Abstract

The properties of many three-dimensional radiographic imaging systems are examined within a common analytic framework. It is found, by performing an important coordinate transformation, that the projection data of these systems can be transformed to a form amenable to analysis by the centralslice theorem. Therefore, a clear relationship between the measured data set and the three-dimensional Fourier transform of the object can be established. For the Fourier aperture system, each measurement in the detector plane gives directly one point in the three-dimensional Fourier transform of the object. The limited view angle of these systems manifests itself in the incomplete collection of the Fourier transform of the object. This “missing cones” region in the Fourier space producesa point-spread function that has long-range conical ridges radiating from the central core. It is shown that degradations in linear reconstructions of extended objects are not as disastrous as might have been expected.

© 1979 Optical Society of America

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  1. C. Chou and H. H. Barrett, "Gamma-ray imaging in Fourier space," Opt. Lett. 3, 187–189 (1978).
  2. L. Mertz, "Applicability of the rotation collimator to nuclear medicine,"medicine," Opt. Commun. 12, 216–219 (1974).
  3. L. T. Chang, B. Macdonald, and V. Perez-Mendez, "Axial tomography and three-dimensional image reconstruction," IEEE Trans. Nucl. S ci., NS-23, 568–572 (1976).
  4. W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, Jr., "Coded aperture imaging of the heart," J. Nucl. Med. 19, 730–730 (1978).
  5. R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, "A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera," J. Nucl. Med. 19, 648–654 (1978).
  6. J. G. Colsher, "Iterative three-dimensional image reconstruction from tomographic projections," Comp. Graph. Img. Proc. 6, 513–537 (1977).
  7. K. C. Tam, V. Perez-Mendez, and B. Macdonald, "Three-dimensional object reconstruction in emission and transmission tomography with limited angular input," IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
  8. B. R. Frieden, "Optical transfer of the three-dimensional object," J. Opt. Soc. Am. 57, 56–66 (1967).
  9. A. Lohmann, "Three-dimensional properties of wave fields," Optik (Stuttgart) 51, 105–117 (1978).
  10. R. N. Bracewell and S. J. Wernecke, "Image reconstruction over a finite field of view," J. Opt. Soc. Am. 65, 1342–1346 (1975).
  11. T. Ionoye, "Image reconstructing with limited angle projection data," IEEE Trans. Nucl. Sci., NS-26, 2666–2669 (1979).
  12. A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735–742 (1975).
  13. A good review of this topic can be found in B. R. Frieden, "Image Enhancement and Restoration," Chap. 5 of Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, Heidelberg, 1975).
  14. J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," Opt. Lett. 3, 27–29 (1978).
  15. R. G. Pridham, "Imaging of volume object by shadow casting," Ph.D. thesis, University of Rhode Island, 1977, Appendix M (unpublished).

1979 (2)

K. C. Tam, V. Perez-Mendez, and B. Macdonald, "Three-dimensional object reconstruction in emission and transmission tomography with limited angular input," IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).

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

1978 (5)

A. Lohmann, "Three-dimensional properties of wave fields," Optik (Stuttgart) 51, 105–117 (1978).

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, Jr., "Coded aperture imaging of the heart," J. Nucl. Med. 19, 730–730 (1978).

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, "A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera," J. Nucl. Med. 19, 648–654 (1978).

J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," Opt. Lett. 3, 27–29 (1978).

C. Chou and H. H. Barrett, "Gamma-ray imaging in Fourier space," Opt. Lett. 3, 187–189 (1978).

1977 (1)

J. G. Colsher, "Iterative three-dimensional image reconstruction from tomographic projections," Comp. Graph. Img. Proc. 6, 513–537 (1977).

1976 (1)

L. T. Chang, B. Macdonald, and V. Perez-Mendez, "Axial tomography and three-dimensional image reconstruction," IEEE Trans. Nucl. S ci., NS-23, 568–572 (1976).

1975 (2)

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735–742 (1975).

R. N. Bracewell and S. J. Wernecke, "Image reconstruction over a finite field of view," J. Opt. Soc. Am. 65, 1342–1346 (1975).

1974 (1)

L. Mertz, "Applicability of the rotation collimator to nuclear medicine,"medicine," Opt. Commun. 12, 216–219 (1974).

1967 (1)

Barrett, H. H.

Bracewell, R. N.

Chang, L. T.

L. T. Chang, B. Macdonald, and V. Perez-Mendez, "Axial tomography and three-dimensional image reconstruction," IEEE Trans. Nucl. S ci., NS-23, 568–572 (1976).

Chou, C.

Colsher, J. G.

J. G. Colsher, "Iterative three-dimensional image reconstruction from tomographic projections," Comp. Graph. Img. Proc. 6, 513–537 (1977).

Fienup, J. R.

Frieden, B. R.

B. R. Frieden, "Optical transfer of the three-dimensional object," J. Opt. Soc. Am. 57, 56–66 (1967).

A good review of this topic can be found in B. R. Frieden, "Image Enhancement and Restoration," Chap. 5 of Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, Heidelberg, 1975).

Ionoye, T.

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

Keyes, Jr., J. W.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, Jr., "Coded aperture imaging of the heart," J. Nucl. Med. 19, 730–730 (1978).

Kirch, D.

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, "A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera," J. Nucl. Med. 19, 648–654 (1978).

Koral, K. F.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, Jr., "Coded aperture imaging of the heart," J. Nucl. Med. 19, 730–730 (1978).

LeFree, M.

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, "A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera," J. Nucl. Med. 19, 648–654 (1978).

Leonard, P. F.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, Jr., "Coded aperture imaging of the heart," J. Nucl. Med. 19, 730–730 (1978).

Lohmann, A.

A. Lohmann, "Three-dimensional properties of wave fields," Optik (Stuttgart) 51, 105–117 (1978).

Macdonald, B.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, "Three-dimensional object reconstruction in emission and transmission tomography with limited angular input," IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).

L. T. Chang, B. Macdonald, and V. Perez-Mendez, "Axial tomography and three-dimensional image reconstruction," IEEE Trans. Nucl. S ci., NS-23, 568–572 (1976).

Mayans, R.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, Jr., "Coded aperture imaging of the heart," J. Nucl. Med. 19, 730–730 (1978).

Mertz, L.

L. Mertz, "Applicability of the rotation collimator to nuclear medicine,"medicine," Opt. Commun. 12, 216–219 (1974).

Papoulis, A.

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735–742 (1975).

Perez-Mendez, V.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, "Three-dimensional object reconstruction in emission and transmission tomography with limited angular input," IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).

L. T. Chang, B. Macdonald, and V. Perez-Mendez, "Axial tomography and three-dimensional image reconstruction," IEEE Trans. Nucl. S ci., NS-23, 568–572 (1976).

Pridham, R. G.

R. G. Pridham, "Imaging of volume object by shadow casting," Ph.D. thesis, University of Rhode Island, 1977, Appendix M (unpublished).

Rogers, W. L.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, Jr., "Coded aperture imaging of the heart," J. Nucl. Med. 19, 730–730 (1978).

Steele, P.

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, "A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera," J. Nucl. Med. 19, 648–654 (1978).

Tam, K. C.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, "Three-dimensional object reconstruction in emission and transmission tomography with limited angular input," IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).

Vogel, R. A.

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, "A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera," J. Nucl. Med. 19, 648–654 (1978).

Wernecke, S. J.

Comp. Graph. Img. Proc. (1)

J. G. Colsher, "Iterative three-dimensional image reconstruction from tomographic projections," Comp. Graph. Img. Proc. 6, 513–537 (1977).

IEEE Trans. Circuits Syst. (1)

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735–742 (1975).

IEEE Trans. Nucl. Sci. (3)

K. C. Tam, V. Perez-Mendez, and B. Macdonald, "Three-dimensional object reconstruction in emission and transmission tomography with limited angular input," IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).

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

L. T. Chang, B. Macdonald, and V. Perez-Mendez, "Axial tomography and three-dimensional image reconstruction," IEEE Trans. Nucl. S ci., NS-23, 568–572 (1976).

J. Nucl. Med. (2)

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, Jr., "Coded aperture imaging of the heart," J. Nucl. Med. 19, 730–730 (1978).

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, "A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera," J. Nucl. Med. 19, 648–654 (1978).

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

L. Mertz, "Applicability of the rotation collimator to nuclear medicine,"medicine," Opt. Commun. 12, 216–219 (1974).

Opt. Lett. (2)

Optik (1)

A. Lohmann, "Three-dimensional properties of wave fields," Optik (Stuttgart) 51, 105–117 (1978).

Other (2)

R. G. Pridham, "Imaging of volume object by shadow casting," Ph.D. thesis, University of Rhode Island, 1977, Appendix M (unpublished).

A good review of this topic can be found in B. R. Frieden, "Image Enhancement and Restoration," Chap. 5 of Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, Heidelberg, 1975).

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