Advertisement

A Novel Convex Relaxation for Non-binary Discrete Tomography

  • Jan KuskeEmail author
  • Paul Swoboda
  • Stefania Petra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)

Abstract

We present a novel convex relaxation and a corresponding inference algorithm for the non-binary discrete tomography problem, that is, reconstructing discrete-valued images from few linear measurements. In contrast to state of the art approaches that split the problem into a continuous reconstruction problem for the linear measurement constraints and a discrete labeling problem to enforce discrete-valued reconstructions, we propose a joint formulation that addresses both problems simultaneously, resulting in a tighter convex relaxation. For this purpose a constrained graphical model is set up and evaluated using a novel relaxation optimized by dual decomposition. We evaluate our approach experimentally and show superior solutions both mathematically (tighter relaxation) and experimentally in comparison to previously proposed relaxations.

Keywords

Linear Programming Relaxation Convex Relaxation Bundle Method High Order Factor Restricted Isometry Property 
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.

References

  1. 1.
  2. 2.
    Batenburg, K.J.: An evolutionary algorithm for discrete tomography. Discret. Appl. Math. 151(1), 36–54 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Batenburg, K.J.: A network flow algorithm for reconstructing binary images from continuous X-rays. JMIV 30(3), 231–248 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Batenburg, K.J., Sijbers, J.: DART: a practical reconstruction algorithm for discrete tomography. IEEE TIP 20(9), 2542–2553 (2011)MathSciNetGoogle Scholar
  5. 5.
    Bussieck, M., Hassler, H., Woeginger, G.J., Zimmermann, U.T.: Fast algorithms for the maximum convolution problem. Oper. Res. Lett. 15, 1–5 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carvalho, B.M., Herman, G.T., Matej, S., Salzberg, C., Vardi, E.: Binary tomography for triplane cardiography. In: Kuba, A., Šáamal, M., Todd-Pokropek, A. (eds.) IPMI 1999. LNCS, vol. 1613, pp. 29–41. Springer, Heidelberg (1999). doi: 10.1007/3-540-48714-X_3 CrossRefGoogle Scholar
  7. 7.
    Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Birkhäuser, Basel (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gouillart, E., Krzakala, F., Mzard, M., Zdeborov, L.: Belief-propagation reconstruction for discrete tomography. Inverse Prob. 29(3), 035003 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kappes, J.H., Petra, S., Schnörr, C., Zisler, M.: TomoGC: binary tomography by constrained graphcuts. In: Gall, J., Gehler, P., Leibe, B. (eds.) GCPR 2015. LNCS, vol. 9358, pp. 262–273. Springer, Cham (2015). doi: 10.1007/978-3-319-24947-6_21 CrossRefGoogle Scholar
  10. 10.
    Keiper, S., Kutyniok, G., Lee, D.G., Pfander, G.E.: Compressed sensing for finite-valued signals. ArXiv e-prints, September 2016Google Scholar
  11. 11.
    Liao, H.Y., Herman, G.T.: Automated estimation of the parameters of Gibbs priors to be used in binary tomography. Discret. Appl. Math. 139(1–3), 149–170 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mohammad-Djafari, A.: Gauss-Markov-Potts priors for images in computer tomography resulting to joint optimal reconstruction and segmentation. Int. J. Tomogr. Stat. 11(W09), 76–92 (2008)MathSciNetGoogle Scholar
  13. 13.
    Roux, S., Leclerc, H., Hild, F.: Efficient binary tomographic reconstruction. JMIV 49(2), 335–351 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Schüle, T., Schnörr, C., Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and D.C. programming. Discret. Appl. Math. 151, 229–243 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sontag, D., Globerson, A., Jaakkola, T.: Introduction to dual decomposition for inference. In: Optimization for Machine Learning. MIT Press (2011)Google Scholar
  16. 16.
    Tarlow, D., Swersky, K., Zemel, R.S., Adams, R.P., Frey, B.J.: Fast exact inference for recursive cardinality models. In: UAI (2012)Google Scholar
  17. 17.
    Weber, S., Schnörr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. In: IWCIA (2003)Google Scholar
  18. 18.
    Werner, T.: A linear programming approach to max-sum problem: a review. IEEE TPAMI 29(7), 1165–1179 (2007)CrossRefGoogle Scholar
  19. 19.
    Zisler, M., Petra, S., Schnörr, C., Schnörr, C.: Discrete tomography by continuous multilabeling subject to projection constraints. In: Rosenhahn, B., Andres, B. (eds.) GCPR 2016. LNCS, vol. 9796, pp. 261–272. Springer, Cham (2016). doi: 10.1007/978-3-319-45886-1_21 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.MIG, Institute of Applied MathematicsHeidelberg UniversityHeidelbergGermany
  2. 2.Institute of Science and Technology (IST) AustriaKlosterneuburgAustria

Personalised recommendations