Intertwined Digital Rays in Discrete Radon Projections Pooled over Adjacent Prime Sized Arrays

  • Imants Svalbe
  • Andrew Kingston
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2886)


Digital projections are image intensity sums taken along directed rays that sample whole pixel values at periodic locations along the ray. For 2D square arrays with sides of prime length, the Discrete Radon Transform (DRT) is very efficient at reconstructing digital images from their digital projections. The periodic gaps in digital rays complicate the use of the DRT for efficient reconstruction of tomographic images from real projection data, where there are no gaps along the projection direction. A new approach to bridge this gap problem is to pool DRT digital projections obtained over a variety of prime sized arrays. The digital gaps are then partially filled by a staggered overlap of discrete sample positions to better approximate a continuous projection ray. This paper identifies primes that have similar and distinct DRT pixel sampling patterns for the rays in digital projections. The projections are effectively pooled by combining several images, each reconstructed at a fixed scale, but using projections that are interpolated over different prime sized arrays. The basis for the pooled image reconstruction approach is outlined and we demonstrate the principle of this mechanism works.


Discrete Radon transform tomographic image reconstruction 


  1. 1.
    Beylkin, G.: Discrete Radon Transform. IEEE Trans. on Acoustics, Speech and Signal Processing ASSP-35(2), 162–172 (1987)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Svalbe, I.: Image Operations in Discrete Radon Space, DICTA, Melbourne, Australia, January 21-22, pp. 285-290 (2002)Google Scholar
  3. 3.
    Matus, F., Flusser, J.: Image Representation via a Finite Radon Transform. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(10), 996–1106 (1993)CrossRefGoogle Scholar
  4. 4.
    Svalbe, I.: Digital Projections in Prime and Composite Arrays, IWCIA, Philadelphia, Also see Electronic Notes in Theoretical Computer Science (August 2001),
  5. 5.
    Svalbe, I.: Sampling Properties of the Discrete Radon Transform, Discrete Applied Mathematics (2003) (accepted for publication)Google Scholar
  6. 6.
    Salzberg, P., Figueroa, R.: Tomography on the 3D-Torus and Crystals. In: Herman, G.T., Kuba, A. (eds.) Discrete Tomography: Foundations, Algorithms and Applications, ch. 19, Birkhauser, Boston (1999)Google Scholar
  7. 7.
    Svalbe, I., van der Spek, D.: Reconstruction of Tomographic Images Using Analog Projections and the Digital Radon Transform. Linear Algebra and its Applications 339, 125–145 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kingston, A.: k-space Representation of the Discrete Radon Transform, PhD. Thesis, School of Physics and Materials Engineering, Monash University (2003) (in preparation)Google Scholar
  9. 9.
    Svalbe, I., Kingston, A.: Farey Sequences and Discrete Radon Transform Projection Angles. In: IWCIA 2003, May 14-16, Palermo, Italy (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Imants Svalbe
    • 1
  • Andrew Kingston
    • 1
  1. 1.Center for X-ray Physics and Imaging, School of Physics and Materials EngineeringMonash UniversityAUS

Personalised recommendations