Abstract

Superresolved image reconstruction is demonstrated by use of multiple images taken through atmospheric turbulence under photon-limited conditions. An iterative reconstruction algorithm applies estimate-maximize techniques to a series of short-exposure images of the desired object scene along with the corresponding image sequence of a guide star. Simulations show that estimates of the Fourier components both below and above the diffraction limit are improved at successive iterations. The estimated images give finer detail of the original object than does the diffraction-limited image. Effects of photon-noise levels on restoration performance are investigated, and a modification to the reconstruction algorithm is derived that accounts for the effects of CCD read noise.

© 1998 Optical Society of America

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References

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  1. G. Ayers, J. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef]
  2. A. W. Lohmann, G. Weigelt, B. Winnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  3. J. Meng, G. Aitken, “Triple-correlation and Knox–Thompson stellar image reconstruction at high signal levels,” J. Opt. Soc. Am. A 12, 284–290 (1995).
    [CrossRef]
  4. P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox–Thompson algorithm,” Opt. Eng. (Bellingham) 47, 91–96 (1983).
  5. T. J. Schulz, “Multiframe blind deconvolution of astronomic images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [CrossRef]
  6. D. G. Sheppard, B. R. Hunt, M. W. Marcellin, “Iterative multiframe superresolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A 15, 978–992 (1998).
    [CrossRef]
  7. D. Gavel, J. Morris, R. Vernon, “Systematic design and analysis of laser-guide-star adaptive-optics systems for large telescopes,” J. Opt. Soc. Am. A 11, 914–924 (1994).
    [CrossRef]
  8. J. Shamir, D. Crowe, J. Beletic, “Improved compensation of atmospheric turbulence effects by multiple mirror systems,” Appl. Opt. 32, 4618–4628 (1993).
    [CrossRef] [PubMed]
  9. B. Welsh, C. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [CrossRef]
  10. B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [CrossRef]
  11. I. A. De La Rue, B. L. Ellerbroek, “A study of multiple guide stars to improve the performance of laser guide star adaptive optical systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998), paper 67.
    [CrossRef]
  12. D. R. Gerwe, M. A. Plonus, “Image restoration of multiple noisy images by use of a priori knowledge of the anisoplanatic point-spread function,” Opt. Lett. 23, 83–85 (1998).
    [CrossRef]
  13. T. J. Holmes, “Maximum-likelihood image restoration for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  14. C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” J. Opt. Soc. Am. A 11, 156–163 (1994).
  15. D. Fried, “Analysis of the CLEAN algorithm and implications for superresolution,” J. Opt. Soc. Am. A 12, 853–860 (1995).
    [CrossRef]
  16. P. J. Sementilli, B. R. Hunt, M. S. Nadar, “Analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993).
    [CrossRef]
  17. A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).
  18. A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
    [CrossRef]
  19. D. Gerwe, M. Plonus, B. Elsebelgy, “Speckle imaging of coherent sources,” in Laser Radar Technology and Applications, G. W. Kamerman, ed. Proc. SPIE2748, 258–271 (1996).
    [CrossRef]
  20. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–122 (1982).
    [CrossRef]
  21. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  22. D. Dayton, S. Sandven, “Hybrid blind deconvolution with high photon noise,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 347–352 (1994).
    [CrossRef]
  23. H. Lantéri, M. Barilli, H. Beaumont, C. Aime, P. Gaucherel, H. Touma, “Comparison of several algorithms for blind deconvolution. Analysis of noise effects,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 275–287 (1995).
  24. A. Glindemann, R. Lane, J. Dainty, “Simulation of time-evolving speckle patterns using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
    [CrossRef]
  25. R. Lane, A. Glindeman, J. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  26. M. Charnotskii, V. Myakinon, V. Zavorotny, “Observation of superresolution in nonisoplanatic imaging through turbulence,” J. Opt. Soc. Am. A 7, 1345–1350 (1990).
    [CrossRef]
  27. D. Gerwe, M. Plonus, B. Elsebelgy, “Long exposure imaging through weak turbulence,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 226–242 (1995).
  28. C. A. Haniff, “Diffraction and resolving power,” J. Opt. Soc. Am. 34, 931–936 (1964).
  29. B. R. Hunt, P. J. Sementilli, “Description of a Poisson imagery superresolution algorithm,” Astron. Soc. Pacific Conf. Ser. 25, 196–199 (1992).

1998 (2)

1995 (2)

1994 (3)

1993 (4)

1992 (2)

R. Lane, A. Glindeman, J. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

B. R. Hunt, P. J. Sementilli, “Description of a Poisson imagery superresolution algorithm,” Astron. Soc. Pacific Conf. Ser. 25, 196–199 (1992).

1991 (2)

B. Welsh, C. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
[CrossRef]

1990 (1)

1988 (2)

1983 (2)

A. W. Lohmann, G. Weigelt, B. Winnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox–Thompson algorithm,” Opt. Eng. (Bellingham) 47, 91–96 (1983).

1982 (1)

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

1977 (1)

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

1972 (1)

1964 (1)

C. A. Haniff, “Diffraction and resolving power,” J. Opt. Soc. Am. 34, 931–936 (1964).

Aime, C.

H. Lantéri, M. Barilli, H. Beaumont, C. Aime, P. Gaucherel, H. Touma, “Comparison of several algorithms for blind deconvolution. Analysis of noise effects,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 275–287 (1995).

Aitken, G.

Ayers, G.

Barilli, M.

H. Lantéri, M. Barilli, H. Beaumont, C. Aime, P. Gaucherel, H. Touma, “Comparison of several algorithms for blind deconvolution. Analysis of noise effects,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 275–287 (1995).

Beaumont, H.

H. Lantéri, M. Barilli, H. Beaumont, C. Aime, P. Gaucherel, H. Touma, “Comparison of several algorithms for blind deconvolution. Analysis of noise effects,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 275–287 (1995).

Beletic, J.

Charnotskii, M.

Crowe, D.

Dainty, J.

A. Glindemann, R. Lane, J. Dainty, “Simulation of time-evolving speckle patterns using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
[CrossRef]

R. Lane, A. Glindeman, J. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

G. Ayers, J. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
[CrossRef]

Dayton, D.

D. Dayton, S. Sandven, “Hybrid blind deconvolution with high photon noise,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 347–352 (1994).
[CrossRef]

De La Rue, I. A.

I. A. De La Rue, B. L. Ellerbroek, “A study of multiple guide stars to improve the performance of laser guide star adaptive optical systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998), paper 67.
[CrossRef]

Dempster, A.

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Ellerbroek, B. L.

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

I. A. De La Rue, B. L. Ellerbroek, “A study of multiple guide stars to improve the performance of laser guide star adaptive optical systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998), paper 67.
[CrossRef]

Elsebelgy, B.

D. Gerwe, M. Plonus, B. Elsebelgy, “Speckle imaging of coherent sources,” in Laser Radar Technology and Applications, G. W. Kamerman, ed. Proc. SPIE2748, 258–271 (1996).
[CrossRef]

D. Gerwe, M. Plonus, B. Elsebelgy, “Long exposure imaging through weak turbulence,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 226–242 (1995).

Fried, D.

Gardner, C.

Gaucherel, P.

H. Lantéri, M. Barilli, H. Beaumont, C. Aime, P. Gaucherel, H. Touma, “Comparison of several algorithms for blind deconvolution. Analysis of noise effects,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 275–287 (1995).

Gavel, D.

Gerwe, D.

D. Gerwe, M. Plonus, B. Elsebelgy, “Speckle imaging of coherent sources,” in Laser Radar Technology and Applications, G. W. Kamerman, ed. Proc. SPIE2748, 258–271 (1996).
[CrossRef]

D. Gerwe, M. Plonus, B. Elsebelgy, “Long exposure imaging through weak turbulence,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 226–242 (1995).

Gerwe, D. R.

Glindeman, A.

R. Lane, A. Glindeman, J. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Glindemann, A.

A. Glindemann, R. Lane, J. Dainty, “Simulation of time-evolving speckle patterns using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
[CrossRef]

Haniff, C. A.

C. A. Haniff, “Diffraction and resolving power,” J. Opt. Soc. Am. 34, 931–936 (1964).

Holmes, T. J.

Hunt, B. R.

Katsaggelos, A. K.

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
[CrossRef]

Laird, N.

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Lane, R.

A. Glindemann, R. Lane, J. Dainty, “Simulation of time-evolving speckle patterns using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
[CrossRef]

R. Lane, A. Glindeman, J. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Lantéri, H.

H. Lantéri, M. Barilli, H. Beaumont, C. Aime, P. Gaucherel, H. Touma, “Comparison of several algorithms for blind deconvolution. Analysis of noise effects,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 275–287 (1995).

Lay, K. T.

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
[CrossRef]

Lohmann, A. W.

Marcellin, M. W.

Matson, C. L.

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” J. Opt. Soc. Am. A 11, 156–163 (1994).

Meng, J.

Morris, J.

Myakinon, V.

Nadar, M. S.

Nisenson, P.

P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox–Thompson algorithm,” Opt. Eng. (Bellingham) 47, 91–96 (1983).

Papaliolios, C.

P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox–Thompson algorithm,” Opt. Eng. (Bellingham) 47, 91–96 (1983).

Plonus, M.

D. Gerwe, M. Plonus, B. Elsebelgy, “Long exposure imaging through weak turbulence,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 226–242 (1995).

D. Gerwe, M. Plonus, B. Elsebelgy, “Speckle imaging of coherent sources,” in Laser Radar Technology and Applications, G. W. Kamerman, ed. Proc. SPIE2748, 258–271 (1996).
[CrossRef]

Plonus, M. A.

Richardson, W. H.

Rubin, D.

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Sandven, S.

D. Dayton, S. Sandven, “Hybrid blind deconvolution with high photon noise,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 347–352 (1994).
[CrossRef]

Schulz, T. J.

Sementilli, P. J.

P. J. Sementilli, B. R. Hunt, M. S. Nadar, “Analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993).
[CrossRef]

B. R. Hunt, P. J. Sementilli, “Description of a Poisson imagery superresolution algorithm,” Astron. Soc. Pacific Conf. Ser. 25, 196–199 (1992).

Shamir, J.

Shepp, L. A.

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

Sheppard, D. G.

Touma, H.

H. Lantéri, M. Barilli, H. Beaumont, C. Aime, P. Gaucherel, H. Touma, “Comparison of several algorithms for blind deconvolution. Analysis of noise effects,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 275–287 (1995).

Vardi, Y.

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

Vernon, R.

Weigelt, G.

Welsh, B.

Winnitzer, B.

Zavorotny, V.

Appl. Opt. (2)

Astron. Soc. Pacific Conf. Ser. (1)

B. R. Hunt, P. J. Sementilli, “Description of a Poisson imagery superresolution algorithm,” Astron. Soc. Pacific Conf. Ser. 25, 196–199 (1992).

IEEE Trans. Med. Imag. (1)

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

IEEE Trans. Signal Process. (1)

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
[CrossRef]

J. Mod. Opt. (1)

A. Glindemann, R. Lane, J. Dainty, “Simulation of time-evolving speckle patterns using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
[CrossRef]

C. A. Haniff, “Diffraction and resolving power,” J. Opt. Soc. Am. 34, 931–936 (1964).

J. Opt. Soc. Am. A (11)

M. Charnotskii, V. Myakinon, V. Zavorotny, “Observation of superresolution in nonisoplanatic imaging through turbulence,” J. Opt. Soc. Am. A 7, 1345–1350 (1990).
[CrossRef]

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

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” J. Opt. Soc. Am. A 11, 156–163 (1994).

D. Fried, “Analysis of the CLEAN algorithm and implications for superresolution,” J. Opt. Soc. Am. A 12, 853–860 (1995).
[CrossRef]

P. J. Sementilli, B. R. Hunt, M. S. Nadar, “Analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993).
[CrossRef]

B. Welsh, C. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

J. Meng, G. Aitken, “Triple-correlation and Knox–Thompson stellar image reconstruction at high signal levels,” J. Opt. Soc. Am. A 12, 284–290 (1995).
[CrossRef]

T. J. Schulz, “Multiframe blind deconvolution of astronomic images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
[CrossRef]

D. G. Sheppard, B. R. Hunt, M. W. Marcellin, “Iterative multiframe superresolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A 15, 978–992 (1998).
[CrossRef]

D. Gavel, J. Morris, R. Vernon, “Systematic design and analysis of laser-guide-star adaptive-optics systems for large telescopes,” J. Opt. Soc. Am. A 11, 914–924 (1994).
[CrossRef]

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

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Opt. Eng. (Bellingham) (1)

P. Nisenson, C. Papaliolios, “Effects of photon noise on speckle image reconstruction with the Knox–Thompson algorithm,” Opt. Eng. (Bellingham) 47, 91–96 (1983).

Opt. Lett. (2)

Waves Random Media (1)

R. Lane, A. Glindeman, J. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other (5)

D. Gerwe, M. Plonus, B. Elsebelgy, “Long exposure imaging through weak turbulence,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 226–242 (1995).

D. Dayton, S. Sandven, “Hybrid blind deconvolution with high photon noise,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 347–352 (1994).
[CrossRef]

H. Lantéri, M. Barilli, H. Beaumont, C. Aime, P. Gaucherel, H. Touma, “Comparison of several algorithms for blind deconvolution. Analysis of noise effects,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2580, 275–287 (1995).

D. Gerwe, M. Plonus, B. Elsebelgy, “Speckle imaging of coherent sources,” in Laser Radar Technology and Applications, G. W. Kamerman, ed. Proc. SPIE2748, 258–271 (1996).
[CrossRef]

I. A. De La Rue, B. L. Ellerbroek, “A study of multiple guide stars to improve the performance of laser guide star adaptive optical systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998), paper 67.
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

A sequence of images of stellar objects to be viewed and a corresponding image sequence of a guide star are captured by use of two shuttered CCD arrays. The light from the guide star is separated by a dichroic mirror with any remaining laser light absorbed by a narrow-band filter.

Fig. 2
Fig. 2

The probability that a photon emitted within object cell Oj during exposure time t will be detected at detector cell Ii is given by element pjit of the discretized anisoplanatic point-spread function.

Fig. 3
Fig. 3

Fast-Fourier-transform implementation of convolutions in Eq. (8), which is the isoplanatic variation of the iterative maximum-likelihood-estimation algorithm given by Eq. (6).

Fig. 4
Fig. 4

On the left is a sample of the computer-generated point-spread functions used in the simulations. On the right is an example of the simulated noisy exposures. Only the center 64 ×64 pixels of each image are shown.

Fig. 5
Fig. 5

Star scene (a) consists of six stars; one star has a dim cloud around it. The dimmest star is 30% as bright as the brightest star, and the cloud is only 5% as bright. When the average number of photons detected from the brightest star is 100 per 5-ms exposure, the average total for the entire field of view is 705 photons per each 5-ms exposure period. When the intensity of the galaxy scene (b) is set such that an average of 100 photons are detected from the brightest cell at each exposure, an average total of 167,980 photons are detected from the entire galaxy scene during each exposure.

Fig. 6
Fig. 6

Amplitudes along the horizontal axis of several two-dimensional Fourier transforms: of the original star scene object of Fig. 5(a), of the 5000th-iteration estimated image, and of the diffraction-limited image. The estimated image is seen to contain significant levels of energy in spectral ranges well beyond the diffraction limit.

Fig. 7
Fig. 7

Amplitudes along the horizontal axis of several two-dimensional Fourier transforms: of the original galaxy scene object Fig. 5(b), of the 5000th-iteration estimated image, and of the diffraction-limited image. The estimated image is seen to contain significant levels of energy in spectral ranges well beyond the diffraction limit.

Fig. 8
Fig. 8

Algorithm’s performance in iteratively estimating the star scene of Fig. 5(a). (a) Relative error energies in the spatial domain, for the full spectrum in the Fourier domain, and for spectral components above the diffraction limit in the Fourier domain are shown as a function of iteration number. The algorithm is seen to iteratively improve its estimate of spectral components above the diffraction limit. (b) The Strehl ratio is also improved at each iteration. (Strehl ratios are greater than one owing to the superresolution effect.)

Fig. 9
Fig. 9

Algorithm’s performance in iteratively estimating the galaxy scene of Fig. 5(b). As in Fig. 8, the Strehl ratio and estimates of spectral components above and below the diffraction limit are improved at each iteration. (Strehl ratios are greater than one owing to the superresolution effect.)

Fig. 10
Fig. 10

The estimated image contains finer detail than the diffraction-limited image in comparison with the original object. In the Fourier domain the diffraction-limited image contains zero power at spectral frequencies outside the limit dictated by the imaging aperture. The restoration algorithm is seen to successfully estimate frequency components well beyond this limit. Images have been gamma brightened by a factor of 5 to enhance contrast and make dim features visible.

Fig. 11
Fig. 11

As in Fig. 10, the estimated image contains finer detail than the diffraction-limited image. The spatial-domain images have been gamma brightened by a factor of 2 and the Fourier domain images by a factor of 8 to enhance contrast and make dim features visible.

Fig. 12
Fig. 12

In this figure the averaged SNR of the 5000th-iteration estimates of the star scene and the galaxy scene are plotted as a function of spatial frequency, calculated according to Eq. (12). Also shown is the SNR of the corresponding diffraction-limited image. The dashed vertical line marks the diffraction limit. It is very clear from this data that the algorithm is able to successfully estimate spatial-frequency components of the source at frequencies both below and above the diffraction limit, thus confirming superresolution.

Fig. 13
Fig. 13

Restoration performance as a function of number of iterations at different object-scene intensity levels. Intensities are described in terms of the average photon-detection rate of photons from the brightest object-scene cell. The measure of error used in (a) is mean square error=[j(λjq-λj)2]1/2/jλj2)1/2, where λjq is the current estimate of the true emission rate λj. (Strehl ratios are greater than one owing to the superresolution effect.)

Fig. 14
Fig. 14

Estimation at the 500th iteration with 83,990 photons per exposure. With an average of five photons per pixel, the algorithm still achieved considerable improvement in image quality. This image was gamma brightened by a factor of 2.

Equations (18)

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

f(s|λ, p)=jit exp(-λjit)λjitsjit/(sjit!),
ln f(s|λq+1, p)|λq, p, I
=jit(-λjitq+1+sjit ln λjitq+1-ln sjit!)λq, p, I.
sjit|λq, p=λjitq,
sjit|λq, p, I=sjit|λq, p, Iit=jsjit=Iitλjitqjλjitq.
jit-λjitq+1+Iitλjitqjλjitqln λjitq+1-ln sjit!|λq, p, I,
λjq+1=λjqitpjitit Iitjλjqpjitpjit.
f[s|ϕ(λ)]=b(s)exp[ϕ(λ)·t(s)T]/a[ϕ(λ)],
λq+1=λq  t(It  gtq)pt,gtq=λqpt,
Φn(κ¯)=0.033Cn2|κ¯2+(2π/Lo)2|-11/6 exp-κ¯2(5.92/lo)2,
P˜f(t+Δt)=P˜ft+αP˜fr1+α2,α=2R ffmax,
j:|f¯jq|>fdiff(λ˜jq-λ˜j)21/2j:|f¯jq|>fdiffλ˜j2,1/2
SNRq(f)=-20 logfjSf|λ˜jq-λ˜j|fjSf|λ˜j|,
Sf={fj: f-Δf|f¯j|f+Δf}.
Iit=jsjit+dit.
ln f(s|λq+1, p)|λq, p, I, d,
sjit|λq, p, I, d=sjit|λq, p, Iit=jsjit+dit,d=Iitλjitqjλjitq+di.
λjq+1=λjqitpjitit Iitjλjqpjit+dipjit.

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