Growth of Discrete Projection Ghosts Created by Iteration

  • Imants Svalbe
  • Shekhar Chandra
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)


Ghost images contain specially aligned pixels with intensities that are designed to sum to zero when projected at any of a pre-selected set of discrete angles. Ghost images find use in synthesizing the content of missing rows of image or projection space from data that contains some deliberate level of information redundancy. Here we examine the properties of ghost images that are constructed through a process of iterated convolution. An initial ghost is propagated by cumulative displacements into other discrete directions to expand the range of angles that have zero-sum projections. The discrete projection scheme used here is the finite Radon transform (FRT). We examine these accumulating ghosts to quantify the growth of their dynamic range of their pixel values and the spread of their spatial extent. After N propagations, a pair of points with intensity ±1 can replicate to produce a maximum total intensity of 2 N . For the discrete projections of the FRT, we show that column-oriented iterations better suppress the range and rate of growth of ghost image values. After N row-based iterations, the peak pixel values of FRT ghost images grow approximately as 20.8N . After N column-based iterations, the peak pixel values of FRT ghost images grow approximately as 20.7N . The slower rate of expansion of pixel values for column iteration comes at the expense of fragmenting the compactness of the set of FRT projection angles that are chosen to sum to zero.


Image Space Projection Angle Ghost Image Projected View Cumulative Displacement 
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  1. 1.
    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: X-rays characterising some classes of discrete sets. Linear Algebra and its Applications 339, 3–21 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chandra, S.S., Normand, N., Kingston, A., Guédon, J.P., Svalbe, I.: Fast Mojette Transform for Discrete Tomography. Elsevier Signal Processing Submitted June (in Review), Available on (2010),
  3. 3.
    Chandra, S.S., Svalbe, I.: A Fast Number Theoretic Finite Radon Transform. In: Proceedings of the Digital Image Computing Techniques and Applications Melbourne (December 2009),
  4. 4.
    Chandra, S., Svalbe, I.D., Guédon, J.-P.: An exact, non-iterative mojette inversion technique utilising ghosts. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 401–412. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Gardner, R.J., Gritzman, P.: Discrete tomography: determination of finite sets by x-rays. Trans. Amer. Math. Soc. 349(6), 2271–2295 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grigoryan, A.M.: New algorithms for calculating the discrete Fourier transforms. J. Vichislit. Matem. i Mat. Fiziki 25(9), 1407–1412 (1986)zbMATHGoogle Scholar
  7. 7.
    Guédon, J.P., Normand, N., Kingston, A., Parrein, B., Servières, M., Évenou, P., Svalbe, I., Autrusseau, F., Hamon, T., Bizais, Y., Coeurjolly, D., Boulos, F., Grail, E.: The Mojette Transform: Theory and Applications. ISTE-Wiley, Chichester (2009)Google Scholar
  8. 8.
    Herman, G.T., Davidi, R.: Image reconstruction form a small number of projections. Inverse Problems 24, 17 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Katz, M.: Questions of Uniqueness and Resolution in Reconstruction from Projections. Lecture Notes in Biomathematics. Springer, Heidelberg (1977)Google Scholar
  10. 10.
    Kingston, A., Svalbe, I.: Projective transforms on periodic discrete image arrays. Advances in Imaging and Electron Physics 139, 75–177 (2006)CrossRefGoogle Scholar
  11. 11.
    Louis, A.K.: Picture reconstruction from projections in restricted range. Mathematical Methods in the Applied Sciences 2, 209–220 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Matúš, F., Flusser, J.: Image Representation via a Finite Radon Transform. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(10), 996–1006 (1993), CrossRefGoogle Scholar
  13. 13.
    Normand, N., Svalbe, I.D., Parrein, B., Kingston, A.M.: Erasure coding with the finite Radon transform. In: IEEE Wireless Communications & Networking Conference, Sydney (April 2010),
  14. 14.
    Svalbe, I., Normand, N.: Properties of minimal ghosts. In: Accepted for presentation at DGCI Nancy, France, April 2011 (2010)Google Scholar
  15. 15.
    Svalbe, I., Normand, N., Nazareth, N., Chandra, S.: On constructing minimal ghosts. In: APRS Conference, DICTA 2010, pp. 1–3 (December 2010)Google Scholar
  16. 16.
    Svalbe, I.: Exact, scaled image rotation using the Finite Radon Transform. Pattern Recognition Letters (2010) (in press),
  17. 17.
    Zopf, S.: Construction of switching components. In: Kuba, A., Nyul, L.G., Palagyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 157–168. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Imants Svalbe
    • 1
  • Shekhar Chandra
    • 1
  1. 1.School of PhysicsMonash UniversityAustralia

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