Abstract

Imaging technologies working at very low light levels acquire data by counting the number of photons impinging on each pixel. Especially in cases with, on average, less than one photocount per pixel, the resulting images are heavily corrupted by Poissonian noise. To tackle this problem, we use methods from Bayesian statistics to retrodict the spatial intensity distribution responsible for the photocount measurements. Unlike the usual photon-limited image denoising algorithms, we calculate the full probability distributions for the intensities at each pixel. The knowledge of these probability distributions helps to assess the validity of results from image analysis using data corrupted by Poisson noise with low photon-count numbers and dark counts.

© 2015 Optical Society of America

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References

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  1. R. Molina, J. Núñez, F. J. Cortijo, and J. Mateos, “Image restoration in astronomy: a Bayesian perspective,” IEEE Signal Process. Mag. 18(2), 11–29 (2001).
    [Crossref]
  2. M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
    [Crossref]
  3. A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
    [Crossref]
  4. P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6, 5913 (2015).
  5. T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
    [Crossref]
  6. S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: applications to photon-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
    [Crossref]
  7. D. Shin, A. Kirmani, V. K. Goyal, and J. H. Shapiro, “Photon-efficient computational 3D and reflectivity imaging with single-photon detectors,” arXiv:1406.1761 (2014).
  8. S. Watanabe, “Symmetry of physical laws. Part III. Prediction and retrodiction,” Rev. Mod. Phys. 27, 179–186 (1955).
    [Crossref]
  9. Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, “Time symmetry in the quantum process of measurement,” Phys. Rev. 134, B1410 (1964).
    [Crossref]
  10. D. T. Pegg and S. M. Barnett, “Retrodiction in quantum optics,” J. Opt. B 1, 442–445 (1999).
    [Crossref]
  11. S. M. Barnett, L. S. Phillips, and D. T. Pegg, “Imperfect photodetection as projection onto mixed states,” Opt. Commun. 158, 45–49 (1998).
    [Crossref]
  12. A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
    [Crossref]
  13. M. Sonnleitner, “Statue of Lord Kelvin, Kelvingrove Park, Glasgow” (2015).
  14. O. Jedrkiewicz, R. Loudon, and J. Jeffers, “Retrodiction for optical attenuators, amplifiers, and detectors,” Phys. Rev. A 70, 033805 (2004).
    [Crossref]
  15. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973), Vol. 2.
  16. Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform based analysis of depth images constructed using sparse single-photon data,” arXiv:1507.02511 (2015).
  17. C. Kervrann, J. Boulanger, and P. Coupé, “Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal,” in Scale Space and Variational Methods in Computer Vision (Springer, 2007), pp. 520–532.
  18. C.-A. Deledalle, F. Tupin, and L. Denis, “Poisson NL means: unsupervised non local means for Poisson noise,” in 17th IEEE International Conference on Image Processing (IEEE, 2010), pp. 801–804.
  19. E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge University, 2003).
  20. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
    [Crossref]

2015 (1)

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6, 5913 (2015).

2014 (1)

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

2009 (2)

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: applications to photon-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[Crossref]

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

2007 (1)

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

2005 (1)

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[Crossref]

2004 (1)

O. Jedrkiewicz, R. Loudon, and J. Jeffers, “Retrodiction for optical attenuators, amplifiers, and detectors,” Phys. Rev. A 70, 033805 (2004).
[Crossref]

2001 (1)

R. Molina, J. Núñez, F. J. Cortijo, and J. Mateos, “Image restoration in astronomy: a Bayesian perspective,” IEEE Signal Process. Mag. 18(2), 11–29 (2001).
[Crossref]

1999 (1)

D. T. Pegg and S. M. Barnett, “Retrodiction in quantum optics,” J. Opt. B 1, 442–445 (1999).
[Crossref]

1998 (1)

S. M. Barnett, L. S. Phillips, and D. T. Pegg, “Imperfect photodetection as projection onto mixed states,” Opt. Commun. 158, 45–49 (1998).
[Crossref]

1982 (1)

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

1964 (1)

Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, “Time symmetry in the quantum process of measurement,” Phys. Rev. 134, B1410 (1964).
[Crossref]

1955 (1)

S. Watanabe, “Symmetry of physical laws. Part III. Prediction and retrodiction,” Rev. Mod. Phys. 27, 179–186 (1955).
[Crossref]

Aharonov, Y.

Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, “Time symmetry in the quantum process of measurement,” Phys. Rev. 134, B1410 (1964).
[Crossref]

Altmann, Y.

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform based analysis of depth images constructed using sparse single-photon data,” arXiv:1507.02511 (2015).

Asaki, T. J.

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

Aspden, R. S.

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6, 5913 (2015).

Barnett, S. M.

D. T. Pegg and S. M. Barnett, “Retrodiction in quantum optics,” J. Opt. B 1, 442–445 (1999).
[Crossref]

S. M. Barnett, L. S. Phillips, and D. T. Pegg, “Imperfect photodetection as projection onto mixed states,” Opt. Commun. 158, 45–49 (1998).
[Crossref]

Bell, J. E.

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6, 5913 (2015).

Bergmann, P. G.

Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, “Time symmetry in the quantum process of measurement,” Phys. Rev. 134, B1410 (1964).
[Crossref]

Bertero, M.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

Boccacci, P.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

Boulanger, J.

C. Kervrann, J. Boulanger, and P. Coupé, “Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal,” in Scale Space and Variational Methods in Computer Vision (Springer, 2007), pp. 520–532.

Boyd, R. W.

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6, 5913 (2015).

Buades, A.

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[Crossref]

Buller, G. S.

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform based analysis of depth images constructed using sparse single-photon data,” arXiv:1507.02511 (2015).

Chartrand, R.

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

Colaço, A.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

Coll, B.

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[Crossref]

Cortijo, F. J.

R. Molina, J. Núñez, F. J. Cortijo, and J. Mateos, “Image restoration in astronomy: a Bayesian perspective,” IEEE Signal Process. Mag. 18(2), 11–29 (2001).
[Crossref]

Coupé, P.

C. Kervrann, J. Boulanger, and P. Coupé, “Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal,” in Scale Space and Variational Methods in Computer Vision (Springer, 2007), pp. 520–532.

Deledalle, C.-A.

C.-A. Deledalle, F. Tupin, and L. Denis, “Poisson NL means: unsupervised non local means for Poisson noise,” in 17th IEEE International Conference on Image Processing (IEEE, 2010), pp. 801–804.

Denis, L.

C.-A. Deledalle, F. Tupin, and L. Denis, “Poisson NL means: unsupervised non local means for Poisson noise,” in 17th IEEE International Conference on Image Processing (IEEE, 2010), pp. 801–804.

Desiderà, G.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

Goyal, V. K.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

D. Shin, A. Kirmani, V. K. Goyal, and J. H. Shapiro, “Photon-efficient computational 3D and reflectivity imaging with single-photon detectors,” arXiv:1406.1761 (2014).

Jaynes, E. T.

E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge University, 2003).

Jedrkiewicz, O.

O. Jedrkiewicz, R. Loudon, and J. Jeffers, “Retrodiction for optical attenuators, amplifiers, and detectors,” Phys. Rev. A 70, 033805 (2004).
[Crossref]

Jeffers, J.

O. Jedrkiewicz, R. Loudon, and J. Jeffers, “Retrodiction for optical attenuators, amplifiers, and detectors,” Phys. Rev. A 70, 033805 (2004).
[Crossref]

Kervrann, C.

C. Kervrann, J. Boulanger, and P. Coupé, “Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal,” in Scale Space and Variational Methods in Computer Vision (Springer, 2007), pp. 520–532.

Kirmani, A.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

D. Shin, A. Kirmani, V. K. Goyal, and J. H. Shapiro, “Photon-efficient computational 3D and reflectivity imaging with single-photon detectors,” arXiv:1406.1761 (2014).

Le, T.

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

Lebowitz, J. L.

Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, “Time symmetry in the quantum process of measurement,” Phys. Rev. 134, B1410 (1964).
[Crossref]

Lefkimmiatis, S.

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: applications to photon-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[Crossref]

Loudon, R.

O. Jedrkiewicz, R. Loudon, and J. Jeffers, “Retrodiction for optical attenuators, amplifiers, and detectors,” Phys. Rev. A 70, 033805 (2004).
[Crossref]

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973), Vol. 2.

Maragos, P.

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: applications to photon-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[Crossref]

Mateos, J.

R. Molina, J. Núñez, F. J. Cortijo, and J. Mateos, “Image restoration in astronomy: a Bayesian perspective,” IEEE Signal Process. Mag. 18(2), 11–29 (2001).
[Crossref]

McCarthy, A.

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform based analysis of depth images constructed using sparse single-photon data,” arXiv:1507.02511 (2015).

McLaughlin, S.

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform based analysis of depth images constructed using sparse single-photon data,” arXiv:1507.02511 (2015).

Molina, R.

R. Molina, J. Núñez, F. J. Cortijo, and J. Mateos, “Image restoration in astronomy: a Bayesian perspective,” IEEE Signal Process. Mag. 18(2), 11–29 (2001).
[Crossref]

Morel, J.-M.

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[Crossref]

Morris, P. A.

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6, 5913 (2015).

Núñez, J.

R. Molina, J. Núñez, F. J. Cortijo, and J. Mateos, “Image restoration in astronomy: a Bayesian perspective,” IEEE Signal Process. Mag. 18(2), 11–29 (2001).
[Crossref]

Padgett, M. J.

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6, 5913 (2015).

Papandreou, G.

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: applications to photon-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[Crossref]

Pegg, D. T.

D. T. Pegg and S. M. Barnett, “Retrodiction in quantum optics,” J. Opt. B 1, 442–445 (1999).
[Crossref]

S. M. Barnett, L. S. Phillips, and D. T. Pegg, “Imperfect photodetection as projection onto mixed states,” Opt. Commun. 158, 45–49 (1998).
[Crossref]

Phillips, L. S.

S. M. Barnett, L. S. Phillips, and D. T. Pegg, “Imperfect photodetection as projection onto mixed states,” Opt. Commun. 158, 45–49 (1998).
[Crossref]

Ren, X.

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform based analysis of depth images constructed using sparse single-photon data,” arXiv:1507.02511 (2015).

Shapiro, J. H.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

D. Shin, A. Kirmani, V. K. Goyal, and J. H. Shapiro, “Photon-efficient computational 3D and reflectivity imaging with single-photon detectors,” arXiv:1406.1761 (2014).

Shepp, L. A.

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

Shin, D.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

D. Shin, A. Kirmani, V. K. Goyal, and J. H. Shapiro, “Photon-efficient computational 3D and reflectivity imaging with single-photon detectors,” arXiv:1406.1761 (2014).

Sonnleitner, M.

M. Sonnleitner, “Statue of Lord Kelvin, Kelvingrove Park, Glasgow” (2015).

Tupin, F.

C.-A. Deledalle, F. Tupin, and L. Denis, “Poisson NL means: unsupervised non local means for Poisson noise,” in 17th IEEE International Conference on Image Processing (IEEE, 2010), pp. 801–804.

Vardi, Y.

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

Venkatraman, D.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

Vicidomini, G.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

Watanabe, S.

S. Watanabe, “Symmetry of physical laws. Part III. Prediction and retrodiction,” Rev. Mod. Phys. 27, 179–186 (1955).
[Crossref]

Wong, F. N.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

IEEE Signal Process. Mag. (1)

R. Molina, J. Núñez, F. J. Cortijo, and J. Mateos, “Image restoration in astronomy: a Bayesian perspective,” IEEE Signal Process. Mag. 18(2), 11–29 (2001).
[Crossref]

IEEE Trans. Image Process. (1)

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: applications to photon-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[Crossref]

IEEE Trans. Med. Imaging (1)

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

Inverse Probl. (1)

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurring with Poisson data: from cells to galaxies,” Inverse Probl. 25, 123006 (2009).
[Crossref]

J. Math. Imaging Vis. (1)

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

J. Opt. B (1)

D. T. Pegg and S. M. Barnett, “Retrodiction in quantum optics,” J. Opt. B 1, 442–445 (1999).
[Crossref]

Multiscale Model. Simul. (1)

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[Crossref]

Nat. Commun. (1)

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6, 5913 (2015).

Opt. Commun. (1)

S. M. Barnett, L. S. Phillips, and D. T. Pegg, “Imperfect photodetection as projection onto mixed states,” Opt. Commun. 158, 45–49 (1998).
[Crossref]

Phys. Rev. (1)

Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, “Time symmetry in the quantum process of measurement,” Phys. Rev. 134, B1410 (1964).
[Crossref]

Phys. Rev. A (1)

O. Jedrkiewicz, R. Loudon, and J. Jeffers, “Retrodiction for optical attenuators, amplifiers, and detectors,” Phys. Rev. A 70, 033805 (2004).
[Crossref]

Rev. Mod. Phys. (1)

S. Watanabe, “Symmetry of physical laws. Part III. Prediction and retrodiction,” Rev. Mod. Phys. 27, 179–186 (1955).
[Crossref]

Science (1)

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

Other (7)

D. Shin, A. Kirmani, V. K. Goyal, and J. H. Shapiro, “Photon-efficient computational 3D and reflectivity imaging with single-photon detectors,” arXiv:1406.1761 (2014).

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973), Vol. 2.

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform based analysis of depth images constructed using sparse single-photon data,” arXiv:1507.02511 (2015).

C. Kervrann, J. Boulanger, and P. Coupé, “Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal,” in Scale Space and Variational Methods in Computer Vision (Springer, 2007), pp. 520–532.

C.-A. Deledalle, F. Tupin, and L. Denis, “Poisson NL means: unsupervised non local means for Poisson noise,” in 17th IEEE International Conference on Image Processing (IEEE, 2010), pp. 801–804.

E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge University, 2003).

M. Sonnleitner, “Statue of Lord Kelvin, Kelvingrove Park, Glasgow” (2015).

Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1. Comparison of the probabilities for the intensity retrodictions with and without prior information on the expected mean photocount number m¯. Broken lines show p(λ|m,m¯) for m¯=0.9 (dashed–dotted) and m¯=5 (dashed). The solid line shows p(λ|m) using the flat prior p(λ), which is equivalent to m¯. The red lines are for measurement m=0, and green for m=2; the respective expectation values are indicated by the short vertical lines in the respective style; η=0.2. We see how information about m¯ shifts the retrodictive distributions of λ to lower values.
Fig. 2.
Fig. 2. (a) Original image [13] and (b) image distorted by artificial Poisson noise serving as the measured image for the upcoming examples. The colors indicate (a) the true intensity values λi and (b) the measurement values mi, i=1,,N. Detection efficiency η=0.05; average number of photocounts m¯0.4; no. of pixels N=120×180.
Fig. 3.
Fig. 3. Mean values of single-pixel retrodictions of the data from Fig. 2(b). (a) Mean value for intensity retrodiction E(λi|mi,m¯) as given in Eq. (8); (b) corresponding image νiE(τ|mi,νi) from transmission retrodiction, cf. Eq. (12). The illumination is assumed to be constant with νi=40i=1,,N. Note that these images are generated using only values from the individual pixels and hence cannot be less noisy than the raw data.
Fig. 4.
Fig. 4. Illustration of the mixed retrodiction model. (a) In single-pixel intensity retrodiction, described in Section 2, the number of photocounts at each pixel is used to calculate a probability distribution for λ at this pixel; (b) the probability distribution obtained after assuming that measurements at two different pixels, m and m, came from the same intensities, cf. Eq. (13); (c) mixed retrodiction as outlined in Section 3.B then uses predefined weights W,W, reflecting the assumed probabilities for the cases λ=λ;λ=λ;. Hence, assumed correlations between the neighboring pixels are used to update the probability distribution p(λ|).
Fig. 5.
Fig. 5. Expectation values from mixed retrodiction E(λ|{mj,Wij}j=1,,N,m¯) using the data from Fig. 2(b) and weights from (a) local averaging [cf. Eq. (21)] and (b) nonlocal averaging [Eq. (24)]. The chosen width σ=2.5 in both cases, and tolerance t=0.5 for the nonlocal-means-inspired retrodiction.
Fig. 6.
Fig. 6. (a) Expected error for intensity retrodiction using the mean value Eerr(λt,m¯) [cf. Eq. (25)] (red solid line) or the maximum likelihood EerrL(λt,m¯) given in Eq. (27) (magenta solid). The dashed lines give the expected margin of error ±Verr(λt,m¯) for both cases. Note that for m¯=ηλt the expected error Eerr crosses zero, though the variance keeps growing λt. (b) Average experimental error obtained by comparing retrodicted images with the original from Fig. 2(a). Error from intensity retrodiction (red circles) behaves as expected; error from transmission retrodiction (green diamonds) shows that this method is optimal only for τ1/2, i.e., λtν/2 (see Fig. 3). The mixed retrodictions described in Section 3 (blue triangles, local weights; upside-down cyan triangles, nonlocal weights) give better average results; see also Fig. 5 and Table 1.
Fig. 7.
Fig. 7. Retrodicted probability p(λ|) for a specific pixel from Fig. 2(b) with m=0, m¯0.4, η=0.05, for different retrodiction models: single-pixel intensity retrodiction (red, expectation value E5.9), single-pixel transmission retrodiction (green, E13.7, ν=40), and the multipixel method as given in Section 3 with local averaging (blue dashed, E3.4) and nonlocal means (cyan dashed, E2.7). The true value λt2.4 is indicated by the black dashed line.
Fig. 8.
Fig. 8. Given the measurement values from Fig. 2, one can expect that the local intensities are larger than the values for (a) single-pixel intensity retrodiction, (b) transmission retrodiction, and mixed retrodiction inspired by (c) local or (d) nonlocal averaging. The chosen evidence level is ev(λiξ)=20 [cf. Eq. (30)]; other parameters are as in Figs. 3 and 5.

Tables (1)

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Table 1. Peak Signal-to-Noise Ratio for Different Retrodiction Models as Compared to the True Image Given in Fig. 2(a)

Equations (30)

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p(m|n)=(nm)ηm(1η)nm.
Pois(n;λ)eλλnn!.
p(λ|m)=p(m|λ)p(λ)p(m).
Gam(x;α,β)xα1βαex/βΓ(α),
p(m|m¯)=m¯m(m¯+1)m+1.
p(λ|m¯)=ηm¯eηλ/m¯.
p(λ|m,m¯)=Gam(λ;m+1,(η(1+1/m¯))1),
E(λ|m,m¯)=m+1η(1+1/m¯).
p(n|τ,ν)=ninc=n(nincn)τn(1τ)nincnp(ninc|ν)=Pois(n;τν).
Gami(x;α,β)xα1βαex/βΓ(α,1/β),
Γ(n,x)0xettn1dt.
E(τ|m,ν)=1ηνΓ(m+2,ην)Γ(m+1,ην).
p(λ|m,m,m¯)=Gam(λ;m+m+1,(η(2+1/m¯))1).
p(m|λ,λ,W)=Pois(m;ηWλ+η(1W)λ).
p(m|λ,W,m¯)=0dλp(m|λ,λ,W)p(λ|m¯)=k=0mGeom(mk;(1W)m¯)Pois(k;ηWλ),
p(λ|m,W,m¯)k=0mGam(λ;k+1,[η(W+1/m¯)]1)×Geom(mk;(1W)m¯)Geom(k;Wm¯).
p(λ|m,m,W,W,m¯)p(m|λ,W,m¯)p(λ|m,W,m¯)=k=0mk=0mGam(λ;k+k+1,[η(W+W+1/m¯)]1)×(k+kk)(m¯W)k(m¯W)k(m¯(W+W)+1)k+k+1×Geom(mk;(1W)m¯)Geom(mk;(1W)m¯).
E(λ|m,m,W,W,m¯)k=0mk=0mk+k+1η(W+W+1/m¯)×(k+kk)(m¯W)k(m¯W)k(m¯(W+W)+1)k+k+1×Geom(mk;(1W)m¯)Geom(mk;(1W)m¯).
(k+k+kk,k,k)(k+k+k)!k!k!k!,
p(λ|{mj,Wij}j=1,,N,m¯)jip(mj|λ,Wij,m¯)p(λ|mi,Wii,m¯).
Dσ(i,j)={1(dij/σ)2fordijσ0else,
E(Δ|ml,mk,m¯)=n=0mk(ml+mknml)n+12ml+mkn+1η¯n=0ml(ml+mknmk)mln+12ml+mkn+1η¯,
Δ^i,j1k=1NDσ(i,k)k=1iNiE(Δ|mi+k,mj+k,m¯)Dσ(i,i+k),
Wij=exp(Δ^i,j2/(2t2)).
Eerr(λt,m¯)E(E(λ|m,m¯)λt|λt)=m=0(m+1η¯λt)Pois(m;ηλt)=1η(1+1/m¯)(1ηλtm¯).
Verr(λt,m¯)E((E(λ|m,m¯)λt)2|λt)Eerr2(λt,m¯)=λtη(1+1/m¯)2.
EerrL(λt,m¯)E(L(λ|m,m¯)λt|λt)=m=0(mη¯λt)Pois(m;ηλt)=λt(1η/η¯).
PSNR(λt,x)=10log10(maxi(λt,i)2i=1N(λt,ixi)2/N),
Pr(λi<ξ|mi,m¯)=0ξp(λ|mi,m¯)dλ=Γ(mi+1;η¯ξ)(m+1)!.
ev(λiξ|mi,m¯)=10log10Pr(λiξ|mi,m¯)Pr(λi<ξ|mi,m¯)

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