Abstract

An arbitrary source function cannot be determined fully from projection data that are limited in number and range of viewing angle. There exists a null subspace in the Hilbert space of possible source functions about which the available projection measurements provide no information. The null-space components of deterministic solutions are usually zero, giving rise to unavoidable artifacts. It is demonstrated that these artifacts may be reduced by a Bayesian maximum a posteriori (MAP) reconstruction method that permits the use of significant a priori information. Since normal distributions are assumed for the a priori and measurement-error probability densities, the MAP reconstruction method presented here is equivalent to the minimum-variance linear estimator with nonstationary mean and covariance ensemble characterizations. A more comprehensive Bayesian approach is suggested in which the ensemble mean and covariance specifications are adjusted on the basis of the measurements.

© 1983 Optical Society of America

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  1. K. M. Hanson, "CT reconstruction from limited projection angles," Proc. Soc. Photo.-Opt. Instrum. Eng. 374, 166–173 (1982).
  2. K. M. Hanson and G. W. Wecksung, "Bayesian approach to limited-angle CT reconstruction," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical Society of America, Washington, D.C., 1983), pp. FA6–FA14.
  3. K. M. Hanson, "Limited-angle CT reconstruction using a priori information," in Proceedings of the First IEEE Computer Society International Symposium on Medical Imaging and Image Interpretation (Institute of Electronics and Electrical Engineers, New York, 1982), pp. 527–533.
    [CrossRef]
  4. S. Twomey, "Information content in remote sensing," Appl. Opt. 13, 942–945 (1974).
    [CrossRef] [PubMed]
  5. S. Twomey and H. B. Howell, "Some aspects of the optical estimation of microstructure in fog and cloud," Appl. Opt. 6, 2125–2131 (1967).
    [CrossRef] [PubMed]
  6. S. Twomey, "The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurement," J. Franklin Inst. 279, 95–109 (1965).
    [CrossRef]
  7. L. A. Shepp and B. F. Logan, "The Fourier reconstruction of a head section," IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
    [CrossRef]
  8. R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471–481 (1970).
    [CrossRef] [PubMed]
  9. P. Gilbert, "Iterative methods for the three-dimensional reconstruction of an object from projections," J. Theor. Biol. 36, 105–117 (1972).
    [CrossRef] [PubMed]
  10. M. Goitein, "Three-dimensional density reconstruction from a series of two-dimensional projections," Nucl. Instrum. Methods 101, 509 (1972).
    [CrossRef]
  11. G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273–294 (1976).
    [CrossRef] [PubMed]
  12. H. B. Buonocore, W. R. Brody, and A. Macovski, "Natural pixel decomposition for two-dimensional image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
    [CrossRef]
  13. H. B. Buonocore, W. R. Brody, and A. Macovski, "Fast minimum variance estimator for limited-angle CT image reconstruction," Med. Phys. 8, 695–702 (1981).
    [CrossRef] [PubMed]
  14. B. F. Logan and L. A. Shepp, "Optical reconstruction of a function from its projections," Duke Math. J. 42, 645–659 (1975).
    [CrossRef]
  15. A. V. Lakshminarayan and A. Lent, "The simultaneous iterative reconstruction technique as a least-squares method," Proc. Soc. Photo.-Opt. Instrum. Eng. 96, 108–116 (1976).
  16. K. T. Smith, P. L. Solomon, 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]
  17. C. Hamaker and D. C. Solmon, "The angles between the null spaces of x rays," J. Math. Anal. Appl. 62, 1–23 (1978).
    [CrossRef]
  18. M. B. Katz, "Questions of uniqueness and resolution in reconstruction from projections," in Lecture Notes in Biomathematics, S. Levin, ed. (Springer-Verlag, Berlin, 1979).
  19. A. K. Louis, "Ghosts in tomography—the null space of the Radon transform," Math. Meth. Appl. Sci. 3, 1–10 (1981).
    [CrossRef]
  20. B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Digest of the International Workshop on Physics and Engineering in Medical Imaging (Optical Society of America, Washington, D.C., 1982), pp. 188–192.
    [CrossRef]
  21. G. T. Herman and A. Lent, "A computer implementation of a Bayesian analysis of image reconstruction," Inf. Control 31, 364–384 (1976).
    [CrossRef]
  22. B. R. Hunt, "Bayesian methods in nonlinear digital image restoration," IEEE Trans. Comput. C-26, 219–229 (1977).
    [CrossRef]
  23. G. T. Herman, H. Hurwitz, A. Lent, and H. Lung, "On the Bayesian approach to image reconstruction," Inf. Control 42, 60–71 (1979).
    [CrossRef]
  24. A. P. Sage and J. L. Melsa, Estimation Theory with Applications to Communications and Control (Krieger, Melbourne, Fla., 1979), p. 175.
  25. 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, J. F. Greenleaf, and G. T. Herman, eds., Proc. IFIP, TC-4 Working Conf., Haifa, Israel, August 1978 (North-Holland, Amsterdam, 1979), pp. 219–233.
  26. S. L. Wood and M. Morf, "A fast implementation of a minimum variance estimator for computerized tomography image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 56–68 (1981).
    [CrossRef]
  27. M. H. Buonocore, "Fast minimum variance estimators for limited-angle CT image reconstruction," Tech. Rep. 81-3, Advanced Imaging Techniques Laboratory, Department of Radiology (Stanford University, Stanford, Calif., 1981).
  28. K. C. Tam and V. Perez-Mendez, "Tomographic imaging with limited-angle input," J. Opt. Soc. Am. 71, 582–592 (1981).
  29. K. C. Tam and V. Perez-Mendez, "Limits to image reconstruction from restricted angular input," IEEE Trans. Nucl. Sci. NS-28, 179–183 (1981).
    [CrossRef]
  30. F. A. Grunbaum, "A study of Fourier space methods for limited-angle image reconstruction," Numerical Funct. Anal. Optim. 2, 31–42 (1980).
    [CrossRef]
  31. 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]
  32. G. Minerbo, "MENT: a maximum entropy algorithm for reconstructing a source from projection data," Comput. Graphics Image Process. 10, 48–68 (1979).
    [CrossRef] [PubMed]
  33. T. Inouye, "Image reconstruction with limited-angle projection data," IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).
    [CrossRef]
  34. T. Inouye, "Image reconstruction with limited-view angle projections,"in Digest of the International Workshop on Physics and Engineering in Medical Imaging (Optical Society of America, Washington, D.C., 1982), pp. 165–168.
  35. H. J. Trussell and B. R. Hunt, "Improved methods of maximum a posteriori restoration," IEEE Trans. Comput. C-27, 57–62 (1979).
    [CrossRef]
  36. H. J. Trussell, "Notes on linear image restoration by maximizing the a posteriori probability," IEEE Trans. Comput. C-27, 57–62 (1978).
    [CrossRef]
  37. T. M. Cannon, H. J. Trussell, and B. R. Hunt, "Comparison of image restoration methods," Appl. Opt. 17, 3384–3390 (1978).

1982 (1)

K. M. Hanson, "CT reconstruction from limited projection angles," Proc. Soc. Photo.-Opt. Instrum. Eng. 374, 166–173 (1982).

1981 (7)

K. C. Tam and V. Perez-Mendez, "Limits to image reconstruction from restricted angular input," IEEE Trans. Nucl. Sci. NS-28, 179–183 (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]

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

H. B. Buonocore, W. R. Brody, and A. Macovski, "Natural pixel decomposition for two-dimensional image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

H. B. Buonocore, W. R. Brody, and A. Macovski, "Fast minimum variance estimator for limited-angle CT image reconstruction," Med. Phys. 8, 695–702 (1981).
[CrossRef] [PubMed]

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

S. L. Wood and M. Morf, "A fast implementation of a minimum variance estimator for computerized tomography image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 56–68 (1981).
[CrossRef]

1980 (1)

F. A. Grunbaum, "A study of Fourier space methods for limited-angle image reconstruction," Numerical Funct. Anal. Optim. 2, 31–42 (1980).
[CrossRef]

1979 (4)

G. Minerbo, "MENT: a maximum entropy algorithm for reconstructing a source from projection data," Comput. Graphics Image Process. 10, 48–68 (1979).
[CrossRef] [PubMed]

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

H. J. Trussell and B. R. Hunt, "Improved methods of maximum a posteriori restoration," IEEE Trans. Comput. C-27, 57–62 (1979).
[CrossRef]

G. T. Herman, H. Hurwitz, A. Lent, and H. Lung, "On the Bayesian approach to image reconstruction," Inf. Control 42, 60–71 (1979).
[CrossRef]

1978 (3)

C. Hamaker and D. C. Solmon, "The angles between the null spaces of x rays," J. Math. Anal. Appl. 62, 1–23 (1978).
[CrossRef]

H. J. Trussell, "Notes on linear image restoration by maximizing the a posteriori probability," IEEE Trans. Comput. C-27, 57–62 (1978).
[CrossRef]

T. M. Cannon, H. J. Trussell, and B. R. Hunt, "Comparison of image restoration methods," Appl. Opt. 17, 3384–3390 (1978).

1977 (2)

B. R. Hunt, "Bayesian methods in nonlinear digital image restoration," IEEE Trans. Comput. C-26, 219–229 (1977).
[CrossRef]

K. T. Smith, P. L. Solomon, 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)

A. V. Lakshminarayan and A. Lent, "The simultaneous iterative reconstruction technique as a least-squares method," Proc. Soc. Photo.-Opt. Instrum. Eng. 96, 108–116 (1976).

G. T. Herman and A. Lent, "A computer implementation of a Bayesian analysis of image reconstruction," Inf. Control 31, 364–384 (1976).
[CrossRef]

G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef] [PubMed]

1975 (1)

B. F. Logan and L. A. Shepp, "Optical reconstruction of a function from its projections," Duke Math. J. 42, 645–659 (1975).
[CrossRef]

1974 (2)

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

S. Twomey, "Information content in remote sensing," Appl. Opt. 13, 942–945 (1974).
[CrossRef] [PubMed]

1972 (2)

P. Gilbert, "Iterative methods for the three-dimensional reconstruction of an object from projections," J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

M. Goitein, "Three-dimensional density reconstruction from a series of two-dimensional projections," Nucl. Instrum. Methods 101, 509 (1972).
[CrossRef]

1970 (1)

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

1967 (1)

1965 (1)

S. Twomey, "The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurement," J. Franklin Inst. 279, 95–109 (1965).
[CrossRef]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Brody, W. R.

H. B. Buonocore, W. R. Brody, and A. Macovski, "Natural pixel decomposition for two-dimensional image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

H. B. Buonocore, W. R. Brody, and A. Macovski, "Fast minimum variance estimator for limited-angle CT image reconstruction," Med. Phys. 8, 695–702 (1981).
[CrossRef] [PubMed]

B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Digest of the International Workshop on Physics and Engineering in Medical Imaging (Optical Society of America, Washington, D.C., 1982), pp. 188–192.
[CrossRef]

Buonocore, H. B.

H. B. Buonocore, W. R. Brody, and A. Macovski, "Natural pixel decomposition for two-dimensional image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

H. B. Buonocore, W. R. Brody, and A. Macovski, "Fast minimum variance estimator for limited-angle CT image reconstruction," Med. Phys. 8, 695–702 (1981).
[CrossRef] [PubMed]

Buonocore, M. H.

M. H. Buonocore, "Fast minimum variance estimators for limited-angle CT image reconstruction," Tech. Rep. 81-3, Advanced Imaging Techniques Laboratory, Department of Radiology (Stanford University, Stanford, Calif., 1981).

Cannon, T. M.

Gilbert, P.

P. Gilbert, "Iterative methods for the three-dimensional reconstruction of an object from projections," J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

Goitein, M.

M. Goitein, "Three-dimensional density reconstruction from a series of two-dimensional projections," Nucl. Instrum. Methods 101, 509 (1972).
[CrossRef]

Gordon, R.

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Grunbaum, F. A.

F. A. Grunbaum, "A study of Fourier space methods for limited-angle image reconstruction," Numerical Funct. Anal. Optim. 2, 31–42 (1980).
[CrossRef]

Hamaker, C.

C. Hamaker and D. C. Solmon, "The angles between the null spaces of x rays," J. Math. Anal. Appl. 62, 1–23 (1978).
[CrossRef]

Hanson, K. M.

K. M. Hanson, "CT reconstruction from limited projection angles," Proc. Soc. Photo.-Opt. Instrum. Eng. 374, 166–173 (1982).

K. M. Hanson and G. W. Wecksung, "Bayesian approach to limited-angle CT reconstruction," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical Society of America, Washington, D.C., 1983), pp. FA6–FA14.

K. M. Hanson, "Limited-angle CT reconstruction using a priori information," in Proceedings of the First IEEE Computer Society International Symposium on Medical Imaging and Image Interpretation (Institute of Electronics and Electrical Engineers, New York, 1982), pp. 527–533.
[CrossRef]

Herman, G. T.

G. T. Herman, H. Hurwitz, A. Lent, and H. Lung, "On the Bayesian approach to image reconstruction," Inf. Control 42, 60–71 (1979).
[CrossRef]

G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef] [PubMed]

G. T. Herman and A. Lent, "A computer implementation of a Bayesian analysis of image reconstruction," Inf. Control 31, 364–384 (1976).
[CrossRef]

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Hirama, M.

Howell, H. B.

Hunt, B. R.

H. J. Trussell and B. R. Hunt, "Improved methods of maximum a posteriori restoration," IEEE Trans. Comput. C-27, 57–62 (1979).
[CrossRef]

T. M. Cannon, H. J. Trussell, and B. R. Hunt, "Comparison of image restoration methods," Appl. Opt. 17, 3384–3390 (1978).

B. R. Hunt, "Bayesian methods in nonlinear digital image restoration," IEEE Trans. Comput. C-26, 219–229 (1977).
[CrossRef]

Hurwitz, H.

G. T. Herman, H. Hurwitz, A. Lent, and H. Lung, "On the Bayesian approach to image reconstruction," Inf. Control 42, 60–71 (1979).
[CrossRef]

Ikeda, O.

Inouye, T.

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

T. Inouye, "Image reconstruction with limited-view angle projections,"in Digest of the International Workshop on Physics and Engineering in Medical Imaging (Optical Society of America, Washington, D.C., 1982), pp. 165–168.

Katz, M. B.

M. B. Katz, "Questions of uniqueness and resolution in reconstruction from projections," in Lecture Notes in Biomathematics, S. Levin, ed. (Springer-Verlag, Berlin, 1979).

Lakshminarayan, A. V.

A. V. Lakshminarayan and A. Lent, "The simultaneous iterative reconstruction technique as a least-squares method," Proc. Soc. Photo.-Opt. Instrum. Eng. 96, 108–116 (1976).

Lent, A.

G. T. Herman, H. Hurwitz, A. Lent, and H. Lung, "On the Bayesian approach to image reconstruction," Inf. Control 42, 60–71 (1979).
[CrossRef]

A. V. Lakshminarayan and A. Lent, "The simultaneous iterative reconstruction technique as a least-squares method," Proc. Soc. Photo.-Opt. Instrum. Eng. 96, 108–116 (1976).

G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef] [PubMed]

G. T. Herman and A. Lent, "A computer implementation of a Bayesian analysis of image reconstruction," Inf. Control 31, 364–384 (1976).
[CrossRef]

Linzer, M.

Logan, B. F.

B. F. Logan and L. A. Shepp, "Optical reconstruction of a function from its projections," Duke Math. J. 42, 645–659 (1975).
[CrossRef]

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

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]

Lung, H.

G. T. Herman, H. Hurwitz, A. Lent, and H. Lung, "On the Bayesian approach to image reconstruction," Inf. Control 42, 60–71 (1979).
[CrossRef]

Macovski, A.

H. B. Buonocore, W. R. Brody, and A. Macovski, "Natural pixel decomposition for two-dimensional image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

H. B. Buonocore, W. R. Brody, and A. Macovski, "Fast minimum variance estimator for limited-angle CT image reconstruction," Med. Phys. 8, 695–702 (1981).
[CrossRef] [PubMed]

B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Digest of the International Workshop on Physics and Engineering in Medical Imaging (Optical Society of America, Washington, D.C., 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, J. F. Greenleaf, and G. T. Herman, eds., Proc. IFIP, TC-4 Working Conf., Haifa, Israel, August 1978 (North-Holland, Amsterdam, 1979), pp. 219–233.

Medoff, B. P.

B. P. Medoff, W. R. Brody, and A. Macovski, "Image reconstruction from limited data," in Digest of the International Workshop on Physics and Engineering in Medical Imaging (Optical Society of America, Washington, D.C., 1982), pp. 188–192.
[CrossRef]

Melsa, J. L.

A. P. Sage and J. L. Melsa, Estimation Theory with Applications to Communications and Control (Krieger, Melbourne, Fla., 1979), p. 175.

Minerbo, G.

G. Minerbo, "MENT: a maximum entropy algorithm for reconstructing a source from projection data," Comput. Graphics Image Process. 10, 48–68 (1979).
[CrossRef] [PubMed]

Morf, M.

S. L. Wood and M. Morf, "A fast implementation of a minimum variance estimator for computerized tomography image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 56–68 (1981).
[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, J. F. Greenleaf, and G. T. Herman, eds., Proc. IFIP, TC-4 Working Conf., Haifa, Israel, August 1978 (North-Holland, Amsterdam, 1979), pp. 219–233.

Norton, S. J

Perez-Mendez, V.

K. C. Tam and V. Perez-Mendez, "Limits to image reconstruction from restricted angular input," IEEE Trans. Nucl. Sci. NS-28, 179–183 (1981).
[CrossRef]

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

Sage, A. P.

A. P. Sage and J. L. Melsa, Estimation Theory with Applications to Communications and Control (Krieger, Melbourne, Fla., 1979), p. 175.

Sato, T.

Shepp, L. A.

B. F. Logan and L. A. Shepp, "Optical reconstruction of a function from its projections," Duke Math. J. 42, 645–659 (1975).
[CrossRef]

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

Smith, K. T.

K. T. Smith, P. L. Solomon, 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.

C. Hamaker and D. C. Solmon, "The angles between the null spaces of x rays," J. Math. Anal. Appl. 62, 1–23 (1978).
[CrossRef]

Solomon, P. L.

K. T. Smith, P. L. Solomon, 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]

Tam, K. C.

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

K. C. Tam and V. Perez-Mendez, "Limits to image reconstruction from restricted angular input," IEEE Trans. Nucl. Sci. NS-28, 179–183 (1981).
[CrossRef]

Trussell, H. J.

H. J. Trussell and B. R. Hunt, "Improved methods of maximum a posteriori restoration," IEEE Trans. Comput. C-27, 57–62 (1979).
[CrossRef]

H. J. Trussell, "Notes on linear image restoration by maximizing the a posteriori probability," IEEE Trans. Comput. C-27, 57–62 (1978).
[CrossRef]

T. M. Cannon, H. J. Trussell, and B. R. Hunt, "Comparison of image restoration methods," Appl. Opt. 17, 3384–3390 (1978).

Twomey, S.

Wagner, S. L.

K. T. Smith, P. L. Solomon, 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]

Wecksung, G. W.

K. M. Hanson and G. W. Wecksung, "Bayesian approach to limited-angle CT reconstruction," in Digest of the Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Optical Society of America, Washington, D.C., 1983), pp. FA6–FA14.

Wood, S. L.

S. L. Wood and M. Morf, "A fast implementation of a minimum variance estimator for computerized tomography image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 56–68 (1981).
[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, J. F. Greenleaf, and G. T. Herman, eds., Proc. IFIP, TC-4 Working Conf., Haifa, Israel, August 1978 (North-Holland, Amsterdam, 1979), pp. 219–233.

Appl. Opt. (4)

Bull. Am. Math. Soc. (1)

K. T. Smith, P. L. Solomon, 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. Biol. Med. (1)

G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef] [PubMed]

Comput. Graphics Image Process. (1)

G. Minerbo, "MENT: a maximum entropy algorithm for reconstructing a source from projection data," Comput. Graphics Image Process. 10, 48–68 (1979).
[CrossRef] [PubMed]

Duke Math. J. (1)

B. F. Logan and L. A. Shepp, "Optical reconstruction of a function from its projections," Duke Math. J. 42, 645–659 (1975).
[CrossRef]

IEEE Trans. Biomed. Eng. (2)

H. B. Buonocore, W. R. Brody, and A. Macovski, "Natural pixel decomposition for two-dimensional image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

S. L. Wood and M. Morf, "A fast implementation of a minimum variance estimator for computerized tomography image reconstruction," IEEE Trans. Biomed. Eng. BME-28, 56–68 (1981).
[CrossRef]

IEEE Trans. Comput. (3)

H. J. Trussell and B. R. Hunt, "Improved methods of maximum a posteriori restoration," IEEE Trans. Comput. C-27, 57–62 (1979).
[CrossRef]

H. J. Trussell, "Notes on linear image restoration by maximizing the a posteriori probability," IEEE Trans. Comput. C-27, 57–62 (1978).
[CrossRef]

B. R. Hunt, "Bayesian methods in nonlinear digital image restoration," IEEE Trans. Comput. C-26, 219–229 (1977).
[CrossRef]

IEEE Trans. Nucl. Sci. (3)

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

K. C. Tam and V. Perez-Mendez, "Limits to image reconstruction from restricted angular input," IEEE Trans. Nucl. Sci. NS-28, 179–183 (1981).
[CrossRef]

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

Inf. Control (2)

G. T. Herman and A. Lent, "A computer implementation of a Bayesian analysis of image reconstruction," Inf. Control 31, 364–384 (1976).
[CrossRef]

G. T. Herman, H. Hurwitz, A. Lent, and H. Lung, "On the Bayesian approach to image reconstruction," Inf. Control 42, 60–71 (1979).
[CrossRef]

J. Franklin Inst. (1)

S. Twomey, "The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurement," J. Franklin Inst. 279, 95–109 (1965).
[CrossRef]

J. Math. Anal. Appl. (1)

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Figures (7)

Fig. 1
Fig. 1

(a) Source distribution used for the first example. (b) ART reconstruction and (c) MENT reconstruction obtained using 11 views covering 90° in projection angle. Unconstrained ART was used, whereas MENT has an implicit nonnegativity constraint.

Fig. 2
Fig. 2

Reconstructions using the a priori information that the unknown source function is a fuzzy annulus with known radius and width. (b) MAP reconstruction was obtained using a flat annulus for f ¯ and the variance image (a) as the diagonal entries of Rf (nondiagonal entries assumed to be zero). (c) The FAIR result was based on a model of the image consisting of 18 Gaussian functions distributed around the circle. The use of a priori knowledge significantly the artifacts present in the deterministic reconstructioning in Fig. 1.

Fig. 3
Fig. 3

The angular dependence of the maximum values along various radii for the ART, MAP, and FAIR reconstructions in Figs. 1 and 2 compared with that for the original function, Fig. 1(a), quantitatively demonstrates the improvement afforded by MAP and FAIR.

Fig. 4
Fig. 4

Reconstructions of (a) source distribution that does not conform to the annular assumption obtained from 11 views subtending 90° using (b) ART, (c) MAP, and (d) FAIR algorithms. Both MAP and FAIR tend to move the added source outside the annulus onto the annulus. However, they provide indications in the reconstructions that there is some exterior activity.

Fig. 5
Fig. 5

Angular dependence of the maximum values in the MAP and the FAIR reconstructions of Fig. 4.

Fig. 6
Fig. 6

Reconstructions of the source in Fig. 1(a) from 11 noisy projections using (a) ART, (b) MAP, and (c) FAIR algorithms show that the latter two algorithms are tolerant of noise.

Fig. 7
Fig. 7

Reconstruction from the same data as used in Fig. 6 obtained by employing MAP as the second step in the FAIR procedure. This global Bayesian approach yields the best estimate of the original function and provides flexibility in the use of a priori information.

Equations (9)

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

g i = h i ( x , y ) f ( x , y ) d x d y ,
f ^ ( x , y ) = i = 1 N a i h i ( x , y ) .
P ( f g ) = P ( g f ) P ( f ) P ( g ) ,
R f - 1 ( f ¯ - f ) + H T R n - 1 ( g - H f ) = 0 ,
f o = f ¯ ,
f n + 1 = f n + c n r n ,
r n = f ¯ - f n + R f H T R n - 1 ( g - H f n ) ,
c n = r n T s n s n T s n ,
s n = ( I + R f H T R n - 1 H ) r n ,