Sets of Uniqueness and Additivity in Integer Lattices

  • Peter C. Fishburn
  • Lawrence A. Shepp
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


A mathematical formulation is provided for inversion problems in which local structures of finite subsets of integer lattices are to be deduced from point counts in prescribed linear manifolds of an n-dimensional space. Notions of uniqueness and additivity for finite lattice sets are defined and characterized by point configurations and by aspects of fractional subsets of the lattice. The latter feature leads to analysis by interior point linear programming, which appears to be a very effective as well as efficient approximation approach to discrete inversion problems.


Interior Point Method Extreme Solution Linear Manifold Integer Lattice Linear Programming Approach 
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]
    L. A. Shepp and B. F. Logan, “The Fourier reconstruction of a head,” IEEE Trans. Nucl. Sci. NS, 21, 21–43 (1974).CrossRefGoogle Scholar
  2. [2]
    L. A. Shepp and J. B. Kruskal, “Computerized tomography: the new medical X-ray technology,” Amer. Math. Monthly, 85, 420–439 (1978).CrossRefGoogle Scholar
  3. [3]
    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,” Phys. Rev. Lett. 71, 4150–4153 (1993).CrossRefPubMedGoogle Scholar
  4. [4]
    P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, “The discrete Radon transform and its approximate inversion via linear programming,” Discrete Appl Math. 75, 39–61 (1997).CrossRefGoogle Scholar
  5. [5]
    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
  6. [6]
    R. A. Crowther, D. J. DeRosier and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. Royal Soc. London Ser. A 317,319–340 (1970).CrossRefGoogle Scholar
  7. [7]
    R. Gordon and G. T. Herman, “Reconstruction of pictures from their projections,” Commun. ACM,14, 759–768 (1971).CrossRefGoogle Scholar
  8. [8]
    R. J. Gardner and P. McMullen, “On Hammer’s X-ray problem,” J. London Math. Soc. 21, 171–175 (1980).CrossRefGoogle Scholar
  9. [9]
    G. Bianchi and M. Longinetti, “Reconstructing plane sets from projections,” Discrete Comput. Geom. 5, 223–242 (1990).CrossRefGoogle Scholar
  10. [10]
    P. Gritzmann and M. Nivat, Eds., “Discrete Tomography: Algorithms and Complexity,” Dagstuhl Seminar Report 165, Internat. Beg. Forsch. Infor., Schloss Dagstuhl, Germany, 1997.Google Scholar
  11. [11]
    P. C. Fishburn, Mathematics of Decision Theory (Mouton, Paris), 1972.Google Scholar
  12. [12]
    P. C. Fishburn, “Finite linear qualitative probability,” J. Math. Psychol. 40, 64–77 (1996).CrossRefGoogle Scholar
  13. [13]
    P. C. Fishburn, “Failure of cancellation conditions for additive linear orders,” J. Combin. Designs, 5, 353–365 (1997).CrossRefGoogle Scholar
  14. [14]
    D. Scott and P. Suppes, “Foundational aspects of theories of measurement,” J. Symbolic Logic 23, 113–128 (1958).CrossRefGoogle Scholar
  15. [15]
    P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,” Int. J. Imaging Sci. Tech. 9, 101–109 (1998).CrossRefGoogle Scholar
  16. [16]
    R. W. Irwing and M. R. Jerrum, “Three-dimensional statistical data security problems,” SIAM J. Comput. 23, 170–184 (1994).CrossRefGoogle Scholar
  17. [17]
    R. Aharoni, G. T. Herman, and A. Kuba, “Binary vectors partially determined by linear equation systems,” Discrete Math. 171, 1–16 (1997).CrossRefGoogle Scholar
  18. [18]
    I. Adler and R. D. C. Monteiro, “Limiting behavior of the affine scaling continuous trajectories for linear programming,” Math. Programming 50, 29–51 (1991).CrossRefGoogle Scholar
  19. [19]
    R. J. Vanderbei, An interior point code for quadratic programming, SOR 94–15, Princeton University, Princeton, NJ, 1994.Google Scholar
  20. [20]
    Y. Vardi and D. Lee, “Discrete Radon transform and its approximate inversion via the EM algorithm,” Int. J. Imaging Sci. Tech. 9, 155–173 (1998).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Peter C. Fishburn
    • 1
  • Lawrence A. Shepp
    • 2
  1. 1.AT&T Labs-ResearchFlorham ParkUSA
  2. 2.Department of StatisticsRutgers UniversityPiscatawayUSA

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