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German Conference on Pattern Recognition

GCPR 2016: Pattern Recognition pp 261–272Cite as

Discrete Tomography by Continuous Multilabeling Subject to Projection Constraints

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Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9796))

Abstract

We present a non-convex variational approach to non-binary discrete tomography which combines non-local projection constraints with a continuous convex relaxation of the multilabeling problem. Minimizing this non-convex energy is achieved by a fixed point iteration which amounts to solving a sequence of convex problems, with guaranteed convergence to a critical point. A competitive numerical evaluation using standard test-datasets demonstrates a significantly improved reconstruction quality for noisy measurements from a small number of projections.

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References

  1. Aarle, W., Palenstijn, W., Beenhouwer, J., Altantzis, T., Bals, S., Batenburg, K., Sijbers, J.: The ASTRA toolbox: a platform for advanced algorithm development in electron tomography. Ultramicroscopy 157, 35–47 (2015)

    Article  Google Scholar 

  2. Batenburg, K., Sijbers, J.: DART: a practical reconstruction algorithm for discrete tomography. IEEE Trans. Image Proc. 20(9), 2542–2553 (2011)

    Article  MathSciNet  Google Scholar 

  3. Bushberg, J., Seibert, J., Leidholdt, E., Boone, J.: The Essential Physics of Medical Imaging, 3rd edn. Wolters Kluwer, Philadelphia (2011)

    Google Scholar 

  4. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Denitiu, A., Petra, S., Schnörr, C., Schnörr, C.: Phase transitions and cosparse tomographic recovery of compound solid bodies from few projections. Fundamenta Informaticae 135, 73–102 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Goris, B., Broek, W., Batenburg, K., Mezerji, H., Bals, S.: Electron tomography based on a total variation minimization reconstruction technique. Ultramicroscopy 113, 120–130 (2012)

    Article  Google Scholar 

  7. Hanke, R., Fuchs, T., Uhlmann, N.: X-ray based methods for non-destructive testing and material characterization. Nucl. Instrum. Methods Phys. Res. Sect. A Accelerators, Spectrometers, Detectors Assoc. Equip. 591(1), 14–18 (2008)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  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, Heidelberg (2015)

    Chapter  Google Scholar 

  10. Klann, E., Ramlau, R.: Regularization properties of mumford-shah-type functionals with perimeter and norm constraints for linear ill-posed problems. SIAM J. Imaging Sci. 6(1), 413–436 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lellmann, J., Becker, F., Schnörr, C.: Convex optimization for multi-class image labeling with a novel family of total variation based regularizers. In: Proceedings of the IEEE Conference on Computer Vision (ICCV 2009) Kyoto, Japan, vol. 1, pp. 646–653 (2009)

    Google Scholar 

  12. Maeda, S., Fukuda, W., Kanemura, A., Ishii, S.: Maximum a posteriori X-ray computed tomography using graph cuts. J. Phys. Conf. Ser. 233, 012023 (2010)

    Article  Google Scholar 

  13. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Natterer, F.: The Mathematics of Computerized Tomography, 1 edn. Society for Industrial and Applied Mathematics, Philadelphia (2001). http://epubs.siam.org/doi/abs/10.1137/1.9780898719284

  15. Dinh, T.P., Bernoussi, S.: Algorithms for solving a class of nonconvex optimization problems. methods of subgradients. In: Hiriart-Urruty, J.B. (ed.) Fermat Days 85: Mathematics for Optimization. North-Holland Mathematics Studies, vol. 129, pp. 249–271. North-Holland, Amsterdam (1986)

    Chapter  Google Scholar 

  16. Dinh, T., Hoai An, L.: Convex analysis approach to D.C. programming: theory, algorithms and applications. Acta Math. Vietnamica 22(1), 289–355 (1997)

    MathSciNet  MATH  Google Scholar 

  17. Pock, T., Chambolle, A., Cremers, D., Bischof, H.: A convex relaxation approach for computing minimal partitions. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2009, pp. 810–817, June 2009

    Google Scholar 

  18. Potts, R.B.: Some generalized order-disorder transformations. Math. Proc. Camb. Phil. Soc. 48, 106–109 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ramlau, R., Ring, W.: A mumford-shah level-set approach for the inversion and segmentation of X-ray tomography data. J. Comput. Phys. 221(2), 539–557 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, vol. 317. Springer, Heidelberg (2009)

    MATH  Google Scholar 

  21. Schüle, T., Schnörr, C., Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and D.C. programming. Discrete Appl. Math. 151(13), 229–243 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sidky, E.Y., Pan, X.: Image Reconstruction in Circular Cone-Beam Computed Tomography by Constrained, Total-Variation Minimization. Phys. Med. Biol. 53(17), 4777 (2008)

    Article  Google Scholar 

  23. Storath, M., Weinmann, A., Frikel, J., Unser, M.: Joint image reconstruction and segmentation using the potts model. Inverse Probl. 31(2), 025003 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Toland, J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66(2), 399–415 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tuysuzoglu, A., Karl, W., Stojanovic, I., Castanon, D., Unlu, M.: Graph-cut based discrete-valued image reconstruction. IEEE Trans. Image Process. 24(5), 1614–1627 (2015)

    Article  MathSciNet  Google Scholar 

  26. Varga, L., Balázs, P., Nagy, A.: An energy minimization reconstruction algorithm for multivalued discrete tomography. In: 3rd International Symposium on Computational Modeling of Objects Represented in Images, Rome, Italy, Proceedings, pp. 179–185. Taylor & Francis (2012)

    Google Scholar 

  27. Weber, S.: Discrete tomography by convex-concave regularization using linear and quadratic optimization. Ph.D. thesis, Ruprecht-Karls-Universität, Heidelberg, Germany (2009)

    Google Scholar 

  28. 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.) DGCI 2006. LNCS, vol. 4245, pp. 146–156. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  29. Weber, S., Schnörr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. Electron. Notes Discrete Math. 12, 243–254 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zach, C., Gallup, D., Frahm, J., Niethammer, M.: Fast global labeling for real-time stereo using multiple plane sweeps. In: Proceedings of the Vision, Modeling, and Visualization Conference 2008, VMV 2008, Konstanz, Germany, 8–10 October 2008, pp. 243–252 (2008)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Heidelberg University, Heidelberg, Germany

    Matthias Zisler, Stefania Petra, Claudius Schnörr & Christoph Schnörr

Authors
  1. Matthias Zisler
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  2. Stefania Petra
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  3. Claudius Schnörr
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  4. Christoph Schnörr
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Corresponding author

Correspondence to Matthias Zisler .

Editor information

Editors and Affiliations

  1. University of Hannover, Hannover, Germany

    Bodo Rosenhahn

  2. Max Planck Institute for Informatics, Saarbrücken, Germany

    Bjoern Andres

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Zisler, M., Petra, S., Schnörr, C., Schnörr, C. (2016). Discrete Tomography by Continuous Multilabeling Subject to Projection Constraints. In: Rosenhahn, B., Andres, B. (eds) Pattern Recognition. GCPR 2016. Lecture Notes in Computer Science(), vol 9796. Springer, Cham. https://doi.org/10.1007/978-3-319-45886-1_21

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  • DOI: https://doi.org/10.1007/978-3-319-45886-1_21

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45885-4

  • Online ISBN: 978-3-319-45886-1

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