Abstract

We develop reconstruction algorithms for local cone-beam tomography for use with generalized scanning trajectories. The algorithms are grounded theoretically in a recently developed chord-based theory for exact image reconstruction and principles of lambda tomography. Being chord based, they are distinct mathematically and conceptually from conventional local tomography reconstruction algorithms. The salient feature of our algorithms is that they permit reconstruction of discontinuities in the profiles of the object function along chords. By consideration of all possible chords, a 3D image that describes the locations of object discontinuities can be reconstructed. Results from microlocal analysis are applied for understanding the object features that can be reconstructed stably by use of the algorithms. A computer-simulation study is conducted to demonstrate the algorithms and compare their performance with an existing algorithm.

© 2007 Optical Society of America

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  1. A. Katsevich, "Theoretically exact FBP-type inversion algorithm for spiral CT," in The Sixth International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, Asilomar, Calif., Oct. 30-Nov. 2, 2001, pp. 3-6.
  2. Y. Zou and X. Pan, "Exact image reconstruction on PI-line from minimum data in helical cone-beam CT," Phys. Med. Biol. 49, 941-959 (2004).
    [CrossRef] [PubMed]
  3. Y. Zou and X. Pan, "Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT," Phys. Med. Biol. 49, 2717-2731 (2004).
    [CrossRef] [PubMed]
  4. J. D. Pack, F. Noo, and R. Clackdoyle, "Cone-beam reconstruction using the backprojection of locally filtered projections," IEEE Trans. Med. Imaging 24, 2317-2336 (2005).
    [CrossRef]
  5. J. D. Pack and F. Noo, "Cone-beam reconstruction using 1D filtering along the projection of M-lines," Inverse Probl. 21, 1105-1120 (2005).
    [CrossRef]
  6. T. Zhuang, S. Leng, B. E. Nett, and G. Chen, "Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data," Phys. Med. Biol. 49, 5489-5503 (2004).
    [CrossRef]
  7. Y. Ye and G. Wang, "Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve," Med. Phys. 32, 42-48 (2005).
    [CrossRef] [PubMed]
  8. E. Y. Sidky, Y. Zou, and X. Pan, "Minimum-data filtered-backprojection algorithm for helical cone-beam computed tomography," Phys. Med. Biol. 50, 1643-1657 (2004).
    [CrossRef]
  9. X. Pan, Y. Zou, and D. Xia, "Peripheral and central ROI-image reconstruction from and data-redundancy exploitation in truncated fan-beam data," Med. Phys. 32, 673-684 (2005).
    [CrossRef] [PubMed]
  10. Y. Zou, X. Pan, and E. Y. Sidky, "Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan," Phys. Med. Biol. 50, 13-27 (2005).
    [CrossRef] [PubMed]
  11. Y. Zou, X. Pan, and E. Y. Sidky, "Theory and algorithms for image reconstruction on chords and within regions of interest," J. Opt. Soc. Am. A 22, 2372-2384 (2005).
    [CrossRef]
  12. F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).
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    [CrossRef]
  14. A. Faridani, E. L. Ritman, and K. T. Smith, "Examples of local tomography," SIAM J. Appl. Math. 52, 1193-1198 (1992).
    [CrossRef]
  15. A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography (CRC Press, 1996).
  16. A. Faridani, K. Buglione, P. Huabsomboon, O. Iancu, and J. McGrath, "Introduction to local tomography," in Contemporary Mathematics: Radon Transforms and Tomography (American Mathematical Society, 2001).
    [CrossRef]
  17. A. Faridani and E. L. Ritman, "High-resolution computed tomography from efficient sampling," Inverse Probl. 16, 635-650 (2000).
    [CrossRef]
  18. W. Spyra, A. Faridani, E. Ritman, and K. Smith, "Computed tomographic imaging of the coronary tree--use of local tomography," IEEE Trans. Med. Imaging 9, 1-4 (1990).
    [CrossRef] [PubMed]
  19. M. A. Anastasio, D. Shi, X. Pan, C. A. Pelizzari, and P. Munro, "A preliminary investigation of local tomography for megavoltage CT imaging," Med. Phys. 30, 2969-2980 (2003).
    [CrossRef] [PubMed]
  20. A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, "Local tomography II," SIAM J. Appl. Math. 57, 1095-1127 (1997).
    [CrossRef]
  21. G. Wang, D. Snyder, and M. Vannier, "Local computed tomography via iterative deblurring," Scanning 18, 582-588 (1996).
    [PubMed]
  22. A. Louis and P. Maass, "Contour reconstruction in 3D X-ray CT," IEEE Trans. Med. Imaging 12, 764-769 (1993).
    [CrossRef] [PubMed]
  23. A. Katsevich, "Cone beam local tomography," SIAM J. Appl. Math. 59, 2224-2246 (1999).
    [CrossRef]
  24. I.-R. Lan, "On an operator associated to a restricted x-ray transform," Ph.D. thesis (Oregon State University, 1999).
  25. A. I. Katsevich, "Improved cone beam local tomography," Inverse Probl. 22, 627-643 (2006).
    [CrossRef]
  26. F. B. Hildebrand, Advanced Calculus for Applications, 2nd ed. (Prentice Hall, 1976).
  27. E. T. Quinto, "Singularities of the X-ray transform and limited data tomography in R2," SIAM J. Math. Anal. 24, 1215-1225 (1993).
    [CrossRef]
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  29. X. Pan, D. Xia, Y. Zou, and L. Yu, "A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans," Phys. Med. Biol. 49, 4349-4369 (2004).
    [CrossRef] [PubMed]
  30. A. Grigis and J. Sjostrand, Microlocal Analysis for Differential Operators: An Introduction (Cambridge U. Press, 1994).
  31. E. T. Quinto, "Radon transforms, differential equations, and microlocal analysis," in Contemporary Mathematics: Radon Transforms and Tomography (American Mathematical Society, 2001), Vol. 278, pp. 57-68.
    [CrossRef]
  32. L. A. Shepp and B. F. Logan, "The Fourier reconstruction of a head section," IEEE Trans. Nucl. Sci. NS-21, 21-43 (1974).
  33. A. Katsevich, "Image reconstruction for the circle and line trajectory," Phys. Med. Biol. 49, 5059-5072 (2004).
    [CrossRef] [PubMed]
  34. P. E. Danielsson, P. Edholm, J. Eriksson, and M. Seger, "Towards exact 3D-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement and a new completeness condition," in Proceedings of the 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, D.W.Townsend and P.E.Kinahan, eds., Pittsburgh, Pa. (1997), pp. 141-144.
  35. K. C. Tam, S. Samarasekera, and F. Sauer, "Exact cone-beam CT with a spiral scan," Phys. Med. Biol. 43, 1015-1024 (1998).
    [CrossRef] [PubMed]
  36. H. H. Barrett and K. J. Myer, Foundations of Image Science (Wiley, 2004).

2006 (1)

A. I. Katsevich, "Improved cone beam local tomography," Inverse Probl. 22, 627-643 (2006).
[CrossRef]

2005 (6)

Y. Zou, X. Pan, and E. Y. Sidky, "Theory and algorithms for image reconstruction on chords and within regions of interest," J. Opt. Soc. Am. A 22, 2372-2384 (2005).
[CrossRef]

J. D. Pack, F. Noo, and R. Clackdoyle, "Cone-beam reconstruction using the backprojection of locally filtered projections," IEEE Trans. Med. Imaging 24, 2317-2336 (2005).
[CrossRef]

J. D. Pack and F. Noo, "Cone-beam reconstruction using 1D filtering along the projection of M-lines," Inverse Probl. 21, 1105-1120 (2005).
[CrossRef]

X. Pan, Y. Zou, and D. Xia, "Peripheral and central ROI-image reconstruction from and data-redundancy exploitation in truncated fan-beam data," Med. Phys. 32, 673-684 (2005).
[CrossRef] [PubMed]

Y. Zou, X. Pan, and E. Y. Sidky, "Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan," Phys. Med. Biol. 50, 13-27 (2005).
[CrossRef] [PubMed]

Y. Ye and G. Wang, "Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve," Med. Phys. 32, 42-48 (2005).
[CrossRef] [PubMed]

2004 (6)

E. Y. Sidky, Y. Zou, and X. Pan, "Minimum-data filtered-backprojection algorithm for helical cone-beam computed tomography," Phys. Med. Biol. 50, 1643-1657 (2004).
[CrossRef]

T. Zhuang, S. Leng, B. E. Nett, and G. Chen, "Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data," Phys. Med. Biol. 49, 5489-5503 (2004).
[CrossRef]

Y. Zou and X. Pan, "Exact image reconstruction on PI-line from minimum data in helical cone-beam CT," Phys. Med. Biol. 49, 941-959 (2004).
[CrossRef] [PubMed]

Y. Zou and X. Pan, "Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT," Phys. Med. Biol. 49, 2717-2731 (2004).
[CrossRef] [PubMed]

A. Katsevich, "Image reconstruction for the circle and line trajectory," Phys. Med. Biol. 49, 5059-5072 (2004).
[CrossRef] [PubMed]

X. Pan, D. Xia, Y. Zou, and L. Yu, "A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans," Phys. Med. Biol. 49, 4349-4369 (2004).
[CrossRef] [PubMed]

2003 (1)

M. A. Anastasio, D. Shi, X. Pan, C. A. Pelizzari, and P. Munro, "A preliminary investigation of local tomography for megavoltage CT imaging," Med. Phys. 30, 2969-2980 (2003).
[CrossRef] [PubMed]

2000 (1)

A. Faridani and E. L. Ritman, "High-resolution computed tomography from efficient sampling," Inverse Probl. 16, 635-650 (2000).
[CrossRef]

1999 (1)

A. Katsevich, "Cone beam local tomography," SIAM J. Appl. Math. 59, 2224-2246 (1999).
[CrossRef]

1998 (1)

K. C. Tam, S. Samarasekera, and F. Sauer, "Exact cone-beam CT with a spiral scan," Phys. Med. Biol. 43, 1015-1024 (1998).
[CrossRef] [PubMed]

1997 (1)

A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, "Local tomography II," SIAM J. Appl. Math. 57, 1095-1127 (1997).
[CrossRef]

1996 (1)

G. Wang, D. Snyder, and M. Vannier, "Local computed tomography via iterative deblurring," Scanning 18, 582-588 (1996).
[PubMed]

1993 (2)

A. Louis and P. Maass, "Contour reconstruction in 3D X-ray CT," IEEE Trans. Med. Imaging 12, 764-769 (1993).
[CrossRef] [PubMed]

E. T. Quinto, "Singularities of the X-ray transform and limited data tomography in R2," SIAM J. Math. Anal. 24, 1215-1225 (1993).
[CrossRef]

1992 (2)

A. Faridani, E. L. Ritman, and K. T. Smith, "Local tomography," SIAM J. Appl. Math. 52, 459-484 (1992).
[CrossRef]

A. Faridani, E. L. Ritman, and K. T. Smith, "Examples of local tomography," SIAM J. Appl. Math. 52, 1193-1198 (1992).
[CrossRef]

1990 (1)

W. Spyra, A. Faridani, E. Ritman, and K. Smith, "Computed tomographic imaging of the coronary tree--use of local tomography," IEEE Trans. Med. Imaging 9, 1-4 (1990).
[CrossRef] [PubMed]

1974 (1)

L. A. Shepp and B. F. Logan, "The Fourier reconstruction of a head section," IEEE Trans. Nucl. Sci. NS-21, 21-43 (1974).

Anastasio, M. A.

M. A. Anastasio, D. Shi, X. Pan, C. A. Pelizzari, and P. Munro, "A preliminary investigation of local tomography for megavoltage CT imaging," Med. Phys. 30, 2969-2980 (2003).
[CrossRef] [PubMed]

Barrett, H. H.

H. H. Barrett and K. J. Myer, Foundations of Image Science (Wiley, 2004).

Buglione, K.

A. Faridani, K. Buglione, P. Huabsomboon, O. Iancu, and J. McGrath, "Introduction to local tomography," in Contemporary Mathematics: Radon Transforms and Tomography (American Mathematical Society, 2001).
[CrossRef]

Chen, G.

T. Zhuang, S. Leng, B. E. Nett, and G. Chen, "Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data," Phys. Med. Biol. 49, 5489-5503 (2004).
[CrossRef]

Clackdoyle, R.

J. D. Pack, F. Noo, and R. Clackdoyle, "Cone-beam reconstruction using the backprojection of locally filtered projections," IEEE Trans. Med. Imaging 24, 2317-2336 (2005).
[CrossRef]

Danielsson, P. E.

P. E. Danielsson, P. Edholm, J. Eriksson, and M. Seger, "Towards exact 3D-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement and a new completeness condition," in Proceedings of the 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, D.W.Townsend and P.E.Kinahan, eds., Pittsburgh, Pa. (1997), pp. 141-144.

Edholm, P.

P. E. Danielsson, P. Edholm, J. Eriksson, and M. Seger, "Towards exact 3D-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement and a new completeness condition," in Proceedings of the 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, D.W.Townsend and P.E.Kinahan, eds., Pittsburgh, Pa. (1997), pp. 141-144.

Eriksson, J.

P. E. Danielsson, P. Edholm, J. Eriksson, and M. Seger, "Towards exact 3D-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement and a new completeness condition," in Proceedings of the 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, D.W.Townsend and P.E.Kinahan, eds., Pittsburgh, Pa. (1997), pp. 141-144.

Faridani, A.

A. Faridani and E. L. Ritman, "High-resolution computed tomography from efficient sampling," Inverse Probl. 16, 635-650 (2000).
[CrossRef]

A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, "Local tomography II," SIAM J. Appl. Math. 57, 1095-1127 (1997).
[CrossRef]

A. Faridani, E. L. Ritman, and K. T. Smith, "Local tomography," SIAM J. Appl. Math. 52, 459-484 (1992).
[CrossRef]

A. Faridani, E. L. Ritman, and K. T. Smith, "Examples of local tomography," SIAM J. Appl. Math. 52, 1193-1198 (1992).
[CrossRef]

W. Spyra, A. Faridani, E. Ritman, and K. Smith, "Computed tomographic imaging of the coronary tree--use of local tomography," IEEE Trans. Med. Imaging 9, 1-4 (1990).
[CrossRef] [PubMed]

A. Faridani, K. Buglione, P. Huabsomboon, O. Iancu, and J. McGrath, "Introduction to local tomography," in Contemporary Mathematics: Radon Transforms and Tomography (American Mathematical Society, 2001).
[CrossRef]

Finch, D. V.

A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, "Local tomography II," SIAM J. Appl. Math. 57, 1095-1127 (1997).
[CrossRef]

Grigis, A.

A. Grigis and J. Sjostrand, Microlocal Analysis for Differential Operators: An Introduction (Cambridge U. Press, 1994).

Hildebrand, F. B.

F. B. Hildebrand, Advanced Calculus for Applications, 2nd ed. (Prentice Hall, 1976).

Huabsomboon, P.

A. Faridani, K. Buglione, P. Huabsomboon, O. Iancu, and J. McGrath, "Introduction to local tomography," in Contemporary Mathematics: Radon Transforms and Tomography (American Mathematical Society, 2001).
[CrossRef]

Iancu, O.

A. Faridani, K. Buglione, P. Huabsomboon, O. Iancu, and J. McGrath, "Introduction to local tomography," in Contemporary Mathematics: Radon Transforms and Tomography (American Mathematical Society, 2001).
[CrossRef]

Katsevich, A.

A. Katsevich, "Image reconstruction for the circle and line trajectory," Phys. Med. Biol. 49, 5059-5072 (2004).
[CrossRef] [PubMed]

A. Katsevich, "Cone beam local tomography," SIAM J. Appl. Math. 59, 2224-2246 (1999).
[CrossRef]

A. Katsevich, "Theoretically exact FBP-type inversion algorithm for spiral CT," in The Sixth International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, Asilomar, Calif., Oct. 30-Nov. 2, 2001, pp. 3-6.

Katsevich, A. I.

A. I. Katsevich, "Improved cone beam local tomography," Inverse Probl. 22, 627-643 (2006).
[CrossRef]

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

Lan, I.-R.

I.-R. Lan, "On an operator associated to a restricted x-ray transform," Ph.D. thesis (Oregon State University, 1999).

Leng, S.

T. Zhuang, S. Leng, B. E. Nett, and G. Chen, "Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data," Phys. Med. Biol. 49, 5489-5503 (2004).
[CrossRef]

Logan, B. F.

L. A. Shepp and B. F. Logan, "The Fourier reconstruction of a head section," IEEE Trans. Nucl. Sci. NS-21, 21-43 (1974).

Louis, A.

A. Louis and P. Maass, "Contour reconstruction in 3D X-ray CT," IEEE Trans. Med. Imaging 12, 764-769 (1993).
[CrossRef] [PubMed]

Maass, P.

A. Louis and P. Maass, "Contour reconstruction in 3D X-ray CT," IEEE Trans. Med. Imaging 12, 764-769 (1993).
[CrossRef] [PubMed]

McGrath, J.

A. Faridani, K. Buglione, P. Huabsomboon, O. Iancu, and J. McGrath, "Introduction to local tomography," in Contemporary Mathematics: Radon Transforms and Tomography (American Mathematical Society, 2001).
[CrossRef]

Munro, P.

M. A. Anastasio, D. Shi, X. Pan, C. A. Pelizzari, and P. Munro, "A preliminary investigation of local tomography for megavoltage CT imaging," Med. Phys. 30, 2969-2980 (2003).
[CrossRef] [PubMed]

Myer, K. J.

H. H. Barrett and K. J. Myer, Foundations of Image Science (Wiley, 2004).

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

Nett, B. E.

T. Zhuang, S. Leng, B. E. Nett, and G. Chen, "Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data," Phys. Med. Biol. 49, 5489-5503 (2004).
[CrossRef]

Noo, F.

J. D. Pack and F. Noo, "Cone-beam reconstruction using 1D filtering along the projection of M-lines," Inverse Probl. 21, 1105-1120 (2005).
[CrossRef]

J. D. Pack, F. Noo, and R. Clackdoyle, "Cone-beam reconstruction using the backprojection of locally filtered projections," IEEE Trans. Med. Imaging 24, 2317-2336 (2005).
[CrossRef]

Pack, J. D.

J. D. Pack, F. Noo, and R. Clackdoyle, "Cone-beam reconstruction using the backprojection of locally filtered projections," IEEE Trans. Med. Imaging 24, 2317-2336 (2005).
[CrossRef]

J. D. Pack and F. Noo, "Cone-beam reconstruction using 1D filtering along the projection of M-lines," Inverse Probl. 21, 1105-1120 (2005).
[CrossRef]

Pan, X.

Y. Zou, X. Pan, and E. Y. Sidky, "Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan," Phys. Med. Biol. 50, 13-27 (2005).
[CrossRef] [PubMed]

X. Pan, Y. Zou, and D. Xia, "Peripheral and central ROI-image reconstruction from and data-redundancy exploitation in truncated fan-beam data," Med. Phys. 32, 673-684 (2005).
[CrossRef] [PubMed]

Y. Zou, X. Pan, and E. Y. Sidky, "Theory and algorithms for image reconstruction on chords and within regions of interest," J. Opt. Soc. Am. A 22, 2372-2384 (2005).
[CrossRef]

Y. Zou and X. Pan, "Exact image reconstruction on PI-line from minimum data in helical cone-beam CT," Phys. Med. Biol. 49, 941-959 (2004).
[CrossRef] [PubMed]

Y. Zou and X. Pan, "Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT," Phys. Med. Biol. 49, 2717-2731 (2004).
[CrossRef] [PubMed]

E. Y. Sidky, Y. Zou, and X. Pan, "Minimum-data filtered-backprojection algorithm for helical cone-beam computed tomography," Phys. Med. Biol. 50, 1643-1657 (2004).
[CrossRef]

X. Pan, D. Xia, Y. Zou, and L. Yu, "A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans," Phys. Med. Biol. 49, 4349-4369 (2004).
[CrossRef] [PubMed]

M. A. Anastasio, D. Shi, X. Pan, C. A. Pelizzari, and P. Munro, "A preliminary investigation of local tomography for megavoltage CT imaging," Med. Phys. 30, 2969-2980 (2003).
[CrossRef] [PubMed]

Pelizzari, C. A.

M. A. Anastasio, D. Shi, X. Pan, C. A. Pelizzari, and P. Munro, "A preliminary investigation of local tomography for megavoltage CT imaging," Med. Phys. 30, 2969-2980 (2003).
[CrossRef] [PubMed]

Quinto, E. T.

E. T. Quinto, "Singularities of the X-ray transform and limited data tomography in R2," SIAM J. Math. Anal. 24, 1215-1225 (1993).
[CrossRef]

E. T. Quinto, "Radon transforms, differential equations, and microlocal analysis," in Contemporary Mathematics: Radon Transforms and Tomography (American Mathematical Society, 2001), Vol. 278, pp. 57-68.
[CrossRef]

Ramm, A. G.

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

Ritman, E.

W. Spyra, A. Faridani, E. Ritman, and K. Smith, "Computed tomographic imaging of the coronary tree--use of local tomography," IEEE Trans. Med. Imaging 9, 1-4 (1990).
[CrossRef] [PubMed]

Ritman, E. L.

A. Faridani and E. L. Ritman, "High-resolution computed tomography from efficient sampling," Inverse Probl. 16, 635-650 (2000).
[CrossRef]

A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, "Local tomography II," SIAM J. Appl. Math. 57, 1095-1127 (1997).
[CrossRef]

A. Faridani, E. L. Ritman, and K. T. Smith, "Local tomography," SIAM J. Appl. Math. 52, 459-484 (1992).
[CrossRef]

A. Faridani, E. L. Ritman, and K. T. Smith, "Examples of local tomography," SIAM J. Appl. Math. 52, 1193-1198 (1992).
[CrossRef]

Samarasekera, S.

K. C. Tam, S. Samarasekera, and F. Sauer, "Exact cone-beam CT with a spiral scan," Phys. Med. Biol. 43, 1015-1024 (1998).
[CrossRef] [PubMed]

Sauer, F.

K. C. Tam, S. Samarasekera, and F. Sauer, "Exact cone-beam CT with a spiral scan," Phys. Med. Biol. 43, 1015-1024 (1998).
[CrossRef] [PubMed]

Seger, M.

P. E. Danielsson, P. Edholm, J. Eriksson, and M. Seger, "Towards exact 3D-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement and a new completeness condition," in Proceedings of the 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, D.W.Townsend and P.E.Kinahan, eds., Pittsburgh, Pa. (1997), pp. 141-144.

Shepp, L. A.

L. A. Shepp and B. F. Logan, "The Fourier reconstruction of a head section," IEEE Trans. Nucl. Sci. NS-21, 21-43 (1974).

Shi, D.

M. A. Anastasio, D. Shi, X. Pan, C. A. Pelizzari, and P. Munro, "A preliminary investigation of local tomography for megavoltage CT imaging," Med. Phys. 30, 2969-2980 (2003).
[CrossRef] [PubMed]

Sidky, E. Y.

Y. Zou, X. Pan, and E. Y. Sidky, "Theory and algorithms for image reconstruction on chords and within regions of interest," J. Opt. Soc. Am. A 22, 2372-2384 (2005).
[CrossRef]

Y. Zou, X. Pan, and E. Y. Sidky, "Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan," Phys. Med. Biol. 50, 13-27 (2005).
[CrossRef] [PubMed]

E. Y. Sidky, Y. Zou, and X. Pan, "Minimum-data filtered-backprojection algorithm for helical cone-beam computed tomography," Phys. Med. Biol. 50, 1643-1657 (2004).
[CrossRef]

Sjostrand, J.

A. Grigis and J. Sjostrand, Microlocal Analysis for Differential Operators: An Introduction (Cambridge U. Press, 1994).

Smith, K.

W. Spyra, A. Faridani, E. Ritman, and K. Smith, "Computed tomographic imaging of the coronary tree--use of local tomography," IEEE Trans. Med. Imaging 9, 1-4 (1990).
[CrossRef] [PubMed]

Smith, K. T.

A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, "Local tomography II," SIAM J. Appl. Math. 57, 1095-1127 (1997).
[CrossRef]

A. Faridani, E. L. Ritman, and K. T. Smith, "Local tomography," SIAM J. Appl. Math. 52, 459-484 (1992).
[CrossRef]

A. Faridani, E. L. Ritman, and K. T. Smith, "Examples of local tomography," SIAM J. Appl. Math. 52, 1193-1198 (1992).
[CrossRef]

Snyder, D.

G. Wang, D. Snyder, and M. Vannier, "Local computed tomography via iterative deblurring," Scanning 18, 582-588 (1996).
[PubMed]

Spyra, W.

W. Spyra, A. Faridani, E. Ritman, and K. Smith, "Computed tomographic imaging of the coronary tree--use of local tomography," IEEE Trans. Med. Imaging 9, 1-4 (1990).
[CrossRef] [PubMed]

Tam, K. C.

K. C. Tam, S. Samarasekera, and F. Sauer, "Exact cone-beam CT with a spiral scan," Phys. Med. Biol. 43, 1015-1024 (1998).
[CrossRef] [PubMed]

Vannier, M.

G. Wang, D. Snyder, and M. Vannier, "Local computed tomography via iterative deblurring," Scanning 18, 582-588 (1996).
[PubMed]

Wang, G.

Y. Ye and G. Wang, "Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve," Med. Phys. 32, 42-48 (2005).
[CrossRef] [PubMed]

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[PubMed]

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Xia, D.

X. Pan, Y. Zou, and D. Xia, "Peripheral and central ROI-image reconstruction from and data-redundancy exploitation in truncated fan-beam data," Med. Phys. 32, 673-684 (2005).
[CrossRef] [PubMed]

X. Pan, D. Xia, Y. Zou, and L. Yu, "A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans," Phys. Med. Biol. 49, 4349-4369 (2004).
[CrossRef] [PubMed]

Ye, Y.

Y. Ye and G. Wang, "Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve," Med. Phys. 32, 42-48 (2005).
[CrossRef] [PubMed]

Yu, L.

X. Pan, D. Xia, Y. Zou, and L. Yu, "A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans," Phys. Med. Biol. 49, 4349-4369 (2004).
[CrossRef] [PubMed]

Zhuang, T.

T. Zhuang, S. Leng, B. E. Nett, and G. Chen, "Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data," Phys. Med. Biol. 49, 5489-5503 (2004).
[CrossRef]

Zou, Y.

Y. Zou, X. Pan, and E. Y. Sidky, "Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan," Phys. Med. Biol. 50, 13-27 (2005).
[CrossRef] [PubMed]

X. Pan, Y. Zou, and D. Xia, "Peripheral and central ROI-image reconstruction from and data-redundancy exploitation in truncated fan-beam data," Med. Phys. 32, 673-684 (2005).
[CrossRef] [PubMed]

Y. Zou, X. Pan, and E. Y. Sidky, "Theory and algorithms for image reconstruction on chords and within regions of interest," J. Opt. Soc. Am. A 22, 2372-2384 (2005).
[CrossRef]

Y. Zou and X. Pan, "Exact image reconstruction on PI-line from minimum data in helical cone-beam CT," Phys. Med. Biol. 49, 941-959 (2004).
[CrossRef] [PubMed]

Y. Zou and X. Pan, "Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT," Phys. Med. Biol. 49, 2717-2731 (2004).
[CrossRef] [PubMed]

X. Pan, D. Xia, Y. Zou, and L. Yu, "A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans," Phys. Med. Biol. 49, 4349-4369 (2004).
[CrossRef] [PubMed]

E. Y. Sidky, Y. Zou, and X. Pan, "Minimum-data filtered-backprojection algorithm for helical cone-beam computed tomography," Phys. Med. Biol. 50, 1643-1657 (2004).
[CrossRef]

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Inverse Probl. (3)

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A. I. Katsevich, "Improved cone beam local tomography," Inverse Probl. 22, 627-643 (2006).
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[CrossRef]

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

Med. Phys. (3)

Y. Ye and G. Wang, "Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve," Med. Phys. 32, 42-48 (2005).
[CrossRef] [PubMed]

X. Pan, Y. Zou, and D. Xia, "Peripheral and central ROI-image reconstruction from and data-redundancy exploitation in truncated fan-beam data," Med. Phys. 32, 673-684 (2005).
[CrossRef] [PubMed]

M. A. Anastasio, D. Shi, X. Pan, C. A. Pelizzari, and P. Munro, "A preliminary investigation of local tomography for megavoltage CT imaging," Med. Phys. 30, 2969-2980 (2003).
[CrossRef] [PubMed]

Phys. Med. Biol. (8)

X. Pan, D. Xia, Y. Zou, and L. Yu, "A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans," Phys. Med. Biol. 49, 4349-4369 (2004).
[CrossRef] [PubMed]

Y. Zou, X. Pan, and E. Y. Sidky, "Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan," Phys. Med. Biol. 50, 13-27 (2005).
[CrossRef] [PubMed]

E. Y. Sidky, Y. Zou, and X. Pan, "Minimum-data filtered-backprojection algorithm for helical cone-beam computed tomography," Phys. Med. Biol. 50, 1643-1657 (2004).
[CrossRef]

T. Zhuang, S. Leng, B. E. Nett, and G. Chen, "Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data," Phys. Med. Biol. 49, 5489-5503 (2004).
[CrossRef]

Y. Zou and X. Pan, "Exact image reconstruction on PI-line from minimum data in helical cone-beam CT," Phys. Med. Biol. 49, 941-959 (2004).
[CrossRef] [PubMed]

Y. Zou and X. Pan, "Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT," Phys. Med. Biol. 49, 2717-2731 (2004).
[CrossRef] [PubMed]

A. Katsevich, "Image reconstruction for the circle and line trajectory," Phys. Med. Biol. 49, 5059-5072 (2004).
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[CrossRef] [PubMed]

Scanning (1)

G. Wang, D. Snyder, and M. Vannier, "Local computed tomography via iterative deblurring," Scanning 18, 582-588 (1996).
[PubMed]

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A. Katsevich, "Cone beam local tomography," SIAM J. Appl. Math. 59, 2224-2246 (1999).
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A. Faridani, E. L. Ritman, and K. T. Smith, "Examples of local tomography," SIAM J. Appl. Math. 52, 1193-1198 (1992).
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E. T. Quinto, "Singularities of the X-ray transform and limited data tomography in R2," SIAM J. Math. Anal. 24, 1215-1225 (1993).
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P. E. Danielsson, P. Edholm, J. Eriksson, and M. Seger, "Towards exact 3D-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement and a new completeness condition," in Proceedings of the 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, D.W.Townsend and P.E.Kinahan, eds., Pittsburgh, Pa. (1997), pp. 141-144.

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Figures (12)

Fig. 1
Fig. 1

Example of a general scanning trajectory. Each source position r 0 ( s ) on the trajectory is indexed by the path-length parameter s. A flat detector plane is assumed that is located at a distance S ( s ) from the point r 0 ( s ) .

Fig. 2
Fig. 2

A chord line is a line that intersects the scanning trajectory at two points, for example, points r 0 ( s a ) and r 0 ( s b ) . The highlighted segment of this line that is contained between the intersection points is defined as a chord.

Fig. 3
Fig. 3

Local tomography scanning geometry. At each source position r 0 ( s ) on the scanning trajectory that connects the chord end points r 0 ( s a ) and r 0 ( s b ) , the ROI denoted by the shaded region is irradiated. From knowledge of the resulting truncated projection data, the locations of singularities on the chord within the ROI can be reconstructed.

Fig. 4
Fig. 4

Example of a singularity located at position r that has a direction ν s . The plane Π ( r ; ν s ) contains r and is normal to ν s ( r ) . The singularity can be reconstructed stably if and only if Π ( r ; ν s ) intersects the scanning trajectory transversally.

Fig. 5
Fig. 5

Numerical phantom that was taken to represent the true object function in our simulation studies. (a) From left to right, the panels display f ( r ) in the planes x = 0 , y = 2.5 cm , and z = 0 slices. (b) Images of Λ f ( r ) in the planes x = 0 , y = 2.5 cm , and z = 0 .

Fig. 6
Fig. 6

Images reconstructed from the noiseless helical projection data containing longitudinal truncations by use of the local BPF algorithm. (a) Images of the slices x = 0 , y = 2.5 cm , and z = 0 that were reconstructed directly onto a collection of horizontal chords. (b) The corresponding images that were interpolated onto a uniform Cartesian grid.

Fig. 7
Fig. 7

Images reconstructed from the noiseless helical projection data containing longitudinal truncations by use of the Louis–Maass algorithm. (a) Images of the slices x = 0 , y = 2.5 cm , and z = 0 (from left to right) that were reconstructed by use of all the available projection data. (b) The corresponding images reconstructed by use of projection data only within the Tam–Danielson window.

Fig. 8
Fig. 8

Images reconstructed from the noiseless helical projection data containing longitudinal and transverse truncations by use of the (a) local BPF and (b) Louis–Maass algorithms.

Fig. 9
Fig. 9

Images of the slices x = 0 , y = 2.5 cm , and z = 0 (from left to right) reconstructed from the helical projection data containing longitudinal truncations with noise level 100,000 photons/detector element. The images were reconstructed by use of the (a) local BPF and (b) Louis–Maass algorithms. The implementation of the Louis–Maass algorithm used only data within the Tam–Danielson window.

Fig. 10
Fig. 10

Images of the slices x = 0 , y = 2.5 cm , and z = 0 (from left to right) reconstructed from the helical projection data containing longitudinal truncations with noise level 10,000 photons/detector element. The images were reconstructed by use of the (a) local BPF and (b) Louis–Maass algorithms. The implementation of the Louis–Maass algorithm used only data within the Tam–Danielson window.

Fig. 11
Fig. 11

Images reconstructed from noiseless projection data acquired with the circle-and-line scanning geometry. (a) Images of the slices x = 0 , y = 2.5 cm , and z = 0 (from left to right) reconstructed by use of the local BPF algorithm. (b) The corresponding images reconstructed by use of the Louis–Maass algorithm.

Fig. 12
Fig. 12

Images reconstructed from projection data containing a noise level of 100,000 photons/detector element acquired with the circle-and-line scanning geometry. (a) Images of the slices x = 0 , y = 2.5 cm , and z = 0 (from left to right) reconstructed by use of the local BPF algorithm. (b). The corresponding images reconstructed by use of the Louis–Maass algorithm.

Equations (25)

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

D ( r 0 , β ̂ ) = 0 d t f ( r 0 + t β ̂ ) ,
β ̂ = r r 0 r r 0 ,
d r 0 ( s ) d s = 1 .
r = 1 2 [ r 0 ( s a ) + r 0 ( s b ) ] + x c e ̂ c , x c R ,
e ̂ c ( s a , s b ) = r 0 ( s b ) r 0 ( s a ) r 0 ( s b ) r 0 ( s a )
r = 1 2 [ r 0 ( s a ) + r 0 ( s b ) ] + x c e ̂ c , x c [ l , l ] ,
Λ m f ( r ) R 3 d ν ν m F ( ν ) exp [ j 2 π ν r ] ,
Λ x m p ( x ) R d ν ν m P ( ν ) exp [ j 2 π ν x ] ,
f LM ( r ) = 2 Γ d s r r 0 ( s ) 1 D ( r 0 ( s ) , β ̂ ) ,
u = w S ( s ) u d , v = w S ( s ) v d ,
P ( u d , v d , s ) = D ( r 0 ( s ) , β ̂ ) ,
β ̂ = 1 A ( u d , v d ) [ u d e ̂ u ( s ) + v d e ̂ v ( s ) S ( s ) e ̂ w ( s ) ] ,
A ( u d , v d ) = u d 2 + v d 2 + S 2 ( s ) .
q D ( r 0 ( q ) , β ̂ ) q = s = P ( u d , v d , s ) s β ̂ .
f ( r ) = f c ( x c , s a , s b ) ,
g c ( x c , s a , s b ) = s a s b d s sgn [ β ̂ e ̂ w ( s ) ] r ( x c ) r 0 ( s ) s P ( u d , v d , s ) β ̂ ,
P ( u d , v d , s ) s β ̂ = ( d r 0 ( s ) d s e ̂ u ( s ) + u d S ( s ) d r 0 ( s ) d s e ̂ w ( s ) ) A ( u d , v d ) r r 0 ( s ) P ( u d , v d , s ) u d + ( d r 0 ( s ) d s e ̂ v ( s ) + v d S ( s ) d r 0 ( s ) d s e ̂ w ( s ) ) A ( u d , v d ) r r 0 ( s ) P ( u d , v d , s ) v d + P ( u d , v d , s ) s r ,
f c ( x c , s a , s b ) = 1 2 π 2 R d x c x c x c g c ( x c , s a , s b ) ,
F c ( ν c , s a , s b ) = j 2 π sgn ( ν c ) G c ( ν c , s a , s b ) ,
ν c m F c ( ν c , s a , s b ) = j 2 π ν c m sgn ( ν c ) G c ( ν c , s a , s b ) = j 2 π ν c m G c ( ν c , s a , s b ) .
Λ x c m f c ( x c , s a , s b ) = ( 1 ) ( m + 1 ) 2 ( 2 π ) m + 1 m x c m g c ( x c , s a , s b ) , m > 0 , odd .
WF ( s a , s b ) ( f ) { ( r , ν s ( r ) ) ( r , ν s ( r ) ) WF ( f ) } ,
WF ( s a , s b ) ( f ) { ( r , ν s ( r ) ) WF ( s a , s b ) ( f ) ν s ( r ) e ̂ c ( s a , s b ) 0 } .
r 0 ( s ) = ( R cos ( s ) , R sin ( s ) , h 2 π s ) ,
F ϕ ( ν ) C N ( 1 + ν ) N if ν ν ν s ν s < ϵ ,

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