Advertisement

Fourier Inversion of the Mojette Transform

  • Andrew Kingston
  • Heyang Li
  • Nicolas Normand
  • Imants Svalbe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8668)

Abstract

The Mojette transform is a form of discrete Radon transform that maps a 2D image (P ×Q pixels) to a set of I 1D projections. Several fast inversion methods exist that require O(PQI) operations but those methods are ill-conditioned. Several robust (or well-conditioned) inversion methods exist, but they are slow, requiring O(P 2 Q 2 I) operations. Ideally we require an inversion scheme that is both fast and robust to deal with noisy projections. Noisy projection data can arise from data that is corrupted in storage or by errors in data transmission, quantisation errors in image compression, or through noisy acquisition of physical projections, such as in X-ray computed tomography. This paper presents a robust reconstruction method, performed in the Fourier domain, that requires O(P 2 QlogP) operations.

Keywords

Radon transform Mojette transform Fourier inversion tomography 

References

  1. 1.
    Bailey, D., Swarztrauber, P.: The fractional fourier transform and applications. SIAM Review 33, 389–404 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bluestein, L.: A linear filtering approach to the computation of the discrete fourier transform. Northeast Electronics Research and Engineering Meeting Record 10 (1968)Google Scholar
  3. 3.
    Dudgeon, D., Mersereau, R.: Multidimensional Digital Signal Processing. Prentice-Hall (1983)Google Scholar
  4. 4.
    Guédon, J.: The Mojette Transform: theory and applications. ISTE-Wiley (2009)Google Scholar
  5. 5.
    Guédon, J., Barba, D., Burger, N.: Psychovisual image coding via an exact discrete Radon transform. In: Wu, L.T. (ed.) Proceedings of Visual Communication and Image Processing 1995, pp. 562–572 (May 1995)Google Scholar
  6. 6.
    Guédon, J., Normand, N.: The Mojette transform: the first ten years. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 79–91. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Katz, M.: Questions of uniqueness and resolution in reconstruction from projections. Springer (1977)Google Scholar
  8. 8.
    Keiner, J., Kunis, S., Potts, D.: Using NFFT 3 - a software library for various nonequispaced fast Fourier transforms. ACM Trans. Math. Software 36, 1–30 (Article 19) (2009)Google Scholar
  9. 9.
    Kunis, S., Potts, D.: Stability results for scattered data interpolation by trigonometric polynomials. SIAM J. Sci. Comput. 29, 1403–1419 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Normand, N., Guédon, J., Philippe, O., Barba, D.: Controlled redundancy for image coding and high-speed transmission. In: Ansari, R., Smith, M. (eds.) Proceedings of SPIE Visual Communications and Image Processing 1996, vol. 2727, pp. 1070–1081. SPIE (February 1996)Google Scholar
  11. 11.
    Normand, N., Kingston, A., Évenou, P.: A geometry driven reconstruction algorithm for the Mojette transform. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 122–133. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Servières, M., Idier, J., Normand, N., Guédon, J.: Conjugate gradient Mojette reconstruction. In: Fitzpatrick, J., Reinhardt, J. (eds.) Proceedings of SPIE Medical Imaging 2005: Image Processing, vol. 5747, pp. 2067–2074 (April 2005)Google Scholar
  13. 13.
    Verbert, P., Guédon, J.: N-dimensional Mojette transfrom. Application to multiple description. IEEE Discrete Signal Processing 2, 1211–1214 (2002)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andrew Kingston
    • 1
  • Heyang Li
    • 1
  • Nicolas Normand
    • 2
  • Imants Svalbe
    • 3
  1. 1.Dept. Applied Maths, RSPEThe Australian National UniversityCanberraAustralia
  2. 2.IRCCyN, École Polytechnique de l’Université de NantesNantesFrance
  3. 3.School of PhysicsMonash UniversityClaytonAustralia

Personalised recommendations