Abstract

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

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  1. R. J. Jaszczak, K. L. Greer, R. E. Coleman, “SPECT using a specially designed cone beam collimator,” J. Nucl. Med. 29, 1398–1405 (1988).
    [PubMed]
  2. G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
    [Crossref] [PubMed]
  3. R. Clack, M. Defrise, “Cone-beam reconstruction by the use of Radon transform intermediate functions,” J. Opt. Soc. Am. A 11, 580–585 (1993).
    [Crossref]
  4. C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The divergent beam X-ray transform,” Rocky Mt. J. Math. 10, 253–283 (1980).
    [Crossref]
  5. F. Natterer, “Recent developments in X-ray tomography,” Lect. Appl. Math. 30, 177–198 (1994).
  6. A. G. Ramm, 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, F. Natterer, eds. (Springer-Verlag, Berlin, 1991), pp. 66–97.
  8. L. A. Feldkamp, L. C. Davis, J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A 1, 612–619 (1984).
    [Crossref]
  9. B. D. Smith, “Cone-beam tomography: recent advances and tutorial review,” Opt. Eng. 29, 524–534 (1991).
    [Crossref]
  10. A. V. Bronnikov, G. Duifhuis, “Wavelet-based image enhancement in x-ray imaging and tomography,” Appl. Opt. 37, 4437–4448 (1998).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  14. H. H. Barrett, H. Gifford, “Cone-beam tomography with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
    [Crossref] [PubMed]
  15. S. Webb, J. Sutcliffe, L. Burkinshow, A. Horsman, “Tomographic reconstruction from experimentally obtained cone-beam projections,” IEEE Trans. Med. Imaging 6, 67–73 (1987).
    [Crossref] [PubMed]
  16. Z. J. Cao, 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).
    [Crossref]
  17. H. Kudo, T. Saito, “Feasible cone beam scanning methods for exact reconstruction in three-dimensional tomography,” J. Opt. Soc. Am. A 7, 2169–2183 (1990).
    [Crossref] [PubMed]
  18. H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
    [Crossref] [PubMed]
  19. G. Zeng, G. Gullberg, “A cone-beam tomography algorithm for orthogonal circle-and-line orbit,” Phys. Med. Biol. 37, 563–577 (1992).
    [Crossref] [PubMed]
  20. X. Yan, R. M. Leahy, “Cone beam tomography with circular, elliptical and spiral orbits,” Phys. Med. Biol. 37, 493–506 (1992).
    [Crossref]
  21. G. Wang, T.-H. Lin, P. Cheng, D. M. Shinozaki, “A general cone-beam reconstruction formula,” IEEE Trans. Med. Imaging 12, 486–496 (1993).
    [Crossref]
  22. M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
    [Crossref] [PubMed]
  23. C. Axelsson, P. E. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3 log N) time,” Phys. Med. Biol. 39, 477–491 (1994).
    [Crossref] [PubMed]
  24. S. Schaller, T. Flohr, 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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  26. H. Kudo, T. Saito, “Fast and stable cone-beam filtered backprojection method for non-planar orbits,” Phys. Med. Biol. 43, 747–760 (1998).
    [Crossref] [PubMed]
  27. F. Noo, M. Defrise, R. Clackdoyle, “Single-slice rebinning method for helical cone-beam CT,” Phys. Med. Biol. 44, 561–570 (1999).
    [Crossref] [PubMed]
  28. M. Defrise, R. Clack, D. Townsend, “Solution to the three-dimensional image reconstruction problem from two-dimensional parallel projections,” J. Opt. Soc. Am. A 10, 869–877 (1993).
    [Crossref]
  29. G. Gullberg, “The reconstruction of fan-beam data by filtering the back-projection,” Comput. Graph. Image Process. 10, 30–47 (1979).
    [Crossref]
  30. F. Peyrin, “The generalized back-projection theorem for cone beam projection data,” IEEE Trans. Nucl. Sci. 32, 1512–1519 (1985).
    [Crossref]
  31. Z. H. Cho, E. X. Wu, 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, M. Amiel, “Analysis of a cone beam x-ray tomographic system for different scanning modes,” J. Opt. Soc. Am. A 9, 1554–1563 (1992).
    [Crossref]
  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.

1999 (1)

F. Noo, M. Defrise, R. Clackdoyle, “Single-slice rebinning method for helical cone-beam CT,” Phys. Med. Biol. 44, 561–570 (1999).
[Crossref] [PubMed]

1998 (4)

S. Schaller, T. Flohr, 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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

H. Kudo, T. Saito, “Fast and stable cone-beam filtered backprojection method for non-planar orbits,” Phys. Med. Biol. 43, 747–760 (1998).
[Crossref] [PubMed]

A. V. Bronnikov, G. Duifhuis, “Wavelet-based image enhancement in x-ray imaging and tomography,” Appl. Opt. 37, 4437–4448 (1998).
[Crossref]

1994 (6)

F. Natterer, “Recent developments in X-ray tomography,” Lect. Appl. Math. 30, 177–198 (1994).

H. H. Barrett, H. Gifford, “Cone-beam tomography with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
[Crossref] [PubMed]

H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
[Crossref] [PubMed]

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[Crossref] [PubMed]

C. Axelsson, P. E. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3 log N) time,” Phys. Med. Biol. 39, 477–491 (1994).
[Crossref] [PubMed]

Z. H. Cho, E. X. Wu, 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).

1993 (4)

G. Wang, T.-H. Lin, P. Cheng, D. M. Shinozaki, “A general cone-beam reconstruction formula,” IEEE Trans. Med. Imaging 12, 486–496 (1993).
[Crossref]

M. Defrise, R. Clack, D. Townsend, “Solution to the three-dimensional image reconstruction problem from two-dimensional parallel projections,” J. Opt. Soc. Am. A 10, 869–877 (1993).
[Crossref]

Z. J. Cao, 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).
[Crossref]

R. Clack, M. Defrise, “Cone-beam reconstruction by the use of Radon transform intermediate functions,” J. Opt. Soc. Am. A 11, 580–585 (1993).
[Crossref]

1992 (4)

G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
[Crossref] [PubMed]

G. Zeng, G. Gullberg, “A cone-beam tomography algorithm for orthogonal circle-and-line orbit,” Phys. Med. Biol. 37, 563–577 (1992).
[Crossref] [PubMed]

X. Yan, R. M. Leahy, “Cone beam tomography with circular, elliptical and spiral orbits,” Phys. Med. Biol. 37, 493–506 (1992).
[Crossref]

F. Peyrin, R. Goutte, M. Amiel, “Analysis of a cone beam x-ray tomographic system for different scanning modes,” J. Opt. Soc. Am. A 9, 1554–1563 (1992).
[Crossref]

1991 (1)

B. D. Smith, “Cone-beam tomography: recent advances and tutorial review,” Opt. Eng. 29, 524–534 (1991).
[Crossref]

1990 (1)

1988 (1)

R. J. Jaszczak, K. L. Greer, R. E. Coleman, “SPECT using a specially designed cone beam collimator,” J. Nucl. Med. 29, 1398–1405 (1988).
[PubMed]

1987 (1)

S. Webb, J. Sutcliffe, L. Burkinshow, A. Horsman, “Tomographic reconstruction from experimentally obtained cone-beam projections,” IEEE Trans. Med. Imaging 6, 67–73 (1987).
[Crossref] [PubMed]

1985 (2)

F. Peyrin, “The generalized back-projection theorem for cone beam projection data,” IEEE Trans. Nucl. Sci. 32, 1512–1519 (1985).
[Crossref]

B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging 4, 14–25 (1985).
[Crossref] [PubMed]

1984 (1)

1983 (1)

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math. 43, 546–552 (1983).
[Crossref]

1980 (1)

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The divergent beam X-ray transform,” Rocky Mt. J. Math. 10, 253–283 (1980).
[Crossref]

1979 (1)

G. Gullberg, “The reconstruction of fan-beam data by filtering the back-projection,” Comput. Graph. Image Process. 10, 30–47 (1979).
[Crossref]

1961 (1)

A. A. Kirillov, “On a problem of I. M. Gel’fand,” Sov. Math. Dokl. 2, 268–269 (1961).

Amiel, M.

Axelsson, C.

C. Axelsson, P. E. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3 log N) time,” Phys. Med. Biol. 39, 477–491 (1994).
[Crossref] [PubMed]

Barrett, H. H.

H. H. Barrett, H. Gifford, “Cone-beam tomography with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
[Crossref] [PubMed]

Bronnikov, A. V.

Burkinshow, L.

S. Webb, J. Sutcliffe, L. Burkinshow, A. Horsman, “Tomographic reconstruction from experimentally obtained cone-beam projections,” IEEE Trans. Med. Imaging 6, 67–73 (1987).
[Crossref] [PubMed]

Cao, Z. J.

Z. J. Cao, 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).
[Crossref]

Cheng, P.

G. Wang, T.-H. Lin, P. Cheng, D. M. Shinozaki, “A general cone-beam reconstruction formula,” IEEE Trans. Med. Imaging 12, 486–496 (1993).
[Crossref]

Cho, Z. H.

Z. H. Cho, E. X. Wu, 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).

Christian, P. E.

G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
[Crossref] [PubMed]

Clack, R.

Clackdoyle, R.

F. Noo, M. Defrise, R. Clackdoyle, “Single-slice rebinning method for helical cone-beam CT,” Phys. Med. Biol. 44, 561–570 (1999).
[Crossref] [PubMed]

Coleman, R. E.

R. J. Jaszczak, K. L. Greer, R. E. Coleman, “SPECT using a specially designed cone beam collimator,” J. Nucl. Med. 29, 1398–1405 (1988).
[PubMed]

Danielsson, P. E.

C. Axelsson, P. E. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3 log N) time,” Phys. Med. Biol. 39, 477–491 (1994).
[Crossref] [PubMed]

Datz, F. L.

G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
[Crossref] [PubMed]

Davis, L. C.

Defrise, M.

F. Noo, M. Defrise, R. Clackdoyle, “Single-slice rebinning method for helical cone-beam CT,” Phys. Med. Biol. 44, 561–570 (1999).
[Crossref] [PubMed]

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[Crossref] [PubMed]

M. Defrise, R. Clack, D. Townsend, “Solution to the three-dimensional image reconstruction problem from two-dimensional parallel projections,” J. Opt. Soc. Am. A 10, 869–877 (1993).
[Crossref]

R. Clack, M. Defrise, “Cone-beam reconstruction by the use of Radon transform intermediate functions,” J. Opt. Soc. Am. A 11, 580–585 (1993).
[Crossref]

Duifhuis, G.

Feldkamp, L. A.

Flohr, T.

S. Schaller, T. Flohr, 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).
[Crossref] [PubMed]

Gifford, H.

H. H. Barrett, H. Gifford, “Cone-beam tomography with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
[Crossref] [PubMed]

Goutte, R.

Grangeat, P.

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, F. Natterer, eds. (Springer-Verlag, Berlin, 1991), pp. 66–97.

Greer, K. L.

R. J. Jaszczak, K. L. Greer, R. E. Coleman, “SPECT using a specially designed cone beam collimator,” J. Nucl. Med. 29, 1398–1405 (1988).
[PubMed]

Gullberg, G.

G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
[Crossref] [PubMed]

G. Zeng, G. Gullberg, “A cone-beam tomography algorithm for orthogonal circle-and-line orbit,” Phys. Med. Biol. 37, 563–577 (1992).
[Crossref] [PubMed]

G. Gullberg, “The reconstruction of fan-beam data by filtering the back-projection,” Comput. Graph. Image Process. 10, 30–47 (1979).
[Crossref]

Hamaker, C.

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The divergent beam X-ray transform,” Rocky Mt. J. Math. 10, 253–283 (1980).
[Crossref]

Hilal, S. K.

Z. H. Cho, E. X. Wu, 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).

Horsman, A.

S. Webb, J. Sutcliffe, L. Burkinshow, A. Horsman, “Tomographic reconstruction from experimentally obtained cone-beam projections,” IEEE Trans. Med. Imaging 6, 67–73 (1987).
[Crossref] [PubMed]

Jaszczak, R. J.

R. J. Jaszczak, K. L. Greer, R. E. Coleman, “SPECT using a specially designed cone beam collimator,” J. Nucl. Med. 29, 1398–1405 (1988).
[PubMed]

Katsevich, A. I.

A. G. Ramm, A. I. Katsevich, The Radon Transform and Local Tomography (CRC Press, Boca Raton, Fla., 1996).

Kirillov, A. A.

A. A. Kirillov, “On a problem of I. M. Gel’fand,” Sov. Math. Dokl. 2, 268–269 (1961).

Kress, J. W.

Kudo, H.

H. Kudo, T. Saito, “Fast and stable cone-beam filtered backprojection method for non-planar orbits,” Phys. Med. Biol. 43, 747–760 (1998).
[Crossref] [PubMed]

H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
[Crossref] [PubMed]

H. Kudo, T. Saito, “Feasible cone beam scanning methods for exact reconstruction in three-dimensional tomography,” J. Opt. Soc. Am. A 7, 2169–2183 (1990).
[Crossref] [PubMed]

Leahy, R. M.

X. Yan, R. M. Leahy, “Cone beam tomography with circular, elliptical and spiral orbits,” Phys. Med. Biol. 37, 493–506 (1992).
[Crossref]

Lin, T.-H.

G. Wang, T.-H. Lin, P. Cheng, D. M. Shinozaki, “A general cone-beam reconstruction formula,” IEEE Trans. Med. Imaging 12, 486–496 (1993).
[Crossref]

Morgan, H. T.

G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
[Crossref] [PubMed]

Natterer, F.

F. Natterer, “Recent developments in X-ray tomography,” Lect. Appl. Math. 30, 177–198 (1994).

Noo, F.

F. Noo, M. Defrise, R. Clackdoyle, “Single-slice rebinning method for helical cone-beam CT,” Phys. Med. Biol. 44, 561–570 (1999).
[Crossref] [PubMed]

Peyrin, F.

F. Peyrin, R. Goutte, M. Amiel, “Analysis of a cone beam x-ray tomographic system for different scanning modes,” J. Opt. Soc. Am. A 9, 1554–1563 (1992).
[Crossref]

F. Peyrin, “The generalized back-projection theorem for cone beam projection data,” IEEE Trans. Nucl. Sci. 32, 1512–1519 (1985).
[Crossref]

Ramm, A. G.

A. G. Ramm, A. I. Katsevich, The Radon Transform and Local Tomography (CRC Press, Boca Raton, Fla., 1996).

Rowland, S. W.

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.

Saito, T.

H. Kudo, T. Saito, “Fast and stable cone-beam filtered backprojection method for non-planar orbits,” Phys. Med. Biol. 43, 747–760 (1998).
[Crossref] [PubMed]

H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
[Crossref] [PubMed]

H. Kudo, T. Saito, “Feasible cone beam scanning methods for exact reconstruction in three-dimensional tomography,” J. Opt. Soc. Am. A 7, 2169–2183 (1990).
[Crossref] [PubMed]

Schaller, S.

S. Schaller, T. Flohr, 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).
[Crossref] [PubMed]

Seger, M.

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).
[Crossref] [PubMed]

Shinozaki, D. M.

G. Wang, T.-H. Lin, P. Cheng, D. M. Shinozaki, “A general cone-beam reconstruction formula,” IEEE Trans. Med. Imaging 12, 486–496 (1993).
[Crossref]

Smith, B. D.

B. D. Smith, “Cone-beam tomography: recent advances and tutorial review,” Opt. Eng. 29, 524–534 (1991).
[Crossref]

B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging 4, 14–25 (1985).
[Crossref] [PubMed]

Smith, K. T.

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The divergent beam X-ray transform,” Rocky Mt. J. Math. 10, 253–283 (1980).
[Crossref]

Solmon, D. C.

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The divergent beam X-ray transform,” Rocky Mt. J. Math. 10, 253–283 (1980).
[Crossref]

Steffen, P.

S. Schaller, T. Flohr, 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).
[Crossref] [PubMed]

Sutcliffe, J.

S. Webb, J. Sutcliffe, L. Burkinshow, A. Horsman, “Tomographic reconstruction from experimentally obtained cone-beam projections,” IEEE Trans. Med. Imaging 6, 67–73 (1987).
[Crossref] [PubMed]

Townsend, D.

Tsui, B. M. W.

Z. J. Cao, 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).
[Crossref]

Tung, C. H.

G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
[Crossref] [PubMed]

Tuy, H. K.

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math. 43, 546–552 (1983).
[Crossref]

Wagner, S. L.

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The divergent beam X-ray transform,” Rocky Mt. J. Math. 10, 253–283 (1980).
[Crossref]

Wang, G.

G. Wang, T.-H. Lin, P. Cheng, D. M. Shinozaki, “A general cone-beam reconstruction formula,” IEEE Trans. Med. Imaging 12, 486–496 (1993).
[Crossref]

Webb, S.

S. Webb, J. Sutcliffe, L. Burkinshow, A. Horsman, “Tomographic reconstruction from experimentally obtained cone-beam projections,” IEEE Trans. Med. Imaging 6, 67–73 (1987).
[Crossref] [PubMed]

Wu, E. X.

Z. H. Cho, E. X. Wu, 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).

Yan, X.

X. Yan, R. M. Leahy, “Cone beam tomography with circular, elliptical and spiral orbits,” Phys. Med. Biol. 37, 493–506 (1992).
[Crossref]

Zeng, G.

G. Zeng, G. Gullberg, “A cone-beam tomography algorithm for orthogonal circle-and-line orbit,” Phys. Med. Biol. 37, 563–577 (1992).
[Crossref] [PubMed]

G. Gullberg, G. Zeng, F. L. Datz, P. E. Christian, C. H. Tung, H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol. 37, 507–534 (1992).
[Crossref] [PubMed]

Appl. Opt. (1)

Comput. Graph. Image Process. (1)

G. Gullberg, “The reconstruction of fan-beam data by filtering the back-projection,” Comput. Graph. Image Process. 10, 30–47 (1979).
[Crossref]

IEEE Trans. Med. Imaging (7)

Z. H. Cho, E. X. Wu, 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).

S. Schaller, T. Flohr, 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).
[Crossref] [PubMed]

G. Wang, T.-H. Lin, P. Cheng, D. M. Shinozaki, “A general cone-beam reconstruction formula,” IEEE Trans. Med. Imaging 12, 486–496 (1993).
[Crossref]

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[Crossref] [PubMed]

B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging 4, 14–25 (1985).
[Crossref] [PubMed]

S. Webb, J. Sutcliffe, L. Burkinshow, A. Horsman, “Tomographic reconstruction from experimentally obtained cone-beam projections,” IEEE Trans. Med. Imaging 6, 67–73 (1987).
[Crossref] [PubMed]

H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
[Crossref] [PubMed]

IEEE Trans. Nucl. Sci. (2)

Z. J. Cao, 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).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Geometry of cone-beam scanning. The support of function f(r) is shown as a gray ball. The position of the source on the path about the object is given by vector s. The line integral is taken along the dashed line that is parallel to unit vector u. A cone-beam projection of the object is collected by rotating vector u.

Fig. 2
Fig. 2

Numerical phantom used in the simulation studies. The bone is represented by a value of 2.0, and the normal tissue is represented by a value of 1.02, with a maximum density of 1.04 and a minimum density of 1.00. The object is centered on the origin, and its radius is equal to 0.9. The xy planes are shown for z=0.1k-0.55, k=0, 1 ,, 11. Image values between 0.99 and 1.05 are displayed.

Fig. 3
Fig. 3

Reconstructed images of the phantom. The reconstruction was made by using the circular orbit with radius R=3.0.

Fig. 4
Fig. 4

Reconstructed images of the phantom. The reconstruction was made by using the helical orbit with radius R=3.0 and number of turns n=6.

Fig. 5
Fig. 5

Intensity profiles of the phantom and the reconstructed images along the line given by (x, 0.25, -0.35). The profile of the phantom is shown by the dashed lines.

Fig. 6
Fig. 6

Reconstructed images of the disk phantom; planes given by y=0 are shown. The reconstructions were obtained by using the circular (upper row) and helical (lower row) orbits with variable radii R=12.0, 6.0, 3.0, 1.5.

Equations (59)

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g(s,u)=0f(s+tu)dt
f(r)=0for|r|>1.
fˇ(n, s)=R3f(r)δ(s-rn)dr,
γ * fˇ(n , sn)=S2γ(un)g(s,u)du,
s(θ)=[R cos θ, R sin θ, τ(θ)]T,θΘ,
g(s,u) backprojection fs(r) deblurring f(r).
g(s(θ), u)=0f(s(θ)+tu)dt,θΘ,uS2
fˆ(s(θ), u)=-+f(s(θ)+tu)dt,
fˆ(s(θ), u)=g(s(θ), u)+g(s(θ),-u).
fˆ(s(θ), u)=-+R3f(r)δ(r-s(θ)-tu)drdt
=R3f(r)-+δ(r-s(θ)-tu)dtdr.
fs(r)=Θ1|r-s(θ)|fˆs(θ), r-s(θ)|r-s(θ)|dθ.
fs(r)=Θ1|r-s(θ)|gs(θ), r-s(θ)|r-s(θ)|+gs(θ), s(θ)-r|r-s(θ)|dθ
=Θ1|r-s(θ)| gs(θ), r-s(θ)|r-s(θ)|dθ,
gs(θ), s(θ)-r|r-s(θ)|=0.
fs(r)=Θ1|r-s(θ)|R3f(r)-+δr-s(θ)-t r-s(θ)|r-s(θ)|dtdrdθ
=R3f(r)Θ-+δ(r-s(θ)-t(r-s(θ)))dtdθdr,
-+δ(r-s(θ)-t(r-s(θ)))dt
=-+δ(r-r+r-s(θ)-t(r-s(θ)))dt=-+δ(r-r-(t-1)(r-s(θ)))dt=-+δ(r-r+t(r-s(θ)))dt,
R3f(r)hs(r, r-r)dr=fs(r),
hs(r, r0)=Θ-+δ(r0+t(r-s(θ)))dtdθ.
hs(r, r0)=Θ-+δ(x0+t(x-R cos θ))×δ(y0+t(y-R sin θ))×δ(z0+t(z-τ(θ))dtdθ.
-+δ(x0+t(x-R cos θ))δ(y0+t(y-R sin θ))
×δ(z0+t(z-τ(θ)))dt
=1|y-R sin θ|-+δ(x0+t(x-R cos θ))
×δ(z0+t(z-τ(θ))δt+y0y-R sin θdt
=δ((y-R sin θ)x0-(x-R cos θ)y0)
×δz0-z-τ(θ)y-R sin θ y0.
hs(r, r0)=1(x02+y02)1/2Θδz0-z-τ(θ)y-R sin θ y0×δ(y-R sin θ)x0(x02+y02)1/2-(x-R cos θ)y0(x02+y02)1/2dθ,
Θδz0-z-τ(θ)y-R sin θ y0δ((y-R sin θ)
×cos ϕ-(x-R cos θ)sin ϕ)dθ
=1RΘδz0-z-τ(θ)y-R sin θ y0δ(sin(ϕ-θ)+(y cos ϕ-x sin ϕ)/R)dθ=1RΘδz0-z-τ(θ)y-R sin θy0×n=12δ(θ-θm)|cos(ϕ-θm)|dθ=1[R2-(xy0-yx0)2/(x02+y02)]1/2×n=12δz0-z-τ(θm)y-R sin θm y0.
sin(ϕ-θ)+(y cos ϕ-x sin ϕ)/R=0;
θm=arcsiny0(x02+y02)1/2-arcsinxy0-yx0R(x02+y02)1/2+(m-1)π, m=1, 2
|cos(ϕ-θm)|=cosarcsinxy0-yx0R(x02+y02)1/2=1-1R2(xy0-yx0)2x02+y021/2.
hs(r, r0)=1[R2(x02+y02)-(xy0-yx0)2]1/2×m=12δz0-z-τ(θm)y-R sin θm y0.
R1,
θ˜m=arcsiny0(x02+y02)1/2+(m-1)π,
sin(θ˜m)=y0(x02+y02)1/2.
m=12δz0-z-τ(θ˜m)y-R sin θ˜m y0
m=12δz0+1R (x02+y02)[z-τ(θ˜m)]
2δ(z0).
[R2(x02+y02)-(xy0-yx0)2]1/2R(x02+y02)1/2
h˜s(r0)=2Rδ(z0)(x02+y02)1/2,
H˜s(kx, ky)F(kx, ky, z)Fs(kx, ky, z),
H˜s(kx, ky)=2R2π(kx2+ky2)1/2
|r-s(θ)| {R2+[z-τ(θ)]2}1/2.
R2{R2+[z-τ(θ)]2}1/2.
f˜s(r)=R2Θ1{R2+[z-τ(θ)]2}1/2 gr-s(θ)|r-s(θ)|, θdθ.
(a) F˜s(kx, ky, z)=(2π)-1--f˜s(r)×exp[-i(kxx+kyy)]dxdy;
(b) F˜(kx, ky, z)=Is(kx2+ky2)A(kx2+ky2)×F˜s(kx, ky, z);
(c) f˜(r)=(2π)-1--F˜(kx, ky, z)×exp[i(kxx+kyy)]dkxdky,
Is(ρ)=ρ2π
A(ρ)=0.5+0.5 cos(πρ/Ω)forρΩ0forρ>Ω,
τ(θ)=0,
Θ={θ : 0θ<2π}.
τ(θ)=1+1n-1θπn-1,
Θ={θ : λ(z)θ<λ(z)+2π},
λ(z)=πnz-1/(n-1)1+1/(n-1)+1

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