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Network Flow Algorithms for Discrete Tomography

  • K.J. Batenburg
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

There exists an elegant correspondence between the problem of reconstructing a 0-1 lattice image from two of its projections and the problem of finding a maximum flow in a certain graph. In this chapter we describe how network flow algorithms can be used to solve a variety of problems from discrete tomography. First, we describe the network flow approach for two projections and several of its generalizations. Subsequently, we present an algorithm for reconstructing 0-1 images from more than two projections. The approach is extended to the reconstruction of 3D images and images that do not have an intrinsic lattice structure.

Keywords

Original Image Lattice Line Lattice Direction Point Edge Reconstruction Problem 
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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References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs, NJ (1993).Google Scholar
  2. 2.
    Anstee, R.P.: The network flows approach for matrices with given row and column sums. Discr. Math., 44, 125–138 (1983).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Batenburg, K.J.: Reconstructing binary images from discrete X-rays. Tech. Rep. PNA-E0418, CWI, Amsterdam, The Netherlands (2004).Google Scholar
  4. 4.
    Bertsekas, D.P., Tseng, P.: RELAX-IV: A faster version of the RELAX code for solving minimum cost flow problems. LIDS Technical Report LIDS-P-2276, MIT, Cambridge, MA (1994).Google Scholar
  5. 5.
    Dürr, C.: Ryser’s algorithm can be implemented in linear time. http://www.lix.polytechnique.fr/~durr/Xray/Ryser/.
  6. 6.
    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math., 8, 399–404 (1956).zbMATHMathSciNetGoogle Scholar
  7. 7.
    Gale, D.: A theorem on flows in networks. Pac. J. Math., 7, 1073–1082 (1957).zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gardner, R.J., Gritzmann, P., Prangenberg, D.: On the computational complexity of reconstructing lattice sets from their X-rays. Discr. Math., 202, 45–71 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goldberg, A.V.: An efficient implementation of a scaling minimum-cost flow algorithm. J. Algebra, 22, 1–29 (1997).Google Scholar
  10. 10.
    Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. J. ACM, 45, 783–797 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by successive approximation. Math. Oper. Res., 15, 430–466 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gritzmann, P., de Vries, S., Wiegelmann, M.: Approximating binary images from discrete X-rays. SIAM J. Opt., 11, 522–546, (2000).zbMATHCrossRefGoogle Scholar
  13. 13.
    Hajdu, L., Tijdeman, R.: Algebraic aspects of discrete tomography. J. Reine Angew. Math., 534, 119–128 (2001).zbMATHMathSciNetGoogle Scholar
  14. 14.
    Herman, G.T., Kuba, A.: Discrete tomography in medical imaging. Proc. IEEE, 91, 380–385 (2003).CrossRefGoogle Scholar
  15. 16.
    Jinschek, J.R., Calderon, H.A., Batenburg, K.J., Radmilovic, V., Kisielowski, C.: Discrete tomography of Ga and InGa particles from HREM image simulation and exit wave reconstruction. Mat. Res. Soc. Symp. Proc., 839, 4.5.1–4.5.6 (2004).Google Scholar
  16. 17.
    Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. SIAM, Philadelphia, PA (2001).Google Scholar
  17. 18.
    Mohammad-Djafari, A.: Binary polygonal shape image reconstruction from a small number of projections. ELEKTRIK, 5, 127–139 (1997).Google Scholar
  18. 19.
    Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Can. J. Math., 9, 371–377 (1957).zbMATHMathSciNetGoogle Scholar
  19. 20.
    Ryser, H.: Combinatorial Mathematics, Mathematical Association of America, Washington, DC (1963).zbMATHGoogle Scholar
  20. 21.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg, Germany (2003).zbMATHGoogle Scholar
  21. 22.
    Slump, C.H., Gerbrands, J.J.: A network flow approach to reconstruction of the left ventricle from two projections. Comp. Graph. Image Proc., 18, 18–36 (1982).CrossRefGoogle Scholar
  22. 23.
    Tanabe, K.: Projection method for solving a singular system. Num. Math., 17, 203–214 (1971).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • K.J. Batenburg
    • 1
  1. 1.CWI, Amsterdam and Mathematical Institute, Leiden UniversityThe Netherlands

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