Journal of Mathematical Imaging and Vision

, Volume 49, Issue 2, pp 335–351 | Cite as

Efficient Binary Tomographic Reconstruction

  • Stéphane RouxEmail author
  • Hugo Leclerc
  • François Hild


Tomographic reconstruction of a binary image from few projections is considered. A novel heuristic algorithm is proposed, the central element of which is a nonlinear transformation ψ(p)=log(p/(1−p)) of the probability p that a pixel of the sought image be 1-valued. It consists of backprojections based on ψ(p) and iterative corrections. Application of this algorithm to a series of artificial test cases leads to exact binary reconstructions, (i.e., recovery of the binary image for each single pixel) from the knowledge of projection data over a few directions. Images up to 106 pixels are reconstructed in a few seconds. A series of test cases is performed for comparison with previous methods, showing a better efficiency and reduced computation times.


Tomographic reconstruction Discrete reconstruction Binary reconstruction Binary image 



Communication of the raw tomographic data shown in Fig. 4 by E. Gouillart (CNRS/Saint-Gobain, Aubervilliers, France) and C. Zang (RWTH, Aachen, Germany) is gratefully acknowledged. We are also indebted to E. Gouillart for the suggestion that boundary sites may be an appropriate measure of complexity as proposed in Ref. [24]. We also thank an anonymous reviewer for interesting and constructive remarks. This work is supported by the French Agence Nationale de la Recherche through “RUPXCUBE” (ANR-09-BLAN-0009-01) and “EDDAM” (ANR-11-BS09-027) projects.


  1. 1.
    Atkinson, C., Soria, J.: An efficient simultaneous reconstruction technique for tomographic particle image velocimetry. Exp. Fluids 47, 553–568 (2009) CrossRefGoogle Scholar
  2. 2.
    Baruchel, J., Buffière, J.-Y., Cloetens, P., di Michiel, M., Ferrié, E., Ludwig, W., Maire, E., Salvo, L.: Advances in synchrotron radiation microtomography. Scr. Mater. 55, 41–46 (2006) CrossRefGoogle Scholar
  3. 3.
    Basu, S., Bresler, Y.: \({\mathcal{O}}(N^{2} \log_{2}N)\) filtered backprojection reconstruction algorithm for tomography. IEEE Trans. Image Process. 9, 1760–1773 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Basu, S., Bresler, Y.: Error analysis and performance optimization of fast hierarchical backprojection algorithms. IEEE Trans. Image Process. 10, 1103–1117 (2001) CrossRefzbMATHGoogle Scholar
  5. 5.
    Batenburg, K.J.: A network flow algorithm for reconstructing binary images from discrete X-rays. J. Math. Imaging Vis. 27, 175–191 (2007) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Batenburg, K.J.: A network flow algorithm for reconstructing binary images from continuous X-rays. J. Math. Imaging Vis. 30, 231–248 (2008) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Batenburg, K.J., Sijbers, J.: Generic iterative subset algorithms for discrete tomography. Discrete Appl. Math. 157, 438–451 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Byrne, C.L.: Iterative image reconstruction algorithms based on cross-entropy minimization. IEEE Trans. Image Process. 2, 96–103 (1993) CrossRefGoogle Scholar
  9. 9.
    Byrne, C.L.: Erratum and addendum to ‘Iterative image reconstruction algorithms based on cross-entropy minimization’. IEEE Trans. Image Process. 4, 226–227 (1995) Google Scholar
  10. 10.
    Byrne, C.L.: Block-iterative methods for image reconstruction from projections. IEEE Trans. Image Process. 5, 792–794 (1996) CrossRefGoogle Scholar
  11. 11.
    Byrne, C.L.: Iterative algorithms for deblurring and deconvolution with constraints. Inverse Probl. 14, 1455–1467 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52, 489–509 (2006) CrossRefzbMATHGoogle Scholar
  13. 13.
    Carvalho, B.M., Herman, G.T., Matej, S., Salzberg, C., Vardi, E.: Binary tomography for triplane cardiography. In: Kuba, A., et al. (eds.) IPMI’99. LNCS, vol. 1613, pp. 29–41. Springer, Berlin (1999) Google Scholar
  14. 14.
    Censor, Y.: Binary steering in discrete tomography reconstruction with sequential and simultaneous iterative algorithms. Linear Algebra Appl. 339, 111–124 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Darroch, J.N., Ratcliff, D.: Generalized iterative scaling for log linear models. Ann. Math. Stat. 43, 1470–1480 (1972) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Donoho, D.L.: Neighborly polytopes and sparse solution of underdetermined linear equations. Technical Report, Department of Statistics, Stanford University (2004) Google Scholar
  17. 17.
    Donoho, D.L., Tanner, J.: Counting the faces of randomly-projected hypercubes and orthants with applications. Discrete Comput. Geom. 43, 522–541 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Donoho, D.L., Tanner, J.: Precise undersampling theorems. Proc. IEEE 98, 913–924 (2010) CrossRefGoogle Scholar
  19. 19.
    Elsinga, G.E., Scarano, F., Wieneke, B., van Oudheusden, B.W.: Tomographic particle image velocimetry. Exp. Fluids 41, 933–947 (2006) CrossRefGoogle Scholar
  20. 20.
    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) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Gardner, R.J.: Geometric Tomography. Cambridge University Press, New York (2006) CrossRefzbMATHGoogle Scholar
  22. 22.
    Gardner, R.J., Gritzmann, P.: Discrete tomography: determination of finite sets by X-rays. Trans. Am. Math. Soc. 349, 2271–2295 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    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) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Gouillart, E., Krzakala, F., Mezard, M., Zdeborová, L.: Belief propagation reconstruction for discrete tomography. Inverse Probl. 29, 035003 (2013) CrossRefGoogle Scholar
  25. 25.
    Gritzmann, P., de Vries, S., Wiegelmann, M.: Approximating binary images from discrete X-rays. SIAM J. Optim. 11, 522–546 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms and Applications. Bikhäuser, Boston (1999) zbMATHGoogle Scholar
  27. 27.
    Herman, G.T., Kuba, A. (eds.): Advances in Discrete Tomography and Its Applications. Bikhäuser, Basel (2007) zbMATHGoogle Scholar
  28. 28.
    Herman, G.T., Lent, A.: Iterative reconstruction algorithms. Comput. Biol. Med. 6, 273–294 (1976) CrossRefGoogle Scholar
  29. 29.
    Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. SIAM, Philadelphia (2001) CrossRefGoogle Scholar
  30. 30.
    Liao, H.Y., Herman, G.T.: Direct Image Reconstruction-Segmentation as Motivated by Electron Microscopy. In: Herman, G.T., Kuba, A. (eds.): Advances in Discrete Tomography and Its Applications. Bikhäuser, Basel (2007) Google Scholar
  31. 31.
    Manku, G.S., Rajagopalan, S., Lindsay, B.G.: Approximate medians and other quantiles in one pass and with limited memory. In: ACM SIGMOD, vol. 12, pp. 426–435 (1998) Google Scholar
  32. 32.
    Mersereau, R.M.: Direct Fourier transform techniques in 3-D image reconstruction. Comput. Biol. Med. 6, 247–258 (1976) CrossRefGoogle Scholar
  33. 33.
    Needell, D., Ward, R.: Stable image reconstruction using total variation minimization. SIAM J. Imaging Sci. 6, 1035–1058 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Schmidlin, P.: Iterative Separation of Sections in Tomographic Scintigrams. Nucl. Med., vol. 15. Schattauer, Stuttgart (1972) Google Scholar
  35. 35.
    Sina Jafarpour, S., Xu, W., Hassibi, B., Calderbank, A.R.: Efficient and robust compressed sensing using optimized expander graphs. IEEE Trans. Inf. Theory 55, 4299–4308 (2009) CrossRefGoogle Scholar
  36. 36.
    Slump, C.H., Gerbrands, J.J.: A network flow approach to reconstruction of the left ventricle from two projections. Comput. Graph. Image Process. 18, 18–36 (1982) CrossRefGoogle Scholar
  37. 37.
    Varga, L., Balázs, P., Nagy, A.: Direction-dependency of a binary tomographic reconstruction algorithm. In: Barneva, R.P., et al. (eds.) CompIMAGE 2010. Lect. Notes in Comput. Sci., vol. 6026, pp. 242–253 (2010) Google Scholar
  38. 38.
    Weber, S., Schnörr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. Electron. Notes Discrete Math. 12 (2003) Google Scholar
  39. 39.
    Weber, S., Nagy, A., Schüle, T., Schnörr, C., Kuba, A.: A benchmark evaluation of large-scale optimization approaches to binary tomography. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) Lect. Notes in Comput. Sci., vol. 4245, pp. 146–156 (2006) Google Scholar
  40. 40.
    Zang, C.: Private communication (2011) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LMT-Cachan (ENS de Cachan/CNRS/UPMC/PRES UniverSud Paris)CachanFrance

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