Abstract

The Feldkamp–David–Kress (FDK) algorithm is widely adopted for cone-beam reconstruction due to its one-dimensional filtered backprojection structure and parallel implementation. In a reconstruction volume, the conspicuous cone-beam artifact manifests as intensity fall-off along the longitudinal direction (the gantry rotation axis). This effect is inherent to circular cone-beam tomography due to the fact that a cone-beam dataset acquired from circular scanning fails to meet the data sufficiency condition for volume reconstruction. Upon observations of the intensity fall-off phenomenon associated with the FDK reconstruction of a ball phantom, we propose an empirical weight formula to compensate for the fall-off degradation. Specifically, a reciprocal cosine can be used to compensate the voxel values along longitudinal direction during three-dimensional backprojection reconstruction, in particular for boosting the values of voxels at positions with large cone angles. The intensity degradation within the z plane, albeit insignificant, can also be compensated by using the same weight formula through a parameter for radial distance dependence. Computer simulations and phantom experiments are presented to demonstrate the compensation effectiveness of the fall-off effect inherent in circular cone-beam tomography.

© 2008 Optical Society of America

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References

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  1. Z. Chen, R. Ning, and D. Conover, “Accurate perspective projection calculation using a pixel-pyramid model for iterative cone-beam reconstruction,” Proc. SPIE 5030, 728-39 (2003).
  2. Z. Chen and R. Ning, “Pixel-pyramid model for divergent projection geometry,” Opt. Eng. 44, 027002 (2005).
  3. K. Mueller, R. Yagel, and J. J. Wheller, “Anti-aliased three-dimensional cone-beam reconstruction of low-contrast objects with algebraic methods,” IEEE Trans. Med. Imaging 18519-537 (1999).
  4. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1999).
  5. H. Turbell, “Cone-beam reconstruction using filtered backprojection,” Ph.D. thesis (Linkoping University, 2001).
  6. X. Wang and R. Ning, “A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry,” IEEE Trans. Med. Imag. 18, 815-824 (1999).
  7. Z. Chen, R. Ning, Y. Yu, and D. Conover, “3D PSF characterization of circle-plus-arc cone-beam tomography,” Proc. SPIE 5745, 664-675 (2005).
  8. P. Grangeat, “Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform,” in Mathematical Methods in Tomography, Vol. 1497 of Lecture Notes in Mathematics, G.T.Herman, A. K. Louis, and F. Natterer, eds. (Springer-Verlag, 1991), pp. 66-97.
  9. H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math. 43, 546-552 (1983).
    [CrossRef]
  10. G. L. Zeng, R. Clack, and G. T. Gullberg, “Implementation of Tuy's cone-beam inversion formula,” Phys. Med. Biol. 39, 493-507 (1994).
    [CrossRef]
  11. X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
    [CrossRef]
  12. L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A 1, 612-619 (1984).
  13. R. Ning, X. Tang, D. Conover, and R. Yu, “Flat panel detector-based cone beam computed tomography with a circle-plus-two arcs data acquisition orbit: preliminary phantom study,” Med. Phys. 30, 1694-1705 (2003).
    [CrossRef]
  14. R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).
  15. M. Defrise and R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imag. 13, 186-195 (1994).
  16. Z. Chen and R. Ning, “Filling the Radon domain of computed tomography by local convex combination,” Appl. Opt. 42, 7043-7051 (2003).
    [CrossRef]
  17. C. Axelsson and P. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3log⁡N) time,” Phys. Med. Biol. 39, 477-491 (1994).
    [CrossRef]
  18. S. Schaller, T. Flohr, and P. Steffen, “An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction,” IEEE Trans. Med. Imag. 17, 244-250 (1998).
  19. N. J. Dusaussoy, “Voir: A volumetric image reconstruction algorithm based on Fourier techniques for inversion of the 3-D Radon transform,” IEEE Trans. Image Process. 5, 121-131(1996).
    [CrossRef]
  20. D. Gottlieb, B. Gustafsson, and P. Forssen, “On the direct Fourier method for computer tomography,” IEEE Trans. Med. Imag. 19, 223-232 (2000).
  21. Z. Chen and R. Ning, “Super-gridded cone-beam reconstruction and its application to point-spread function calculation,” Appl. Opt. 44, 4615-4624 (2005).
    [CrossRef]
  22. Z. Chen and R. Ning, “Volume fusion of two-circular-orbits cone-beam tomography,” Appl. Opt. 45, 5960-5966(2006).
    [CrossRef]

2006 (1)

2005 (4)

Z. Chen and R. Ning, “Pixel-pyramid model for divergent projection geometry,” Opt. Eng. 44, 027002 (2005).

Z. Chen, R. Ning, Y. Yu, and D. Conover, “3D PSF characterization of circle-plus-arc cone-beam tomography,” Proc. SPIE 5745, 664-675 (2005).

X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
[CrossRef]

Z. Chen and R. Ning, “Super-gridded cone-beam reconstruction and its application to point-spread function calculation,” Appl. Opt. 44, 4615-4624 (2005).
[CrossRef]

2003 (3)

Z. Chen and R. Ning, “Filling the Radon domain of computed tomography by local convex combination,” Appl. Opt. 42, 7043-7051 (2003).
[CrossRef]

Z. Chen, R. Ning, and D. Conover, “Accurate perspective projection calculation using a pixel-pyramid model for iterative cone-beam reconstruction,” Proc. SPIE 5030, 728-39 (2003).

R. Ning, X. Tang, D. Conover, and R. Yu, “Flat panel detector-based cone beam computed tomography with a circle-plus-two arcs data acquisition orbit: preliminary phantom study,” Med. Phys. 30, 1694-1705 (2003).
[CrossRef]

2000 (2)

R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).

D. Gottlieb, B. Gustafsson, and P. Forssen, “On the direct Fourier method for computer tomography,” IEEE Trans. Med. Imag. 19, 223-232 (2000).

1999 (2)

K. Mueller, R. Yagel, and J. J. Wheller, “Anti-aliased three-dimensional cone-beam reconstruction of low-contrast objects with algebraic methods,” IEEE Trans. Med. Imaging 18519-537 (1999).

X. Wang and R. Ning, “A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry,” IEEE Trans. Med. Imag. 18, 815-824 (1999).

1998 (1)

S. Schaller, T. Flohr, and P. Steffen, “An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction,” IEEE Trans. Med. Imag. 17, 244-250 (1998).

1996 (1)

N. J. Dusaussoy, “Voir: A volumetric image reconstruction algorithm based on Fourier techniques for inversion of the 3-D Radon transform,” IEEE Trans. Image Process. 5, 121-131(1996).
[CrossRef]

1994 (3)

G. L. Zeng, R. Clack, and G. T. Gullberg, “Implementation of Tuy's cone-beam inversion formula,” Phys. Med. Biol. 39, 493-507 (1994).
[CrossRef]

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

C. Axelsson and P. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3log⁡N) time,” Phys. Med. Biol. 39, 477-491 (1994).
[CrossRef]

1984 (1)

1983 (1)

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

Axelsson, C.

C. Axelsson and P. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3log⁡N) time,” Phys. Med. Biol. 39, 477-491 (1994).
[CrossRef]

Chen, B.

R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).

Chen, Z.

Z. Chen and R. Ning, “Volume fusion of two-circular-orbits cone-beam tomography,” Appl. Opt. 45, 5960-5966(2006).
[CrossRef]

Z. Chen and R. Ning, “Super-gridded cone-beam reconstruction and its application to point-spread function calculation,” Appl. Opt. 44, 4615-4624 (2005).
[CrossRef]

Z. Chen, R. Ning, Y. Yu, and D. Conover, “3D PSF characterization of circle-plus-arc cone-beam tomography,” Proc. SPIE 5745, 664-675 (2005).

Z. Chen and R. Ning, “Pixel-pyramid model for divergent projection geometry,” Opt. Eng. 44, 027002 (2005).

Z. Chen and R. Ning, “Filling the Radon domain of computed tomography by local convex combination,” Appl. Opt. 42, 7043-7051 (2003).
[CrossRef]

Z. Chen, R. Ning, and D. Conover, “Accurate perspective projection calculation using a pixel-pyramid model for iterative cone-beam reconstruction,” Proc. SPIE 5030, 728-39 (2003).

Clack, R.

G. L. Zeng, R. Clack, and G. T. Gullberg, “Implementation of Tuy's cone-beam inversion formula,” Phys. Med. Biol. 39, 493-507 (1994).
[CrossRef]

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

Conover, D.

Z. Chen, R. Ning, Y. Yu, and D. Conover, “3D PSF characterization of circle-plus-arc cone-beam tomography,” Proc. SPIE 5745, 664-675 (2005).

R. Ning, X. Tang, D. Conover, and R. Yu, “Flat panel detector-based cone beam computed tomography with a circle-plus-two arcs data acquisition orbit: preliminary phantom study,” Med. Phys. 30, 1694-1705 (2003).
[CrossRef]

Z. Chen, R. Ning, and D. Conover, “Accurate perspective projection calculation using a pixel-pyramid model for iterative cone-beam reconstruction,” Proc. SPIE 5030, 728-39 (2003).

R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).

Danielsson, P.

C. Axelsson and P. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3log⁡N) time,” Phys. Med. Biol. 39, 477-491 (1994).
[CrossRef]

Davis, L. C.

Defrise, M.

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

Drapkin, E.

X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
[CrossRef]

Dusaussoy, N. J.

N. J. Dusaussoy, “Voir: A volumetric image reconstruction algorithm based on Fourier techniques for inversion of the 3-D Radon transform,” IEEE Trans. Image Process. 5, 121-131(1996).
[CrossRef]

Feldkamp, L. A.

Flohr, T.

S. Schaller, T. Flohr, and P. Steffen, “An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction,” IEEE Trans. Med. Imag. 17, 244-250 (1998).

Forssen, P.

D. Gottlieb, B. Gustafsson, and P. Forssen, “On the direct Fourier method for computer tomography,” IEEE Trans. Med. Imag. 19, 223-232 (2000).

Gottlieb, D.

D. Gottlieb, B. Gustafsson, and P. Forssen, “On the direct Fourier method for computer tomography,” IEEE Trans. Med. Imag. 19, 223-232 (2000).

Grangeat, P.

P. Grangeat, “Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform,” in Mathematical Methods in Tomography, Vol. 1497 of Lecture Notes in Mathematics, G.T.Herman, A. K. Louis, and F. Natterer, eds. (Springer-Verlag, 1991), pp. 66-97.

Gullberg, G. T.

G. L. Zeng, R. Clack, and G. T. Gullberg, “Implementation of Tuy's cone-beam inversion formula,” Phys. Med. Biol. 39, 493-507 (1994).
[CrossRef]

Gustafsson, B.

D. Gottlieb, B. Gustafsson, and P. Forssen, “On the direct Fourier method for computer tomography,” IEEE Trans. Med. Imag. 19, 223-232 (2000).

Hagiwara, A.

X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
[CrossRef]

Hsieh, J.

X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
[CrossRef]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1999).

Kress, J. W.

Mueller, K.

K. Mueller, R. Yagel, and J. J. Wheller, “Anti-aliased three-dimensional cone-beam reconstruction of low-contrast objects with algebraic methods,” IEEE Trans. Med. Imaging 18519-537 (1999).

Nilsen, R.

X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
[CrossRef]

Ning, R.

Z. Chen and R. Ning, “Volume fusion of two-circular-orbits cone-beam tomography,” Appl. Opt. 45, 5960-5966(2006).
[CrossRef]

Z. Chen and R. Ning, “Super-gridded cone-beam reconstruction and its application to point-spread function calculation,” Appl. Opt. 44, 4615-4624 (2005).
[CrossRef]

Z. Chen, R. Ning, Y. Yu, and D. Conover, “3D PSF characterization of circle-plus-arc cone-beam tomography,” Proc. SPIE 5745, 664-675 (2005).

Z. Chen and R. Ning, “Pixel-pyramid model for divergent projection geometry,” Opt. Eng. 44, 027002 (2005).

Z. Chen, R. Ning, and D. Conover, “Accurate perspective projection calculation using a pixel-pyramid model for iterative cone-beam reconstruction,” Proc. SPIE 5030, 728-39 (2003).

Z. Chen and R. Ning, “Filling the Radon domain of computed tomography by local convex combination,” Appl. Opt. 42, 7043-7051 (2003).
[CrossRef]

R. Ning, X. Tang, D. Conover, and R. Yu, “Flat panel detector-based cone beam computed tomography with a circle-plus-two arcs data acquisition orbit: preliminary phantom study,” Med. Phys. 30, 1694-1705 (2003).
[CrossRef]

R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).

X. Wang and R. Ning, “A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry,” IEEE Trans. Med. Imag. 18, 815-824 (1999).

Ning, Y.

R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).

Schaller, S.

S. Schaller, T. Flohr, and P. Steffen, “An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction,” IEEE Trans. Med. Imag. 17, 244-250 (1998).

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1999).

Steffen, P.

S. Schaller, T. Flohr, and P. Steffen, “An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction,” IEEE Trans. Med. Imag. 17, 244-250 (1998).

Tang, X.

X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
[CrossRef]

R. Ning, X. Tang, D. Conover, and R. Yu, “Flat panel detector-based cone beam computed tomography with a circle-plus-two arcs data acquisition orbit: preliminary phantom study,” Med. Phys. 30, 1694-1705 (2003).
[CrossRef]

R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).

Thibault, J.

X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
[CrossRef]

Turbell, H.

H. Turbell, “Cone-beam reconstruction using filtered backprojection,” Ph.D. thesis (Linkoping University, 2001).

Tuy, H. K.

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

Wang, X.

X. Wang and R. Ning, “A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry,” IEEE Trans. Med. Imag. 18, 815-824 (1999).

Wheller, J. J.

K. Mueller, R. Yagel, and J. J. Wheller, “Anti-aliased three-dimensional cone-beam reconstruction of low-contrast objects with algebraic methods,” IEEE Trans. Med. Imaging 18519-537 (1999).

Yagel, R.

K. Mueller, R. Yagel, and J. J. Wheller, “Anti-aliased three-dimensional cone-beam reconstruction of low-contrast objects with algebraic methods,” IEEE Trans. Med. Imaging 18519-537 (1999).

Yu, R.

R. Ning, X. Tang, D. Conover, and R. Yu, “Flat panel detector-based cone beam computed tomography with a circle-plus-two arcs data acquisition orbit: preliminary phantom study,” Med. Phys. 30, 1694-1705 (2003).
[CrossRef]

R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).

Yu, Y.

Z. Chen, R. Ning, Y. Yu, and D. Conover, “3D PSF characterization of circle-plus-arc cone-beam tomography,” Proc. SPIE 5745, 664-675 (2005).

Zeng, G. L.

G. L. Zeng, R. Clack, and G. T. Gullberg, “Implementation of Tuy's cone-beam inversion formula,” Phys. Med. Biol. 39, 493-507 (1994).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Image Process. (1)

N. J. Dusaussoy, “Voir: A volumetric image reconstruction algorithm based on Fourier techniques for inversion of the 3-D Radon transform,” IEEE Trans. Image Process. 5, 121-131(1996).
[CrossRef]

IEEE Trans. Med. Imag. (5)

D. Gottlieb, B. Gustafsson, and P. Forssen, “On the direct Fourier method for computer tomography,” IEEE Trans. Med. Imag. 19, 223-232 (2000).

R. Ning, B. Chen, R. Yu, D. Conover, X. Tang, and Y. Ning, “Flat panel detector-based cone-beam volume CT angiography imaging: system evaluation,” IEEE Trans. Med. Imag. 19, 9494-963 (2000).

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

X. Wang and R. Ning, “A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry,” IEEE Trans. Med. Imag. 18, 815-824 (1999).

S. Schaller, T. Flohr, and P. Steffen, “An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction,” IEEE Trans. Med. Imag. 17, 244-250 (1998).

IEEE Trans. Med. Imaging (1)

K. Mueller, R. Yagel, and J. J. Wheller, “Anti-aliased three-dimensional cone-beam reconstruction of low-contrast objects with algebraic methods,” IEEE Trans. Med. Imaging 18519-537 (1999).

J. Opt. Soc. Am. A (1)

Med. Phys. (1)

R. Ning, X. Tang, D. Conover, and R. Yu, “Flat panel detector-based cone beam computed tomography with a circle-plus-two arcs data acquisition orbit: preliminary phantom study,” Med. Phys. 30, 1694-1705 (2003).
[CrossRef]

Opt. Eng. (1)

Z. Chen and R. Ning, “Pixel-pyramid model for divergent projection geometry,” Opt. Eng. 44, 027002 (2005).

Phys. Med. Biol. (3)

G. L. Zeng, R. Clack, and G. T. Gullberg, “Implementation of Tuy's cone-beam inversion formula,” Phys. Med. Biol. 39, 493-507 (1994).
[CrossRef]

X. Tang, J. Hsieh, A. Hagiwara, R. Nilsen, J. Thibault, and E. Drapkin, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory,” Phys. Med. Biol. 50, 3889-905 (2005).
[CrossRef]

C. Axelsson and P. Danielsson, “Three-dimensional reconstruction from cone-beam data in O(N3log⁡N) time,” Phys. Med. Biol. 39, 477-491 (1994).
[CrossRef]

Proc. SPIE (2)

Z. Chen, R. Ning, Y. Yu, and D. Conover, “3D PSF characterization of circle-plus-arc cone-beam tomography,” Proc. SPIE 5745, 664-675 (2005).

Z. Chen, R. Ning, and D. Conover, “Accurate perspective projection calculation using a pixel-pyramid model for iterative cone-beam reconstruction,” Proc. SPIE 5030, 728-39 (2003).

SIAM J. Appl. Math. (1)

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

Other (3)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1999).

H. Turbell, “Cone-beam reconstruction using filtered backprojection,” Ph.D. thesis (Linkoping University, 2001).

P. Grangeat, “Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform,” in Mathematical Methods in Tomography, Vol. 1497 of Lecture Notes in Mathematics, G.T.Herman, A. K. Louis, and F. Natterer, eds. (Springer-Verlag, 1991), pp. 66-97.

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Figures (9)

Fig. 1
Fig. 1

Diagram of the cone-beam scanner. The gantry orbit plane defines the midplane of the object domain, and the gantry rotation axis (z axis) defines the longitudinal direction. The cone-beam projections are indexed by the projection angle β. Each detector row and the point source (S) define a fan-beam plane.

Fig. 2
Fig. 2

Illustration of the longitudinal intensity fall-off effect in cone-beam tomography. (a) Cross-sectional image of a ball object, (b) the FDK-reconstructed image. The profiles of the longitudinal scanlines in (a) and (b) are plotted in (a1) and (b1), respectively.

Fig. 3
Fig. 3

Demonstration of the intensity fall-off compensation with different longitudinal degradations. The ball cross section ( y = 0 ) was reconstructed by Eq. (2) with different parameter settings: (a)  c 1 = 0 , (b)  c 1 = 1 , (c)  c 1 = 1.3 , (d)  c 1 = 1.5 . All the images are displayed by window [ 0.95 , 1.05 ] . The vertical lines are added to extract profiles (as shown in Fig. 5).

Fig. 4
Fig. 4

Profiles of the scanlines in Fig. 4. The numbers in the figure legend show the mean voxel errors with respect to the original profile.

Fig. 5
Fig. 5

Influence of parameter c 2 in Eq. (3) on the intensity fall-off compensation. (a)  c 2 = 0 , (b)  c 2 = 0.5 , (c)  c 2 = 1 , (d)  c 2 = 1.5 . All the images are displayed by a window [ 0.95 , 1.05 ] . The scanline profiles from these images are displayed in Fig. 7.

Fig. 6
Fig. 6

Plots of the scanline profiles in Fig. 6. The numbers in the figure legend show the mean voxel errors with respect to the original profile.

Fig. 7
Fig. 7

Simulation with a 3D Shepp–Logan phantom. (a) Slice image of the phantom, (b) image reconstructed by Eq. (1), (c) image reconstructed by Eq. (2), and (d) profiles of the vertical scanlines.

Fig. 8
Fig. 8

Geometry of a cone-beam CT scanner for breast imaging. The numbers are given in millimeters.

Fig. 9
Fig. 9

Experiment results with a gel breast phantom. (a) Sagittal plane reconstructed with FDK reconstruction, (b) the same sagittal plane with compensation reconstruction, (c) plots of the scanlines. Two bright spots are embedded objects in the phantom. The vertical white lines in (a) and (b) are added for marking the scanlines.

Tables (1)

Tables Icon

Table 1 Parameter Settings for Simulations

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

f ( x , y , z ) = 1 2 0 2 π 1 U 2 D D 2 + p 2 + ξ 2 P β ( p , ξ ) h ( t U p ) d p d β t = x cos β + y sin β , s = x sin β + y cos β ξ = D z D s , U = D s D h ( t ) = | ω | exp ( i 2 π ω t ) d ω ,
f ( x , y , z ) = 1 2 0 2 π W ( x , y , z ) U 2 D D 2 + p 2 + ξ 2 P ( p , ξ ; β ) h ( t U p ) d p d β ,
W ( x , y , z ; c 1 , c 2 ) = 1 cos [ c 1 z / ( R c 2 r ) ] with r = x 2 + y 2 + z 2 ,
0 c 1 z / ( R c 2 r ) < π / 2.
W ( x , y , z ; c 1 , 0 ) = W ( z ; c 1 ) = 1 cos ( c 1 z / R ) ,
W ( x , y , 0 ; c 1 , c 2 ) 1 ,
W ( x , y , z ; 0 , c 2 ) 1.

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