Skip to main content
SpringerLink
Log in

A Network Flow Algorithm for Reconstructing Binary Images from Continuous X-rays

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Tomography is a powerful technique to obtain accurate images of the interior of an object in a nondestructive way. Conventional reconstruction algorithms, such as filtered backprojection, require many projections to obtain high quality reconstructions. If the object of interest is known in advance to consist of only a few different materials, corresponding to known image intensities, the use of this prior knowledge in the reconstruction procedure can dramatically reduce the number of required projections.

In previous work we proposed a network flow algorithm for reconstructing a binary image defined on a lattice from its projections. In this paper we propose a new algorithm for the reconstruction of binary images that do not have an intrinsic lattice structure and are defined on a continuous domain, from a small number of their projections.

Our algorithm relies on the fact that the problem of reconstructing an image from only two projections can be formulated as a network flow problem in a graph. We derive this formulation for parallel beam and fan beam tomography and show how it can be used to compute binary reconstructions from more than two projections.

Our algorithm is capable of computing high quality reconstructions from very few projections. We evaluate its performance on both simulated and real experimental projection data and compare it to other reconstruction algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)

    Google Scholar 

  2. Batenburg, K.J.: A new algorithm for 3D binary tomography. Electron. Notes Discrete Math. 20, 247–261 (2005)

    Article  MathSciNet  Google Scholar 

  3. Batenburg, K.J.: A network flow algorithm for reconstructing binary images from discrete X-rays. J. Math. Imaging Vis. 27(2), 175–191 (2007)

    Article  MathSciNet  Google Scholar 

  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 (1994)

  5. Censor, Y.: Binary steering in discrete tomography reconstruction with sequential and simultaneous iterative algorithms. Linear Algebra Appl. 339, 111–124 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fishburn, P., Schwander, P., Shepp, L., Vanderbei, R.: The discrete Radon transform and its approximate inversion via linear programming. Discrete Appl. Math. 75, 39–61 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gale, D.: A theorem on flows in networks. Pac. J. Math. 7, 1073–1082 (1957)

    MATH  MathSciNet  Google Scholar 

  8. Gardner, R.J.: Geometric Tomography, 2nd edn. Cambridge University Press, New York (2006)

    MATH  Google Scholar 

  9. Gardner, R.J., Gritzmann, P., Prangenberg, D.: On the computational complexity of reconstructing lattice sets from their X-rays. Discrete Math. 202, 45–71 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by successive approximations. Math. Oper. Res. 15, 430–466 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms and Applications. Birkhäuser, Basel (1999)

    MATH  Google Scholar 

  12. Herman, G.T., Kuba, A. (eds.): Advances in Discrete Tomography and Its Applications. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  13. ILOG CPLEX: http://www.ilog.com/products/cplex/

  14. 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. MRS Proc. 839, 4.5.1–4.5.6 (2004)

    Google Scholar 

  15. Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. SIAM, Philadelphia (2001)

    Google Scholar 

  16. Natterer, F.: The Mathematics of Computerized Tomography. SIAM, Philadelphia (2001)

    MATH  Google Scholar 

  17. O’Rourke, J.: Computational Geometry in C. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  18. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics series, vol. 24. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  19. Schüle, T., Schnörr, C., Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and D.C. programming. Discrete Appl. Math. 151, 229–243 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Weber, S., Schnörr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. Electron. Notes Discrete Math. 12 (2003)

  21. Weber, S., Schüle, T., Schnörr, C., Hornegger, J.: A linear programming approach to limited angle 3D reconstruction from DSA projections. Methods Inf. Med. 4, 320–326 (2004)

    Google Scholar 

  22. Weber, S., Schüle, T., Hornegger, J., Schnörr, C.: Binary tomography by iterating linear programs from noisy projections. In: Proc. of IWCIA 2004. Lecture Notes in Computer Science vol. 3322, pp. 38–51 (2004)

Download references

Author information

Authors and Affiliations

  1. University of Antwerp, Wilrijk, Belgium

    Kees Joost Batenburg

Authors
  1. Kees Joost Batenburg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kees Joost Batenburg.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Batenburg, K.J. A Network Flow Algorithm for Reconstructing Binary Images from Continuous X-rays. J Math Imaging Vis 30, 231–248 (2008). https://doi.org/10.1007/s10851-007-0053-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-007-0053-2

Keywords

Search

Navigation