Abstract

The problem posed in this paper is that of restoring a Poisson-point-process intensity that has been degraded by a band-limiting filter followed by a truncation of the signal. The approach is to derive a maximum-likelihood estimate from the count data of the degraded point process. The expectation-maximization algorithm is used to realize this estimate, while the derivation of this algorithm is an extension to previous developments by Shepp and Vardi [ IEEE Trans. Med. Imaging MI-2, 113 ( 1982)], Snyder et al. [ IEEE Trans. Nucl. Sci. NS-28, 3575 ( 1981)], and others used for positron-emission tomography. We also extend our own work reported earlier by considering the truncated signal, which is analogous to practical situations in both two- and three-dimensional microscopy in which the image of the specimen has been truncated. Computer simulations with one-dimensional and two-dimensional signals demonstrate such a reconstruction with reasonable success. The plausibility of doing such a reconstruction is explained in that for the noiseless case the transformation characterizing the degradation is invertible.

© 1989 Optical Society of America

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References

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  1. T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  2. D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
    [CrossRef]
  3. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imaging, MI-1, 113–122 (1982).
    [CrossRef]
  4. 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]
  5. K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 302–316 (1984).
  6. D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
    [CrossRef] [PubMed]
  7. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
    [CrossRef] [PubMed]
  8. S. Inoué, Video Microscopy (Plenum, New York, 1986).
  9. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  10. P. A. Jansson, R. H. Hunt, E. K. Plyler, “Resolution enhancement of spectra,”J. Opt. Soc. Am. 60, 596–599 (1970).
    [CrossRef]
  11. D. L. Snyder, Random Point Processes (Wiley, New York, 1978).
  12. T. J. Holmes, Y. H. Liu, “Application of maximum-likelihood image-restoration in quantum-photon limited noncoherent optical imaging systems and their relation to nuclear-medicine imaging,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 109–117 (1988).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979).
  15. L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,”IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
    [CrossRef]
  16. 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).
  17. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
  18. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).

1988

1987

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

1985

1984

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. 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

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]

1982

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

1981

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

1977

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

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

1970

1961

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).

Agard, D. A.

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

Bille, J.

Carson, R.

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. 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).

Erhardt, A.

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).

Holmes, T. J.

T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
[CrossRef]

T. J. Holmes, Y. H. Liu, “Application of maximum-likelihood image-restoration in quantum-photon limited noncoherent optical imaging systems and their relation to nuclear-medicine imaging,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 109–117 (1988).
[CrossRef]

Hunt, R. H.

Inoué, S.

S. Inoué, Video Microscopy (Plenum, New York, 1986).

Jansson, P. A.

Kaufman, L.

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

Komitowski, D.

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. Assist. Tomogr. 8, 302–316 (1984).

Liu, Y. H.

T. J. Holmes, Y. H. Liu, “Application of maximum-likelihood image-restoration in quantum-photon limited noncoherent optical imaging systems and their relation to nuclear-medicine imaging,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 109–117 (1988).
[CrossRef]

Papoulis, A.

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

Plyler, E. K.

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]

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).

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).

Shepp, L. A.

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

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).

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.

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

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

Ter-Pogossian, M. M.

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

Thomas, L. J.

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

Vardi, Y.

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

Zinser, G.

Annu. Rev. Biophys. Bioeng.

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

Appl. Opt.

Bell Syst. Tech. J.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).

IEEE Trans. Med. Imaging

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

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]

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

J. Comput. Assist. Tomogr.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. R. Statist. Soc. 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).

Opt. Acta

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

Other

S. Inoué, Video Microscopy (Plenum, New York, 1986).

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

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

T. J. Holmes, Y. H. Liu, “Application of maximum-likelihood image-restoration in quantum-photon limited noncoherent optical imaging systems and their relation to nuclear-medicine imaging,” in Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.976, 109–117 (1988).
[CrossRef]

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).

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Figures (9)

Fig. 1
Fig. 1

(a) Signal to be restored. This signal represents the point-process intensity λ. Although this is a continuous signal according to our model, it is represented in the simulation by a discrete array with a sample spacing of 32 nm. The x axis on this plot shows the indices of these samples such that the total spatial width of the 64-element array is 2.048 μm. (b) Degraded signal without truncation. This is the noiseless one-dimensional analogy of a diffraction-limited noncoherent image from a system with a numerical aperture of 1.25 and a wavelength of 525 nm. (c) Truncated signal. This signal represents the analogy of collecting an image of an object that is larger than the viewing window.

Fig. 2
Fig. 2

(a) Restored signal from the degraded signal of Fig. 1(b) following 1000 iterations of the algorithm in Ref. 1. (b) Restored signal from the degraded signal of Fig. 1(c) following 1000 iterations of the algorithm in Ref. 1.

Fig. 3
Fig. 3

(a) Initial guess λ ^ ( 0 )(x) used in the reconstruction from Fig. 1(c) with the extended algorithm of Section 2. Reconstructions at 100, 1000, and 10,000 iterations are shown in (b), (c), and (d), respecitively.

Fig. 4
Fig. 4

(a) Different initial guess from that shown in Fig. 3(a). Corresponding reconstructions using the extended algorithm of Section 2 at 100, 1000, and 10,000 iterations are shown in (b), (c), and (d), respectively.

Fig. 5
Fig. 5

Simulation with quantum-photon noise at an average of 4000 photons per cell. (a) Original nondegraded signal. (b) Degraded signal. (c) Initial guess for the first iteration. (d) Restoration at 1000 iterations.

Fig. 6
Fig. 6

Simulation similar to that of Fig. 5 but at an average of 400 photons per cell.

Fig. 7
Fig. 7

Simulation similar to that of Fig. 5 but at an average of 40 photons per cell.

Fig. 8
Fig. 8

Simulation similar to that of Fig. 5 but at an average of 4 photons per cell.

Fig. 9
Fig. 9

Two-dimensional simulation with quantum-photon noise: (a) original nondegraded signal; (b) diffraction-limited, quantum-limited image at an average of 160 photons per pixel over a 64 × 64 image; (c) truncated version of (b); (d) restoration after 50,000 iterations.

Equations (26)

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μ ( u ) = R p ( u - x ) λ ( x ) d x .
N w ( d u ) = { N d ( d u ) d u W 0 otherwise .
OTF ( f ) = { 1 - f ν / ( 2 n ) f < ( 2 n ) / ν 0 otherwise ,
ν = 525 nm
n = 1.25 ,
μ i j = p ( u j - x i ) λ ( x i ) Δ x Δ u
μ j = i μ i j .
L ( λ ) = Pr [ { N d j : all j } λ ] = j μ j N d j exp ( - μ j ) / N d j ! ,
1 = R p ( u - x ) R p ( u - z ) λ ^ ( z ) d z N d ( d u )             for all x ,
N ^ t ( k ) ( d x ) = E [ N t ( d x ) { N w ( d u ) } , λ ( x ) = λ ^ ( k ) ( x ) ] .
d x = d x m ,
d u = d u j ,
N t ( d x ) = N t ( d x m ) = N t m ,
N d ( d u ) = N d ( d u j ) = N d j ,
N w ( d u ) = N w ( d u j ) = N w j ,
N ^ t ( k ) ( d x ) = N t ( k ) ( d x ) = N ^ t m ( k ) ,
λ ( x ) = λ ^ ( k ) ( x ) = λ ,
N ^ t m ( k ) = E [ N t m { N w j } , λ ]
= j E [ N m j N d j , u j W , λ ] + j E [ N m j u j W , λ ]
= j u j W N d j μ m j [ i μ i j ] - 1 + j u j W E [ N d j u j W , λ ] μ m j [ i μ i j ] - 1
= j u j W [ p ( u j - x m ) λ ( x m ) i p ( u j - x i ) λ ( x i ) ] N d j + j u j W [ p ( u j - x m ) λ ( x m ) i p ( u j - x i ) λ ( x i ) ] × E [ N d j u j W , λ ] ,
E [ N d j u j W , λ ] = i p ( u j - x i ) λ ( x i ) .
N ^ t ( k ) ( d x ) = d x R p ( u - x ) λ ^ ( k ) ( x ) p ( u - z ) λ ^ ( k ) ( z ) d z N ^ d ( k ) ( d u ) ,
N ^ d ( k ) ( d u ) = { N d ( d u ) u W R p ( u - z ) λ ^ ( k ) ( z ) d z u W .
λ ^ ( k + 1 ) ( x ) d x = N ^ t ( k ) ( d x ) ,
λ ^ ( k + 1 ) ( d x ) = λ ^ ( k ) ( x ) R p ( u - x ) R p ( u - z ) λ ^ ( k ) ( z ) d z N ^ d ( k ) ( d u ) .

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