Skip to main content
SpringerLink
Log in

Reconstruction of sparse-view tomography via preconditioned Radon sensing matrix

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Computed Tomography (CT) is one of the significant research areas in the field of medical image analysis. As X-rays used in CT image reconstruction are harmful to the human body, it is necessary to reduce the X-ray dosage while also maintaining good quality of CT images. Since medical images have a natural sparsity, one can directly employ compressive sensing (CS) techniques to reconstruct the CT images. In CS, sensing matrices having low coherence (a measure providing correlation among columns) provide better image reconstruction. However, the sensing matrix constructed through the incomplete angular set of Radon projections typically possesses large coherence. In this paper, we attempt to reduce the coherence of the sensing matrix via a square and invertible preconditioner possessing a small condition number, which is obtained through a convex optimization technique. The stated properties of our preconditioner imply that it can be used effectively even in noisy cases. We demonstrate empirically that the preconditioned sensing matrix yields better signal recovery than the original sensing matrix.

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

Similar content being viewed by others

References

  1. Frush, D.P., Donnelly, L.F., Rosen, N.S.: Computed tomography and radiation risks: what pediatric health care providers should know. Pediatrics 112, 951–957 (2003)

    Article  Google Scholar 

  2. Sastry, C.S., Das, P.C.: Wavelet based multilevel backprojection algorithm for parallel and fan beam scanning geometries. Int. J. Wavelets Multiresolution Inf. Process. 4(3), 523–545 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jan, J.: Medical Image Processing, Reconstruction and Restoration: Concepts and Methods. CRC Press, Boca Raton (2005)

    Book  Google Scholar 

  4. Pan, X.C., Sidky, E.Y., Vannier, M.: Why do commercial ct scanners still employ traditional, filtered back-projection for image reconstruction? Inverse Probl. 25(12), 1230009 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beister, M., Kolditz, D., Kalender, W.A.: Iterative reconstruction methods in x-ray CT. Eur. J. Med. Phys. 28, 94–108 (2012)

    Google Scholar 

  6. Jorgensen, J.S., Hansen, P.C., Schmidt, S.: Sparse image reconstruction in computed tomography. Technical University of Denmark, Kgs. Lyngby, PHD-2013 (293)

  7. Chen, C., Xu, G.: A new linearized split bregman iterative algorithm for image reconstruction in sparse view x-ray computed tomography. Comput. Math. Appl. 71(8), 1537–1559 (2016)

    Article  MathSciNet  Google Scholar 

  8. Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal Processing. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  9. Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Birkhuser, Basel (2013)

    Book  MATH  Google Scholar 

  10. Elad, M.: Optimized projections for compressed sensing. IEEE Trans. Signal Process. 55(12), 5695–5702 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lustig, M., Donoho, D., Pauly, J.: Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58, 1182–1195 (2007)

    Article  Google Scholar 

  12. Duarte-Carvajalino, J.M., Sapiro, G.: Learning to sense sparse signals: simultaneous sensing matrix and sparsifying dictionary optimization. IEEE Trans. Image Proc. 18(7), 1395–1408 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Xu, J., Pi, Y., Cao, Z.: Optimized projection matrix for compressive sensing. EURASIP J. Adv. Signal Process. 2010, 560349 (2010)

    Article  Google Scholar 

  14. Vahid, A., Ferdowsi, S., Sanei, S.: A gradient-based alternating minimization approach for optimization of the measurement matrix in compressive sensing. Signal Process. 92, 999–1009 (2012)

    Article  Google Scholar 

  15. Lu, W., Hinamoto, T.: Design of projection matrix for compressive sensing by non-smooth optimization. In: IEEE international symposium on circuits and systems (ISCAS), pp. 1279–1282 (2014)

  16. Natterer, F.: The Mathematics of Computerized Tomography. Wiley, New York, NY (1986)

    MATH  Google Scholar 

  17. Sahiner, B., Yagle, A.E.: Limited angle tomography using wavelets. In: Nuclear science symposium and medical imaging conference, pp. 1912–1916 (1993)

  18. Rantala, M., Vanska, S., Jarvenpaa, S., Kalke, M., Lassas, M., Moberg, J., Siltanen, S.: Wavelet-based reconstruction for limited-angle x-ray tomography. IEEE Trans. Med. Imaging 25(2), 210–217 (2006)

    Article  Google Scholar 

  19. Sidky, E., 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 

  20. Ritschl, L., Bergner, F., Fleischmann, C., Kachelrie, M.: Improved total variation-based CT image reconstruction applied to clinical data. Phys. Med. Biol. 56(6), 1545–1561 (2011)

    Article  Google Scholar 

  21. Prasad, T., Kumar, P.U.P., Sastry, C.S., Jampana, P.V.: Reconstruction of sparse-view tomography via banded matrices. In: Mukherjee, S., et al. (eds.) Computer Vision, Graphics, and Image Processing. ICVGIP 2016. Lecture Notes in Computer Science, vol. 10481, pp. 204–215. Springer, Cham (2017)

    Google Scholar 

  22. Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. IEEE Press, New York, NY (1988)

    MATH  Google Scholar 

  23. Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, C., Yin, W., Jiang, H., Zhang, Y.: Improved total variation-based CT image reconstruction applied to clinical data. Phys. Med. Biol. 56(6), 1545–1561 (2011)

    Article  Google Scholar 

  25. Peaceman, D.W., Rachford, H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, C., Yin, W., Jiang, H., Zhang, Y.: An efficient augmented lagrangian method with applications to total variation minimization. Comput. Optim. Appl. 56(3), 507–530 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kelley, B.T., Madisetti, V.K.: The fast discrete Radon transform-I: theory. IEEE Trans. Image Proc. 2(3), 382–400 (1993)

    Article  Google Scholar 

  28. Lu, Z., Pong, T.K.: Minimizing condition number via convex programming. SIAM J. Matrix Anal. Appl. 32, 1193–1211 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, C., Sun, D., toh, K.-C.: Solving log-determinant optimization problems by a newton-cg primal proximal point algorithm. SIAM J. Optim. 20(6), 2994–3013 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ravishankar, S., Bresler, Y.: Learning sparsifying transforms. IEEE Trans. Signal Proc. 61(5), 1072–1086 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bernstein, D.S.: Matrix Mathematics. Princeton University Press, Princeton (2005)

    Google Scholar 

  32. Pytlak, R.: Conjugate Gradient Algorithms in Nonconvex Optimization. Springer, Berlin (2009)

    MATH  Google Scholar 

  33. Beck, A.: Introduction to Nonlinear Optimization Theory, Algorithms and Applications with MATLAB. SIAM, Philadelphia (2014)

    Book  MATH  Google Scholar 

  34. Jin, A., Yazici, B., Ntziachristos, V.: Light illumination and detection patterns for fluorescence diffuse optical tomography based on compressive sensing. IEEE Trans. Image Proc. 23(6), 2609–2624 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Moreno, J., Jaime, B., Saucedo, S.: Towards no-reference of peak signal to noise ratio. Int. J. Adv. Comput. Sci. Appl. (IJACSA) 4(1), 123–130 (2013)

    Google Scholar 

  36. Wang, Z., Bovik, A.C., Sheikh, H., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)

    Article  Google Scholar 

Download references

Acknowledgements

The second and third authors are thankful to CSIR (No. 25(219)/13/EMR-II), Govt. of India, for its support. Prasad Theeda gratefully acknowledges the support being received from MHRD, Govt. of India.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Hyderabad, Sangareddy, India

    Prasad Theeda, P. U. Praveen Kumar & C. S. Sastry

  2. Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Sangareddy, India

    P. V. Jampana

  3. Department of Computer Science, CHRIST University, Bengaluru, India

    P. U. Praveen Kumar

Authors
  1. Prasad Theeda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. U. Praveen Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. S. Sastry
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. V. Jampana
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Prasad Theeda.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Theeda, P., Kumar, P.U.P., Sastry, C.S. et al. Reconstruction of sparse-view tomography via preconditioned Radon sensing matrix. J. Appl. Math. Comput. 59, 285–303 (2019). https://doi.org/10.1007/s12190-018-1180-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-018-1180-1

Keywords

Mathematics Subject Classification

Search

Navigation