Abstract

A new restoring algorithm, maximum bounded entropy (MBE), has been investigated. It incorporates prior knowledge of both a lower and upper bound in the unknown object. Its outputs are maximum probable estimates of the object under the following conditions: (a) the photons forming the image behave as classical particles; (b) the object is assumed to be biased toward a flat gray scene in the absence of image data; (c) the object is modeled as consisting of high-gradient foreground details riding on top of a smoothly varying background that is not to be restored but rather must be estimated in a separate step; and (d) the image noise is Poisson. The resulting MBE estimator obeys the sum of maximum entropy for the occupied photon sites in the object and maximum entropy for the unoccupied sites. The result is an estimate of the object that obeys an analytic form that functionally cannot take on values outside the known bounds. The algorithm was applied to the problem of reconstructing rod cross sections due to tomographic viewing. This problem is ideal because the object consists only of upper- and lower-bound values. We found that only four projections are needed to provide a good reconstruction and that twenty projections allow for the resolution of a single pixel wide crack in one of the rods.

© 1985 Optical Society of America

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  1. A. C. Schell, “Enhancing the Angular Resolution of Incoherent Sources,” Radio Electron. Eng. 29, 21 (1965).
    [CrossRef]
  2. P. A. Jansson, R. H. Hunt, E. K. Plyler, “Spectral Resolution Enhancement,” J. Opt. Soc. Am. 58, 1665 (1968).
    [CrossRef]
  3. Y. Biraud, “A New Approach for Increasing the Resolving Power by Data Processing,” Astron. Astrophys. 1, 124 (1969).
  4. B. R. Frieden, “Restoring with Maximum Entropy,” J. Opt. Soc. Am. 62, 511 (1972).
    [CrossRef] [PubMed]
  5. W. H. Richardson, “Iterative Method of Image Restoration,” J. Opt. Soc. Am. 62, 55 (1972).
    [CrossRef]
  6. L. B. Lucy, “An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745 (1974).
    [CrossRef]
  7. S. J. Wernecke, L. R. D'Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351 (1977).
    [CrossRef]
  8. S. F. Gull, G. J. Daniell, “Image Reconstruction from Incomplete and Noisy Data,” Nature London 272, 686 (1978).
    [CrossRef]
  9. B. R. Frieden, “Image Restoration Using a Norm of Maximum Information,” Proc. Soc. Photo-Opt. Instrum. Eng. 207, 14 (1979).
  10. A. R. Davies, T. Cochrane, O. M. Al-Faour, “The Numerical Inversion of Truncated Autocorrelation Functions,” Opt. Acta 27, 107 (1980).
    [CrossRef]
  11. R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. 29, 471 (1970).
    [CrossRef] [PubMed]
  12. G. Minerbo, “MENT: A Maximum Entropy Algorithm for Reconstructing a Source from Projection Data,” Comput. Graphics Image Process. 10, 48 (1979).
    [CrossRef]
  13. A. Lent, “A Convergent Algorithm for Maximum Entropy Image Restoration with a Medical X-ray Application,” in SPSE Conference Proceedings, R. Shaw, Ed. (SPSE, Washington, D.C., 1976), pp. 249–257.
  14. G. T. Herman, A. Lent, S. W. Roland, “ART: Mathematics and Applications,” J. Theor. Biol. 42, 1 (1973).
    [CrossRef] [PubMed]
  15. C. F. Barton, “Computerized Axial Tomography for Neutron Radiography of Nuclear Fuel,” Trans. Am. Nucl. Soc. 27, 212 (1977).
  16. B. R. Frieden, “Estimation—A New Role for Maximum Entropy,” in SPSE Conference Proceedings, R. Shaw, Ed. (SPSE, Washington, D.C., 1976), pp. 261–265.
  17. P. A. Jansson, R. H. Hunt, E. K. Plyler, “Spectral Resolution Enhancement,” J. Opt. Soc. Am. 60, 596 (1970).
    [CrossRef]
  18. B. R. Frieden, “Statistical Estimates of Bounded Optical Scenes by the Method of Prior Probabilities',” IEEE Trans. Inf. Theory IT-19, 118 (1973).
    [CrossRef]
  19. R. Kikuchi, B. H. Soffer, “Maximum Entropy Image Restoration. I. The Entropy Expression,” J. Opt. Soc. Am. 67, 1656 (1977).
    [CrossRef]
  20. B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer, New York, 1983).
    [CrossRef]
  21. B. R. Frieden, D. C. Wells, “Restoring with Maximum Entropy. III. Poisson Sources and Backgrounds,” J. Opt. Soc. Am. 68, 93 (1978).
    [CrossRef]
  22. B. R. Frieden, “New Restoring Algorithm for the Preferential Enhancement of Edge Gradients,” J. Opt. Soc. Am. 66, 280 (1976).
    [CrossRef]
  23. N. C. Gallagher, G. L. Wise, “Passband and Stopband Properties of Median Filters,” in Proceedings, 1980 Conference on Information Sciences and Systems (Princeton University, New Jersey, 1980), pp. 303–307.
  24. B. R. Frieden, “Some Statistical Properties of the Median Window,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 219 (1981).
  25. H. H. Barrett, W. Swindell, Radiological Imaging (Academic, New York, 1981).
  26. We thank B. H. Soffer of Hughes Research Laboratories for the basic idea behind this proof.

1981 (1)

B. R. Frieden, “Some Statistical Properties of the Median Window,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 219 (1981).

1980 (1)

A. R. Davies, T. Cochrane, O. M. Al-Faour, “The Numerical Inversion of Truncated Autocorrelation Functions,” Opt. Acta 27, 107 (1980).
[CrossRef]

1979 (2)

G. Minerbo, “MENT: A Maximum Entropy Algorithm for Reconstructing a Source from Projection Data,” Comput. Graphics Image Process. 10, 48 (1979).
[CrossRef]

B. R. Frieden, “Image Restoration Using a Norm of Maximum Information,” Proc. Soc. Photo-Opt. Instrum. Eng. 207, 14 (1979).

1978 (2)

S. F. Gull, G. J. Daniell, “Image Reconstruction from Incomplete and Noisy Data,” Nature London 272, 686 (1978).
[CrossRef]

B. R. Frieden, D. C. Wells, “Restoring with Maximum Entropy. III. Poisson Sources and Backgrounds,” J. Opt. Soc. Am. 68, 93 (1978).
[CrossRef]

1977 (3)

R. Kikuchi, B. H. Soffer, “Maximum Entropy Image Restoration. I. The Entropy Expression,” J. Opt. Soc. Am. 67, 1656 (1977).
[CrossRef]

S. J. Wernecke, L. R. D'Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351 (1977).
[CrossRef]

C. F. Barton, “Computerized Axial Tomography for Neutron Radiography of Nuclear Fuel,” Trans. Am. Nucl. Soc. 27, 212 (1977).

1976 (1)

1974 (1)

L. B. Lucy, “An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745 (1974).
[CrossRef]

1973 (2)

G. T. Herman, A. Lent, S. W. Roland, “ART: Mathematics and Applications,” J. Theor. Biol. 42, 1 (1973).
[CrossRef] [PubMed]

B. R. Frieden, “Statistical Estimates of Bounded Optical Scenes by the Method of Prior Probabilities',” IEEE Trans. Inf. Theory IT-19, 118 (1973).
[CrossRef]

1972 (2)

1970 (2)

P. A. Jansson, R. H. Hunt, E. K. Plyler, “Spectral Resolution Enhancement,” J. Opt. Soc. Am. 60, 596 (1970).
[CrossRef]

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

1969 (1)

Y. Biraud, “A New Approach for Increasing the Resolving Power by Data Processing,” Astron. Astrophys. 1, 124 (1969).

1968 (1)

1965 (1)

A. C. Schell, “Enhancing the Angular Resolution of Incoherent Sources,” Radio Electron. Eng. 29, 21 (1965).
[CrossRef]

Al-Faour, O. M.

A. R. Davies, T. Cochrane, O. M. Al-Faour, “The Numerical Inversion of Truncated Autocorrelation Functions,” Opt. Acta 27, 107 (1980).
[CrossRef]

Barrett, H. H.

H. H. Barrett, W. Swindell, Radiological Imaging (Academic, New York, 1981).

Barton, C. F.

C. F. Barton, “Computerized Axial Tomography for Neutron Radiography of Nuclear Fuel,” Trans. Am. Nucl. Soc. 27, 212 (1977).

Bender, R.

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Biraud, Y.

Y. Biraud, “A New Approach for Increasing the Resolving Power by Data Processing,” Astron. Astrophys. 1, 124 (1969).

Cochrane, T.

A. R. Davies, T. Cochrane, O. M. Al-Faour, “The Numerical Inversion of Truncated Autocorrelation Functions,” Opt. Acta 27, 107 (1980).
[CrossRef]

D'Addario, L. R.

S. J. Wernecke, L. R. D'Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351 (1977).
[CrossRef]

Daniell, G. J.

S. F. Gull, G. J. Daniell, “Image Reconstruction from Incomplete and Noisy Data,” Nature London 272, 686 (1978).
[CrossRef]

Davies, A. R.

A. R. Davies, T. Cochrane, O. M. Al-Faour, “The Numerical Inversion of Truncated Autocorrelation Functions,” Opt. Acta 27, 107 (1980).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Some Statistical Properties of the Median Window,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 219 (1981).

B. R. Frieden, “Image Restoration Using a Norm of Maximum Information,” Proc. Soc. Photo-Opt. Instrum. Eng. 207, 14 (1979).

B. R. Frieden, D. C. Wells, “Restoring with Maximum Entropy. III. Poisson Sources and Backgrounds,” J. Opt. Soc. Am. 68, 93 (1978).
[CrossRef]

B. R. Frieden, “New Restoring Algorithm for the Preferential Enhancement of Edge Gradients,” J. Opt. Soc. Am. 66, 280 (1976).
[CrossRef]

B. R. Frieden, “Statistical Estimates of Bounded Optical Scenes by the Method of Prior Probabilities',” IEEE Trans. Inf. Theory IT-19, 118 (1973).
[CrossRef]

B. R. Frieden, “Restoring with Maximum Entropy,” J. Opt. Soc. Am. 62, 511 (1972).
[CrossRef] [PubMed]

B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer, New York, 1983).
[CrossRef]

B. R. Frieden, “Estimation—A New Role for Maximum Entropy,” in SPSE Conference Proceedings, R. Shaw, Ed. (SPSE, Washington, D.C., 1976), pp. 261–265.

Gallagher, N. C.

N. C. Gallagher, G. L. Wise, “Passband and Stopband Properties of Median Filters,” in Proceedings, 1980 Conference on Information Sciences and Systems (Princeton University, New Jersey, 1980), pp. 303–307.

Gordon, R.

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Gull, S. F.

S. F. Gull, G. J. Daniell, “Image Reconstruction from Incomplete and Noisy Data,” Nature London 272, 686 (1978).
[CrossRef]

Herman, G. T.

G. T. Herman, A. Lent, S. W. Roland, “ART: Mathematics and Applications,” J. Theor. Biol. 42, 1 (1973).
[CrossRef] [PubMed]

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Hunt, R. H.

Jansson, P. A.

Kikuchi, R.

Lent, A.

G. T. Herman, A. Lent, S. W. Roland, “ART: Mathematics and Applications,” J. Theor. Biol. 42, 1 (1973).
[CrossRef] [PubMed]

A. Lent, “A Convergent Algorithm for Maximum Entropy Image Restoration with a Medical X-ray Application,” in SPSE Conference Proceedings, R. Shaw, Ed. (SPSE, Washington, D.C., 1976), pp. 249–257.

Lucy, L. B.

L. B. Lucy, “An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745 (1974).
[CrossRef]

Minerbo, G.

G. Minerbo, “MENT: A Maximum Entropy Algorithm for Reconstructing a Source from Projection Data,” Comput. Graphics Image Process. 10, 48 (1979).
[CrossRef]

Plyler, E. K.

Richardson, W. H.

Roland, S. W.

G. T. Herman, A. Lent, S. W. Roland, “ART: Mathematics and Applications,” J. Theor. Biol. 42, 1 (1973).
[CrossRef] [PubMed]

Schell, A. C.

A. C. Schell, “Enhancing the Angular Resolution of Incoherent Sources,” Radio Electron. Eng. 29, 21 (1965).
[CrossRef]

Soffer, B. H.

Swindell, W.

H. H. Barrett, W. Swindell, Radiological Imaging (Academic, New York, 1981).

Wells, D. C.

Wernecke, S. J.

S. J. Wernecke, L. R. D'Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351 (1977).
[CrossRef]

Wise, G. L.

N. C. Gallagher, G. L. Wise, “Passband and Stopband Properties of Median Filters,” in Proceedings, 1980 Conference on Information Sciences and Systems (Princeton University, New Jersey, 1980), pp. 303–307.

Astron. Astrophys. (1)

Y. Biraud, “A New Approach for Increasing the Resolving Power by Data Processing,” Astron. Astrophys. 1, 124 (1969).

Astron. J. (1)

L. B. Lucy, “An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745 (1974).
[CrossRef]

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 (1979).
[CrossRef]

IEEE Trans. Comput. (1)

S. J. Wernecke, L. R. D'Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351 (1977).
[CrossRef]

IEEE Trans. Inf. Theory (1)

B. R. Frieden, “Statistical Estimates of Bounded Optical Scenes by the Method of Prior Probabilities',” IEEE Trans. Inf. Theory IT-19, 118 (1973).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Theor. Biol. (2)

G. T. Herman, A. Lent, S. W. Roland, “ART: Mathematics and Applications,” J. Theor. Biol. 42, 1 (1973).
[CrossRef] [PubMed]

R. Gordon, R. Bender, G. T. Herman, “Algebraic Reconstruction Techniques for Three-Dimensional Electron Microscopy and X-ray Photography,” J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Nature London (1)

S. F. Gull, G. J. Daniell, “Image Reconstruction from Incomplete and Noisy Data,” Nature London 272, 686 (1978).
[CrossRef]

Opt. Acta (1)

A. R. Davies, T. Cochrane, O. M. Al-Faour, “The Numerical Inversion of Truncated Autocorrelation Functions,” Opt. Acta 27, 107 (1980).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

B. R. Frieden, “Image Restoration Using a Norm of Maximum Information,” Proc. Soc. Photo-Opt. Instrum. Eng. 207, 14 (1979).

B. R. Frieden, “Some Statistical Properties of the Median Window,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 219 (1981).

Radio Electron. Eng. (1)

A. C. Schell, “Enhancing the Angular Resolution of Incoherent Sources,” Radio Electron. Eng. 29, 21 (1965).
[CrossRef]

Trans. Am. Nucl. Soc. (1)

C. F. Barton, “Computerized Axial Tomography for Neutron Radiography of Nuclear Fuel,” Trans. Am. Nucl. Soc. 27, 212 (1977).

Other (6)

B. R. Frieden, “Estimation—A New Role for Maximum Entropy,” in SPSE Conference Proceedings, R. Shaw, Ed. (SPSE, Washington, D.C., 1976), pp. 261–265.

H. H. Barrett, W. Swindell, Radiological Imaging (Academic, New York, 1981).

We thank B. H. Soffer of Hughes Research Laboratories for the basic idea behind this proof.

B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer, New York, 1983).
[CrossRef]

N. C. Gallagher, G. L. Wise, “Passband and Stopband Properties of Median Filters,” in Proceedings, 1980 Conference on Information Sciences and Systems (Princeton University, New Jersey, 1980), pp. 303–307.

A. Lent, “A Convergent Algorithm for Maximum Entropy Image Restoration with a Medical X-ray Application,” in SPSE Conference Proceedings, R. Shaw, Ed. (SPSE, Washington, D.C., 1976), pp. 249–257.

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Figures (3)

Fig. 1
Fig. 1

(a) Point spread function (logarithm of intensities); (b) object (logarithm of intensities); (c) Poisson image, SNR = 10; (d) image filtered into Gaussian form, σ = 2; (e) estimated background B; (f) MBE reconstruction using ρ = 200.

Fig. 2
Fig. 2

(a) Point spread function (logarithm of intensities); (b) object (logarithm of intensities); (c) Poisson image, SNR = 10; (d) image filtered into Gaussian form, σ = 2; (e) MBE reconstruction using ρ = 200; (f) ME reconstruction using ρ = 200; (g) MBE reconstruction using ρ = 300; (h) MBE reconstruction using ρ = 400.

Fig. 3
Fig. 3

(a) Point spread function (logarithm of intensities); (b) object (logarithm of intensities); (c) Poisson image, SNR = 10; (d) image filtered into Gaussian form, σ = 1.5; (e) MBE reconstruction using ρ = 200; (f) MBE reconstruction using ρ = 400; (g) image (c) filtered into Gaussian form, σ = 2; (h) MBE reconstruction using ρ = 400.

Equations (21)

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o n = m n Δ o .
0 o n b , n = 1 , , M .
o n = b 2 = constant , n = 1 , , M ,
P 1 ( o ) = n = 1 M ( b / Δ o ) ! m n ! ( b / Δ o m n ) ! p n m n q n b / Δ o m n , m n = o n Δ o , p n = o n b , q n = 1 p n .
P 1 ( o ) = ( 1 / 2 ) M b / Δ o n = 1 M ( b / Δ o ) ! m n ! ( b / Δ o m n ) ! = const × n = 1 M 1 ( o n / Δ o ) ! ( b / Δ o o n / Δ o ) ! ,
i m = n = 1 M o n s ( s m x n ) + B m + n m .
P ( o , n ) = maximum ,
P ( o , n ) = P 1 ( o ) P 2 ( n | o ) .
P 2 ( i | o ) = m = 1 M a m i m / Δ i exp ( a m ) ( i m / Δ i ) ! P ( n | o ) ,
a m i Δ i = Δ i 1 [ n = 1 M o n s ( x m x n ) + B m ] .
ln P 1 ( o ) + ln P ( n | o ) m = 1 M λ m ( i m i m data ) μ ( Σ o n E ) = maximum
n ( o n / Δ o ) ln ( o n / Δ o ) n ( b / Δ o o n / Δ o ) ln ( b / Δ o o n / δ o ) + ln P ( n | o ) m λ m ( i m i m data ) μ ( Σ o n E ) = maximum .
i m data = [ n = 1 M o n s ( x m x n ) + B m ] exp ( 1 Λ M / ρ ) ,
E = n = 1 M o n ,
o n = b 1 + exp [ Γ + m Λ m s ( x m x n ) ] ,
Γ μ Δ o , Λ m λ m Δ o , ρ Δ o / Δ i .
E n = m + ( b / Δ o m ) + ( E n b / Δ o ) .
P ( E n ) = P z ( m ) P z ( b / Δ o m ) P z ( E n b / Δ o ) ,
P z ( k ) = ( k + z 1 ) ! k ! ( z 1 ) ! p k .
P z ( k ) = p k k ! .
P ( E n ) p n m m ! q n b / Δ o m ( b / Δ o m ) ! .

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