, pp. 121–132 (1996).Google Scholar
  19. [19]
    R. W. Irving and M. R. Jerrum, “Three-dimensional statistical data security problems,” SIAM J. Comput. 23, 170–184 (1994).CrossRefGoogle Scholar
  20. [20]
    P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, “Sets uniquely determined by projections on axes. II. Discrete case,” Discrete Math. 91, 149–159 (1991).CrossRefGoogle Scholar
  21. [21]
    P. C. Fishburn, P. Schwander, L. A. Shepp, and J. Vanderbei, “The discrete Radon transform and its approximate inversion via linear programming,” Discrete Appl. Math. 75, 39–62 (1997).CrossRefGoogle Scholar
  22. [22]
    P. Gritzmann and M. Wiegelmann, “On combinatorial patterns given by cross-characteristics: Uniqueness versus additivity,” in preparation.Google Scholar
  23. [23]
    H. J. Ryser, “Combinatorial properties of matrices of zeros and ones,” Canad. J. Math. 9, 371–377 (1957).CrossRefGoogle Scholar
  24. [24]
    T. Y. Kong and G. T. Herman, “On which grids can tomographic equivalence of binary pictures be characterized in terms of elementary switching operations,” Intern. J. Imaging Syst. Techn. 9, 118–125 (1998).CrossRefGoogle Scholar
  25. [25]
    R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the computational complexity of inverting higher-dimensional discrete X-ray transforms,” in preparation.Google Scholar
  26. [26]
    H. Edelsbrunner and S. S. Skiena, “Probing convex polygons with X-rays,” SIAM. J. Comp. 17, 870–882 (1988).CrossRefGoogle Scholar
  27. [27]
    S. Patch, “Iterative algorithm for discrete tomography,” Intern. J. Imaging Syst. Techn. 9, 132–134 (1998).CrossRefGoogle Scholar
  28. [28]
    Y.-C. Chen and A. Shastri, “On joint realization of (0,1) matrices,” Linear Algebra Appl. 112, 75–85 (1989).CrossRefGoogle Scholar
  29. [29]
    R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the computational complexity of determining polyatomic structures by X-rays,” Theor. Comput. Sci., to appear.Google Scholar
  30. [30]
    M. Chrobak and C. Dürr, “Reconstructing polyatomic structures from discrete X-rays: NP-completeness proof for three atoms,” preprint.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Richard J. Gardner
    • 1
  • Peter Gritzmann
    • 2
  1. 1.Department of MathematicsWestern Washington UniversityBellinghamUSA
  2. 2.Zentrum MathematikTechnische Universität MünchenMünchenGermany

Personalised recommendations