Skip to main content

Approximating Bicolored Images from Discrete Projections

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 6636))

Abstract

We study the problem of reconstructing bicolored images from their discrete projections that is the number of pixels of each color lying on each row and column. The problem is well known to be NP- complete so, we study a restricted case (with bounded projections) and present an approximating algorithm based on a max-flow technique for the general case.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anstee, R.P.: The network flows approach for matrices with given row and column sums. Discrete Math. 44, 125–138 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: X-rays characterizing some classes of discrete sets. Linear Algebra and its Applications 339, 3–21 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Batenburg, K.J.: Network flow algorithms for discrete tomography. In: Herman, G., Kuba, A. (eds.) Advances in Discrete Tomography and its Applications, pp. 175–205. Birkhäuser, Boston (2007)

    Chapter  Google Scholar 

  4. Batenburg, K.J.: An evolutionary algorithm for discrete tomography. Discrete Applied Mathematics 151, 36–54 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brunetti, S., Costa, M.C., Frosini, A., Jarray, F., Picouleau, C.: Reconstruction of binary matrices under adjacency constraints. In: Herman, G., Kuba, A. (eds.) Advances in Discrete Tomography and its Applications, Boston, pp. 125–150 (2007)

    Google Scholar 

  6. Baumann, J., Kiss, Z., Krimmel, S., Kauba, A., Nagy, A., Rodek, L., Schillinger, S., Stephan, J.: Discrete tomography methods for nondestructive testing. In: Herman, G., Kuba, A. (eds.) Advances in Discrete Tomography and its Applications, Boston, pp. 303–331 (2007)

    Google Scholar 

  7. Brocchi, S., Frosini, A., Rinaldi, S.: Solving some instances of the two color problem. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 505–516. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  8. Chrobak, M., Dürr, C.: Reconstructing Polyatomic Structures from X-Rays: NP Completness proof for three Atoms. Theoretical computer Science 259(1), 1–98 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Chrobak, M., Couperus, P., Dürr, C., Woeginger, G.: A note on tiling under tomographic constraints. Theoretical computer Science 290, 2125–2136 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Costa, M.C., De Werra, D., Picouleau, C.: Using graphs for some discrete tomography problems. Discrete Applied Mathematics 154, 35–46 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Costa, M.C., de Werra, D., Picouleau, C., Schindld, D.: Discrete Applied Mathematics 148, 240–245 (2005)

    Article  MathSciNet  Google Scholar 

  12. Dürr, C., Guinez, F., Matamala, M.: Reconstructing 3-colored grids from horizontal and vertical projections is NP-hard, arXiv:0904.3169v1 (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Gardner, R.J., Gritzmann, P., Prangenberg, D.: On the computational complexity of determining polyatomic structures by X-rays. Theoretical Computer Science 233, 91–106 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goldberg, A.V.: An efficient implementation of a scaling minimum-cost flow algorithm. Journal of Algorithms 22, 1–29 (1997)

    Article  MathSciNet  Google Scholar 

  17. Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. Journal ACM 45, 783–797 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hall, P.: A model for learning human vascular anatomy. DIMACS Serie in Discrete Mathematical Problems with Medical Applications 55, 11–27 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Herman, G.T., Kuba, A.: Advances in Discrete Tomography and its Applications. Birkhäuser, Boston (2007)

    Book  MATH  Google Scholar 

  20. Jarray, F.: Solving problems of discrete tomography. Applications in workforce scheduling, Ph.D. Thesis, University of CNAM, Paris (2004)

    Google Scholar 

  21. Jarray, F.: Workforce scheduling and table coloring. In: Proceedings ROADEF 2005, Tours, pp. 92–100 (2004)

    Google Scholar 

  22. Jarray, F.: A lagrangean approach to reconstruct bicolored images from discrete orthogonal projections. Pure mathematics and applications (Linear algebra and computer science) 20(1), 17–25 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Kisielowski, C., Schwander, P., Baumann, F.H., Seibt, M., Kim, Y., Ourmazd, A.: An approach to quantitative high-resolution transmission electron microscopy of crystalline materials. Ultramicroscopy 58, 131–155 (1995)

    Article  Google Scholar 

  24. Onnasch, D.G.W., Prause, G.P.M.: Heart Chamber Reconstruction from Biplane Angiography. In: Herman, G., Kuba, A. (eds.) Discrete Tomography: Foundations, Algorithms and Applications, pp. 385–403. Birkhäuser, Boston (1999)

    Chapter  Google Scholar 

  25. Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9, 371–377 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sachs, J.R.J., Sauer, K.: 3D Reconstruction from sparse Radiographic Data. In: Herman, G., Kuba, A. (eds.) Discrete Tomography: Foundations, Algorithms and Applications, pp. 363–383. Birkhäuser, Boston (1999)

    Chapter  Google Scholar 

  27. Salzberg, P.M., Rivera-vega, P.I., Rodriguez, A.: Network flow model for binary tomography on lattices. Internal journal imaging system technology 9, 145–154 (1998)

    Google Scholar 

  28. Slump, C.H., et al.: A network flow approach to reconstruction of the left ventricle from two projections. Comput. Gr. Im. Proc. 18, 18–36 (1982)

    Article  Google Scholar 

  29. Wang, B., Zhang, F.: On the precise Number of (0,1)-Matrices in \(\mathcal{U}(R,S)\). Discrete Mathematics 187, 211–220 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jarray, F., Tlig, G. (2011). Approximating Bicolored Images from Discrete Projections. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds) Combinatorial Image Analysis. IWCIA 2011. Lecture Notes in Computer Science, vol 6636. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21073-0_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-21073-0_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21072-3

  • Online ISBN: 978-3-642-21073-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics