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Reconstructing Convex Matrices by Integer Programming Approaches

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Journal of Mathematical Modelling and Algorithms in Operations Research

Abstract

We consider the problem of reconstructing two-dimensional convex binary matrices from their row and column sums with adjacent ones. Instead of requiring the ones to occur consecutively in each row and column, we maximize the number of adjacent ones. We reformulate the problem by using integer programming and we develop approximate solutions based on linearization and convexification techniques.

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References

  1. Balázs, P.: A framework for generating some discrete sets with disjoint components by using uniform distributions. Discrete Appl. Math. 406, 15–23 (2008)

    MATH  Google Scholar 

  2. Balázs, P.: Reconstruction of canonical hv-convex discrete sets from horizontal and vertical projections. In: 13th International Workshop on Combinatorial Image Analysis. Lecture Notes in Computer Science, vol. 5852, pp. 280–288. Playa del Carmen, Mexico (2009)

  3. Balázs, P.: A benchmark set for the reconstructin of hv-convex discrete sets. Discrete Appl. Math. 157, 3447–3456 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Balázs, P.: Benchmark collections for the reconstruction of hv-convex discret sets. http://www.inf.u-szeged.hu/pbalazs/benchmark/benchmark.html (2010). Accessed 11 Sept 2009

  5. Billionnet, A., Elloumi, S., Lambert, A.: Extending the QCR method to general mixed-integer programs. Math. Program. 131, 381–401 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Billionnet, A., Elloumi, S., Plateau, M.-C.: Improving the performance of standard solvers for quadratic 0-1 programs by a tight convex reformulation: the QCR method. Discrete Appl. Math. 25, 1185–1197 (2007)

    MathSciNet  Google Scholar 

  7. Chang, S.: The reconstruction of binary patterns from their projections. Commun. ACM 14, 21–25 (1971)

    Article  MATH  Google Scholar 

  8. Chrobak, M., Dürr, C.: Reconstructing hv-convex polyominoes from orthogonal projections. Inf. Process. Lett. 69, 283–289 (1999)

    Article  MATH  Google Scholar 

  9. Dahl, G., Flatberg, T.: Optimization and reconstruction of hv-convex (0, 1)-matrices. Discrete Appl. Math. 151, 93–105 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Del Lungo, A.: Polyominoes defined by two vectors. Theor. Comp. Sci. 127, 187–198 (1994)

    Article  MATH  Google Scholar 

  11. Flajolet, P., Zimmermann, P., Van Custem, B.: A calculs for the random generation of labelled combinatorial structures. Theor. Comp. Sci. 132, 1–35 (1994)

    Article  MATH  Google Scholar 

  12. Fortet, R.: Applications de l’algèbre de boole en recherche opérationnelle. RAIRO Rech. Opér. 4, 5–36 (1959)

    MathSciNet  Google Scholar 

  13. Gale, D.: A theorem on flows in networks. Discrete Math. 187, 1073–1082 (1957)

    MathSciNet  Google Scholar 

  14. Glover, F.: Improved linear integer programming formulation of non linear integer problems. Manage. Sci. 22, 445–460 (1975)

    MathSciNet  Google Scholar 

  15. Haber, R.M.: Term Rank of 0,1 Matrices. Rendiconti del Seminario Matematico della Università di Padova, vol. 30, pp. 24–51 (1960)

  16. Hammer, P.L., Rubin, A.A.: Some remarks on quadratic programming with 0-1 variables. RAIRO Rech. Opér. 3, 67–79 (1970)

    MathSciNet  Google Scholar 

  17. Herman, G.T., Kuba, A.: Discrete Tomography: Foundations, Algorithms and Applications. Birkhauser (1999)

  18. Irving, R.W., Jerrum, M.R.: Three-dimensional statistical data security problems. SIAM J. Comput 23, 170–184 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jarray, F., Costa, M.-C., Picouleau, C.: Approximating hv-convex binary matrices and images from discrete projections. In: 14th Discrete Geometry for Computer Imagery. LNCS, vol. 4992, pp. 413–422. Lyon, France (2008)

    Chapter  Google Scholar 

  20. Kuba, A.: The reconstruction of two-directionally connected binary patterns from their two orthogonal projections. Comput. Vis. Graph. Image Process. 27(3), 249–265 (1984)

    Article  Google Scholar 

  21. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. New York (1988)

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

    Article  Google Scholar 

  23. Pinzani, R., Barcucci, E., Del Lungo, A., Nivat, M.: Reconstructing convex polyominoes from horizontal and vertical projections. Theor. Comp. Sci. 155(2), 321–347 (1996)

    Article  MATH  Google Scholar 

  24. Prause, G.P.M., Onnasch, D.G.W.: Binary reconstruction of the heart chambers from biplane angiographic image sequence. IEEE Trans. Med. Imag. 15, 532–559 (1996)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Shliferstein, A.R., Chien, Y.T.: Switching components and the ambiguity problem in the reconstruction of pictures from their projections. Pattern Recogn. 10, 327–340 (1978)

    Article  MATH  Google Scholar 

  27. Wang, Y.: Characterization of binary patterns and their projections. IEEE Trans. Comput. 24, 1032–1035 (1975)

    Article  MATH  Google Scholar 

  28. Woeginger, G.: The reconstruction of polyominoes from their orthogonal projections. Inf. Process. Lett. 77, 225–229 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. CEDRIC-CNAM, 292 rue St-Martin, 75003, Paris, France

    Alain Billionnet, Fethi Jarray & Ghassen Tlig

  2. Faculté des Sciences de Tunis El Manar, 2092, El Manar II, Tunisie

    Ghassen Tlig & Ezzedine Zagrouba

Authors
  1. Alain Billionnet
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  2. Fethi Jarray
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  3. Ghassen Tlig
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  4. Ezzedine Zagrouba
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Correspondence to Fethi Jarray.

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Billionnet, A., Jarray, F., Tlig, G. et al. Reconstructing Convex Matrices by Integer Programming Approaches. J Math Model Algor 12, 329–343 (2013). https://doi.org/10.1007/s10852-012-9193-5

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  • DOI: https://doi.org/10.1007/s10852-012-9193-5

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