Abstract

Image-reconstruction algorithms implemented on existing computerized tomography (CT) scanners require the collection of line integrals that are evenly spaced over 360 deg. In many practical situations, some of the line integrals are inaccurately measured or are not measured at all. In these limited-data situations, conventional algorithms produce images with severe streak artifacts. Recently, several other image-reconstruction algorithms were suggested, each tailored to a specific type of limited-data problem. These algorithms make minimal use of a priori knowledge about the image; only one has been demonstrated with real x-ray data. We present a new operator framework that treats all types of limited-data image-reconstruction problems in a unified way. From this framework we derive iterative convolution backprojection algorithms that make no restrictions on the location of missing line integrals. All available a priori information is incorporated by constraint operators. The algorithm has been implemented on a commercial CT scanner. We present examples of images reconstructed from real x-ray data in two limited-data situations and demonstrate the use of additional a priori information to reduce streak artifacts further.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
    [PubMed]
  2. G. H. Glover and N. J. Peic, "An algorithm for the reduction of metal clip artifacts in CT reconstructions," Med. Phys. 8,799–807 (1981).
    [CrossRef] [PubMed]
  3. R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: III: Projection completion methods (theory)," Optik 50, 189–204 (1978).
  4. R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: IV: Projection completion methods (computational examples)," Optik 50, 269–278 (1978).
  5. A. Peres, "Tomographic reconstruction from limited angular data," J. Comput. Assisted Tomog. 3, 800–803 (1979).
  6. T. Inouye, "Image reconstruction with limited angle projection data," IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).
  7. T. Inouye, "Image reconstruction with limited view angle projections," in Proceedings of the International Workshop on Physics and Engineering in Medical Imaging (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 165–168.
    [CrossRef]
  8. R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  9. A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  10. T. Sato, S. J. Norton, M. Linzer, O. Ikeda, and M. Hirama, "Tomographic image reconstruction from limited projections using iterative revisions in image and transform spaces," Appl. Opt. 20, 395–399 (1981).
    [CrossRef] [PubMed]
  11. K. C. Tam and V. Perez-Mendez, "Tomographical imaging with limited-angle input," J. Opt. Soc. Am. 71, 582–592 (1981).
    [CrossRef]
  12. K. C. Tam and V. Perez-Mendez, "Limited-angle three dimensional reconstructions using Fourier transform iterations and Radon transform iterations," Opt. Engin. 20, 586–589 (1981).
    [CrossRef]
  13. B. K. P. Horn, "Fan-beam reconstruction methods," Proc. IEEE 67, 1616–1623 (1979).
    [CrossRef]
  14. A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).
    [CrossRef]
  15. D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part 1—Theory," IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [CrossRef]
  16. M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: Part 2—Applications and numerical results," IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
    [CrossRef]
  17. M. Nassi, W. R. Brody, B. P. Medoff, and A. Macovski, "Iterative reconstruction—reprojection: an algorithm for limited data cardiac computed tomography," IEEE Trans. Biomed. Eng. BME-29, 333–341 (1982).
    [CrossRef]
  18. M. Ein-Gal, D. Rosenfeld, and A. Macovski, "The consistency of the shadow: an approach to preprocessing in computerized tomography," in Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections (Optical Society of America, Washington, D.C., 1975), paper WB5.
  19. A. K. Louis, "Ghosts in tomography—the null space of the radon transform," Math. Meth. Appl. Sci. 3, 1–10 (1981).
    [CrossRef]
  20. D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).
  21. B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Proceedings of the International Workshop on Physics and Engineering in Medical Imaging(Institute of Electrical and Electronics Engineers, New York, 1982), pp. 188–192.
    [CrossRef]
  22. M. Ein-Gal, "The shadow transform—an approach to cross sectional imaging," Ph.D. Thesis (Stanford University, Stanford, Calif., 1974).
  23. S. L. Wood, A. Macovski, and M. Morf, "Reconstructions with limited data using estimation theory," in Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv et al., eds. (North-Holland, Amsterdam, 1979), pp. 219–233.
  24. M. H. Buonocore, "Fast minimum variance estimators for limited angle computed tomography image reconstruction," Ph.D. Thesis (Stanford University, Stanford, Calif., 1981).
  25. K. M. Hanson, "Limited angle CT reconstruction using a priori information," in Proceedings of the IEEE Computer Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1982).
  26. R. W. Schafer, R. M. Mersereau, M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).
    [CrossRef]
  27. B. P. Medoff, "Image reconstruction from limited data," Ph.D. Thesis (Stanford University, Stanford, Calif., 1983).

1982 (3)

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part 1—Theory," IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: Part 2—Applications and numerical results," IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

M. Nassi, W. R. Brody, B. P. Medoff, and A. Macovski, "Iterative reconstruction—reprojection: an algorithm for limited data cardiac computed tomography," IEEE Trans. Biomed. Eng. BME-29, 333–341 (1982).
[CrossRef]

1981 (6)

A. K. Louis, "Ghosts in tomography—the null space of the radon transform," Math. Meth. Appl. Sci. 3, 1–10 (1981).
[CrossRef]

T. Sato, S. J. Norton, M. Linzer, O. Ikeda, and M. Hirama, "Tomographic image reconstruction from limited projections using iterative revisions in image and transform spaces," Appl. Opt. 20, 395–399 (1981).
[CrossRef] [PubMed]

K. C. Tam and V. Perez-Mendez, "Tomographical imaging with limited-angle input," J. Opt. Soc. Am. 71, 582–592 (1981).
[CrossRef]

G. H. Glover and N. J. Peic, "An algorithm for the reduction of metal clip artifacts in CT reconstructions," Med. Phys. 8,799–807 (1981).
[CrossRef] [PubMed]

K. C. Tam and V. Perez-Mendez, "Limited-angle three dimensional reconstructions using Fourier transform iterations and Radon transform iterations," Opt. Engin. 20, 586–589 (1981).
[CrossRef]

A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).
[CrossRef]

1979 (2)

A. Peres, "Tomographic reconstruction from limited angular data," J. Comput. Assisted Tomog. 3, 800–803 (1979).

T. Inouye, "Image reconstruction with limited angle projection data," IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

1978 (2)

R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: III: Projection completion methods (theory)," Optik 50, 189–204 (1978).

R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: IV: Projection completion methods (computational examples)," Optik 50, 269–278 (1978).

1977 (1)

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

1975 (1)

A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
[CrossRef]

Bates, R. H. T.

R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: III: Projection completion methods (theory)," Optik 50, 189–204 (1978).

R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: IV: Projection completion methods (computational examples)," Optik 50, 269–278 (1978).

Breiman, R. S.

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

Brody, W. R.

M. Nassi, W. R. Brody, B. P. Medoff, and A. Macovski, "Iterative reconstruction—reprojection: an algorithm for limited data cardiac computed tomography," IEEE Trans. Biomed. Eng. BME-29, 333–341 (1982).
[CrossRef]

B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Proceedings of the International Workshop on Physics and Engineering in Medical Imaging(Institute of Electrical and Electronics Engineers, New York, 1982), pp. 188–192.
[CrossRef]

Buonocore, M. H.

M. H. Buonocore, "Fast minimum variance estimators for limited angle computed tomography image reconstruction," Ph.D. Thesis (Stanford University, Stanford, Calif., 1981).

Ein-Gal, M.

M. Ein-Gal, "The shadow transform—an approach to cross sectional imaging," Ph.D. Thesis (Stanford University, Stanford, Calif., 1974).

M. Ein-Gal, D. Rosenfeld, and A. Macovski, "The consistency of the shadow: an approach to preprocessing in computerized tomography," in Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections (Optical Society of America, Washington, D.C., 1975), paper WB5.

Gerchberg, R. W.

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
[CrossRef]

Glover, G. H.

G. H. Glover and N. J. Peic, "An algorithm for the reduction of metal clip artifacts in CT reconstructions," Med. Phys. 8,799–807 (1981).
[CrossRef] [PubMed]

Guthaner, D. F.

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

Hanson, K. M.

K. M. Hanson, "Limited angle CT reconstruction using a priori information," in Proceedings of the IEEE Computer Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1982).

Harell, G. S.

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

Hirama, M.

Horn, B. K. P.

B. K. P. Horn, "Fan-beam reconstruction methods," Proc. IEEE 67, 1616–1623 (1979).
[CrossRef]

Ikeda, O.

Inouye, T.

T. Inouye, "Image reconstruction with limited angle projection data," IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

T. Inouye, "Image reconstruction with limited view angle projections," in Proceedings of the International Workshop on Physics and Engineering in Medical Imaging (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 165–168.
[CrossRef]

Lent, A.

A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).
[CrossRef]

Lewitt, R. M.

R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: III: Projection completion methods (theory)," Optik 50, 189–204 (1978).

R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: IV: Projection completion methods (computational examples)," Optik 50, 269–278 (1978).

Linzer, M.

Louis, A. K.

A. K. Louis, "Ghosts in tomography—the null space of the radon transform," Math. Meth. Appl. Sci. 3, 1–10 (1981).
[CrossRef]

Luenberger, D. G.

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

Macovski, A.

M. Nassi, W. R. Brody, B. P. Medoff, and A. Macovski, "Iterative reconstruction—reprojection: an algorithm for limited data cardiac computed tomography," IEEE Trans. Biomed. Eng. BME-29, 333–341 (1982).
[CrossRef]

M. Ein-Gal, D. Rosenfeld, and A. Macovski, "The consistency of the shadow: an approach to preprocessing in computerized tomography," in Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections (Optical Society of America, Washington, D.C., 1975), paper WB5.

B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Proceedings of the International Workshop on Physics and Engineering in Medical Imaging(Institute of Electrical and Electronics Engineers, New York, 1982), pp. 188–192.
[CrossRef]

S. L. Wood, A. Macovski, and M. Morf, "Reconstructions with limited data using estimation theory," in Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv et al., eds. (North-Holland, Amsterdam, 1979), pp. 219–233.

Marshall, W. H.

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

Medoff, B. P.

M. Nassi, W. R. Brody, B. P. Medoff, and A. Macovski, "Iterative reconstruction—reprojection: an algorithm for limited data cardiac computed tomography," IEEE Trans. Biomed. Eng. BME-29, 333–341 (1982).
[CrossRef]

B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Proceedings of the International Workshop on Physics and Engineering in Medical Imaging(Institute of Electrical and Electronics Engineers, New York, 1982), pp. 188–192.
[CrossRef]

B. P. Medoff, "Image reconstruction from limited data," Ph.D. Thesis (Stanford University, Stanford, Calif., 1983).

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Morehouse, C. C.

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

Morf, M.

S. L. Wood, A. Macovski, and M. Morf, "Reconstructions with limited data using estimation theory," in Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv et al., eds. (North-Holland, Amsterdam, 1979), pp. 219–233.

Nassi, M.

M. Nassi, W. R. Brody, B. P. Medoff, and A. Macovski, "Iterative reconstruction—reprojection: an algorithm for limited data cardiac computed tomography," IEEE Trans. Biomed. Eng. BME-29, 333–341 (1982).
[CrossRef]

Norton, S. J.

Papoulis, A.

A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Peic, N. J.

G. H. Glover and N. J. Peic, "An algorithm for the reduction of metal clip artifacts in CT reconstructions," Med. Phys. 8,799–807 (1981).
[CrossRef] [PubMed]

Peres, A.

A. Peres, "Tomographic reconstruction from limited angular data," J. Comput. Assisted Tomog. 3, 800–803 (1979).

Perez-Mendez, V.

K. C. Tam and V. Perez-Mendez, "Limited-angle three dimensional reconstructions using Fourier transform iterations and Radon transform iterations," Opt. Engin. 20, 586–589 (1981).
[CrossRef]

K. C. Tam and V. Perez-Mendez, "Tomographical imaging with limited-angle input," J. Opt. Soc. Am. 71, 582–592 (1981).
[CrossRef]

Richards, M. A.

R. W. Schafer, R. M. Mersereau, M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Rosenfeld, D.

M. Ein-Gal, D. Rosenfeld, and A. Macovski, "The consistency of the shadow: an approach to preprocessing in computerized tomography," in Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections (Optical Society of America, Washington, D.C., 1975), paper WB5.

Sato, T.

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Seppi, E. J.

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

Sezan, M. I.

M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: Part 2—Applications and numerical results," IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Stark, H.

M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: Part 2—Applications and numerical results," IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Tam, K. C.

K. C. Tam and V. Perez-Mendez, "Limited-angle three dimensional reconstructions using Fourier transform iterations and Radon transform iterations," Opt. Engin. 20, 586–589 (1981).
[CrossRef]

K. C. Tam and V. Perez-Mendez, "Tomographical imaging with limited-angle input," J. Opt. Soc. Am. 71, 582–592 (1981).
[CrossRef]

Tuy, H.

A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).
[CrossRef]

Webb, H.

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part 1—Theory," IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Wexler, L.

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

Wood, S. L.

S. L. Wood, A. Macovski, and M. Morf, "Reconstructions with limited data using estimation theory," in Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv et al., eds. (North-Holland, Amsterdam, 1979), pp. 219–233.

Youla, D. C.

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part 1—Theory," IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Biomed. Eng. (1)

M. Nassi, W. R. Brody, B. P. Medoff, and A. Macovski, "Iterative reconstruction—reprojection: an algorithm for limited data cardiac computed tomography," IEEE Trans. Biomed. Eng. BME-29, 333–341 (1982).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

IEEE Trans. Med. Imaging (2)

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part 1—Theory," IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: Part 2—Applications and numerical results," IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

T. Inouye, "Image reconstruction with limited angle projection data," IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

J. Comput. Assisted Tomog. (1)

A. Peres, "Tomographic reconstruction from limited angular data," J. Comput. Assisted Tomog. 3, 800–803 (1979).

J. Math. Anal. Appl. (1)

A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," J. Math. Anal. Appl. 83, 554–565 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

Math. Meth. Appl. Sci. (1)

A. K. Louis, "Ghosts in tomography—the null space of the radon transform," Math. Meth. Appl. Sci. 3, 1–10 (1981).
[CrossRef]

Med. Phys. (1)

G. H. Glover and N. J. Peic, "An algorithm for the reduction of metal clip artifacts in CT reconstructions," Med. Phys. 8,799–807 (1981).
[CrossRef] [PubMed]

Opt. Acta (1)

R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Engin. (1)

K. C. Tam and V. Perez-Mendez, "Limited-angle three dimensional reconstructions using Fourier transform iterations and Radon transform iterations," Opt. Engin. 20, 586–589 (1981).
[CrossRef]

Optik (2)

R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: III: Projection completion methods (theory)," Optik 50, 189–204 (1978).

R. M. Lewitt and R. H. T. Bates, "Image reconstruction from projections: IV: Projection completion methods (computational examples)," Optik 50, 269–278 (1978).

Radiology (1)

G. S. Harell, D. F. Guthaner, R. S. Breiman, C. C. Morehouse, E. J. Seppi, W. H. Marshall, and L. Wexler, "Stop-action cardiac computed tomography," Radiology 123, 515–517 (1977).
[PubMed]

Other (11)

T. Inouye, "Image reconstruction with limited view angle projections," in Proceedings of the International Workshop on Physics and Engineering in Medical Imaging (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 165–168.
[CrossRef]

B. K. P. Horn, "Fan-beam reconstruction methods," Proc. IEEE 67, 1616–1623 (1979).
[CrossRef]

M. Ein-Gal, D. Rosenfeld, and A. Macovski, "The consistency of the shadow: an approach to preprocessing in computerized tomography," in Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections (Optical Society of America, Washington, D.C., 1975), paper WB5.

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Proceedings of the International Workshop on Physics and Engineering in Medical Imaging(Institute of Electrical and Electronics Engineers, New York, 1982), pp. 188–192.
[CrossRef]

M. Ein-Gal, "The shadow transform—an approach to cross sectional imaging," Ph.D. Thesis (Stanford University, Stanford, Calif., 1974).

S. L. Wood, A. Macovski, and M. Morf, "Reconstructions with limited data using estimation theory," in Computer Aided Tomography and Ultrasonics in Medicine, J. Raviv et al., eds. (North-Holland, Amsterdam, 1979), pp. 219–233.

M. H. Buonocore, "Fast minimum variance estimators for limited angle computed tomography image reconstruction," Ph.D. Thesis (Stanford University, Stanford, Calif., 1981).

K. M. Hanson, "Limited angle CT reconstruction using a priori information," in Proceedings of the IEEE Computer Society International Symposium (Institute of Electrical and Electronics Engineers, New York, 1982).

R. W. Schafer, R. M. Mersereau, M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).
[CrossRef]

B. P. Medoff, "Image reconstruction from limited data," Ph.D. Thesis (Stanford University, Stanford, Calif., 1983).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Schematic diagram of the iterative limited-data-reconstruction algorithm. The constraints represent a priori information about the image and line-integral data, for example, a known outer boundary, positivity, or a known background-attenuation value.

Fig. 2
Fig. 2

Representation of a typical fan-beam x-ray CT scanner illustrating the bagel problem. The object contains an x-ray-opaque region (a hole). In each projection, line integrals along ray paths that pass through the hole are missing. The bagel problem is to produce a reconstruction of the object in the bagel-shaped region exterior to the hole by using only the measured line integrals along ray paths that do not pass through the hole.

Fig. 3
Fig. 3

The bagel algorithm. The algorithm repeats until a steady state is reached.

Fig. 4
Fig. 4

Full-data reconstruction from measured x-ray data showing a slice through a test object that contains within it three smaller rods. Scan data and reconstruction were obtained using a Varian CT whole-body scanner.

Fig. 5
Fig. 5

Reconstruction illustrating the type of streak artifact generated by convolution backprojection when the central region contains an x-ray-opaque structure.

Fig. 6
Fig. 6

Reconstruction using an initial estimate of the missing line integrals based on an average of nearby measured line integrals. Since the initial estimates are not correct, the reconstructed image has severe artifacts.

Fig. 7
Fig. 7

Result of setting the image in Fig. 6 to zero outside the bagel and setting the hole to an assigned constant value.

Fig. 8
Fig. 8

Reconstruction produced after the first iteration of the bagel algorithm. Line-integral values calculated from the image shown in Fig. 7 are used as estimates of the missing data. Compared with the image shown in Fig. 6, the artifact is somewhat reduced in amplitude.

Fig. 9
Fig. 9

Steady-state reconstruction produced after 20 iterations of the bagel algorithm. Compared with the limited-data reconstruction produced by convolution backprojection (Fig. 5), the background is more uniform and the streaks are greatly reduced in amplitude. The reconstructed values in the hole agree closely with the value assigned to the hole by the iterative process.

Fig. 10
Fig. 10

The residual ghost image, equal to the difference between the steady-state limited-data reconstruction (Fig. 9) and the actual attenuation values in the bagel (Fig. 4). White areas represent positive values; black areas represent negative values.

Fig. 11
Fig. 11

(a) Sinogram representation of line-integral data for the bagel problem. Each horizontal line corresponds to line integrals measured at one source position. The vertically oriented strip in the central region of the sinogram corresponds to missing measurements along ray paths that pass through the hole. (b) Sinogram of the line integrals calculated from the ghost image in Fig. 10. The nonzero line integrals (white areas) are concentrated in the missing-data region (vertical strip). Therefore the ghost is invisible to the measured line integrals.

Fig. 12
Fig. 12

Second experiment. Limited-data reconstruction with severe streak artifact produced by convolution backprojection. In 10% of the projections, the central 20% of the nonzero line integrals are missing.

Fig. 13
Fig. 13

Steady-state image produced by the iterative algorithm using known extent and positivity constraints. There is significant residual streak artifact.

Fig. 14
Fig. 14

A priori knowledge of the background density in a region of the object is incorporated by a constraint operator. This operator resets the central region of the image to the known value at each step of the iteration, as shown above.

Fig. 15
Fig. 15

Steady-state image produced by the iterative algorithm using the known background constraint in addition to known extent and positivity. Compared with using only finite extent and positivity (Fig. 13), there is significant reduction in the residual streak artifact. Note that the small structure in the central part of the object is clearly resolved.

Equations (12)

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

z = [ x y ] .
x ^ = C x x ^ .
R [ x ^ y ] = C I R [ x ^ y ] .
SR [ x ^ y ] = [ x ^ y ] .
SR [ x ^ 1 y ] - SR [ x ^ 2 y ] = [ x ^ 1 y ] - [ x ^ 2 y ] ,
SR [ x ^ 1 - x ^ 2 0 ] = [ x ^ 1 - x ^ 2 0 ] .
G = R [ g 0 ] .
S G = [ g 0 ]
S y G = 0.
[ x ^ y ] = SR [ x ^ y ] = S C I R [ x ^ y ] = S C I R [ C x x ^ y ] .
x ^ = S x C I R [ C x x ^ y ] T ( x ^ ) .
x ^ i + 1 = T ( x ^ i ) = S x C I R [ C x x ^ i y ] .