Abstract

A maximum likelihood based iterative algorithm adapted from nuclear medicine imaging for noncoherent optical imaging was presented in a previous publication with some initial computer-simulation testing. This algorithm is identical in form to that previously derived in a different way by W. H. Richardson, “ Bayesian-Based Iterative Method of Image Restoration,” J. Opt. Soc. Am. 62, 55– 59 ( 1972) and L. B. Lucy, “ An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745– 765 ( 1974). Foreseen applications include superresolution and 3-D fluorescence microscopy. This paper presents further simulation testing of this algorithm and a preliminary experiment with a defocused camera. The simulations show quantified resolution improvement as a function of iteration number, and they show qualitatively the trend in limitations on restored resolution when noise is present in the data. Also shown are results of a simulation in restoring missing-cone information for 3-D imaging. Conclusions are in support of the feasibility of using these methods with real systems, while computational cost and timing estimates indicate that it should be realistic to implement these methods. It is suggested in the Appendix that future extensions to the maximum likelihood based derivation of this algorithm will address some of the limitations that are experienced with the nonextended form of the algorithm presented here.

© 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. L. A. Shepp, Y. Vardi, “Maximum Likelihood Reconstruction for Emission Tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
    [CrossRef]
  3. 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]
  4. K. Lange, R. Carson, “EM Reconstruction Algorithms for Emission and Transmission Tomography,” J. Comput. Assisted Tomogr. 8, 302–316 (1984).
  5. W. H. Richardson, “Bayesian-Based Iterative Method of Image Restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  6. L. B. Lucy, “An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745–765 (1974).
    [CrossRef]
  7. 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,” Soc. Photo-Opt. Instrum. Eng. 976, 109–117 (1988).
  8. T. J. Holmes, “Expectation-Maximization Restoration of Bandlimited, Truncated Point-Process Intensities with Application in Microscopy,” J. Opt. Soc. Am. A, 6, 1006–1014 (1989).
    [CrossRef]
  9. D. G. Politte, D. L. Snyder, “Results of a Comparative Study of a Reconstruction Procedure for Producing Improved Estimates of Radioactivity Distributions in Time-of-Flight Emission Tomography,” IEEE Trans. Nucl. Sci. NS-31, 614–619 (1984).
    [CrossRef]
  10. D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 228–238 (1987).
    [CrossRef]
  11. M. I. Miller, D. L. Snyder, “The Role of Likelihood and Entropy in Incomplete-Data Problems: Applications to Estimating Point-Process Intensities and Toeplitz Constrained Covariances,” Proc. IEEE 75, 892–907 (1987).
    [CrossRef]
  12. M. I. Miller, B. Roysam, J. A. Schrauner, “Quantum-Limited ML Imaging Using Good's Roughness Penalty Implementation on a Massively Parallel Processor,” presented at the IEEE 1988 International Conference on Acoustics, Speech, and Signal Processing (1988), paper M 4.21; IEEE Multidimensional Signal Processing2, 932–935 (1988).
  13. H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
    [CrossRef]
  14. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1979).
  15. B. R. Frieden, “Optical Transfer of 3-Dimensional Object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  16. D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
    [CrossRef]
  17. N. Striebl, “Three-Dimensional Imaging by a Microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  18. J. M. Carazo, J. L. Carrascosa, “Information Recovery in Missing Angular Data Cases: an Approach by the Convex Projections Method in Three Dimensions,” J. Microsc. 145, Pt. 1, 23–43 (1987).
    [CrossRef]
  19. A. Macovski, Medical Imaging Systems (Prentice-Hall, Englewood Cliffs, NJ, 1983).
  20. N. Striebl, “Depth Transfer by an Imaging System,” Opt. Acta 31, 1233–1241 (1984).
    [CrossRef]
  21. F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, NJ, 1987).
  22. Mercury Computer Systems, Lowell, MA 01854.
  23. L. Kaufman, “Implementing and Accelerating the EM Algorithm for Positron Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
    [CrossRef]
  24. K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared Imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
    [CrossRef]
  25. T. Hebert, R. Leahy, “A Generalized EM Algorithm for 3-D Bayesian Reconstruction from Poisson Data Using Gibbs Priors,” IEEE Trans. Med. Imaging MI-8, 194–202 (1989).
    [CrossRef]
  26. D. L. Snyder, M. I. Miller, “The Use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
    [CrossRef]
  27. T. J. Holmes, Y. H. Liu, in preparation.
  28. T. J. Holmes, “Acceleration of Maximum-Likelihood Image Restoration for Fluorescence Microscopy and Other Noncoherent Imagery,” in Technical Digest, Topical Meeting on Quantum-Limited Imaging and Information Processing (Optical Society of America, Washington, DC, 1989), paper C5-1.

1989 (2)

T. J. Holmes, “Expectation-Maximization Restoration of Bandlimited, Truncated Point-Process Intensities with Application in Microscopy,” J. Opt. Soc. Am. A, 6, 1006–1014 (1989).
[CrossRef]

T. Hebert, R. Leahy, “A Generalized EM Algorithm for 3-D Bayesian Reconstruction from Poisson Data Using Gibbs Priors,” IEEE Trans. Med. Imaging MI-8, 194–202 (1989).
[CrossRef]

1988 (3)

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared Imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

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,” Soc. Photo-Opt. Instrum. Eng. 976, 109–117 (1988).

1987 (4)

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 228–238 (1987).
[CrossRef]

M. I. Miller, D. L. Snyder, “The Role of Likelihood and Entropy in Incomplete-Data Problems: Applications to Estimating Point-Process Intensities and Toeplitz Constrained Covariances,” Proc. IEEE 75, 892–907 (1987).
[CrossRef]

J. M. Carazo, J. L. Carrascosa, “Information Recovery in Missing Angular Data Cases: an Approach by the Convex Projections Method in Three Dimensions,” J. Microsc. 145, Pt. 1, 23–43 (1987).
[CrossRef]

L. Kaufman, “Implementing and Accelerating the EM Algorithm for Positron Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

1985 (2)

D. L. Snyder, M. I. Miller, “The Use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

N. Striebl, “Three-Dimensional Imaging by a Microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[CrossRef]

1984 (4)

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef]

N. Striebl, “Depth Transfer by an Imaging System,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

K. Lange, R. Carson, “EM Reconstruction Algorithms for Emission and Transmission Tomography,” J. Comput. Assisted Tomogr. 8, 302–316 (1984).

D. G. Politte, D. L. Snyder, “Results of a Comparative Study of a Reconstruction Procedure for Producing Improved Estimates of Radioactivity Distributions in Time-of-Flight Emission Tomography,” IEEE Trans. Nucl. Sci. NS-31, 614–619 (1984).
[CrossRef]

1983 (1)

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 (1)

L. A. Shepp, Y. Vardi, “Maximum Likelihood Reconstruction for Emission Tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

1974 (1)

L. B. Lucy, “An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745–765 (1974).
[CrossRef]

1972 (1)

1967 (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]

Agard, D. A.

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef]

Capps, R. W.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared Imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Carazo, J. M.

J. M. Carazo, J. L. Carrascosa, “Information Recovery in Missing Angular Data Cases: an Approach by the Convex Projections Method in Three Dimensions,” J. Microsc. 145, Pt. 1, 23–43 (1987).
[CrossRef]

Carrascosa, J. L.

J. M. Carazo, J. L. Carrascosa, “Information Recovery in Missing Angular Data Cases: an Approach by the Convex Projections Method in Three Dimensions,” J. Microsc. 145, Pt. 1, 23–43 (1987).
[CrossRef]

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, NJ, 1979).

Frieden, B. R.

Grasdalen, G. L.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared Imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Hebert, T.

T. Hebert, R. Leahy, “A Generalized EM Algorithm for 3-D Bayesian Reconstruction from Poisson Data Using Gibbs Priors,” IEEE Trans. Med. Imaging MI-8, 194–202 (1989).
[CrossRef]

Hodapp, K.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared Imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Holmes, T. J.

T. J. Holmes, “Expectation-Maximization Restoration of Bandlimited, Truncated Point-Process Intensities with Application in Microscopy,” J. Opt. Soc. Am. A, 6, 1006–1014 (1989).
[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,” Soc. Photo-Opt. Instrum. Eng. 976, 109–117 (1988).

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, in preparation.

T. J. Holmes, “Acceleration of Maximum-Likelihood Image Restoration for Fluorescence Microscopy and Other Noncoherent Imagery,” in Technical Digest, Topical Meeting on Quantum-Limited Imaging and Information Processing (Optical Society of America, Washington, DC, 1989), paper C5-1.

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]

Kaufman, L.

L. Kaufman, “Implementing and Accelerating the EM Algorithm for Positron Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

Lange, K.

K. Lange, R. Carson, “EM Reconstruction Algorithms for Emission and Transmission Tomography,” J. Comput. Assisted Tomogr. 8, 302–316 (1984).

Leahy, R.

T. Hebert, R. Leahy, “A Generalized EM Algorithm for 3-D Bayesian Reconstruction from Poisson Data Using Gibbs Priors,” IEEE Trans. Med. Imaging MI-8, 194–202 (1989).
[CrossRef]

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,” Soc. Photo-Opt. Instrum. Eng. 976, 109–117 (1988).

T. J. Holmes, Y. H. Liu, in preparation.

Lucy, L. B.

L. B. Lucy, “An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745–765 (1974).
[CrossRef]

Macovski, A.

A. Macovski, Medical Imaging Systems (Prentice-Hall, Englewood Cliffs, NJ, 1983).

Miller, M. I.

M. I. Miller, D. L. Snyder, “The Role of Likelihood and Entropy in Incomplete-Data Problems: Applications to Estimating Point-Process Intensities and Toeplitz Constrained Covariances,” Proc. IEEE 75, 892–907 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 228–238 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, “The Use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

M. I. Miller, B. Roysam, J. A. Schrauner, “Quantum-Limited ML Imaging Using Good's Roughness Penalty Implementation on a Massively Parallel Processor,” presented at the IEEE 1988 International Conference on Acoustics, Speech, and Signal Processing (1988), paper M 4.21; IEEE Multidimensional Signal Processing2, 932–935 (1988).

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, NJ, 1987).

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, NJ, 1987).

Politte, D. G.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 228–238 (1987).
[CrossRef]

D. G. Politte, D. L. Snyder, “Results of a Comparative Study of a Reconstruction Procedure for Producing Improved Estimates of Radioactivity Distributions in Time-of-Flight Emission Tomography,” IEEE Trans. Nucl. Sci. NS-31, 614–619 (1984).
[CrossRef]

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]

Richardson, W. H.

Roysam, B.

M. I. Miller, B. Roysam, J. A. Schrauner, “Quantum-Limited ML Imaging Using Good's Roughness Penalty Implementation on a Massively Parallel Processor,” presented at the IEEE 1988 International Conference on Acoustics, Speech, and Signal Processing (1988), paper M 4.21; IEEE Multidimensional Signal Processing2, 932–935 (1988).

Salas, L.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared Imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Schrauner, J. A.

M. I. Miller, B. Roysam, J. A. Schrauner, “Quantum-Limited ML Imaging Using Good's Roughness Penalty Implementation on a Massively Parallel Processor,” presented at the IEEE 1988 International Conference on Acoustics, Speech, and Signal Processing (1988), paper M 4.21; IEEE Multidimensional Signal Processing2, 932–935 (1988).

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.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 228–238 (1987).
[CrossRef]

M. I. Miller, D. L. Snyder, “The Role of Likelihood and Entropy in Incomplete-Data Problems: Applications to Estimating Point-Process Intensities and Toeplitz Constrained Covariances,” Proc. IEEE 75, 892–907 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, “The Use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

D. G. Politte, D. L. Snyder, “Results of a Comparative Study of a Reconstruction Procedure for Producing Improved Estimates of Radioactivity Distributions in Time-of-Flight Emission Tomography,” IEEE Trans. Nucl. Sci. NS-31, 614–619 (1984).
[CrossRef]

Striebl, N.

N. Striebl, “Three-Dimensional Imaging by a Microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[CrossRef]

N. Striebl, “Depth Transfer by an Imaging System,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

Strom, S. E.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared Imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Thomas, L. J.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 228–238 (1987).
[CrossRef]

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum Likelihood Reconstruction for Emission Tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Ann. Rev. Biophys. Bioeng. (1)

D. A. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef]

Astron. J. (1)

L. B. Lucy, “An Iterative Technique for the Rectification of Observed Distributions,” Astron. J. 79, 745–765 (1974).
[CrossRef]

Astrophys. J. (1)

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared Imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

IEEE Trans. Med. Imaging (4)

T. Hebert, R. Leahy, “A Generalized EM Algorithm for 3-D Bayesian Reconstruction from Poisson Data Using Gibbs Priors,” IEEE Trans. Med. Imaging MI-8, 194–202 (1989).
[CrossRef]

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]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography,” IEEE Trans. Med. Imaging MI-6, 228–238 (1987).
[CrossRef]

IEEE Trans. Nucl. Sci. (3)

D. G. Politte, D. L. Snyder, “Results of a Comparative Study of a Reconstruction Procedure for Producing Improved Estimates of Radioactivity Distributions in Time-of-Flight Emission Tomography,” IEEE Trans. Nucl. Sci. NS-31, 614–619 (1984).
[CrossRef]

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, M. I. Miller, “The Use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (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. Microsc. (1)

J. M. Carazo, J. L. Carrascosa, “Information Recovery in Missing Angular Data Cases: an Approach by the Convex Projections Method in Three Dimensions,” J. Microsc. 145, Pt. 1, 23–43 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

N. Striebl, “Depth Transfer by an Imaging System,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

Proc. IEEE (1)

M. I. Miller, D. L. Snyder, “The Role of Likelihood and Entropy in Incomplete-Data Problems: Applications to Estimating Point-Process Intensities and Toeplitz Constrained Covariances,” Proc. IEEE 75, 892–907 (1987).
[CrossRef]

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]

Soc. Photo-Opt. Instrum. Eng. (1)

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,” Soc. Photo-Opt. Instrum. Eng. 976, 109–117 (1988).

Other (7)

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

M. I. Miller, B. Roysam, J. A. Schrauner, “Quantum-Limited ML Imaging Using Good's Roughness Penalty Implementation on a Massively Parallel Processor,” presented at the IEEE 1988 International Conference on Acoustics, Speech, and Signal Processing (1988), paper M 4.21; IEEE Multidimensional Signal Processing2, 932–935 (1988).

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, NJ, 1987).

Mercury Computer Systems, Lowell, MA 01854.

A. Macovski, Medical Imaging Systems (Prentice-Hall, Englewood Cliffs, NJ, 1983).

T. J. Holmes, Y. H. Liu, in preparation.

T. J. Holmes, “Acceleration of Maximum-Likelihood Image Restoration for Fluorescence Microscopy and Other Noncoherent Imagery,” in Technical Digest, Topical Meeting on Quantum-Limited Imaging and Information Processing (Optical Society of America, Washington, DC, 1989), paper C5-1.

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Figures (11)

Fig. 1
Fig. 1

Image restoration algorithm.

Fig. 2
Fig. 2

Overview of the simulation presented in Ref. 1: Top left to top right (a)–(c): (a) Original object to be restored. Total width of the bright object area is 0.82 μm. (b) Diffraction-limited image for a N.A. of 1.25 and wavelength of 0.525 μm. (c) Restored image after 10,000 iterations. Bottom left to bottom right (d)–(f): (d) Real part of the Fourier transform of (a). (e) Real part of the Fourier transform of (b). The flat gray areas indicate removed components. (f) Real part of the Fourier transform of (c).

Fig. 3
Fig. 3

Resolution improvement calculated from the noise-free simulation of Fig. 2.

Fig. 4
Fig. 4

Top left to top right (a)–(d) represent the noise-free case and total photon counts of 640 K, 160 K, and 40 K, respectively. Midleft to midright (e)–(h) are respective restorations at 1000 iterations. Lower left to lower right (i)–(l) are the respective restorations at 10,000 iterations.

Fig. 5
Fig. 5

Simulation with a background object. Clockwise (a), (b), and (c) are the original object, diffraction-limited image, with 640 K total photons and restored image after 1000 iterations. The intensity of the detailed object in (a) is twice that of the background object.

Fig. 6
Fig. 6

Defocused camera experiment: (a) (top left) in-focus object to be restored; (b) (top right) out-of-focus image; (c) (bottom left) calibrated impulse response; (d) (bottom right) restored image after 600 iterations.

Fig. 7
Fig. 7

OTF calculated by way of Eq. (4). The horizontal and vertical coordinates are ρ,η, respectively.

Fig. 8
Fig. 8

Images obtained in the missing-cone simulation: (a) original object; (b) Image obtained from the object in (a), degraded by the OTF of Eq. (4), with 4.1 M total photons; (c) restored image after 1000 iterations.

Fig. 9
Fig. 9

Real part of the Fourier transform of Figs. 8(a)–(c), respectively.

Fig. 10
Fig. 10

Error energies of the real part of the Fourier transform in the missing-cone simulation with 4.1 M total photons.

Fig. 11
Fig. 11

Computation cost and timing estimates.

Tables (2)

Tables Icon

Table I Resolution Improvements

Tables Icon

Table II Error Energies

Equations (8)

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

OTF ( ρ ) = r J 1 ( 2 π r | ρ | ) / | ρ | ,
OTF ( ρ , η ) = { Re { 1 2 π ρ A [ 1 ( 2 η A ρ ) 2 ] 1 / 2 } , ρ 0 , δ ( η ) / ( 8 A 2 ) , ρ = 0 ,
A = 1 / ( 2 N . A . ) .
OTF ( ρ , η ) = 1 λ 2 ρ A Re { [ 1 ( λ ρ A + 2 | η | A ρ ) 2 ] 1 / 2 } ,
min x [ a ( x ) ] = 0.0 ,
E = S [ f ̂ ( x ) f ( x ) ] 2 d x ,
c ( r ) = R 2 λ ̂ ( x ) γ ( x ) d x R 2 [ λ ̂ ( x ) ] 2 d x R 2 [ γ ( x ) ] 2 d x ,
γ ( x ) = o ( x ) * a r ( x ) ,

Metrics