Abstract
We present an iterative reconstruction algorithm for binary tomography, called TomoGC, that solves the reconstruction problem based on a constrained graphical model by a sequence of graphcuts. TomoGC reconstructs objects even if a low number of measurements are only given, which enables shorter observation periods and lower radiation doses in industrial and medical applications. We additionally suggest some modifications of established methods that improve state-of-the-art methods. A comprehensive numerical evaluation demonstrates that the proposed method can reconstruct objects from a small number of projections more accurate and also faster than competitive methods.
Similar content being viewed by others
References
van Aarle, W., Palenstijn, W.J., Beenhouwer, J.D., Altantzis, T., Bals, S., Batenburg, K.J., Sijbers, J.: The ASTRA-toolbox: a platform for advanced algorithm development in electron tomography. Ultramicroscopy (2015)
Batenburg, K.J.: A network flow algorithm for reconstructing binary images from continuous x-rays. J. Math. Imaging Vis. 30(3), 231–248 (2008)
Batenburg, K.J., Sijbers, J.: Generic iterative subset algorithms for discrete tomography. Discrete Appl. Math. 157(3), 438–451 (2009)
Batenburg, K.J., Sijbers, J.: DART: a practical reconstruction algorithm for discrete tomography. IEEE Trans. Image Process. 20(9), 2542–2553 (2011)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)
Bleichrodt, F., Tabak, F., Batenburg, K.J.: SDART: an algorithm for discrete tomography from noisy projections. Comput. Vis. Image Underst. 129, 63–74 (2014)
Bracewell, R.N., Riddle, A.C.: Inversion of fan-beam scans in radio astronomy. Astron. J. 150(2), 427–434 (1967)
Capricelli, T., Combettes, P.: A convex programming algorithm for noisy discrete tomography. In: Advances in Discrete Tomography and its Applications. Birkhäuser, Boston (2007)
Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, vol. 2057. Springer, Heidelberg (2013)
Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM Rev. 23(4), 444–466 (1981)
Censor, Y., Zenios, S.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford University Press, New York (1997)
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)
Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 136–152. Springer, Heidelberg (2005)
Combettes, P.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5–6), 475–504 (2004)
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)
Gorelick, L., Schmidt, F.R., Boykov, Y.: Fast trust region for segmentation. In: 2013 IEEE Conference on Computer Vision and Pattern Recognition, Portland, OR, USA, 23–28 June, 2013, pp. 1714–1721 (2013)
Goris, B., Van den Broek, W., Batenburg, K., Mezerji, H., Bals, S.: Electron tomography based on a total variation minimization reconstruction techniques. Ultramicroscopy 113, 120–130 (2012)
Gouillart, E., Krzakala, F., Mezard, M., Zdeborova, L.: Belief-propagation reconstruction for discrete tomography. Inverse Prob. 29(3), 035003 (2013)
Gregor, J., Benson, T.: Computational analysis and improvement of SIRT. IEEE Trans. Med. Imaging 27(7), 918–924 (2008)
Gustavsson, E., Patriksson, M., Strömberg, A.B.: Primal convergence from dual subgradient methods for convex optimization. Math. Program. 150(2), 365–390 (2015)
Hanke, R., Fuchs, T., Uhlmann, N.: X-ray based methods for non-destructive testing and material characterization. Nucl. Instrum. Meth. Phys. Res. Sect. A: Accelerators, Spectrometers, Detectors Associated Equipment 591(1), 14–18 (2008). Radiation Imaging Detectors 2007 Proceedings of the 9th International Workshop on Radiation Imaging Detectors
Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46, 105–122 (1990)
Kolmogorov, V., Zabin, R.: What energy functions can be minimized via graph cuts? IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 147–159 (2004)
Kolmogorov, V.: Convergent tree-reweighted message passing for energy minimization. IEEE Trans. Pattern Anal. Mach. Intell. 28(10), 1568–1583 (2006)
Kolmogorov, V., Rother, C.: Minimizing nonsubmodular functions with graph cuts-a review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1274–1279 (2007)
Komodakis, N., Paragios, N., Tziritas, G.: MRF energy minimization and beyond via dual decomposition. IEEE Trans. Pattern Anal. Mach. Intell. 33(3), 531–552 (2011)
Lim, Y., Jung, K., Kohli, P.: Efficient energy minimization for enforcing label statistics. IEEE Trans. Pattern Anal. Mach. Intell. 36(9), 1893–1899 (2014)
Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1(3), 1–108 (2013)
Smith-Bindman, R., Lipson, J., Marcus, R., et al.: Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer. Arch. Intern. Med. 169(22), 2078–2086 (2009)
Raj, A., Singh, G., Zabih, R.: MRF’s for MRI’s: Bayesian reconstruction of MR images via graph cuts. In: 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2006), New York, NY, USA, 17–22 June 2006, pp. 1061–1068 (2006)
Sidky, E.Y., Jakob, H., Jörgensen, J.H., Pan, X.: Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm. Phys. Med. Biol. 57(10), 3065 (2012)
Storath, M., Weinmann, A., Frikel, J., Unser, M.: Joint image reconstruction and segmentation using the Potts model. Inverse Prob. 31(2), 025003 (2015)
Tang, M., Ben Ayed, I., Boykov, Y.: Pseudo-bound optimization for binary energies. In: Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T. (eds.) ECCV 2014, Part V. LNCS, vol. 8693, pp. 691–707. Springer, Heidelberg (2014)
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)
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)
Weber, S., Schnörr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. Electron. Notes Discrete Math. 12, 243–254 (2003)
Xiao, L., Johansson, M., Boyd, S.: Simultaneous routing and resource allocation via dual decomposition. IEEE Trans. Commun. 52(7), 1136–1144 (2004)
Acknowledgements
Financial support of our research work by the DFG, grant GRK 1653. is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Kappes, J.H., Petra, S., Schnörr, C., Zisler, M. (2015). TomoGC: Binary Tomography by Constrained GraphCuts. In: Gall, J., Gehler, P., Leibe, B. (eds) Pattern Recognition. DAGM 2015. Lecture Notes in Computer Science(), vol 9358. Springer, Cham. https://doi.org/10.1007/978-3-319-24947-6_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-24947-6_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24946-9
Online ISBN: 978-3-319-24947-6
eBook Packages: Computer ScienceComputer Science (R0)