Abstract

We consider the problem of estimating one nonblurred and cleaned image from a sequence of P randomly translated images corrupted with Poisson noise. We develop a new algorithm based on maximum-likelihood (ML) estimation for two unknown parameters: the reconstructed image itself and the set of translations of the low-light-level images. We demonstrate that the ML reconstructed image is proportional to the sum of the low-light-level images after correcting for the unknown movement and that its entropy is minimal. The images of the sequence are matched together by means of an iterative minimum-entropy algorithm, where a systematic search under displacements for the images is performed. We develop a fast version of this algorithm, and we present results for simulated images and experimental data. The probability of good matching of a low-level image sequence is estimated numerically when the light level of the images in the sequence decreases, corresponding to small numbers of photons detected (down to 20) in each image of the sequence. We compare these results with those obtained when the low-light-level images are matched to a known reference, i.e., the linear correlation method, and with those from the optimal one, when the noise has a Poisson distribution. This approach is applied to astronomical images that are acquired by photocounting from a balloon-borne ultraviolet imaging telescope.

© 1998 Optical Society of America

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References

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  1. R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
    [CrossRef]
  2. J. A. Morse, “A method for correcting aspect solution errors in ROSAT HRI observations of compact sources,” Publ. Astron. Soc. Pac. 106, 675–682 (1994).
    [CrossRef]
  3. R. E. Sequeira, J. A. Gubner, B. E. A. Sala, “Image detection using low-level illumination,” IEEE Trans. Image Process. 2, 18–26 (1994).
    [CrossRef]
  4. D. L. Snyder, T. L. Schulz, “High resolution imaging at low-light levels through weak turbulence,” J. Opt. Soc. Am. A 7, 1251–1265 (1990).
    [CrossRef]
  5. B. D. Hall, D. Reinhard, R. Monot, “Optimum rate for a CCD detector,” Rev. Sci. Instrum. 66, 2668–2671 (1995).
    [CrossRef]
  6. G. M. Morris, “Scene matching using photon-limited images,” J. Opt. Soc. Am. A 1, 482–488 (1984).
    [CrossRef]
  7. D. S. Lalush, M. W. Tsui, “The importance of preconditioners in fast Poisson-based iterative reconstruction algorithms for SPECT,” in IEEE Nuclear Science Symposium and Medical Imaging Conference Record, P. A. Moonier, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N. Y., 1995), Vol. 3, pp. 1326–1330.
  8. J. M. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
    [CrossRef]
  9. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thomson and triple correlation through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
    [CrossRef]
  10. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  11. A. Vander Lugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  12. B. V. K. Vijaya Kumar, F. M. Dickey, J. M. Delaurentis, “Correlation filters minimizing peak location errors,” J. Opt. Soc. Am. A 9, 678–682 (1992).
    [CrossRef]
  13. J. D. Downie, J. F. Walkup, “Optimal correlation filters for images with signal-dependent noise,” J. Opt. Soc. Am. A 11, 1599–1609 (1994).
    [CrossRef]
  14. M. Guillaume, Th. Amouroux, Ph. Refregier, B. Milliard, A. Llebaria, “Optimal correlation at low photon levels: study for astronomical images,” Opt. Lett. 22, 322–324 (1997).
    [CrossRef] [PubMed]
  15. B. J. Slocumb, D. L. Snyder, “Maximum likelihood estimation applied to quantum limited optical position sensing,” in Acquisition, Tracking, and Pointing IV, S. Gowrinathan, ed., Proc. SPIE1304, 165–176 (1990).
    [CrossRef]
  16. T. J. Schulz, D. L. Snyder, “Imaging a randomly moving object from quantum-limited data: applications to image recovery from second- and third-order autocorrelations,” J. Opt. Soc. Am. A 8, 801–807 (1991).
    [CrossRef]
  17. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  18. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  19. R. G. Lane, “Methods for maximum-likelihood deconvolution,” J. Opt. Soc. Am. A 13, 1992–1998 (1996).
    [CrossRef]
  20. A. Prades, J. Nunez, F. Perez, V. Pala, R. Arbiol, “Aerial photography restoration using MLE algorithm,” in ISPRS Commission III Symposium: Spatial Information from Digital Photogrammetry and Computer Vision, H. Ebner, C. Helpke, K. Eder, eds., Proc. SPIE2357, 683–688 (1994).
    [CrossRef]
  21. B. Milliard, J. Donas, M. Laget, “A 40-cm UV (2000 Å) balloon-borne imaging telescope: results and current work,” Adv. Space Res. 11, 135–137 (1991).
    [CrossRef]
  22. M. Laget, D. Burgarella, B. Milliard, J. Donas, “UV-(2000 Å) imaging of globular clusters,” Astron. Astrophys. 259, 510–521 (1992).
  23. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [CrossRef]
  24. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
    [CrossRef]
  25. J. L. Nieto, A. Llebaria, S. di Serego Alighieri, “Photo-counting detectors in time-resolved imaging mode: image recentering and selection algorithms,” Astron. Astrophys. 178, 301–306 (1987).
  26. C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989).
  27. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024–2036 (1989).
    [CrossRef]

1997

1996

1995

R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
[CrossRef]

B. D. Hall, D. Reinhard, R. Monot, “Optimum rate for a CCD detector,” Rev. Sci. Instrum. 66, 2668–2671 (1995).
[CrossRef]

1994

J. A. Morse, “A method for correcting aspect solution errors in ROSAT HRI observations of compact sources,” Publ. Astron. Soc. Pac. 106, 675–682 (1994).
[CrossRef]

R. E. Sequeira, J. A. Gubner, B. E. A. Sala, “Image detection using low-level illumination,” IEEE Trans. Image Process. 2, 18–26 (1994).
[CrossRef]

J. D. Downie, J. F. Walkup, “Optimal correlation filters for images with signal-dependent noise,” J. Opt. Soc. Am. A 11, 1599–1609 (1994).
[CrossRef]

1993

1992

B. V. K. Vijaya Kumar, F. M. Dickey, J. M. Delaurentis, “Correlation filters minimizing peak location errors,” J. Opt. Soc. Am. A 9, 678–682 (1992).
[CrossRef]

M. Laget, D. Burgarella, B. Milliard, J. Donas, “UV-(2000 Å) imaging of globular clusters,” Astron. Astrophys. 259, 510–521 (1992).

1991

B. Milliard, J. Donas, M. Laget, “A 40-cm UV (2000 Å) balloon-borne imaging telescope: results and current work,” Adv. Space Res. 11, 135–137 (1991).
[CrossRef]

T. J. Schulz, D. L. Snyder, “Imaging a randomly moving object from quantum-limited data: applications to image recovery from second- and third-order autocorrelations,” J. Opt. Soc. Am. A 8, 801–807 (1991).
[CrossRef]

1990

1989

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024–2036 (1989).
[CrossRef]

1988

1987

J. L. Nieto, A. Llebaria, S. di Serego Alighieri, “Photo-counting detectors in time-resolved imaging mode: image recentering and selection algorithms,” Astron. Astrophys. 178, 301–306 (1987).

1984

1974

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1972

1964

A. Vander Lugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

1948

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Amouroux, Th.

Arbiol, R.

A. Prades, J. Nunez, F. Perez, V. Pala, R. Arbiol, “Aerial photography restoration using MLE algorithm,” in ISPRS Commission III Symposium: Spatial Information from Digital Photogrammetry and Computer Vision, H. Ebner, C. Helpke, K. Eder, eds., Proc. SPIE2357, 683–688 (1994).
[CrossRef]

Ayers, G. R.

Burgarella, D.

M. Laget, D. Burgarella, B. Milliard, J. Donas, “UV-(2000 Å) imaging of globular clusters,” Astron. Astrophys. 259, 510–521 (1992).

Dainty, J. C.

Delaurentis, J. M.

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024–2036 (1989).
[CrossRef]

di Serego Alighieri, S.

J. L. Nieto, A. Llebaria, S. di Serego Alighieri, “Photo-counting detectors in time-resolved imaging mode: image recentering and selection algorithms,” Astron. Astrophys. 178, 301–306 (1987).

Dickey, F. M.

Donas, J.

M. Laget, D. Burgarella, B. Milliard, J. Donas, “UV-(2000 Å) imaging of globular clusters,” Astron. Astrophys. 259, 510–521 (1992).

B. Milliard, J. Donas, M. Laget, “A 40-cm UV (2000 Å) balloon-borne imaging telescope: results and current work,” Adv. Space Res. 11, 135–137 (1991).
[CrossRef]

Downie, J. D.

Edmonds, P. D.

R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
[CrossRef]

Gilliland, R. L.

R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
[CrossRef]

Gubner, J. A.

R. E. Sequeira, J. A. Gubner, B. E. A. Sala, “Image detection using low-level illumination,” IEEE Trans. Image Process. 2, 18–26 (1994).
[CrossRef]

Guillaume, M.

Hall, B. D.

B. D. Hall, D. Reinhard, R. Monot, “Optimum rate for a CCD detector,” Rev. Sci. Instrum. 66, 2668–2671 (1995).
[CrossRef]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Laget, M.

M. Laget, D. Burgarella, B. Milliard, J. Donas, “UV-(2000 Å) imaging of globular clusters,” Astron. Astrophys. 259, 510–521 (1992).

B. Milliard, J. Donas, M. Laget, “A 40-cm UV (2000 Å) balloon-borne imaging telescope: results and current work,” Adv. Space Res. 11, 135–137 (1991).
[CrossRef]

Lalush, D. S.

D. S. Lalush, M. W. Tsui, “The importance of preconditioners in fast Poisson-based iterative reconstruction algorithms for SPECT,” in IEEE Nuclear Science Symposium and Medical Imaging Conference Record, P. A. Moonier, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N. Y., 1995), Vol. 3, pp. 1326–1330.

Lane, R. G.

Llebaria, A.

M. Guillaume, Th. Amouroux, Ph. Refregier, B. Milliard, A. Llebaria, “Optimal correlation at low photon levels: study for astronomical images,” Opt. Lett. 22, 322–324 (1997).
[CrossRef] [PubMed]

J. L. Nieto, A. Llebaria, S. di Serego Alighieri, “Photo-counting detectors in time-resolved imaging mode: image recentering and selection algorithms,” Astron. Astrophys. 178, 301–306 (1987).

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Milliard, B.

M. Guillaume, Th. Amouroux, Ph. Refregier, B. Milliard, A. Llebaria, “Optimal correlation at low photon levels: study for astronomical images,” Opt. Lett. 22, 322–324 (1997).
[CrossRef] [PubMed]

M. Laget, D. Burgarella, B. Milliard, J. Donas, “UV-(2000 Å) imaging of globular clusters,” Astron. Astrophys. 259, 510–521 (1992).

B. Milliard, J. Donas, M. Laget, “A 40-cm UV (2000 Å) balloon-borne imaging telescope: results and current work,” Adv. Space Res. 11, 135–137 (1991).
[CrossRef]

Monot, R.

B. D. Hall, D. Reinhard, R. Monot, “Optimum rate for a CCD detector,” Rev. Sci. Instrum. 66, 2668–2671 (1995).
[CrossRef]

Morris, G. M.

Morse, J. A.

J. A. Morse, “A method for correcting aspect solution errors in ROSAT HRI observations of compact sources,” Publ. Astron. Soc. Pac. 106, 675–682 (1994).
[CrossRef]

Nieto, J. L.

J. L. Nieto, A. Llebaria, S. di Serego Alighieri, “Photo-counting detectors in time-resolved imaging mode: image recentering and selection algorithms,” Astron. Astrophys. 178, 301–306 (1987).

Northcott, J. M.

Northcott, M. J.

Nunez, J.

A. Prades, J. Nunez, F. Perez, V. Pala, R. Arbiol, “Aerial photography restoration using MLE algorithm,” in ISPRS Commission III Symposium: Spatial Information from Digital Photogrammetry and Computer Vision, H. Ebner, C. Helpke, K. Eder, eds., Proc. SPIE2357, 683–688 (1994).
[CrossRef]

Pala, V.

A. Prades, J. Nunez, F. Perez, V. Pala, R. Arbiol, “Aerial photography restoration using MLE algorithm,” in ISPRS Commission III Symposium: Spatial Information from Digital Photogrammetry and Computer Vision, H. Ebner, C. Helpke, K. Eder, eds., Proc. SPIE2357, 683–688 (1994).
[CrossRef]

Perez, F.

A. Prades, J. Nunez, F. Perez, V. Pala, R. Arbiol, “Aerial photography restoration using MLE algorithm,” in ISPRS Commission III Symposium: Spatial Information from Digital Photogrammetry and Computer Vision, H. Ebner, C. Helpke, K. Eder, eds., Proc. SPIE2357, 683–688 (1994).
[CrossRef]

Petro, L.

R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
[CrossRef]

Prades, A.

A. Prades, J. Nunez, F. Perez, V. Pala, R. Arbiol, “Aerial photography restoration using MLE algorithm,” in ISPRS Commission III Symposium: Spatial Information from Digital Photogrammetry and Computer Vision, H. Ebner, C. Helpke, K. Eder, eds., Proc. SPIE2357, 683–688 (1994).
[CrossRef]

Refregier, Ph.

Reinhard, D.

B. D. Hall, D. Reinhard, R. Monot, “Optimum rate for a CCD detector,” Rev. Sci. Instrum. 66, 2668–2671 (1995).
[CrossRef]

Richardson, W. H.

Saha, A.

R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
[CrossRef]

Sala, B. E. A.

R. E. Sequeira, J. A. Gubner, B. E. A. Sala, “Image detection using low-level illumination,” IEEE Trans. Image Process. 2, 18–26 (1994).
[CrossRef]

Schulz, T. J.

Schulz, T. L.

Sequeira, R. E.

R. E. Sequeira, J. A. Gubner, B. E. A. Sala, “Image detection using low-level illumination,” IEEE Trans. Image Process. 2, 18–26 (1994).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Shara, M. M.

R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
[CrossRef]

Slocumb, B. J.

B. J. Slocumb, D. L. Snyder, “Maximum likelihood estimation applied to quantum limited optical position sensing,” in Acquisition, Tracking, and Pointing IV, S. Gowrinathan, ed., Proc. SPIE1304, 165–176 (1990).
[CrossRef]

Snyder, D. L.

Therrien, C. W.

C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989).

Tsui, M. W.

D. S. Lalush, M. W. Tsui, “The importance of preconditioners in fast Poisson-based iterative reconstruction algorithms for SPECT,” in IEEE Nuclear Science Symposium and Medical Imaging Conference Record, P. A. Moonier, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N. Y., 1995), Vol. 3, pp. 1326–1330.

Vander Lugt, A.

A. Vander Lugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Vijaya Kumar, B. V. K.

Walkup, J. F.

Adv. Space Res.

B. Milliard, J. Donas, M. Laget, “A 40-cm UV (2000 Å) balloon-borne imaging telescope: results and current work,” Adv. Space Res. 11, 135–137 (1991).
[CrossRef]

Astron. Astrophys.

M. Laget, D. Burgarella, B. Milliard, J. Donas, “UV-(2000 Å) imaging of globular clusters,” Astron. Astrophys. 259, 510–521 (1992).

J. L. Nieto, A. Llebaria, S. di Serego Alighieri, “Photo-counting detectors in time-resolved imaging mode: image recentering and selection algorithms,” Astron. Astrophys. 178, 301–306 (1987).

Astron. J.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Astrophys. J.

R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
[CrossRef]

Bell Syst. Tech. J.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024–2036 (1989).
[CrossRef]

IEEE Trans. Image Process.

R. E. Sequeira, J. A. Gubner, B. E. A. Sala, “Image detection using low-level illumination,” IEEE Trans. Image Process. 2, 18–26 (1994).
[CrossRef]

IEEE Trans. Inf. Theory

A. Vander Lugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Publ. Astron. Soc. Pac.

J. A. Morse, “A method for correcting aspect solution errors in ROSAT HRI observations of compact sources,” Publ. Astron. Soc. Pac. 106, 675–682 (1994).
[CrossRef]

Rev. Sci. Instrum.

B. D. Hall, D. Reinhard, R. Monot, “Optimum rate for a CCD detector,” Rev. Sci. Instrum. 66, 2668–2671 (1995).
[CrossRef]

Other

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

D. S. Lalush, M. W. Tsui, “The importance of preconditioners in fast Poisson-based iterative reconstruction algorithms for SPECT,” in IEEE Nuclear Science Symposium and Medical Imaging Conference Record, P. A. Moonier, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N. Y., 1995), Vol. 3, pp. 1326–1330.

B. J. Slocumb, D. L. Snyder, “Maximum likelihood estimation applied to quantum limited optical position sensing,” in Acquisition, Tracking, and Pointing IV, S. Gowrinathan, ed., Proc. SPIE1304, 165–176 (1990).
[CrossRef]

A. Prades, J. Nunez, F. Perez, V. Pala, R. Arbiol, “Aerial photography restoration using MLE algorithm,” in ISPRS Commission III Symposium: Spatial Information from Digital Photogrammetry and Computer Vision, H. Ebner, C. Helpke, K. Eder, eds., Proc. SPIE2357, 683–688 (1994).
[CrossRef]

C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989).

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Figures (10)

Fig. 1
Fig. 1

(a) Reference image of the sky, (b) one low-level image of the sequence with 224 photons and λ=0.0008.

Fig. 2
Fig. 2

Evolution of the reduced entropy relative to the number of iterations for P=2000 images. The mean number of photons in one image is 28 for λ=0.0001.

Fig. 3
Fig. 3

Evolution of the reduced entropy Er for various translation probability laws of the sequence P(jp) and standard deviations σ.

Fig. 4
Fig. 4

Values of reduced entropy after convergence of the algorithm, relative to the number of images P in the sequence, for various values of λ.

Fig. 5
Fig. 5

Probability of good translation estimation calculated as the ratio of perfectly matched low-level images, relative to the number of images P in the sequence, for various values of λ.

Fig. 6
Fig. 6

Probability of good translation estimation calculated as the proportion of matched images, relative to λ, for various values of the number of images P in the sequence. The curves denoted by Δ and × show the results obtained when the reference is known for linear correlation with reference and for correlation with logarithm of reference, respectively.

Fig. 7
Fig. 7

Standard deviation error σ in the reconstructed image plotted relative to the number of images in the sequence for various values of λ.

Fig. 8
Fig. 8

(a) Image reconstructed without matching of the sequence, (b) image reconstructed with minimum entropy.

Fig. 9
Fig. 9

Image reconstructed with the experimental data Messier 3: (a) without correction, (b) after correction.

Fig. 10
Fig. 10

Section of the most luminous star in Messier 3: correction for various temporal resolution images. The intensity is the number of photons collected at the corresponding pixel.

Equations (20)

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

P[sp(i)|μp(i)]=exp[-μp(i)] [μp(i)]sp(i)sp(i)!.
μ(i)=ΔTu(i),r(i)=Tu(i),μ(i)=TΔTr(i),
L (r(i), J¯)=p=1PP[sp(i)|r(i), jp],
J¯=(j1, j2,, jP).
l(r(i), J¯)=p=1P-ΔTTr(i)+sp(i+jp)×lnΔTTr(i)-ln[sp(i+jp)!].
l(r(i), J¯)r(i)=0,
rML(i, J¯)=TPΔTp=1Psp(i+jp).
δ l(jp)=i=1Nsp(i+jp)ln[r(i)].
l(r, J¯)=pi ln[P(s(i)/r, jp)],
J¯ML=arg maxJ¯[l(r, J¯)].
J¯ML=arg maxJ¯pisp(i+jp)ln[rML(i, J¯)]-rML(i, J¯)-ln[sp(i+jp)!].
J¯ML=arg minJ¯-irML(i, J¯)ln[rML(i, J¯)].
E[rML(J¯)]=-irML(i, J¯)ln[rML(i, J¯)]
Var(k, p)=iDp[Apk(i)+sp(i+jpk-Δ j)]×ln[Apk(i)+sp(i+jpk-Δ j)]-Apk(i)ln[Apk(i)],
E[r(J¯k)]=-i=1N[Apk(i)+sp(i+jpk)]×ln[Apk(i)+sp(i+jpk)],
E[r(J¯k)]=-i=1N[Apk(i)+sp(i+jpk-Δ j)]×ln[Apk(i)+sp(i+jpk-Δ j)],
E[r(J¯k)]=-iDp[Apk(i)+sp(i+jpk-Δ j)]×ln[Apk(i)+sp(i+jpk-Δ j)]-iDpApk(i)ln[Apk(i)]
E[r(J¯k)]=-iDp[Apk(i)+sp(i+jpk-Δ j)]×ln[Apk(i)+sp(i+jpk-Δ j)]-i=1NApk(i)ln[Apk(i)]+iDpApk(i)ln[Apk(i)],
Var(k, p)=iDp{[Apk(i)+sp(i+jpk-Δ j)]×ln[Apk(i)+sp(i+jpk-Δ j)]-Apk(i)ln[Apk(i)]}.
Δ jopt=arg minΔ j{E[r(J¯k)]}=arg maxΔ j[Var(k, p)],

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