Super-Resolution and Sparse View CT Reconstruction

  • Guangming ZangEmail author
  • Mohamed Aly
  • Ramzi Idoughi
  • Peter Wonka
  • Wolfgang Heidrich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11220)


We present a flexible framework for robust computed tomography (CT) reconstruction with a specific emphasis on recovering thin 1D and 2D manifolds embedded in 3D volumes. To reconstruct such structures at resolutions below the Nyquist limit of the CT image sensor, we devise a new 3D structure tensor prior, which can be incorporated as a regularizer into more traditional proximal optimization methods for CT reconstruction. As a second, smaller contribution, we also show that when using such a proximal reconstruction framework, it is beneficial to employ the Simultaneous Algebraic Reconstruction Technique (SART) instead of the commonly used Conjugate Gradient (CG) method in the solution of the data term proximal operator. We show empirically that CG often does not converge to the global optimum for tomography problem even though the underlying problem is convex. We demonstrate that using SART provides better reconstruction results in sparse-view settings using fewer projection images. We provide extensive experimental results for both contributions on both simulated and real data. Moreover, our code will also be made publicly available.


Super resolution Proximal optimization Tomography 



This work was supported by KAUST as part of VCC Center Competitive Funding.

Supplementary material

474218_1_En_9_MOESM1_ESM.pdf (3.3 mb)
Supplementary material 1 (pdf 3391 KB)


  1. 1.
    Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. SIAM, Philadelphia (2001)Google Scholar
  2. 2.
    Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections. Springer Science & Business Media, New York (2009)Google Scholar
  3. 3.
    Clinthorne, N.H., Pan, T.S., Chiao, P.C., Rogers, W., Stamos, J.: Preconditioning methods for improved convergence rates in iterative reconstructions. IEEE Trans. Med. Img. 12(1), 78–83 (1993)CrossRefGoogle Scholar
  4. 4.
    Elbakri, I.A., Fessler, J.A.: Efficient and accurate likelihood for iterative image reconstruction in x-ray computed tomography. In: Medical Imaging, International Society for Optics and Photonics, pp. 1839–1850 (2003)Google Scholar
  5. 5.
    Xu, J., Tsui, B.M.: Quantifying the importance of the statistical assumption in statistical X-ray CT image reconstruction. IEEE Trans. Med. Img. 33(1), 61–73 (2014)CrossRefGoogle Scholar
  6. 6.
    Sidky, E.Y., Jørgensen, 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)CrossRefGoogle Scholar
  7. 7.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom. 60(1), 259–268 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)CrossRefGoogle Scholar
  9. 9.
    Parikh, N., Boyd, S.: Proximal algorithms. Foundations and Trends in Optimization (2013)Google Scholar
  10. 10.
    Zang, G., Idoughi, R., Tao, R., Lubineau, G., Wonka, P., Heidrich, W.: Space-time tomography for continuously deforming objects. ACM Trans. Graph. 37(4), 100 (2018)CrossRefGoogle Scholar
  11. 11.
    Mory, C., et al.: ECG-gated C-arm computed tomography using L1 regularization. In: EUSIPCO, IEEE (2012)Google Scholar
  12. 12.
    Rit, S., Oliva, M.V., Brousmiche, S., Labarbe, R., Sarrut, D., Sharp, G.C.: The reconstruction toolkit (RTK), an open-source cone-beam CT reconstruction toolkit based on the insight toolkit (ITK). J. Phys.: Conf. Ser. (2014)Google Scholar
  13. 13.
    Andersen, A., Kak, A.C.: Simultaneous algebraic reconstruction technique (SART): a superior implementation of the art algorithm. Ultrason. Imaging 6(1), 81–94 (1984)CrossRefGoogle Scholar
  14. 14.
    Andersen, A.H.: Algebraic reconstruction in CT from limited views. IEEE Trans. Med. Img. (1989)Google Scholar
  15. 15.
    Mueller, K., Yagel, R., Wheller, J.J.: Fast implementations of algebraic methods for three-dimensional reconstruction from cone-beam data. IEEE Trans. Med. Img. 18(6), 538–548 (1999)CrossRefGoogle Scholar
  16. 16.
    Mueller, K., Yagel, R.: Rapid 3-D cone-beam reconstruction with the simultaneous algebraic reconstruction technique (sart) using 2-D texture mapping hardware. IEEE Trans. Med. Img. 19, 1227–1237 (2000)CrossRefGoogle Scholar
  17. 17.
    Feldkamp, L., Davis, L., Kress, J.: Practical cone-beam algorithm. J. Opt. Soc. Am. 1(6), 612–619 (1984)CrossRefGoogle Scholar
  18. 18.
    Pan, X., Sidky, E.Y., Vannier, M.: Why do commercial ct scanners still employ traditional, filtered back-projection for image reconstruction? Inverse Probl. 25(12), 123009 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (art) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29(3), 471–481 (1970)CrossRefGoogle Scholar
  20. 20.
    Lent, A.: A convergent algorithm for maximum entropy image restoration, with a medical X-ray application. Image Analysis and Evaluation, pp. 249–257 (1977)Google Scholar
  21. 21.
    Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Img. 1(2), 113–122 (1982)CrossRefGoogle Scholar
  22. 22.
    Censor, Y.: Finite series-expansion reconstruction methods. Proc. IEEE (1983)Google Scholar
  23. 23.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer Science & Business Media, Berlin (2011)CrossRefGoogle Scholar
  24. 24.
    Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Fixed-Point Algorithms for Inverse Problems in Science and EngineeringGoogle Scholar
  25. 25.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer Science & Business Media, Berlin (2006)Google Scholar
  26. 26.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ramani, S., Fessler, J.A.: A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction. IEEE Trans. Med. Img. (2012)Google Scholar
  28. 28.
    Nien, H., Fessler, J.: Fast x-ray CT image reconstruction using a linearized augmented lagrangian method with ordered subsets. IEEE Trans. Med. Img. (2015)Google Scholar
  29. 29.
    Esser, E., Zhang, X., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015–1046 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Erdogan, H., Fessler, J.A.: Ordered subsets algorithms for transmission tomography. Phys. Med. Biol. 44(11), 2835 (1999)CrossRefGoogle Scholar
  31. 31.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Adler, J., Öktem, O.: Learned primal-dual reconstruction. IEEE Trans. Med. Img. 37(6), 1348–1357 (2018)CrossRefGoogle Scholar
  33. 33.
    Bai, T., Yan, H., Jia, X., Jiang, S., Wang, G., Mou, X.: Z-index parameterization for volumetric CT image reconstruction via 3-D dictionary learning. IEEE Trans. Med. Img. 36(12), 2466–2478 (2017)CrossRefGoogle Scholar
  34. 34.
    Chen, H., Zhang, Y., Chen, Y., Zhang, J., Zhang, W., Sun, H., Lv, Y., Liao, P., Zhou, J., Wang, G.: Learn: learned experts assessment-based reconstruction network for sparse-data CT. IEEE Trans. Med. Img. (2018)Google Scholar
  35. 35.
    Wu, D., Kim, K., El Fakhri, G., Li, Q.: Iterative low-dose ct reconstruction with priors trained by artificial neural network. IEEE Trans. Med. Img. 36(12), 2479–2486 (2017)CrossRefGoogle Scholar
  36. 36.
    Klukowska, J., Davidi, R., Herman, G.T.: Snark09-a software package for reconstruction of 2D images from 1D projections. Comput. Methods Programs Biomed. 110(3), 424–440 (2013)CrossRefGoogle Scholar
  37. 37.
    Palenstijn, W.J., Batenburg, K.J., Sijbers, J.: The astra tomography toolbox. In: 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE, vol. 2013. (2013)Google Scholar
  38. 38.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)Google Scholar
  39. 39.
    Björck, A.: Numerical methods for least squares problems. SIAM J. Imaging Sci. (1996)Google Scholar
  40. 40.
    Aly, M., Zang, G., Heidrich, W., Wonka, P.: Trex: A tomography reconstruction proximal framework for robust sparse view X-ray applications. arXiv preprint arXiv:1606.03601 (2016)
  41. 41.
    Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)Google Scholar
  42. 42.
    Boyd, S.: Ee364b: Convex optimization II. Course Notes.
  43. 43.
    Weickert, J.: Anisotropic Diffusion in Image Processing, vol. 1. Teubner Stuttgart (1998)Google Scholar
  44. 44.
    Lefkimmiatis, S., Roussos, A., Unser, M., Maragos, P.: Convex generalizations of total variation based on the structure tensor with applications to inverse problems. In: International Conference on Scale Space and Variational Methods in Computer Vision (2013)Google Scholar
  45. 45.
    Lefkimmiatis, S., Roussos, A., Maragos, P., Unser, M.: Structure tensor total variation. SIAM J. Imaging Sci. (2015)Google Scholar
  46. 46.
    Lefkimmiatis, S., Ward, J.P., Unser, M.: Hessian schatten-norm regularization for linear inverse problems. IEEE Trans. Image Proc. 22(5), 1873–1888 (2013)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Gregson, J., Krimerman, M., Hullin, M.B., Heidrich, W.: Stochastic tomography and its applications in 3D imaging of mixing fluids. ACM Trans. Graph. (2012)Google Scholar
  48. 48.
    Forsyth, D., Ponce, J.: Computer Vision: A Modern Approach. Prentice Hall, Upper Saddle River (2002)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Guangming Zang
    • 1
    Email author
  • Mohamed Aly
    • 1
  • Ramzi Idoughi
    • 1
  • Peter Wonka
    • 1
  • Wolfgang Heidrich
    • 1
  1. 1.King Abdullah University of Science and TechnologyThuwalSaudi Arabia

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