Skip to main content
SpringerLink
Log in

Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We propose a scaled gradient projection algorithm for the reconstruction of 3D X-ray tomographic images from limited data. The problem arises from the discretization of an ill-posed integral problem and, due to the incompleteness of the data, has infinite possible solutions. Hence, by following a regularization approach, we formulate the reconstruction problem as the nonnegatively constrained minimization of an objective function given by the sum of a fit-to-data term and a smoothed differentiable Total Variation function. The problem is challenging for its very large size and because a good reconstruction is required in a very short time. For these reasons, we propose to use a gradient projection method, accelerated by exploiting a scaling strategy for defining gradient-based descent directions and generalized Barzilai–Borwein rules for the choice of the step-lengths. The numerical results on a 3D phantom are very promising since they show the ability of the scaling strategy to accelerate the convergence in the first iterations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beister, M., Kolditz, D., Kalender, W.: Iterative reconstruction methods in X-ray CT. Phys. Medica 28, 94–108 (2012)

    Article  Google Scholar 

  4. Bertero, M., Lantéri, H., Zanni, L.: Iterative image reconstruction: a point of view. In: Censor, Y., et al. (eds.) Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), pp. 37–63. Birkhauser-Verlag, Basel (2008)

    Google Scholar 

  5. Bertsekas, D.: Convex Optimization Theory. Supplementary Chapter 6 on Convex Optimization Algorithms. Athena Scientific, Belmont (2009)

    Google Scholar 

  6. Birgin, E.G., Martinez, J.M., Raydan, M.: Inexact spectral projected gradient methods on convex sets. IMA J. Numer. Anal. 23, 539–559 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonettini, S., Landi, G., Piccolomini, L.E., Zanni, L.: Scaling techniques for gradient projection-type methods in astronomical image deblurring. Int. J. Comput. Math. 90(1), 9–29 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bonettini, S., Porta, F., Ruggiero, V.: A variable metric inertial method for convex optimization. SIAM J. Sci. Comput. 31(4), A2558–A2584 (2016)

    Article  MATH  Google Scholar 

  9. Bonettini, S., Prato, M.: New convergence results for the scaled gradient projection method. Inverse Probl. 31(9), 1196–1211 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bonettini, S., Zanella, R., Zanni, L.: A scaled gradient projection method for constrained image deblurring. Inverse Probl. 25(1), 015002 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brenner, D.J., Hall, E.: Computed tomography: an increasing source of radiation exposure. N. Engl. J. Med. 357, 2277–2284 (2007)

    Article  Google Scholar 

  12. Coli, V.L., Piccolomini, E.L., Morotti, E., Zanni, L.: A fast gradient projection method for 3D image reconstruction from limited tomographic data. J. Phys. Conf. Ser. 904, 012013 (2017)

    Article  Google Scholar 

  13. Coli, V.L., Ruggiero, V., Zanni, L.: Scaled first-order methods for a class of large-scale constrained least square problems. In: Sergeyev, Y.D., Kvasov, D.E., Dell’Accio, F., Mukhametzhanov, M.S. (eds.) Numerical Computations: Theory and Algorithms (NUMTA-2016), pp. 040002-1–040002-4. AIP Publishing, Melville (2016)

  14. De Asmundis, R., di Serafino, D., Hager, W., Toraldo, G., Zhang, H.: An efficient gradient method using the Yuan steplength. Comput. Optim. Appl. 59(3), 541–563 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Defrise, M., Vanhove, C., Liu, X.: An algorithm for total variation regularization in high-dimensional linear problems. Inverse Probl. 52, 329–356 (2011)

    MathSciNet  MATH  Google Scholar 

  16. di Serafino, D., Ruggiero, V., Toraldo, G., Zanni, L.: On the steplength selection in gradient methods for unconstrained optimization. Appl. Math. Comput. 318, 176–195 (2018)

  17. Feldkamp, L., Davis, L., Kress, J.: Practical cone-beam algorithm. J. Opt. Soc. Am. 1, 612–619 (1984)

    Article  Google Scholar 

  18. Frassoldati, G., Zanni, L., Zanghirati, G.: New adaptive stepsize selections in gradient methods. J. Ind. Manag. Optim. 4(2), 299–312 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gonzaga, C., Schneider, R.M.: On the steepest descent algorithm for quadratic functions. Comput. Optim. Appl. 63(2), 523–542 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Graff, C., Sidky, E.: Compressive sensing in medical imaging. Appl. Opt. 54(8), C23–C44 (2015)

    Article  Google Scholar 

  21. Jensen, T.L., Jørgensen, J.H., Hansen, P.C., Jensen, S.H.: Implementation of an optimal first-order method for strongly convex total variation regularization. BIT Numer. Math. 52, 329–356 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jørgensen, J.H., Jensen, T.L., Hansen, P.C., Jensen, S.H., Sidky, E.Y., Pan, X.: Accelerated gradient methods for total-variation-based CT image reconstruction. In: 11th Fully 3D Image Reconstruction in Radiology and Nuclear Medicins, pp. 435–438 (2011)

  23. Kim, D., Pal, D., Thibault, J., Fessler, J.A.: Accelerating ordered subsets image reconstruction for X-ray CT using spatially nonuniform optimization transfer. IEEE Trans. Med. Imaging 32(11), 1965–1978 (2013)

    Article  Google Scholar 

  24. Kim, D., Ramani, S., Fessler, J.A.: Combining ordered subsets and momentum for accelerated X-rays CT imaging reconstruction. IEEE Trans. Med. Imaging 34(1), 167–178 (2015)

    Article  Google Scholar 

  25. Lange, K., Hunter, D., Yang, I.: Optimization transfer using surrogate objective functions. J. Comput. Graph. Stat. 9(1), 1–20 (2000)

    MathSciNet  Google Scholar 

  26. Lantéri, H., Roche, M., Aime, C.: Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms. Inverse Probl. 18(5), 1397–1419 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Loli Piccolomini, E., Morotti, E.: A fast TV-based iterative algorithm for digital breast tomosynthesis image reconstruction. J. Algorithms Comput. Technol. 10(4), 277–289 (2016)

    Article  MathSciNet  Google Scholar 

  28. Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course. Applied Optimization. Kluwer Academic Publ., Dordrecht (2004)

    Book  MATH  Google Scholar 

  29. Nesterov, Y.: Gradient methods for minimizing composite functions. Math. Program. 140, 125–161 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Porta, F., Prato, M., Zanni, L.: A new steplength selection for scaled gradient methods with application to image deblurring. J. Sci. Comput. 65(3), 895–919 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rangayyan, R., Dhawan, A., Gordon, R.: Algorithms for limited-view computed tomography: an annotated bibliography and a challenge. Appl. Opt. 24(23), 4000–4012 (1985)

    Article  Google Scholar 

  32. Rose, S., Andersen, M., Sidky, E.Y., Pan, X.: Noise properties of CT images reconstructed by use of constrained total-variation, data-discrepancy minimization. Med. Phys. 42(5), 2690–2698 (2015)

    Article  Google Scholar 

  33. Siddon, R.L.: Fast calculation of the exact radiological path for a three-dimensional CT array. Med. Phys. 12(2), 252–255 (1985)

    Article  Google Scholar 

  34. 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), 3065–3091 (2012)

    Article  Google Scholar 

  35. Sidky, E.Y., Jørgensen, J.H., Pan, X.: First-order convex feasibility for x-ray CT. Med. Phys. 40(3), 3115–1–15 (2013)

    Article  Google Scholar 

  36. Sidky, E.Y., Kao, C.M., Pan, X.: Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT. J. X-ray Sci. Technol. 14(2), 119–139 (2006)

    Google Scholar 

  37. Sidky, E.Y., Pan, X.: Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization. Phys. Med. Biol. 53, 4777–4807 (2008)

    Article  Google Scholar 

  38. Sidky, E.Y., Pan, X., Reiser, I.S., Nishikawa, R.M.: Enhanced imaging of microcalcifications in digital breast tomosynthesis through improved image-reconstruction algorithms. Med. Phys. 36(11), 4920–4932 (2009)

    Article  Google Scholar 

  39. Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002)

    Book  MATH  Google Scholar 

  40. Yu, H., Wang, G.: A soft-threshold filtering approach for reconstruction from a limited number of projections. Phys. Med. Biol. 55, 3905–3916 (2010)

    Article  Google Scholar 

  41. Zhou, B., Gao, L., Dai, Y.H.: Gradient methods with adaptive step-sizes. Comput. Optim. Appl. 35(1), 69–86 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been partially supported by the Italian Institute GNCS - INdAM and by the FAR2015 project of the University of Modena and Reggio Emilia, Italy.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Bologna, Italy

    E. Loli Piccolomini

  2. Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Modena, Italy

    V. L. Coli & L. Zanni

  3. Department of Mathematics, University of Padova, Padova, Italy

    E. Morotti

Authors
  1. E. Loli Piccolomini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. L. Coli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Morotti
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. L. Zanni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Loli Piccolomini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Piccolomini, E.L., Coli, V.L., Morotti, E. et al. Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm. Comput Optim Appl 71, 171–191 (2018). https://doi.org/10.1007/s10589-017-9961-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-017-9961-2

Keywords

Search

Navigation