Abstract

By using the Feldkamp–Davis–Kress (FDK) algorithm, we can efficiently produce a digital volume, called the FDK volume, from cone-beam data acquired along a circular scan orbit. Due to the insufficiency of the cone-beam data set, the FDK volume suffers from nonuniform reproduction exactness. Specifically, the midplane (on the scan-orbit plane) can be exactly reproduced, and the reproduction exactness of off-midplanes decreases as the distance from the midplane increases. We describe the longitudinal falling-off degradation by a hatlike function and the spatial distribution over the object domain by an exactness volume. With two orthogonal circular scan orbits, we can reconstruct two FDK volumes and generate two exactness volumes. We propose a volume fusion scheme to combine the two FDK volumes into a single volume. Let Va and Vb denote the two FDK volumes, let Ea and Eb denote the exactness volumes for orbits Γa and Γb, respectively, then the volume fusion is defined by Vab=VaWa+VbWb, with Wa=Ea/(Ea+Eb) and Wb=1Wa. In the result, the overall reproduction exactness of Vab is expected to outperform that of Va, or Vb, or (Va+Vb)/2. In principle, this volume-fusion scheme is applicable for general cone-beam tomography with multiple nonorthogonal and noncircular orbits.

© 2006 Optical Society of America

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References

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  1. L. A. Feldkamp, L. C. Davis, and J. W. Kress, "Practical cone-beam algorithm," J. Opt. Soc. Am. A 1, 612-619 (1984).
    [CrossRef]
  2. 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).
    [CrossRef] [PubMed]
  3. 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.
    [CrossRef]
  4. B. D. Smith and C. C. Peck, "Implementations, comparisons, and an investigation of heuristic techniques for cone-beam tomography," IEEE Trans. Med. Imaging 15, 519-531 (1996).
    [CrossRef] [PubMed]
  5. H. K. Tuy, "An inversion formula for cone-beam reconstruction," SIAM J. Appl. Math. 43, 546-552 (1983).
    [CrossRef]
  6. 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] [PubMed]
  7. 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. Imaging 19, 949-963 (2000).
    [CrossRef] [PubMed]
  8. Z. Chen and R. Ning, "Three-dimensional PSF measurement of cone-beam CT system by iterative edge-blurring algorithm," Phys. Med. Biol. 49, 1865-1880 (2004).
    [CrossRef] [PubMed]
  9. G. L. Zeng and G. T. Gullberg, "A cone-beam tomography algorithm for orthogonal circle-and-line orbit," Phys. Med. Biol. 37, 563-577 (1992).
    [CrossRef] [PubMed]
  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] [PubMed]
  11. X. Wang and R. Ning, "A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry," IEEE Trans. Med. Imaging 18, 815-824 (1999).
    [CrossRef] [PubMed]
  12. D. Finch, "Cone-beam reconstruction with sources on a curve," SIAM J. Appl. Math. 665-673 (1985).
  13. F. Noo, R. Clack, T. A. White, and T. J. Roney, "The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography," Phys. Med. Biol. 43, 797-810 (1998).
    [CrossRef] [PubMed]
  14. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).
  15. C. Axelsson and P. Danielsson, "Three-dimensional reconstruction from cone-beam data in O(N3 log N) time," Phys. Med. Biol. 39, 477-491 (1994).
    [CrossRef] [PubMed]
  16. S. Schaller, T. Flohr, and P. Steffen, "An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction," IEEE Trans. Med. Imaging 17, 244-250 (1998).
    [CrossRef] [PubMed]
  17. N. J. Dusaussoy, "VOIR: a volumetric image reconstruction algorithm based on Fourier techniques for inversion for the 3-D Radon transform," IEEE Trans. Image Process. 5, 121-131 (1996).
    [CrossRef] [PubMed]
  18. X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone-beam projection data," IEEE Trans. Med. Imaging 10, 462-472 (1991).
    [CrossRef] [PubMed]
  19. Z. Chen and R. Ning, "Filling the radon domain of computed tomography by local convex combination," Appl. Opt. 42, 7043-7051 (2003).
    [CrossRef] [PubMed]
  20. Z. Chen, R. Ning, Y. Yu, and D. Conover, "3D PSF characterization of circle-plus-arc cone-beam tomography," in Proc. SPIE 5745, 664-675 (2005).
    [CrossRef]

2005

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

2004

Z. Chen and R. Ning, "Three-dimensional PSF measurement of cone-beam CT system by iterative edge-blurring algorithm," Phys. Med. Biol. 49, 1865-1880 (2004).
[CrossRef] [PubMed]

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] [PubMed]

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

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. Imaging 19, 949-963 (2000).
[CrossRef] [PubMed]

1999

X. Wang and R. Ning, "A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry," IEEE Trans. Med. Imaging 18, 815-824 (1999).
[CrossRef] [PubMed]

1998

F. Noo, R. Clack, T. A. White, and T. J. Roney, "The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography," Phys. Med. Biol. 43, 797-810 (1998).
[CrossRef] [PubMed]

S. Schaller, T. Flohr, and P. Steffen, "An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction," IEEE Trans. Med. Imaging 17, 244-250 (1998).
[CrossRef] [PubMed]

1996

N. J. Dusaussoy, "VOIR: a volumetric image reconstruction algorithm based on Fourier techniques for inversion for the 3-D Radon transform," IEEE Trans. Image Process. 5, 121-131 (1996).
[CrossRef] [PubMed]

B. D. Smith and C. C. Peck, "Implementations, comparisons, and an investigation of heuristic techniques for cone-beam tomography," IEEE Trans. Med. Imaging 15, 519-531 (1996).
[CrossRef] [PubMed]

1994

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).
[CrossRef] [PubMed]

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] [PubMed]

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

1992

G. L. Zeng and G. T. Gullberg, "A cone-beam tomography algorithm for orthogonal circle-and-line orbit," Phys. Med. Biol. 37, 563-577 (1992).
[CrossRef] [PubMed]

1991

X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone-beam projection data," IEEE Trans. Med. Imaging 10, 462-472 (1991).
[CrossRef] [PubMed]

1985

D. Finch, "Cone-beam reconstruction with sources on a curve," SIAM J. Appl. Math. 665-673 (1985).

1984

1983

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(N3 log N) time," Phys. Med. Biol. 39, 477-491 (1994).
[CrossRef] [PubMed]

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. Imaging 19, 949-963 (2000).
[CrossRef] [PubMed]

Chen, Z.

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

Z. Chen and R. Ning, "Three-dimensional PSF measurement of cone-beam CT system by iterative edge-blurring algorithm," Phys. Med. Biol. 49, 1865-1880 (2004).
[CrossRef] [PubMed]

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

Clack, R.

F. Noo, R. Clack, T. A. White, and T. J. Roney, "The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography," Phys. Med. Biol. 43, 797-810 (1998).
[CrossRef] [PubMed]

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] [PubMed]

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).
[CrossRef] [PubMed]

Conover, D.

Z. Chen, R. Ning, Y. Yu, and D. Conover, "3D PSF characterization of circle-plus-arc cone-beam tomography," in Proc. SPIE 5745, 664-675 (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] [PubMed]

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. Imaging 19, 949-963 (2000).
[CrossRef] [PubMed]

Danielsson, P.

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

Davis, L. C.

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).

Defrise, M.

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).
[CrossRef] [PubMed]

Dusaussoy, N. J.

N. J. Dusaussoy, "VOIR: a volumetric image reconstruction algorithm based on Fourier techniques for inversion for the 3-D Radon transform," IEEE Trans. Image Process. 5, 121-131 (1996).
[CrossRef] [PubMed]

Feldkamp, L. A.

Finch, D.

D. Finch, "Cone-beam reconstruction with sources on a curve," SIAM J. Appl. Math. 665-673 (1985).

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. Imaging 17, 244-250 (1998).
[CrossRef] [PubMed]

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.
[CrossRef]

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] [PubMed]

G. L. Zeng and G. T. Gullberg, "A cone-beam tomography algorithm for orthogonal circle-and-line orbit," Phys. Med. Biol. 37, 563-577 (1992).
[CrossRef] [PubMed]

Kress, J. W.

Leahy, R. M.

X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone-beam projection data," IEEE Trans. Med. Imaging 10, 462-472 (1991).
[CrossRef] [PubMed]

Ning, R.

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

Z. Chen and R. Ning, "Three-dimensional PSF measurement of cone-beam CT system by iterative edge-blurring algorithm," Phys. Med. Biol. 49, 1865-1880 (2004).
[CrossRef] [PubMed]

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] [PubMed]

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

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. Imaging 19, 949-963 (2000).
[CrossRef] [PubMed]

X. Wang and R. Ning, "A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry," IEEE Trans. Med. Imaging 18, 815-824 (1999).
[CrossRef] [PubMed]

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. Imaging 19, 949-963 (2000).
[CrossRef] [PubMed]

Noo, F.

F. Noo, R. Clack, T. A. White, and T. J. Roney, "The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography," Phys. Med. Biol. 43, 797-810 (1998).
[CrossRef] [PubMed]

Peck, C. C.

B. D. Smith and C. C. Peck, "Implementations, comparisons, and an investigation of heuristic techniques for cone-beam tomography," IEEE Trans. Med. Imaging 15, 519-531 (1996).
[CrossRef] [PubMed]

Roney, T. J.

F. Noo, R. Clack, T. A. White, and T. J. Roney, "The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography," Phys. Med. Biol. 43, 797-810 (1998).
[CrossRef] [PubMed]

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. Imaging 17, 244-250 (1998).
[CrossRef] [PubMed]

Smith, B. D.

B. D. Smith and C. C. Peck, "Implementations, comparisons, and an investigation of heuristic techniques for cone-beam tomography," IEEE Trans. Med. Imaging 15, 519-531 (1996).
[CrossRef] [PubMed]

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. Imaging 17, 244-250 (1998).
[CrossRef] [PubMed]

Tang, X.

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] [PubMed]

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. Imaging 19, 949-963 (2000).
[CrossRef] [PubMed]

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. Imaging 18, 815-824 (1999).
[CrossRef] [PubMed]

White, T. A.

F. Noo, R. Clack, T. A. White, and T. J. Roney, "The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography," Phys. Med. Biol. 43, 797-810 (1998).
[CrossRef] [PubMed]

Yan, X.

X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone-beam projection data," IEEE Trans. Med. Imaging 10, 462-472 (1991).
[CrossRef] [PubMed]

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] [PubMed]

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. Imaging 19, 949-963 (2000).
[CrossRef] [PubMed]

Yu, Y.

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

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] [PubMed]

G. L. Zeng and G. T. Gullberg, "A cone-beam tomography algorithm for orthogonal circle-and-line orbit," Phys. Med. Biol. 37, 563-577 (1992).
[CrossRef] [PubMed]

Appl. Opt.

IEEE Trans. Image Process.

N. J. Dusaussoy, "VOIR: a volumetric image reconstruction algorithm based on Fourier techniques for inversion for the 3-D Radon transform," IEEE Trans. Image Process. 5, 121-131 (1996).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging

X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone-beam projection data," IEEE Trans. Med. Imaging 10, 462-472 (1991).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

B. D. Smith and C. C. Peck, "Implementations, comparisons, and an investigation of heuristic techniques for cone-beam tomography," IEEE Trans. Med. Imaging 15, 519-531 (1996).
[CrossRef] [PubMed]

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. Imaging 19, 949-963 (2000).
[CrossRef] [PubMed]

X. Wang and R. Ning, "A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry," IEEE Trans. Med. Imaging 18, 815-824 (1999).
[CrossRef] [PubMed]

S. Schaller, T. Flohr, and P. Steffen, "An efficient Fourier method for 3D Radon inversion in exact cone-beam CT reconstruction," IEEE Trans. Med. Imaging 17, 244-250 (1998).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Med. Phys.

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] [PubMed]

Phys. Med. Biol.

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

F. Noo, R. Clack, T. A. White, and T. J. Roney, "The dual-ellipse cross vertex path for exact reconstruction of long objects in cone-beam tomography," Phys. Med. Biol. 43, 797-810 (1998).
[CrossRef] [PubMed]

Z. Chen and R. Ning, "Three-dimensional PSF measurement of cone-beam CT system by iterative edge-blurring algorithm," Phys. Med. Biol. 49, 1865-1880 (2004).
[CrossRef] [PubMed]

G. L. Zeng and G. T. Gullberg, "A cone-beam tomography algorithm for orthogonal circle-and-line orbit," Phys. Med. Biol. 37, 563-577 (1992).
[CrossRef] [PubMed]

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] [PubMed]

Proc. SPIE

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

SIAM

D. Finch, "Cone-beam reconstruction with sources on a curve," SIAM J. Appl. Math. 665-673 (1985).

SIAM J. Appl. Math.

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

Other

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.
[CrossRef]

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).

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Figures (6)

Fig. 1
Fig. 1

Diagram of the two-circular-orbit cone-beam tomography. Circular orbit Γ a lies at the transaxial plane (z = 0), and Γ b lies at the coronal plane (y = 0).

Fig. 2
Fig. 2

Demonstration of cone-beam artifacts. (a) Disk phantom consisting of eight identical ellipsoids and one ball, and (b) cone-beam reconstruction by the FDK algorithm, with a horizontal circular orbit ( Γ a in Fig. 1) with a cone angle of 30°. The images correspond to the coronal slice (y = 0) of the phantom.

Fig. 3
Fig. 3

Three degradation functions for characterizing the intensity falling-off behavior of cone-beam tomography. For the horizontal circular cone-beam scan, all these functions assume a maximum value 1 at z = 0 and consistently decrease as | z | increases.

Fig. 4
Fig. 4

Demonstration of the volume-fusion scheme for two FDK volumes. The coronal slices of the volumes are displayed. (a) FDK volume V a (with orbit Γ a ), (b) FDK volume V b (with orbit Γ b ), and the composite volumes (c) V a b = ( V a + V b ) / 2 and (d) V a b = ( W a V a + W b V b ) . Due to the symmetry of the 3D phantom in Fig. 2, only the upper parts are displayed. In these figures, one vertical line and one horizontal line are added to extract scan-line profiles, as shown in Fig. 5.

Fig. 5
Fig. 5

Profiles corresponding to two scan lines in Fig. 4. The scan-line profiles from different volumes are distinguished by the plot line style: dotted curve ( V a ) , dashed–dotted curve ( V b ) , dashed curve [ ( V a +  V b ) / 2 ] , thick solid curve ( W a V a + W b V b ) , and thin solid curve [original phantom, in Fig. 2(a)]. The profiles corresponding to vertical scan lines are collected in (a) by s 1 ( t ) , and profiles corresponding to horizontal scan lines are in (b) by s 2 ( t ) , where t serves as the voxel index along the scan lines.

Fig. 6
Fig. 6

Effect of degradation functions on the overall reproduction errors of the volume fusion of two FDK volumes. (a) High-order cosine function: { cos [ z π / ( 2 z max ) ] } n ; (b) Gaussian function: exp [ z 2 / ( 2 σ 2 ) ] .

Tables (1)

Tables Icon

Table 1 Evaluation of Reproduction Performance

Equations (31)

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

Ω = { ( x , y , z ) [ x max , x max ] [ y max , y max ] [ z max , z max ] } .
g ( α , θ ) = 0 f [ ϕ ( θ ) + t α ] d t , θ Γ ,
ϕ ( θ ) = [ ϕ 1 ( θ ) , ϕ 2 ( θ ) , ϕ 3 ( θ ) ] T ,
f ( x , y , z ) = Γ [ g α [ ( α , θ ) ϕ ( θ ) ] h α ( α , θ ) ] | α = x ϕ ( θ ) d θ
h ( α , θ ) = i ( 2 π ) 4 M R 3 sgn [ ϕ ( θ ) β ] exp [ i ( α β ) ] d β ,
ϕ ( θ ) = ( R   cos   θ , R   sin   θ , 0 ) T , θ Γ = [ 0 , 2 π ] .
ϕ ( θ ) = ( R   sin   θ , R cos θ , 0 ) T .
M ϕ ( θ ) = [ ϕ ( θ ) , 0 , 0 ] T ,
M = [ sin   θ cos   θ 0 cos   θ sin   θ 0 0 0 1 ] .
g r ( α ^ , θ ) = g ( M α , θ ) ;
f ^ ( x , y , z ) = R 4 π 2 × 0 2 π lim ε 0 [ g r ( α ^ , θ )  E ε ˙ α ^ 1 ( α ^ 1 ) ] | α ^ = M [ x - ϕ ( θ ) ] d θ ,
E ε ( α ^ 1 ) = { 1 / ε 2 , | α ^ 1 | < ε 1 / α ^ 1 2 , | α ^ 1 | ε .
g ˜ ( z 1 , z 2 , θ ) = lim ε 0 D z 1 2 + z 2     2 + D 2 × g ( z 1 , z 2 , θ ) E ε ( z 1 z 1 ) d z 1 ,
f ^ ( x , y , z ) = R D 4 π 2 0 2 π g ˜ ( z 1 , z 2 , θ ) ( x   cos   θ + y sin θ R ) 2 d θ ,
( x , y , z ) Ω .
Γ = Γ a Γ b = ( R cos θ a , R sin θ a ,   0 ) ( 0 , R cos θ b , R sin θ b ) , θ a [ 0 , 2 π ] , θ b [ 0 , 2 π ] .
V a = FDK { f ( x , y , z ) , Γ a } ,
V b = FDK { f ( x , y , z ) , Γ b } ,
d ( z ) = ( z max | z | ) / z max , z [ z max , z max ] ,
E a ( x , y , z ) = d ( z ) .
E b ( x , y , z ) = d ( y ) .
d ( z ; n ) = { cos [ z π / ( 2 z max ) ] } n ,
d ( z ; σ ) = exp [ z 2 / ( 2 σ 2 ) ] ,
V a b = V a W a + V b W b .
W a = E a E a + E b ,
W b = E b E a + E b ,
W a + W b = 1 ,
W a ( x , y , z ) + W b ( x , y , z ) = 1 , ( x , y , z ) Ω ,
ε λ = mean { | V λ ( x , y , z ) V 0 ( x , y , z ) | , ( x , y , z ) Ω } , λ { a , b , a b , . . . } ,
ε a b < ε λ , λ = { a , b } ,
ε W a V a + W b V b < ε ( V a + V b ) / 2 .

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