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Reconstruction of sparse-view tomography via preconditioned Radon sensing matrix

  • Prasad Theeda
  • P. U. Praveen Kumar
  • C. S. Sastry
  • P. V. Jampana
Original Research

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.

Keywords

Computed tomography Radon transform Compressive sensing Sparse approximation Preconditioning TVAL3 

Mathematics Subject Classification

94A12 65F50 42A15 68U10 

Notes

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.

References

  1. 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)CrossRefGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Jan, J.: Medical Image Processing, Reconstruction and Restoration: Concepts and Methods. CRC Press, Boca Raton (2005)CrossRefGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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. 6.
    Jorgensen, J.S., Hansen, P.C., Schmidt, S.: Sparse image reconstruction in computed tomography. Technical University of Denmark, Kgs. Lyngby, PHD-2013 (293)Google Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal Processing. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Birkhuser, Basel (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Elad, M.: Optimized projections for compressed sensing. IEEE Trans. Signal Process. 55(12), 5695–5702 (2007)MathSciNetCrossRefGoogle Scholar
  11. 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)CrossRefGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Xu, J., Pi, Y., Cao, Z.: Optimized projection matrix for compressive sensing. EURASIP J. Adv. Signal Process. 2010, 560349 (2010)CrossRefGoogle Scholar
  14. 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)CrossRefGoogle Scholar
  15. 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)Google Scholar
  16. 16.
    Natterer, F.: The Mathematics of Computerized Tomography. Wiley, New York, NY (1986)zbMATHGoogle Scholar
  17. 17.
    Sahiner, B., Yagle, A.E.: Limited angle tomography using wavelets. In: Nuclear science symposium and medical imaging conference, pp. 1912–1916 (1993)Google Scholar
  18. 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)CrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 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. 22.
    Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. IEEE Press, New York, NY (1988)zbMATHGoogle Scholar
  23. 23.
    Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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)CrossRefGoogle Scholar
  25. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kelley, B.T., Madisetti, V.K.: The fast discrete Radon transform-I: theory. IEEE Trans. Image Proc. 2(3), 382–400 (1993)CrossRefGoogle Scholar
  28. 28.
    Lu, Z., Pong, T.K.: Minimizing condition number via convex programming. SIAM J. Matrix Anal. Appl. 32, 1193–1211 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ravishankar, S., Bresler, Y.: Learning sparsifying transforms. IEEE Trans. Signal Proc. 61(5), 1072–1086 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Bernstein, D.S.: Matrix Mathematics. Princeton University Press, Princeton (2005)Google Scholar
  32. 32.
    Pytlak, R.: Conjugate Gradient Algorithms in Nonconvex Optimization. Springer, Berlin (2009)zbMATHGoogle Scholar
  33. 33.
    Beck, A.: Introduction to Nonlinear Optimization Theory, Algorithms and Applications with MATLAB. SIAM, Philadelphia (2014)CrossRefzbMATHGoogle Scholar
  34. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 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. 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)CrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  • Prasad Theeda
    • 1
  • P. U. Praveen Kumar
    • 1
    • 3
  • C. S. Sastry
    • 1
  • P. V. Jampana
    • 2
  1. 1.Department of MathematicsIndian Institute of Technology HyderabadSangareddyIndia
  2. 2.Department of Chemical EngineeringIndian Institute of Technology HyderabadSangareddyIndia
  3. 3.Department of Computer ScienceCHRIST UniversityBengaluruIndia

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