OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 11 — Nov. 1, 2000
  • pp: 1993–2000

Cone-beam reconstruction by backprojection and filtering

Andrei V. Bronnikov  »View Author Affiliations

JOSA A, Vol. 17, Issue 11, pp. 1993-2000 (2000)

View Full Text Article

Acrobat PDF (466 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A new analytical method for tomographic image reconstruction from cone-beam projections acquired on the source orbits lying on a cylinder is presented. By application of a weighted cone-beam backprojection, the reconstruction problem is reduced to an image-restoration problem characterized by a shift-variant point-spread function that is given analytically. Assuming that the source is relatively far from the imaged object, a formula for an approximate shift-invariant inverse filter is derived; the filter is presented in the Fourier domain. Results of numerical experiments with circular and helical orbits are considered.

© 2000 Optical Society of America

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(110.6960) Imaging systems : Tomography

Andrei V. Bronnikov, "Cone-beam reconstruction by backprojection and filtering," J. Opt. Soc. Am. A 17, 1993-2000 (2000)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. R. J. Jaszczak, K. L. Greer, and R. E. Coleman, “SPECT using a specially designed cone beam collimator,” J. Nucl. Med. 29, 1398–1405 (1988).
  2. G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, and H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
  3. R. Clack and M. Defrise, “Cone-beam reconstruction by the use of Radon transform intermediate functions,” J. Opt. Soc. Am. A 11, 580–585 (1993).
  4. C. Hamaker, K. T. Smith, D. C. Solmon, and S. L. Wagner, “The divergent beam X-ray transform,” Rocky Mt. J. Math. 10, 253–283 (1980).
  5. F. Natterer, “Recent developments in X-ray tomography,” Lect. Appl. Math. 30, 177–198 (1994).
  6. A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography (CRC Press, Boca Raton, Fla., 1996).
  7. P. Grangeat, “Mathematical framework of cone-beam 3D reconstruction via the first derivative of the Radon transform,” Vol. 1497 of Lecture Notes in Mathematics, G. T. Herman, A. K. Louis, and F. Natterer, eds. (Springer-Verlag, Berlin, 1991), pp. 66–97.
  8. L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A 1, 612–619 (1984).
  9. B. D. Smith, “Cone-beam tomography: recent advances and tutorial review,” Opt. Eng. 29, 524–534 (1991).
  10. A. V. Bronnikov and G. Duifhuis, “Wavelet-based image enhancement in x-ray imaging and tomography,” Appl. Opt. 37, 4437–4448 (1998).
  11. A. A. Kirillov, “On a problem of I. M. Gel’fand,” Sov. Math. Dokl. 2, 268–269 (1961).
  12. H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math. 43, 546–552 (1983).
  13. B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging 4, 14–25 (1985).
  14. H. H. Barrett and H. Gifford, “Cone-beam tomography with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
  15. S. Webb, J. Sutcliffe, L. Burkinshow, and A. Horsman, “Tomographic reconstruction from experimentally obtained cone-beam projections,” IEEE Trans. Med. Imaging 6, 67–73 (1987).
  16. Z. J. Cao and B. M. W. Tsui, “A fully three-dimensional reconstruction algorithm with the nonstationary filter for improved single-orbit cone beam SPECT,” IEEE Trans. Nucl. Sci. 40, 280–287 (1993).
  17. H. Kudo and T. Saito, “Feasible cone beam scanning methods for exact reconstruction in three-dimensional tomography,” J. Opt. Soc. Am. A 7, 2169–2183 (1990).
  18. H. Kudo and T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
  19. G. Zeng and G. Gullberg, “A cone-beam tomography algorithm for orthogonal circle-and-line orbit,” Phys. Med. Biol. 37, 563–577 (1992).
  20. X. Yan and R. M. Leahy, “Cone beam tomography with circular, elliptical and spiral orbits,” Phys. Med. Biol. 37, 493–506 (1992).
  21. G. Wang, T.-H. Lin, P. Cheng, and D. M. Shinozaki, “A general cone-beam reconstruction formula,” IEEE Trans. Med. Imaging 12, 486–496 (1993).
  22. M. Defrise and R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
  23. C. Axelsson and P. E. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3 log N) time,” Phys. Med. Biol. 39, 477–491 (1994).
  24. S. Schaller, T. Flohr, and P. Steffen, “An efficient Fourier method for 3-D Radon inversion in exact cone-beam CT reconstruction,” IEEE Trans. Med. Imaging 17, 244–250 (1998).
  25. M. Seger, “Three-dimensional reconstruction from cone-beam data using an efficient Fourier technique combined with a special interpolation filter,” Phys. Med. Biol. 43, 951–959 (1998).
  26. H. Kudo and T. Saito, “Fast and stable cone-beam filtered backprojection method for non-planar orbits,” Phys. Med. Biol. 43, 747–760 (1998).
  27. F. Noo, M. Defrise, and R. Clackdoyle, “Single-slice rebinning method for helical cone-beam CT,” Phys. Med. Biol. 44, 561–570 (1999).
  28. M. Defrise, R. Clack, and D. Townsend, “Solution to the three-dimensional image reconstruction problem from two-dimensional parallel projections,” J. Opt. Soc. Am. A 10, 869–877 (1993).
  29. G. Gullberg, “The reconstruction of fan-beam data by filtering the back-projection,” Comput. Graph. Image Process. 10, 30–47 (1979).
  30. F. Peyrin, “The generalized back-projection theorem for cone beam projection data,” IEEE Trans. Nucl. Sci. 32, 1512–1519 (1985).
  31. Z. H. Cho, E. X. Wu, and S. K. Hilal, “Weighted backprojection approach to cone beam 3D projection reconstruction for truncated spherical detection geometry,” IEEE Trans. Med. Imaging 13, 111–121 (1994).
  32. F. Peyrin, R. Goutte, and M. Amiel, “Analysis of a cone beam x-ray tomographic system for different scanning modes,” J. Opt. Soc. Am. A 9, 1554–1563 (1992).
  33. S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 9–79.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited