Early Results on General Vertex Sets and Truncated Projections in Cone-Beam Tomography

  • Rolf Clackdoyle
  • Michel Defrise
  • Frédéric Noo
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 110)


This paper summarizes a talk titled “Some recent results in cone-beam tomography” which was presented at the IMA workshop “Computational Radiology and Imaging: Therapy and Diagnostics” March 17–21, 1997.

A cone-beam projection of some object is a collection of rays-sums through the object where all the rays converge in a single “vertex point” in space.Usually this vertex point is outside the object and is often assumed that from each vertex-point, every non-zero ray-sum through the object is available.If some of these ray-sums are not available, the cone-beam projection is called a truncated projection.

Several algorithms are available to reconstruct the object from its cone-beam projections,under the assumptions that the vertex point travels along a suitable path in space and that no projection is truncated data will be discussed,and an algorithm that is able to reconstruct from a discrete,unordered set of vertex points (but with no truncated projections) will be presented.

Images obtained from these algorithms will be presented for the cases of computer -simulated data, and for data taken from a large-area CT scanner.


Single Photon Emission Compute Tomography Vertex Point Dynamic Single Photon Emission Compute Tomography Vertex Path Computational Radiology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. H. Barrett, Dipole-sheet transform, J. Opt. Soc. Am., 72, 468–475, 1982.CrossRefGoogle Scholar
  2. [2]
    R. Clack, M. Defrise, Overview of reconstruction algorithms for exact cone-beam tomography, SPIE Proceedings series, 2299, Mathematical Methods in Medical Imaging III, San Diego, CA, 1994Google Scholar
  3. [3]
    M. Defrise, R. Clack, A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection, IEEE Trans. Med. Imag., 13, 186–195, 1994.CrossRefGoogle Scholar
  4. [4]
    L. A. Feldkamp, L. C. Davis, J. W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Amer. A, A6, 612–619, 1984.CrossRefGoogle Scholar
  5. [5]
    P. Grangeat, Analyse d’un système d’imagerie 3D par reconstruction à partir de radiographies X en géométrie conique, Ph.D. Thesis, Ecole Nationale Supérieure des Télécommunications, France, 1987.Google Scholar
  6. [6]
    P. Grangeat, Mathematical framework of cone-beam 3D reconstruction via the first derivative of the Radon transform, Mathematical methods in tomography, Herman, Louis and Natterer (eds.), Lecture Notes in Mathematics 1497, 66–97, Springer-Verlag, Berlin 1991.Google Scholar
  7. [7]
    W. P. Klein, H. H. Barrett, I. W. Pang, D. D. Patton, M. M. Rogulski, J. D. Sain, FASTSPECT: Electrical and mechanical design of a high-resolution dynamic SPECT imager, Conference Record of the 1995 Nuclear Science Symposium and Medical Imaging Conference, San Francisco, CA, 931–933, 1996.Google Scholar
  8. [8]
    H. Kudo, T. Saito, Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits, IEEE Trans. Med. Imag., 13, 196–211, 1994.CrossRefGoogle Scholar
  9. [9]
    H. Kudo, T. Saito, An extended completeness condition for exact cone-beam reconstruction and its application, IEEE Conference Record of the 1994 Nuclear Science Symposium and Medical Imaging Conference, Norfolk, VA., 1710–1714, 1995.Google Scholar
  10. [10]
    E. J. Morton, S. Webb, J. E. Bateman, L. J Clarke, C. G. Shelton, Three-dimensional x-ray microtomography for medical and biological applications, Phys. Med. Biol., 35, 805–820, 1990.CrossRefGoogle Scholar
  11. [11]
    F. Natterer, The Mathematics of Computerized Tomography, ed. Wiley: New York, 1986.zbMATHGoogle Scholar
  12. [12]
    F. Noo, R. Clack, M. Defrise, Cone-beam Reconstruction from General Discrete Vertex Sets, IEEE Conference Record of the 1996 Nuclear Science Symposium and Medical Imaging Conference, Anaheim, CA., 1496–1500, 1997.Google Scholar
  13. [13]
    F. Noo, R. Clack, M. Defrise, Cone-beam Reconstruction from General Discrete Vertex Sets using Radon Rebinning Algorithms, IEEE Trans. Nucl. Sci., 44, 1309–1316, 1997.CrossRefGoogle Scholar
  14. [14]
    F. Noo, R. Clack, T. J. Roney, T. A. White, Symmetrical vertex paths for exact reconstruction in cone-beam C.T., Phys. Med. Biol., 43, 797–810, 1998.Google Scholar
  15. [15]
    F. Noo, M. Defrise, R. Clack, T. J. Roney, T. A. White, S. G. Galbraith, Stable and Efficient Shift-Variant Algorithm for Circle-plus-Lines Orbits in Cone-Beam C.T., Proceedings ICIP-96: 1996 International Conference on Image Processing, Lausanne, September 1996, P. Delogne (ed). IEEE, Ceuterick, Leuven, Belgium, 539–542, 1996.Google Scholar
  16. [16]
    F. Noo, M. Defrise, R. Clack, Direct reconstruction of cone-beam data acquired with a vertex path containing a circle, IEEE Trans. Imag. Proc. 7, 854–867, 1998.CrossRefGoogle Scholar
  17. [17]
    T. J. Roney, S. G. Galbraith, T. A. White, M. O’reilly, R. Clack, M. DE-Frise, F. Noo, Feasibility and applications of cone-beam X-ray imaging for containerized waste, Proceedings for the 4th Nondestructive Assay and Nondestructive Examination Waste Characterization Conference (Salt Lake City, 1995, 295–324, 1995.Google Scholar
  18. [18]
    B. D. Smith, Image reconstruction from cone-beam projections: Necessary and sufficient conditions and reconstruction methods, IEEE Trans. Med. Imag., 4, 14–25, 1985.CrossRefGoogle Scholar
  19. [19]
    H. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43, 546–522, 1983.MathSciNetGoogle Scholar
  20. [20]
    Y. Weng, G. L. Zeng, G. T. Gullberg, A reconstruction algorithm for helical cone-beam SPECT, IEEE Trans. Nuc. Sci., 40, 1092–1101, 1993.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Rolf Clackdoyle
    • 1
  • Michel Defrise
    • 2
  • Frédéric Noo
    • 3
  1. 1.University of UtahUSA
  2. 2.Free University of BrusselsBelgium
  3. 3.University of LiègeBelgium

Personalised recommendations