Abstract

Noncoherent optical-imaging systems are identified as potential applications for the maximum-likelihood image-restoration methods that are currently being studied for various modalities of nuclear-medicine imaging. An analogy between the quantum-photon measurements of such an optical system and that of a gamma camera allow for this new application. Results of a computer simulation are presented that support its feasibility. One important property revealed by this simulation is that the maximum-likelihood method demonstrates the ability to extrapolate the Fourier spectrum of a band-limited signal. This ability can be partially understood in that this algorithm, similar to some of the other spectral-extrapolation algorithms, constrains the solution to nonnegative values. This observation has implications on the potential of superresolution, the restoration of images from a defocused optical system, and three-dimensional imaging with a microscope.

© 1988 Optical Society of America

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References

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  1. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. B 39, 1–37 (1977).
  2. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
    [CrossRef]
  3. K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomogr. 8, 302–316 (1984).
  4. L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
    [CrossRef]
  5. E. Tanaka, “A fast reconstruction algorithm for stationary positron emission tomography based on a modified EM algorithm,” IEEE Trans. Med. Imaging MI-6, 98–105 (1987).
    [CrossRef]
  6. D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
    [CrossRef]
  7. M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 769–778 (1985).
    [CrossRef]
  8. J. D. Bronzino, Biomedical Engineering and Instrumentation (PWS, Boston, Mass., 1986).
  9. A. Macovski, Medical Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1983).
  10. D. L. Snyder, Washington University, St. Louis, Missouri 63130 (personal communication).
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    [CrossRef] [PubMed]
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  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  17. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979).
  18. D. L. Snyder, Random Point Process (Wiley, New York, 1978).
  19. P. Bremaud, Point Processes and Queues—Martingale Dynamics (Springer-Verlag, New York, 1980).
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    [CrossRef] [PubMed]
  21. Y. Zou, C. K. Rushforth, “Least-squares reconstruction of spatially limited objects using smoothness and non-negativity constraints,” Appl. Opt. 21, 1249–1252 (1982).
    [CrossRef]
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    [CrossRef]
  23. S. J. Howard, “Fast algorithm for implementing the minimum-negativity constraint for Fourier spectrum extrapolation,” Appl. Opt. 25, 1670–1675 (1986).
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    [CrossRef]
  26. M. I. Sezan, H. Stark, “Image restoration by convex projections in the presence of noise,” Appl. Opt. 22, 2781–2789 (1983).
    [CrossRef] [PubMed]
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  30. D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
    [CrossRef] [PubMed]
  31. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  32. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
  33. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

1987 (2)

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

E. Tanaka, “A fast reconstruction algorithm for stationary positron emission tomography based on a modified EM algorithm,” IEEE Trans. Med. Imaging MI-6, 98–105 (1987).
[CrossRef]

1986 (2)

1985 (5)

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[CrossRef]

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 769–778 (1985).
[CrossRef]

D. J. Arndt-Jovin, M. Robert-Nicoud, S. J. Kaufman, T. M. Jovin, “Fluorescence digital imaging microscopy in cell biology,” Science 230, 247–256 (1985).
[CrossRef] [PubMed]

Y. Tsuchiya, E. Inuzuka, T. Kurono, M. Hosoda, “Photon-counting image acquisition system and its applications,” J. Imag. Technol. 11, 1084–1088 (1985).

J. Maeda, “Restoration of bandlimited images by an iterative damped least-squares method with adaptive regularization,” Appl. Opt. 24, 1421–1425 (1985).
[CrossRef] [PubMed]

1984 (2)

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomogr. 8, 302–316 (1984).

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef] [PubMed]

1983 (3)

M. I. Sezan, H. Stark, “Image restoration by convex projections in the presence of noise,” Appl. Opt. 22, 2781–2789 (1983).
[CrossRef] [PubMed]

D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

R. J. Mammone, “Spectral extrapolation of constrained signals,” J. Opt. Soc. Am. 73, 1476–1480 (1983).
[CrossRef]

1982 (3)

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. B 39, 1–37 (1977).

1974 (1)

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

1969 (1)

1968 (1)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

1909 (1)

G. I. Taylor, “Interference fringes with feeble light,” Proc. Cambridge Philos. Soc. 15, 114–115 (1909).

Agard, D. A.

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef] [PubMed]

Arndt-Jovin, D. J.

D. J. Arndt-Jovin, M. Robert-Nicoud, S. J. Kaufman, T. M. Jovin, “Fluorescence digital imaging microscopy in cell biology,” Science 230, 247–256 (1985).
[CrossRef] [PubMed]

Bremaud, P.

P. Bremaud, Point Processes and Queues—Martingale Dynamics (Springer-Verlag, New York, 1980).

Bronzino, J. D.

J. D. Bronzino, Biomedical Engineering and Instrumentation (PWS, Boston, Mass., 1986).

Carson, R.

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomogr. 8, 302–316 (1984).

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979).

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. B 39, 1–37 (1977).

Gerchberg, R. W.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Hosoda, M.

Y. Tsuchiya, E. Inuzuka, T. Kurono, M. Hosoda, “Photon-counting image acquisition system and its applications,” J. Imag. Technol. 11, 1084–1088 (1985).

Howard, S. J.

Inuzuka, E.

Y. Tsuchiya, E. Inuzuka, T. Kurono, M. Hosoda, “Photon-counting image acquisition system and its applications,” J. Imag. Technol. 11, 1084–1088 (1985).

Jovin, T. M.

D. J. Arndt-Jovin, M. Robert-Nicoud, S. J. Kaufman, T. M. Jovin, “Fluorescence digital imaging microscopy in cell biology,” Science 230, 247–256 (1985).
[CrossRef] [PubMed]

Kaufman, L.

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

Kaufman, S. J.

D. J. Arndt-Jovin, M. Robert-Nicoud, S. J. Kaufman, T. M. Jovin, “Fluorescence digital imaging microscopy in cell biology,” Science 230, 247–256 (1985).
[CrossRef] [PubMed]

Kurono, T.

Y. Tsuchiya, E. Inuzuka, T. Kurono, M. Hosoda, “Photon-counting image acquisition system and its applications,” J. Imag. Technol. 11, 1084–1088 (1985).

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. B 39, 1–37 (1977).

Lange, K.

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomogr. 8, 302–316 (1984).

Louden, R.

R. Louden, The Quantum Theory of Light (Clarendon, Oxford, 1983).

Macovski, A.

A. Macovski, Medical Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Maeda, J.

Mammone, R. J.

Mandel, L.

Meinel, E. S.

Miller, M. I.

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 769–778 (1985).
[CrossRef]

Miller, T. R.

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 769–778 (1985).
[CrossRef]

Murata, K.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

Pfleegor, R. L.

Politte, D. G.

D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Robert-Nicoud, M.

D. J. Arndt-Jovin, M. Robert-Nicoud, S. J. Kaufman, T. M. Jovin, “Fluorescence digital imaging microscopy in cell biology,” Science 230, 247–256 (1985).
[CrossRef] [PubMed]

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. B 39, 1–37 (1977).

Rushforth, C. K.

Sezan, M. I.

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Snyder, D.

D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

Snyder, D. L.

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 769–778 (1985).
[CrossRef]

D. L. Snyder, Washington University, St. Louis, Missouri 63130 (personal communication).

D. L. Snyder, Random Point Process (Wiley, New York, 1978).

Stark, H.

Stokseth, P. A.

Streibl, N.

Tanaka, E.

E. Tanaka, “A fast reconstruction algorithm for stationary positron emission tomography based on a modified EM algorithm,” IEEE Trans. Med. Imaging MI-6, 98–105 (1987).
[CrossRef]

Taylor, G. I.

G. I. Taylor, “Interference fringes with feeble light,” Proc. Cambridge Philos. Soc. 15, 114–115 (1909).

Tsuchiya, Y.

Y. Tsuchiya, E. Inuzuka, T. Kurono, M. Hosoda, “Photon-counting image acquisition system and its applications,” J. Imag. Technol. 11, 1084–1088 (1985).

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Zou, Y.

Annu. Rev. Biophys. Bioeng. (1)

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef] [PubMed]

Appl. Opt. (5)

IEEE Trans. Med. Imaging (3)

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

E. Tanaka, “A fast reconstruction algorithm for stationary positron emission tomography based on a modified EM algorithm,” IEEE Trans. Med. Imaging MI-6, 98–105 (1987).
[CrossRef]

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

IEEE Trans. Nucl. Sci. (2)

D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 769–778 (1985).
[CrossRef]

J. Comput. Assisted Tomogr. (1)

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomogr. 8, 302–316 (1984).

J. Imag. Technol. (1)

Y. Tsuchiya, E. Inuzuka, T. Kurono, M. Hosoda, “Photon-counting image acquisition system and its applications,” J. Imag. Technol. 11, 1084–1088 (1985).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. R. Statist. Soc. B (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. B 39, 1–37 (1977).

Opt. Acta (1)

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

Proc. Cambridge Philos. Soc. (1)

G. I. Taylor, “Interference fringes with feeble light,” Proc. Cambridge Philos. Soc. 15, 114–115 (1909).

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Science (1)

D. J. Arndt-Jovin, M. Robert-Nicoud, S. J. Kaufman, T. M. Jovin, “Fluorescence digital imaging microscopy in cell biology,” Science 230, 247–256 (1985).
[CrossRef] [PubMed]

Other (10)

R. Louden, The Quantum Theory of Light (Clarendon, Oxford, 1983).

J. D. Bronzino, Biomedical Engineering and Instrumentation (PWS, Boston, Mass., 1986).

A. Macovski, Medical Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1983).

D. L. Snyder, Washington University, St. Louis, Missouri 63130 (personal communication).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979).

D. L. Snyder, Random Point Process (Wiley, New York, 1978).

P. Bremaud, Point Processes and Queues—Martingale Dynamics (Springer-Verlag, New York, 1980).

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

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Figures (11)

Fig. 1
Fig. 1

Illustration showing the analogy between (a) a conventional gamma camera and (b) a noncoherent optical-imaging system.

Fig. 2
Fig. 2

(a) The simulated object to be restored and (b) its dimensions. The dark areas indicate a relative intensity of 0, and the bright areas indicate a relative intensity of 1.0. In Figs. 29 the aspect ratio of the images is about 1.3, such that the rectangular area designated in Fig. 2(b) actually represents a square.

Fig. 3
Fig. 3

The real part of the Fourier transform, calculated by the FFT, of the object shown in Fig. 1. The range of intensities represents numbers from −158.1 to 426.0.

Fig. 4
Fig. 4

The OTF for a numerical aperture of 1.25. The range of intensities represents numbers from 0 to 1.0. The constant dark areas indicate a value of 0.

Fig. 5
Fig. 5

The real part of the Fourier transform of the observed image. The range of intensities represents numbers from −84.3 to 426. The constant gray areas outside of the central region indicate a value of 0.

Fig. 6
Fig. 6

The simulated observed image showing the classical diffraction limitation in resolution.

Fig. 7
Fig. 7

Restored images at (a) 160, (b) 1600, (c) 5900, and (d) 10,000 iterations. The intensity ranges in these images represent numbers (a) from 1.9 × 10−9 to 1.27, (b) from 3.4 × 10−30 to 1.70, (c) from 1.4 × 10−64 to 1.73, and (d) from 0.0 to 1.78.

Fig. 8
Fig. 8

Real part of the Fourier transform of Fig. 7(c).

Fig. 9
Fig. 9

Preliminary results with simulated quantum-photon noise. Diffraction-limited images at SNR’s of (a) 10, (c) 40, and (e) 160. (b), (d), (f) restored images at 2000 iterations for the SNR’s of (a), (c), and (e), respectively.

Fig. 10
Fig. 10

Error energies of the space-domain images.

Fig. 11
Fig. 11

Error energies of the Fourier transforms.

Tables (1)

Tables Icon

Table 1 Error Energies for a, the Space-Domain Images; b, the Fourier Transform of These Images; c, the Portions of This Transform Lying Outside the Passband of the OTF

Equations (16)

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

λ ̂ ( k + 1 ) ( x ) = R 2 p ̂ x ( k ) ( x u ) N ( d u )
p ̂ x ( k ) ( x u ) = p ( x u ) λ ̂ ( k ) ( x ) [ r 2 p ( z u ) λ ̂ ( k ) ( z ) d z ] 1 ,
[ u 1 , u 1 + d u 1 ) × [ u 2 , u 2 , d u 2 ) ,
u = ( u 1 , u 2 ) .
SNR = 1 N all i m i 2 σ i 2 ,
SNR = 1 N all i m i ,
64 × 64 × 10 = 40,960.
40,960 / ( 128 × 128 ) = 2.5 ,
E = A [ g ( y ) ĝ ( y ) ] 2 d y ,
λ ( x ) = 0 if | x 1 | > x max or | x 2 | > x max ,
B = { u | u 1 | < u max , | u 2 | < u max } ,
u = ( u 1 , u 2 ) ,
Λ finite ( u ) | u 1 | u max | u 2 | u max = Λ ( u ) | u 1 | u max | u 2 | u max
Λ finite ( u ) | u 1 | u max | u 2 | u max = M ( u ) / OTF ( u ) | u 1 | u max | u 2 | u max ,
u max = u cut / 2 ,
OTF ( u ) = 0 if | u | > u cut .

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