Abstract

The theory of tomographical imaging with limited angular input, from which two reconstruction algorithms are derived, is discussed. The existence of missing information because of incomplete angular coverage is demonstrated, and an iteration algorithm to recover this information from a priori knowledge of the finite extent of the object is developed. Smoothing algorithms to stabilize reconstructions in the presence of noise are given. The effects of digitization and finite truncation of the reconstruction region in numerical computation are also analyzed. It is shown that the limited-angle problem is governed by a set of eigenvalues whose spectrum is determined by the imaging angle and the finite extent of the object. The distortion on a point source caused by the missing information is calculated; from the results some properties of the iteration scheme, such as spatial uniformity, are derived.

© 1981 Optical Society of America

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  1. J. S. Robertson and et al., “32-crystal positron transverse section detector,” in Tomographic Imaging in Nuclear Medicine, G. S. Freedman, ed. (Society of Nuclear Medicine, New York, 1973), pp. 142–153.
  2. G. N. Hounsfield, “Computerized transverse axial scanning (tomography): Part I. description of system,” Br. J. Radiol. 46, 1016–1022 (1973).
    [Crossref] [PubMed]
  3. E. C. McCullough and et al., “Performance evaluation and quality assurance of computer tomography scanners, with illustrations from the EMI, ACTA and Delta scanners,” Radiology 120, 173–188 (1976).
    [PubMed]
  4. C. B. Lim and et al., “Characteristics of multiwire proportional chambers for positron imaging,” IEEE Trans. Nucl. Sci. NS-21, 85–88 (1974).
    [Crossref]
  5. D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. BME-19, 20–28 (1972).
    [Crossref]
  6. K. C. Tam, Limited-Angle Imaging in Positron Cameras: Theory and Practice, Ph.D. thesis (University of California, Berkeley, Calif., 1979), Sec. 3.1.2.
    [Crossref]
  7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Chap. 8.
  8. G. Chu and K. C. Tam, “Three-dimensional imaging in the positron camera using Fourier techniques,” Phys. Med. Biol. 22, 245–265 (1977).
    [Crossref] [PubMed]
  9. K. C. Tam and et al., “Three-dimensional reconstructions in planar positron cameras using Fourier deconvolution of generalized tomograms,” IEEE Trans. Nucl. Sci. NS-25, 152–159 (1978).
    [Crossref]
  10. L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Three-dimensional image reconstruction using pinhole arrays,” IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
    [Crossref]
  11. F. Riesz and B. Sz.- Nagy, Functional Analysis (Frederick Ungar, New York, 1955), Chap. 6.
  12. B. K. Vainshtein and S. S. Orlov, “General theory of direct 3-D reconstruction,” in Techniques of Three-Dimensional Reconstruction (Brookhaven National Laboratory, Upton, New York, 1974).
  13. R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
    [Crossref]
  14. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [Crossref]
  15. B. Macdonald, Lawrence Berkeley Laboratory, Berkeley, California, personal communication.
  16. A. Klug and R. A. Crowther, “Three-dimensional image reconstruction from the viewpoint of information theory,” Nature 238, 435–440 (1972).
    [Crossref]
  17. J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am. 54, 931–936 (1964).
    [Crossref]
  18. D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
    [Crossref]
  19. K. T. Smith, D. C. Solmon, and S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
    [Crossref]
  20. D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43 (1961).
    [Crossref]
  21. C. W. Barnes, “Object restoration in a diffraction-limited imaging system,” J. Opt. Soc. Am. 56, 575–578 (1966).
    [Crossref]
  22. T. Inouye, “Image reconstruction with limited angle projection data,” IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).
  23. K. C. Tam, B. Macdonald, and V. Perez-Mendez, “3-D object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
    [Crossref]
  24. K. C. Tam and V. Perez-Mendez, “Limited-angle 3-D reconstructions using Fourier transform iterations and Radon transform iterations,” Opt. Eng. (1981) (to be published).
    [Crossref]
  25. K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl Sci. NS-27, 445–458 (1980).
    [Crossref]
  26. M. Y. Chiu and et al., “Three-dimensional radiographic imaging with a restricted view angle,” J. Opt. Soc. Am. 69, 1323–1333 (1979).
    [Crossref]
  27. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Amer. 62, 511–518 (1972).
    [Crossref]
  28. G. Minerbo, “MENT: A maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graphics Image Processing 10, 48–68 (1979).
    [Crossref]
  29. K. C. Tam and V. Perez-Mendez, “Limits to image reconstruction from restricted-angular input,” IEEE Trans. Nucl. Sci. NS-28, (1981).
  30. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 44.
  31. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1952), p. 67.
  32. A. Lent and H. Tuy, “An iterative method for the extrapolation of band-limited functions,” tech. rep. no. MIPG 35 (State University of New York at Buffalo, Buffalo, N.Y., 1979).
  33. F. A. Grünbaum, “A study of Fourier space methods for limited angle image reconstruction,” Numer. Funct. Anal. Optimiz. 2, 31 (1980).
    [Crossref]
  34. R. W. Gerchberg, “Super-resolutions through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [Crossref]
  35. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [Crossref]
  36. D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745 (1965).
    [Crossref]

1981 (1)

K. C. Tam and V. Perez-Mendez, “Limits to image reconstruction from restricted-angular input,” IEEE Trans. Nucl. Sci. NS-28, (1981).

1980 (2)

F. A. Grünbaum, “A study of Fourier space methods for limited angle image reconstruction,” Numer. Funct. Anal. Optimiz. 2, 31 (1980).
[Crossref]

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl Sci. NS-27, 445–458 (1980).
[Crossref]

1979 (4)

M. Y. Chiu and et al., “Three-dimensional radiographic imaging with a restricted view angle,” J. Opt. Soc. Am. 69, 1323–1333 (1979).
[Crossref]

G. Minerbo, “MENT: A maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graphics Image Processing 10, 48–68 (1979).
[Crossref]

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

K. C. Tam, B. Macdonald, and V. Perez-Mendez, “3-D object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

1978 (1)

K. C. Tam and et al., “Three-dimensional reconstructions in planar positron cameras using Fourier deconvolution of generalized tomograms,” IEEE Trans. Nucl. Sci. NS-25, 152–159 (1978).
[Crossref]

1977 (2)

G. Chu and K. C. Tam, “Three-dimensional imaging in the positron camera using Fourier techniques,” Phys. Med. Biol. 22, 245–265 (1977).
[Crossref] [PubMed]

K. T. Smith, D. C. Solmon, and S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

1976 (3)

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[Crossref]

E. C. McCullough and et al., “Performance evaluation and quality assurance of computer tomography scanners, with illustrations from the EMI, ACTA and Delta scanners,” Radiology 120, 173–188 (1976).
[PubMed]

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Three-dimensional image reconstruction using pinhole arrays,” IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[Crossref]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

1974 (2)

R. W. Gerchberg, “Super-resolutions through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

C. B. Lim and et al., “Characteristics of multiwire proportional chambers for positron imaging,” IEEE Trans. Nucl. Sci. NS-21, 85–88 (1974).
[Crossref]

1973 (1)

G. N. Hounsfield, “Computerized transverse axial scanning (tomography): Part I. description of system,” Br. J. Radiol. 46, 1016–1022 (1973).
[Crossref] [PubMed]

1972 (3)

D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. BME-19, 20–28 (1972).
[Crossref]

A. Klug and R. A. Crowther, “Three-dimensional image reconstruction from the viewpoint of information theory,” Nature 238, 435–440 (1972).
[Crossref]

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Amer. 62, 511–518 (1972).
[Crossref]

1970 (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

1966 (1)

1965 (1)

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745 (1965).
[Crossref]

1964 (1)

1962 (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

1961 (1)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43 (1961).
[Crossref]

Barnes, C. W.

Chang, L. T.

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Three-dimensional image reconstruction using pinhole arrays,” IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[Crossref]

Chiu, M. Y.

Chu, G.

G. Chu and K. C. Tam, “Three-dimensional imaging in the positron camera using Fourier techniques,” Phys. Med. Biol. 22, 245–265 (1977).
[Crossref] [PubMed]

Crowther, R. A.

A. Klug and R. A. Crowther, “Three-dimensional image reconstruction from the viewpoint of information theory,” Nature 238, 435–440 (1972).
[Crossref]

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

DeRosier, D. J.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Chap. 8.

Frieden, B. R.

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Amer. 62, 511–518 (1972).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolutions through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Grant, D. G.

D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. BME-19, 20–28 (1972).
[Crossref]

Grünbaum, F. A.

F. A. Grünbaum, “A study of Fourier space methods for limited angle image reconstruction,” Numer. Funct. Anal. Optimiz. 2, 31 (1980).
[Crossref]

Harris, J. L.

Hounsfield, G. N.

G. N. Hounsfield, “Computerized transverse axial scanning (tomography): Part I. description of system,” Br. J. Radiol. 46, 1016–1022 (1973).
[Crossref] [PubMed]

Inouye, T.

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

Klug, A.

A. Klug and R. A. Crowther, “Three-dimensional image reconstruction from the viewpoint of information theory,” Nature 238, 435–440 (1972).
[Crossref]

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

Lent, A.

A. Lent and H. Tuy, “An iterative method for the extrapolation of band-limited functions,” tech. rep. no. MIPG 35 (State University of New York at Buffalo, Buffalo, N.Y., 1979).

Lim, C. B.

C. B. Lim and et al., “Characteristics of multiwire proportional chambers for positron imaging,” IEEE Trans. Nucl. Sci. NS-21, 85–88 (1974).
[Crossref]

Macdonald, B.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl Sci. NS-27, 445–458 (1980).
[Crossref]

K. C. Tam, B. Macdonald, and V. Perez-Mendez, “3-D object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Three-dimensional image reconstruction using pinhole arrays,” IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[Crossref]

B. Macdonald, Lawrence Berkeley Laboratory, Berkeley, California, personal communication.

McCullough, E. C.

E. C. McCullough and et al., “Performance evaluation and quality assurance of computer tomography scanners, with illustrations from the EMI, ACTA and Delta scanners,” Radiology 120, 173–188 (1976).
[PubMed]

Minerbo, G.

G. Minerbo, “MENT: A maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graphics Image Processing 10, 48–68 (1979).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Chap. 8.

Nagy, B. Sz.-

F. Riesz and B. Sz.- Nagy, Functional Analysis (Frederick Ungar, New York, 1955), Chap. 6.

Orlov, S. S.

B. K. Vainshtein and S. S. Orlov, “General theory of direct 3-D reconstruction,” in Techniques of Three-Dimensional Reconstruction (Brookhaven National Laboratory, Upton, New York, 1974).

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 44.

Perez-Mendez, V.

K. C. Tam and V. Perez-Mendez, “Limits to image reconstruction from restricted-angular input,” IEEE Trans. Nucl. Sci. NS-28, (1981).

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl Sci. NS-27, 445–458 (1980).
[Crossref]

K. C. Tam, B. Macdonald, and V. Perez-Mendez, “3-D object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Three-dimensional image reconstruction using pinhole arrays,” IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[Crossref]

K. C. Tam and V. Perez-Mendez, “Limited-angle 3-D reconstructions using Fourier transform iterations and Radon transform iterations,” Opt. Eng. (1981) (to be published).
[Crossref]

Phillips, D. L.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

Pollak, H. O.

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43 (1961).
[Crossref]

Riesz, F.

F. Riesz and B. Sz.- Nagy, Functional Analysis (Frederick Ungar, New York, 1955), Chap. 6.

Robertson, J. S.

J. S. Robertson and et al., “32-crystal positron transverse section detector,” in Tomographic Imaging in Nuclear Medicine, G. S. Freedman, ed. (Society of Nuclear Medicine, New York, 1973), pp. 142–153.

Slepian, D.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745 (1965).
[Crossref]

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43 (1961).
[Crossref]

Smith, K. T.

K. T. Smith, D. C. Solmon, and S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

Solmon, D. C.

K. T. Smith, D. C. Solmon, and S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[Crossref]

Sonnenblick, E.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745 (1965).
[Crossref]

Tam, K. C.

K. C. Tam and V. Perez-Mendez, “Limits to image reconstruction from restricted-angular input,” IEEE Trans. Nucl. Sci. NS-28, (1981).

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl Sci. NS-27, 445–458 (1980).
[Crossref]

K. C. Tam, B. Macdonald, and V. Perez-Mendez, “3-D object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

K. C. Tam and et al., “Three-dimensional reconstructions in planar positron cameras using Fourier deconvolution of generalized tomograms,” IEEE Trans. Nucl. Sci. NS-25, 152–159 (1978).
[Crossref]

G. Chu and K. C. Tam, “Three-dimensional imaging in the positron camera using Fourier techniques,” Phys. Med. Biol. 22, 245–265 (1977).
[Crossref] [PubMed]

K. C. Tam, Limited-Angle Imaging in Positron Cameras: Theory and Practice, Ph.D. thesis (University of California, Berkeley, Calif., 1979), Sec. 3.1.2.
[Crossref]

K. C. Tam and V. Perez-Mendez, “Limited-angle 3-D reconstructions using Fourier transform iterations and Radon transform iterations,” Opt. Eng. (1981) (to be published).
[Crossref]

Tuy, H.

A. Lent and H. Tuy, “An iterative method for the extrapolation of band-limited functions,” tech. rep. no. MIPG 35 (State University of New York at Buffalo, Buffalo, N.Y., 1979).

Vainshtein, B. K.

B. K. Vainshtein and S. S. Orlov, “General theory of direct 3-D reconstruction,” in Techniques of Three-Dimensional Reconstruction (Brookhaven National Laboratory, Upton, New York, 1974).

Wagner, S. L.

K. T. Smith, D. C. Solmon, and S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1952), p. 67.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1952), p. 67.

Bell Syst. Tech. J. (2)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43 (1961).
[Crossref]

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745 (1965).
[Crossref]

Br. J. Radiol. (1)

G. N. Hounsfield, “Computerized transverse axial scanning (tomography): Part I. description of system,” Br. J. Radiol. 46, 1016–1022 (1973).
[Crossref] [PubMed]

Bull. Am. Math. Soc. (1)

K. T. Smith, D. C. Solmon, and S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

Comput. Graphics Image Processing (1)

G. Minerbo, “MENT: A maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graphics Image Processing 10, 48–68 (1979).
[Crossref]

IEEE Trans. Biomed. Eng. (1)

D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. BME-19, 20–28 (1972).
[Crossref]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

IEEE Trans. Nucl Sci. (1)

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl Sci. NS-27, 445–458 (1980).
[Crossref]

IEEE Trans. Nucl. Sci. (6)

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

K. C. Tam, B. Macdonald, and V. Perez-Mendez, “3-D object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

K. C. Tam and V. Perez-Mendez, “Limits to image reconstruction from restricted-angular input,” IEEE Trans. Nucl. Sci. NS-28, (1981).

C. B. Lim and et al., “Characteristics of multiwire proportional chambers for positron imaging,” IEEE Trans. Nucl. Sci. NS-21, 85–88 (1974).
[Crossref]

K. C. Tam and et al., “Three-dimensional reconstructions in planar positron cameras using Fourier deconvolution of generalized tomograms,” IEEE Trans. Nucl. Sci. NS-25, 152–159 (1978).
[Crossref]

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Three-dimensional image reconstruction using pinhole arrays,” IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[Crossref]

J. Assoc. Comput. Mach. (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

J. Math. Anal. Appl. (1)

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Amer. (1)

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Amer. 62, 511–518 (1972).
[Crossref]

Nature (1)

A. Klug and R. A. Crowther, “Three-dimensional image reconstruction from the viewpoint of information theory,” Nature 238, 435–440 (1972).
[Crossref]

Numer. Funct. Anal. Optimiz. (1)

F. A. Grünbaum, “A study of Fourier space methods for limited angle image reconstruction,” Numer. Funct. Anal. Optimiz. 2, 31 (1980).
[Crossref]

Opt. Acta (1)

R. W. Gerchberg, “Super-resolutions through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Phys. Med. Biol. (1)

G. Chu and K. C. Tam, “Three-dimensional imaging in the positron camera using Fourier techniques,” Phys. Med. Biol. 22, 245–265 (1977).
[Crossref] [PubMed]

Proc. R. Soc. London A (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London A 317, 319–340 (1970).
[Crossref]

Radiology (1)

E. C. McCullough and et al., “Performance evaluation and quality assurance of computer tomography scanners, with illustrations from the EMI, ACTA and Delta scanners,” Radiology 120, 173–188 (1976).
[PubMed]

Other (10)

J. S. Robertson and et al., “32-crystal positron transverse section detector,” in Tomographic Imaging in Nuclear Medicine, G. S. Freedman, ed. (Society of Nuclear Medicine, New York, 1973), pp. 142–153.

F. Riesz and B. Sz.- Nagy, Functional Analysis (Frederick Ungar, New York, 1955), Chap. 6.

B. K. Vainshtein and S. S. Orlov, “General theory of direct 3-D reconstruction,” in Techniques of Three-Dimensional Reconstruction (Brookhaven National Laboratory, Upton, New York, 1974).

K. C. Tam, Limited-Angle Imaging in Positron Cameras: Theory and Practice, Ph.D. thesis (University of California, Berkeley, Calif., 1979), Sec. 3.1.2.
[Crossref]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Chap. 8.

B. Macdonald, Lawrence Berkeley Laboratory, Berkeley, California, personal communication.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 44.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1952), p. 67.

A. Lent and H. Tuy, “An iterative method for the extrapolation of band-limited functions,” tech. rep. no. MIPG 35 (State University of New York at Buffalo, Buffalo, N.Y., 1979).

K. C. Tam and V. Perez-Mendez, “Limited-angle 3-D reconstructions using Fourier transform iterations and Radon transform iterations,” Opt. Eng. (1981) (to be published).
[Crossref]

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Figures (17)

Fig. 1
Fig. 1

Two-dimensional imaging devices.

Fig. 2
Fig. 2

Three-dimensional imaging devices.

Fig. 3
Fig. 3

Point-response function defined by an area element.

Fig. 4
Fig. 4

A two-dimensional point-response function ϕ0(r) and its optical transfer function Φ0(k).

Fig. 5
Fig. 5

A pyramidal three-dimensional point-response function ϕ0(r) and its optical transfer function Φ0(k).

Fig. 6
Fig. 6

Space invariance of the point-response function in the x dimension. If the edges of the detection cones generated at every point in the object do not intersect the vertical edges of the reconstruction region, the point-response function is space invariant in the x dimension.

Fig. 7
Fig. 7

Undeterminacy in the matrix method.

Fig. 8
Fig. 8

Fourier-transform-iteration scheme for filling in missing-cone Fourier components.

Fig. 9
Fig. 9

Radon-transform-iteration scheme for filling in missing projections.

Fig. 10
Fig. 10

Schematic representations of the allowed cone and the object extent.

Fig. 11
Fig. 11

Eigenvalues of BA for a two-dimensional problem for various half-angles of the allowed cone.

Fig. 12
Fig. 12

Root-mean-square error of the reconstructed image of a two-dimensional phantom as a function of the number of iterations. The half-angle of the allowed cone is tan−1 (0.5).

Fig. 13
Fig. 13

Root-mean-square error of the reconstructed image of a two-dimensional phantom after 20 iterations as a function of the half-angle of the allowed cone.

Fig. 14
Fig. 14

Positive and negative density distributions of a point source whose missing-cone Fourier components have been set to zero. The half-angle of the allowed cone is tan−1 (0.5).

Fig. 15
Fig. 15

A 11 × 11 square boundary representing the finite extent of an object within a 32 × 32 reconstruction area.

Fig. 16
Fig. 16

Eigenvalues of the zeroth-order prolate-spheroidal equation.

Fig. 17
Fig. 17

Iteration scheme to stabilize the matrix method.

Equations (46)

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ϕ ( r ) = ρ ( r ) ϕ 0 ( r , r ) d 3 r .
ϕ ( r ) = ρ ( r ) ϕ 0 ( r - r ) d 3 r .
Φ ( k ) = Φ 0 ( k ) R ( k ) ,
Φ 0 ( k ) = ϕ 0 ( r ) exp ( 2 π i k · r ) d 3 r , Φ ( k ) = ϕ ( r ) exp ( 2 π i k · r ) d 3 r , R ( k ) = ρ ( r ) exp ( 2 π i k · r ) d 3 r .
ρ ( r ) = R ( k ) exp ( - 2 π i k · r ) d 3 k ,
R ( k ) = { Φ ( k ) Φ 0 ( k )             if Φ 0 ( k ) 0 undetermined             if Φ 0 ( k ) = 0 [ since here Eq . ( 3 a ) becomes 0 = 0 ]
φ ( k x , z ) = - φ 0 ( k x , z - z ) p ( k x , z ) d z ,
φ 0 ( k x , z ) = - ϕ 0 ( x , z ) exp ( 2 π i k x x ) d x , φ ( k x , z ) = - ϕ ( x , z ) exp ( 2 π i k x x ) d x , p ( k x , z ) = - ρ ( x , z ) exp ( 2 π i k x x ) d x ,
ρ ( x , z ) = - p ( k x , z ) exp ( - 2 π i k x x ) d k x ,
p ( k x , z ) = i - g i * ( k x , z ) φ ( k x , z ) d z α i g i ( k x , z ) .
r ( x , z ) r ( θ , z ) ,
θ = tan - 1 ( x / z ) ,             z = z .
ϕ 0 ( θ , z ) = F ( θ ) angle subtended by d x at origin π 1 d x = F ( θ ) π cos 2 θ z .
ϕ 0 ( θ , z ) z = F ( θ ) cos 2 θ π .
Φ 0 ( k x , k z ) = - d z - d x ϕ 0 ( x , z ) exp [ 2 π i ( k x x + k z z ) ] .
Φ 0 ( k x , k z ) = { δ ( z ) π - θ 0 θ 0 F ( θ ) d θ if k x = 0 1 π k x F ( θ i ) cos 2 θ i if k x 0 ,
- φ 0 ( k x , z - z ) exp ( - 2 π i k z z ) d z = exp ( - 2 π i k z z ) Φ 0 ( k x , k z ) .
p ( k x , z ) = - C ( k x , k z ) exp ( - 2 π i k z z ) d k z ,
C ( k x , k z ) = { - φ ( k x , z ) exp ( 2 π i k z z ) d z Φ 0 ( k x , k z ) = Φ ( k x , k z ) Φ 0 ( k x , k z ) if Φ 0 ( k x , k z ) 0 undetermined if Φ 0 ( k x , k z ) = 0 .
[ 2 ρ ( r ) ] 2 d 3 r = minimum .
R ( k ) = Φ ( k ) Φ 0 ( k ) + γ ( 2 π ) 4 k 4 Φ 0 ( k ) ,
Y = A X ,
Y = { A + γ G [ ( 2 π k x ) 2 I + C Δ z 2 ] 2 } X .
c i j = { - 2 i = j 1 i = j ± 1 0 otherwise .
m π D 2 d .
R ( k ) = i a i Ψ i ( k ) ,
A f = χ A f , B f = F - 1 χ B F f ,
χ A ( k ) = { 1 k R a 0 k R a , χ B ( x ) = { 1 x R b 0 x R b .
E t ( n ) ( k ) = - i a i ( 1 - λ i ) n Ψ i ( k ) ,             0 < λ i < 1 ,
σ = [ i , k [ reconstruction ( i , k ) - phantom ( i , k ) ] 2 number of pixels ] 1 / 2 .
ρ ( x , z ) = 1 π 2 ( tan θ 0 z 2 - x 2 tan θ 0 )             for ( x , z ) ( 0 , 0 ) .
Φ 0 ( k x , k z ) = - d z - d x ϕ 0 ( x , z ) exp [ 2 π i ( k x x + k z z ) ] , = - d z - π π d θ ϕ 0 ( θ , z ) exp [ 2 π i z × ( k x tan θ + k z ) ] z sec 2 θ , = - θ 0 θ 0 d θ F ( θ ) π - d z exp [ 2 π i z ( k x tan θ + k z ) ] .
- exp [ 2 π i z ( k x tan θ + k z ) ] d z = δ ( k x tan θ + k z ) ;
Φ 0 ( k x , k z ) = - θ 0 θ 0 F ( θ ) π δ ( k x tan θ + k z ) d θ , = { δ ( k z ) π - θ 0 θ 0 F ( θ ) d θ if k x = 0 F ( θ i ) cos 2 θ i π k x if k x 0 ,
ϕ 0 ( s , z ) z 2 = H ( s / z ) ,
Φ 0 ( w , k z ) = d z d 2 s ϕ 0 ( s , z ) exp [ 2 π i ( w · s + k z z ) ] = H ( t ) δ ( w · t + k z ) d 2 t .
Φ 0 ( w , k z ) = { δ ( k z ) H ( t ) d 2 t if w = 0 H ( - k z / w , t 2 ) d t 2 if w 0 .
ϕ 0 ( x , y , z ) > 0 whenever 0 x z tan θ 0 and 0 y z tan θ 0 ,
Φ 0 ( k x , k y , k z ) > 0 if ( k x + k y ) > k z / tan θ 0 when k x , k y > 0 or ( k x + k y ) k z / tan θ 0 when k x = 0 or k y = 0 = 0             otherwise .
s Δ z ( z ) = n = - δ ( z - n Δ z ) .
Φ 0 ( z ) ( k x , k z ) = Φ 0 ( k x , k z ) * S Δ k z ( k z ) ,
S Δ k z ( k z ) = Δ k z n = - δ ( k z - n Δ k z ) ,             Δ k z = 1 Δ z .
φ 0 ( k x , z - z ) = - θ 0 θ 0 F ( θ ) π exp [ 2 π i k x tan θ ( z - z ) ] d θ .
z 1 z 2 d z z 1 z 2 d z φ 0 ( k x , z - z ) f ( z ) f * ( z ) = z 1 z 2 d z z 1 z 2 d z - θ 0 θ 0 d θ F ( θ ) π × exp [ 2 π i k x tan θ ( z - z ) ] f ( z ) f * ( z ) = 1 π - θ 0 θ 0 F ( θ ) | z 1 z 2 exp [ ( 2 π i k x tan θ ) z ] f ( z ) d z | 2 d θ .
I ( k x tan θ ) = z 1 z 2 exp [ ( 2 π i k x tan θ ) z ] f ( z ) d z = 0
g ( z ) = z 1 z 2 φ 0 ( k x , z - z ) f ( z ) d z