Tomographic Equivalence and Switching Operations

  • T. Yung Kong
  • Gabor T. Herman
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


A binary picture on an arbitrary grid is a mapping f from the set of all grid points to {0,1} such that f (x) = 1 for only finitely many grid points x. If two binary pictures f 1 and f 2 on the same grid have the property that for every grid line P the sets {p ∈ l| f 1(p)= 1} and{ p ∈ l|f 2(p)= 1} contain exactly the same number of grid points, then we say that f 1 and f 2 are tomographically equivalent. Given a binary picture f on the usual 2-dimensional square grid, there may exist an upright rectangle R (of any size) whose sides are grid lines, such that f = 1 at two diagonally opposite corner points of R and f= 0 at the other two corner points. If so, then we call the process of changing the value of the picture f from 1 to 0 and 0 to 1 at the four corner points of R (without changing the value of f at any other grid point) a rectangular 4-switch. Ryser showed in the 1950s that two binary pictures on the square grid are tomographically equivalent if and only if one picture can be transformed to the other by a finite sequence of rectangular 4-switches. We present a few different versions of this theorem, describe an application, and also give a proof of the result. We then show that the result has no analog on grids that have grid lines in three or more directions (such as the 3-dimensional cubic grid),because on such grids one can find for every integer L two tomographically equivalent binary pictures that differ at more than L grid points and are not tomographically equivalent to any other binary picture.


Grid Point Induction Hypothesis Corner Point Grid Line Switching Operation 
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  1. [1]
    H. J. Ryser, “Combinatorial properties of matrices of zeros and ones,” Can. J. Mathematics,9, 371–377 (1957).CrossRefGoogle Scholar
  2. [2]
    H. J. Ryser, Combinatorial Mathematics (Mathematical Association of America, Washington, DC), 1963, Chapter 6.Google Scholar
  3. [3]
    G. Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods (Springer-Verlag, Berlin), 1995.CrossRefGoogle Scholar
  4. [4]
    T. H. Cormen, C. E. Leiserson, and R. L. Rivest, Introduction to Algorithms (MIT Press, Cambridge, Massachusetts), 1990.Google Scholar
  5. [5]
    A. Kuba, “Determination of the structure of the class A(R, S) of (0,1)-matrices,” Acta Cybernetica,9, 121–132 (1989).Google Scholar
  6. [6]
    R. Aharoni, G. T. Herman, and A. Kuba, “Binary vectors partially determined by linear equation systems,” Discrete Mathematics, 171, 1–16 (1997).CrossRefGoogle Scholar
  7. [7]
    R. P. Anstee, “The network flows approach for matrices with given row and column sums,” Discrete Mathematics, 44, 125–138 (1983).CrossRefGoogle Scholar
  8. [8]
    T. Y. Kong and G. T. Herman, “On which grids can tomographic equivalence of binary pictures be characterized in terms of elementary switching operations?” Int. J. Imaging Syst. Technol. 9, 118–125 (1998).CrossRefGoogle Scholar
  9. [9]
    R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the computational complexity of reconstructing lattice sets from their X-rays,” to appear in Discrete Mathematics. Google Scholar
  10. [10]
    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 Syst. Technol. 9, 101–109 (1998).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • T. Yung Kong
    • 1
  • Gabor T. Herman
    • 2
  1. 1.Department of Computer ScienceQueens College, CUNYFlushingUSA
  2. 2.Medical Image Processing Group, Department of RadiologyUniversity of PennsylvaniaPhiladelphiaUSA

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