Abstract

I present new lower and upper bounds on the minimum probability of error (MPE) in Bayesian multihypothesis testing that follow from an exact integral of a version of the statistical entropy of the posterior distribution, or equivocation. I also show that these bounds are exponentially tight and thus achievable in the asymptotic limit of many conditionally independent and identically distributed measurements. I then relate the minimum mean-squared error (MMSE) and the MPE by means of certain elementary error probability integrals. In the second half of the paper, I compare the MPE and MMSE for the problem of locating a single point source with subdiffractive uncertainty. The source-strength threshold needed to achieve a desired degree of source localization seems to be far more modest than the well established threshold for the different optical super-resolution problem of disambiguating two point sources with subdiffractive separation.

© 2012 Optical Society of America

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References

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  1. N. Santhi and A. Vardy, “On an improvement over Rényi’s equivocation bound,” in Proceedings of the 44th Annual Allerton Conference on Communication, Control, and Computing 2006 (Curran, 2006), pp. 1118–1124.
  2. T. Routtenberg and J. Tabrikian, “General class of lower bounds on the probability of error in multiple hypothesis testing,” http://arxiv.org/abs/1005.2880v1 (2010).
  3. H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493–507 (1952).
    [CrossRef]
  4. T. Kailath, “The divergence and Bhattacharya distance in signal selection,” IEEE Trans. Commun. Technol. 15, 52–60 (1967).
    [CrossRef]
  5. A. Rényi, “On the amount of missing information and the Neyman-Pearson lemma,” Festschrift for J. Neyman (Wiley, 1966), pp. 281–288.
  6. M. Hellman and J. Raviv, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory 16, 368–372 (1970).
    [CrossRef]
  7. C. Leang and D. Johnson, “On the asymptotics of M-hypothesis Bayesian detection,” IEEE Trans. Inf. Theory 43, 280–282 (1997).
    [CrossRef]
  8. C. Helstrom, “Detection and resolution of incoherent objects by a background-limited optical system,” J. Opt. Soc. Am. 59, 164–175 (1969).
    [CrossRef]
  9. M. Shahram and P. Milanfar, “Imaging below the diffraction limit: a statistical analysis,” IEEE Trans. Image Process. 13, 677–689 (2004).
    [CrossRef]
  10. L. Lucy, “Statistical limits to superresolution,” Astron. Astrophys. 261, 706–710 (1992).
  11. S. Smith, “Statistical resolution limits and the complexified Cramér-Rao bound,” IEEE Trans. Signal Process. 53, 1597–1609(2005).
    [CrossRef]
  12. D. Fried, “Analysis of CLEAN algorithm and implications for superresolution,” J. Opt. Soc. Am. A 12, 853–860 (1995).
    [CrossRef]
  13. P. Magain, F. Courbin, and S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
    [CrossRef]
  14. R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE 7094, 709405 (2008).
    [CrossRef]
  15. H. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, 1968).
  16. T. Cover and J. Thomas, Elements of Information Theory(Wiley, 1991).
  17. D. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge University, 2005), Sec. 4.5.
  18. F. Li, M. Ng, R. Plemmons, S. Prasad, and Q. Zhang, “Hyperspectral image segmentation, deblurring, and spectral analysis for material identification,” Proc. SPIE 7701, 770103 (2010).
    [CrossRef]
  19. J. Seidler, “Bounds on the mean-square error and the quality of domain decisions based on mutual information,” IEEE Trans. Inf. Theory 17, 655–665 (1971).
    [CrossRef]
  20. M. Feder and N. Merhav, “Relations between entropy and error probability,” IEEE Trans. Inf. Theory 40, 259–266(1994).
    [CrossRef]
  21. S. Ho and S. Verdu, “On the interplay between conditional entropy and error probability,” IEEE Trans. Inf. Theory 56, 5930–5942 (2010).
    [CrossRef]
  22. Technically, the approximations implicit in Eqs. (37) and (38) are accurate for data points x that are not too close to the boundaries of the decision regions. However, since probability distributions like P(x|m) and P(m|x) always occur inside integrals over decision regions, such inaccuracies near the decision boundaries do not materially affect the smallness of integrals like Eq. (44).
  23. It will be convenient to denote the photon data in this problem by the symbol k, rather than x.
  24. R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
    [CrossRef]
  25. M. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching (SHRImP),” Proc. Natl. Acad. Sci. USA 101, 6462–6465 (2004).
    [CrossRef]
  26. S. Narravula, Department of Physics and Astronomy, University of New Mexico, Albuquerque, N.M. 87131, and S. Prasad are preparing a manuscript to be called “A simple upper bound on the minimum mean squared error of Bayesian estimation.” (sprasad@unm.edu)
  27. A. M. Walker, “On the asymptotic behaviour of posterior distributions,” J. R. Stat. Soc. B 31, 80–88 (1969).
  28. C. Rushforth and R. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  29. B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectra,” J. Opt. Soc. Am. 57, 1013–1019 (1967).
    [CrossRef]
  30. C. Matson and D. Tyler, “Primary and secondary superresolution by data inversion,” Opt. Express 14, 456–473 (2006).
    [CrossRef]
  31. S. Prasad and X. Luo, “Support-assisted optical superresolution of low-resolution image sequences: the one-dimensional problem,” Opt. Express 17, 23213–23233 (2009).
    [CrossRef]
  32. M. Bertero and C. Mol, “Super-resolution by data inversion,” Prog. Opt. 36, 129–178 (1996).
    [CrossRef]
  33. J. Ianniello, “Large and small error performance limits for multipath time delay estimation,” IEEE Trans. Acoust. Speech Signal Process. 34, 245–251 (1986).
    [CrossRef]

2010 (2)

F. Li, M. Ng, R. Plemmons, S. Prasad, and Q. Zhang, “Hyperspectral image segmentation, deblurring, and spectral analysis for material identification,” Proc. SPIE 7701, 770103 (2010).
[CrossRef]

S. Ho and S. Verdu, “On the interplay between conditional entropy and error probability,” IEEE Trans. Inf. Theory 56, 5930–5942 (2010).
[CrossRef]

2009 (1)

2008 (1)

R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE 7094, 709405 (2008).
[CrossRef]

2006 (1)

2005 (1)

S. Smith, “Statistical resolution limits and the complexified Cramér-Rao bound,” IEEE Trans. Signal Process. 53, 1597–1609(2005).
[CrossRef]

2004 (2)

M. Shahram and P. Milanfar, “Imaging below the diffraction limit: a statistical analysis,” IEEE Trans. Image Process. 13, 677–689 (2004).
[CrossRef]

M. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching (SHRImP),” Proc. Natl. Acad. Sci. USA 101, 6462–6465 (2004).
[CrossRef]

2002 (1)

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef]

1998 (1)

P. Magain, F. Courbin, and S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

1997 (1)

C. Leang and D. Johnson, “On the asymptotics of M-hypothesis Bayesian detection,” IEEE Trans. Inf. Theory 43, 280–282 (1997).
[CrossRef]

1996 (1)

M. Bertero and C. Mol, “Super-resolution by data inversion,” Prog. Opt. 36, 129–178 (1996).
[CrossRef]

1995 (1)

1994 (1)

M. Feder and N. Merhav, “Relations between entropy and error probability,” IEEE Trans. Inf. Theory 40, 259–266(1994).
[CrossRef]

1992 (1)

L. Lucy, “Statistical limits to superresolution,” Astron. Astrophys. 261, 706–710 (1992).

1986 (1)

J. Ianniello, “Large and small error performance limits for multipath time delay estimation,” IEEE Trans. Acoust. Speech Signal Process. 34, 245–251 (1986).
[CrossRef]

1971 (1)

J. Seidler, “Bounds on the mean-square error and the quality of domain decisions based on mutual information,” IEEE Trans. Inf. Theory 17, 655–665 (1971).
[CrossRef]

1970 (1)

M. Hellman and J. Raviv, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory 16, 368–372 (1970).
[CrossRef]

1969 (2)

C. Helstrom, “Detection and resolution of incoherent objects by a background-limited optical system,” J. Opt. Soc. Am. 59, 164–175 (1969).
[CrossRef]

A. M. Walker, “On the asymptotic behaviour of posterior distributions,” J. R. Stat. Soc. B 31, 80–88 (1969).

1968 (1)

1967 (2)

B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectra,” J. Opt. Soc. Am. 57, 1013–1019 (1967).
[CrossRef]

T. Kailath, “The divergence and Bhattacharya distance in signal selection,” IEEE Trans. Commun. Technol. 15, 52–60 (1967).
[CrossRef]

1952 (1)

H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493–507 (1952).
[CrossRef]

Bertero, M.

M. Bertero and C. Mol, “Super-resolution by data inversion,” Prog. Opt. 36, 129–178 (1996).
[CrossRef]

Chernoff, H.

H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493–507 (1952).
[CrossRef]

Courbin, F.

P. Magain, F. Courbin, and S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

Cover, T.

T. Cover and J. Thomas, Elements of Information Theory(Wiley, 1991).

Feder, M.

M. Feder and N. Merhav, “Relations between entropy and error probability,” IEEE Trans. Inf. Theory 40, 259–266(1994).
[CrossRef]

Fried, D.

Frieden, B. R.

Gordon, M.

M. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching (SHRImP),” Proc. Natl. Acad. Sci. USA 101, 6462–6465 (2004).
[CrossRef]

Ha, T.

M. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching (SHRImP),” Proc. Natl. Acad. Sci. USA 101, 6462–6465 (2004).
[CrossRef]

Harris, R.

Hellman, M.

M. Hellman and J. Raviv, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory 16, 368–372 (1970).
[CrossRef]

Helstrom, C.

Hier, R.

R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE 7094, 709405 (2008).
[CrossRef]

Ho, S.

S. Ho and S. Verdu, “On the interplay between conditional entropy and error probability,” IEEE Trans. Inf. Theory 56, 5930–5942 (2010).
[CrossRef]

Ianniello, J.

J. Ianniello, “Large and small error performance limits for multipath time delay estimation,” IEEE Trans. Acoust. Speech Signal Process. 34, 245–251 (1986).
[CrossRef]

Johnson, D.

C. Leang and D. Johnson, “On the asymptotics of M-hypothesis Bayesian detection,” IEEE Trans. Inf. Theory 43, 280–282 (1997).
[CrossRef]

Kailath, T.

T. Kailath, “The divergence and Bhattacharya distance in signal selection,” IEEE Trans. Commun. Technol. 15, 52–60 (1967).
[CrossRef]

Larson, D.

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef]

Leang, C.

C. Leang and D. Johnson, “On the asymptotics of M-hypothesis Bayesian detection,” IEEE Trans. Inf. Theory 43, 280–282 (1997).
[CrossRef]

Li, F.

F. Li, M. Ng, R. Plemmons, S. Prasad, and Q. Zhang, “Hyperspectral image segmentation, deblurring, and spectral analysis for material identification,” Proc. SPIE 7701, 770103 (2010).
[CrossRef]

Lucy, L.

L. Lucy, “Statistical limits to superresolution,” Astron. Astrophys. 261, 706–710 (1992).

Luo, X.

MacKay, D.

D. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge University, 2005), Sec. 4.5.

Magain, P.

P. Magain, F. Courbin, and S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

Matson, C.

Merhav, N.

M. Feder and N. Merhav, “Relations between entropy and error probability,” IEEE Trans. Inf. Theory 40, 259–266(1994).
[CrossRef]

Milanfar, P.

M. Shahram and P. Milanfar, “Imaging below the diffraction limit: a statistical analysis,” IEEE Trans. Image Process. 13, 677–689 (2004).
[CrossRef]

Mol, C.

M. Bertero and C. Mol, “Super-resolution by data inversion,” Prog. Opt. 36, 129–178 (1996).
[CrossRef]

Narravula, S.

S. Narravula, Department of Physics and Astronomy, University of New Mexico, Albuquerque, N.M. 87131, and S. Prasad are preparing a manuscript to be called “A simple upper bound on the minimum mean squared error of Bayesian estimation.” (sprasad@unm.edu)

Ng, M.

F. Li, M. Ng, R. Plemmons, S. Prasad, and Q. Zhang, “Hyperspectral image segmentation, deblurring, and spectral analysis for material identification,” Proc. SPIE 7701, 770103 (2010).
[CrossRef]

Plemmons, R.

F. Li, M. Ng, R. Plemmons, S. Prasad, and Q. Zhang, “Hyperspectral image segmentation, deblurring, and spectral analysis for material identification,” Proc. SPIE 7701, 770103 (2010).
[CrossRef]

Prasad, S.

F. Li, M. Ng, R. Plemmons, S. Prasad, and Q. Zhang, “Hyperspectral image segmentation, deblurring, and spectral analysis for material identification,” Proc. SPIE 7701, 770103 (2010).
[CrossRef]

S. Prasad and X. Luo, “Support-assisted optical superresolution of low-resolution image sequences: the one-dimensional problem,” Opt. Express 17, 23213–23233 (2009).
[CrossRef]

S. Narravula, Department of Physics and Astronomy, University of New Mexico, Albuquerque, N.M. 87131, and S. Prasad are preparing a manuscript to be called “A simple upper bound on the minimum mean squared error of Bayesian estimation.” (sprasad@unm.edu)

Puetter, R.

R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE 7094, 709405 (2008).
[CrossRef]

Raviv, J.

M. Hellman and J. Raviv, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory 16, 368–372 (1970).
[CrossRef]

Rényi, A.

A. Rényi, “On the amount of missing information and the Neyman-Pearson lemma,” Festschrift for J. Neyman (Wiley, 1966), pp. 281–288.

Routtenberg, T.

T. Routtenberg and J. Tabrikian, “General class of lower bounds on the probability of error in multiple hypothesis testing,” http://arxiv.org/abs/1005.2880v1 (2010).

Rushforth, C.

Santhi, N.

N. Santhi and A. Vardy, “On an improvement over Rényi’s equivocation bound,” in Proceedings of the 44th Annual Allerton Conference on Communication, Control, and Computing 2006 (Curran, 2006), pp. 1118–1124.

Seidler, J.

J. Seidler, “Bounds on the mean-square error and the quality of domain decisions based on mutual information,” IEEE Trans. Inf. Theory 17, 655–665 (1971).
[CrossRef]

Selvin, P. R.

M. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching (SHRImP),” Proc. Natl. Acad. Sci. USA 101, 6462–6465 (2004).
[CrossRef]

Shahram, M.

M. Shahram and P. Milanfar, “Imaging below the diffraction limit: a statistical analysis,” IEEE Trans. Image Process. 13, 677–689 (2004).
[CrossRef]

Smith, S.

S. Smith, “Statistical resolution limits and the complexified Cramér-Rao bound,” IEEE Trans. Signal Process. 53, 1597–1609(2005).
[CrossRef]

Sohy, S.

P. Magain, F. Courbin, and S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

Tabrikian, J.

T. Routtenberg and J. Tabrikian, “General class of lower bounds on the probability of error in multiple hypothesis testing,” http://arxiv.org/abs/1005.2880v1 (2010).

Thomas, J.

T. Cover and J. Thomas, Elements of Information Theory(Wiley, 1991).

Thompson, R.

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef]

Tyler, D.

Van Trees, H.

H. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, 1968).

Vardy, A.

N. Santhi and A. Vardy, “On an improvement over Rényi’s equivocation bound,” in Proceedings of the 44th Annual Allerton Conference on Communication, Control, and Computing 2006 (Curran, 2006), pp. 1118–1124.

Verdu, S.

S. Ho and S. Verdu, “On the interplay between conditional entropy and error probability,” IEEE Trans. Inf. Theory 56, 5930–5942 (2010).
[CrossRef]

Walker, A. M.

A. M. Walker, “On the asymptotic behaviour of posterior distributions,” J. R. Stat. Soc. B 31, 80–88 (1969).

Webb, W.

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef]

Zhang, Q.

F. Li, M. Ng, R. Plemmons, S. Prasad, and Q. Zhang, “Hyperspectral image segmentation, deblurring, and spectral analysis for material identification,” Proc. SPIE 7701, 770103 (2010).
[CrossRef]

Ann. Math. Stat. (1)

H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Stat. 23, 493–507 (1952).
[CrossRef]

Astron. Astrophys. (1)

L. Lucy, “Statistical limits to superresolution,” Astron. Astrophys. 261, 706–710 (1992).

Astrophys. J. (1)

P. Magain, F. Courbin, and S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

Biophys. J. (1)

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

J. Ianniello, “Large and small error performance limits for multipath time delay estimation,” IEEE Trans. Acoust. Speech Signal Process. 34, 245–251 (1986).
[CrossRef]

IEEE Trans. Commun. Technol. (1)

T. Kailath, “The divergence and Bhattacharya distance in signal selection,” IEEE Trans. Commun. Technol. 15, 52–60 (1967).
[CrossRef]

IEEE Trans. Image Process. (1)

M. Shahram and P. Milanfar, “Imaging below the diffraction limit: a statistical analysis,” IEEE Trans. Image Process. 13, 677–689 (2004).
[CrossRef]

IEEE Trans. Inf. Theory (5)

M. Hellman and J. Raviv, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory 16, 368–372 (1970).
[CrossRef]

C. Leang and D. Johnson, “On the asymptotics of M-hypothesis Bayesian detection,” IEEE Trans. Inf. Theory 43, 280–282 (1997).
[CrossRef]

J. Seidler, “Bounds on the mean-square error and the quality of domain decisions based on mutual information,” IEEE Trans. Inf. Theory 17, 655–665 (1971).
[CrossRef]

M. Feder and N. Merhav, “Relations between entropy and error probability,” IEEE Trans. Inf. Theory 40, 259–266(1994).
[CrossRef]

S. Ho and S. Verdu, “On the interplay between conditional entropy and error probability,” IEEE Trans. Inf. Theory 56, 5930–5942 (2010).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. Smith, “Statistical resolution limits and the complexified Cramér-Rao bound,” IEEE Trans. Signal Process. 53, 1597–1609(2005).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

J. R. Stat. Soc. B (1)

A. M. Walker, “On the asymptotic behaviour of posterior distributions,” J. R. Stat. Soc. B 31, 80–88 (1969).

Opt. Express (2)

Proc. Natl. Acad. Sci. USA (1)

M. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching (SHRImP),” Proc. Natl. Acad. Sci. USA 101, 6462–6465 (2004).
[CrossRef]

Proc. SPIE (2)

R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE 7094, 709405 (2008).
[CrossRef]

F. Li, M. Ng, R. Plemmons, S. Prasad, and Q. Zhang, “Hyperspectral image segmentation, deblurring, and spectral analysis for material identification,” Proc. SPIE 7701, 770103 (2010).
[CrossRef]

Prog. Opt. (1)

M. Bertero and C. Mol, “Super-resolution by data inversion,” Prog. Opt. 36, 129–178 (1996).
[CrossRef]

Other (9)

H. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, 1968).

T. Cover and J. Thomas, Elements of Information Theory(Wiley, 1991).

D. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge University, 2005), Sec. 4.5.

Technically, the approximations implicit in Eqs. (37) and (38) are accurate for data points x that are not too close to the boundaries of the decision regions. However, since probability distributions like P(x|m) and P(m|x) always occur inside integrals over decision regions, such inaccuracies near the decision boundaries do not materially affect the smallness of integrals like Eq. (44).

It will be convenient to denote the photon data in this problem by the symbol k, rather than x.

S. Narravula, Department of Physics and Astronomy, University of New Mexico, Albuquerque, N.M. 87131, and S. Prasad are preparing a manuscript to be called “A simple upper bound on the minimum mean squared error of Bayesian estimation.” (sprasad@unm.edu)

N. Santhi and A. Vardy, “On an improvement over Rényi’s equivocation bound,” in Proceedings of the 44th Annual Allerton Conference on Communication, Control, and Computing 2006 (Curran, 2006), pp. 1118–1124.

T. Routtenberg and J. Tabrikian, “General class of lower bounds on the probability of error in multiple hypothesis testing,” http://arxiv.org/abs/1005.2880v1 (2010).

A. Rényi, “On the amount of missing information and the Neyman-Pearson lemma,” Festschrift for J. Neyman (Wiley, 1966), pp. 281–288.

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Figures (9)

Fig. 1.
Fig. 1.

Schematic diagram of the optical setup.

Fig. 2.
Fig. 2.

Plot of Km(0) (in units of ηK/2) versus m for M=8 and l=1,2,3,4,5.

Fig. 3.
Fig. 3.

MPE for source localization versus image-pixel width (in units of θ0) for (a) M=4, (b) M=8, and (c) M=12, for SNR=10, 103, and 105 in the Gaussian read-noise model.

Fig. 4.
Fig. 4.

MPE for source localization versus image-pixel width (in units of θ0) for (a) M=4, (b) M=8, and (c) M=12, for SNR=10, 103, and 105 in the Poisson shot-noise model.

Fig. 5.
Fig. 5.

MMSE for resolving the source position versus image-pixel width (in units of θ0) for (a) M=4, (b) M=8, and (c) M=12, for three different source strengths, K=10, 103, and 105, in the Poisson shot-noise model.

Fig. 6.
Fig. 6.

Log-log plot of the MMSE (solid curves) and CRB (dashed line) versus source strength for a critical-width image pixel (L=θ0) and for five different values of M, namely 4, 8, 12, 16, and 20. The dotted horizontal lines are drawn at heights (θ0/M)2/12, corresponding to the variance of the uniform distribution over a subpixel of width θ0/M.

Fig. 7.
Fig. 7.

Plot of the source-strength threshold, Kmin, versus the degree of OSR, M, for the one-pixel imager (square symbols) and the three-pixel imager (plus symbols).

Fig. 8.
Fig. 8.

MMSE UBs for resolving the source position versus image-pixel width (in units of θ0) for M=8 and (a) K=10 and (b) K=103. Plotted here are the linear UBs (solid curves) for one, three, five, and seven-pixel imagers, the MMSE (dotted curve) for the one-pixel imager, and quadratic UBs (dashed curves) for the one- and three-pixel imagers.

Fig. 9.
Fig. 9.

Log-log plot of the quadratic UB on the MMSE (solid curve) and CRB (dashed line) versus source strength for the three-pixel imager. As in Fig. 6, the dotted horizontal lines are drawn at heights (θ0/M)2/12, corresponding to the variance of the uniform distribution over a subpixel of width θ0/M.

Equations (74)

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

j=argmaxm{P(x|m)σm}
PM(max)=E(max{P(m|x)m=1,,M}).
Pe(min)=1E(max{P(m|x)m=1,,M}).
χ(n;x)=P¯(x)n[m=1MPn(m|x)]1/n.
Pe(min)=1E[χ(;X)]
=E[1dnχ(n;X)n].
Zn(x)=m=1MPn(m|x),
χ(n;x)n=χ(n;x)[1n2lnZn(x)+1nZn(x)m=1MPn(m|x)lnP(m|x)].
Pn(m|x)=1Zn(x)Pn(m|x).
χ(n;x)n=1n2χ(n;x)m=1MPn(m|x)lnPn(m|x)=1n2χ(n;x)Hn(H|x),
Hn(H|X=x)=m=1MPn(m|x)lnPn(m|x),
Pe(min)=1Ex{exp[1dnn2Hn(H|X=x)]}.
Pe(min)1exp[1dnn2Hn(H|X)],
Pe(min)=E[1dnn2χ(n;X)Hn(H|X)],
Pe,LB(min)Pe(min)Pe,UB(min),
Pe,LB(min)=E[χ(;X)1dnn2Hn(X)];
Pe,UB(min)=E{1dnn2Hn(X)}.
Pe,LB(min)=E[χ(;X)lnχ(;X)],
Pe,UB(min)=E[lnχ(;X)]=E{ln[1/χ(;X)]},
P(xN|m)=n=1NP(xn|m),
P(xN)=m=1Mσmn=1NP(xn|m),
P(m|xN)=σmn=1NP(xn|m)m=1Mσmn=1NP(xn|m)=1[1+mmσmσmn=1NP(xn|m)P(xn|m)].
m*=argmaxmP(m|xN)
χ(;xN)=max{P(m|xN)|m=1,,M}=P(m*|xN)=11+mm*rmm*,
rmm*=n=1N[P(xn|m)P(xn|m*)]=P(xN|m)P(xN|m*).
Pe(min)E[mm*rmm*].
P(xN)=1Mm=1MP(xN|m),
Pe(min)1Mm*=1MdxNP(xN|m*)mm*rmm*=1Mm1=1Mm2m1Em1(rm2m1),
m1m2m1Rm1dxNmin[P(xN|m1),P(xN|m2)]m1m2m1Rm1dxNPs(xN|m1)P1s(xN|m2)],for anys[0,1].
maxm2m1dxNPs(xN|m1)P1s(xN|m2)],
[dxPs(x|m1)P1s(x|m2)]N.
limN1NlnPe(min)maxm2m1infs(0,1)ln[dxPs(x|m1)P1s(x|m2)]minm2m1{infs(0,1)ln[dxPs(x|m1)P1s(x|m2)]}=minm2m1C(m1,m2).
ΔPe=Pe(min)Pe,LB(min)ΔPe(max)Pe,UB(min)Pe,LB(min)=E{[1χ(;XN)]ln[1/χ(;XN)]}.
ΔPeE[mm*rmm*]2.
mm*rmm*11+mm*rmm*ln(1+mm*rmm*)mm*rmm*[1(1mm*rmm*)2]=O(mm*rmm*)2,
MMSE=σP2σMMSEE2=E(θ2)E(θ^MMSE2),
θ^MMSE(x)=mθmP(m|x).
P(m|x)=P(x|m)σmmP(x|m)σm=11+mmP(x|m)σmP(x|m)σm1mmP(x|m)σmP(x|m)σm,
P(m|x)P(x|m)σmP(x|m)σm1.
MMSE=mσmθm2mθm2P2(m|x)P(x)dxmmmθmθmP(m|x)P(m|x)P(x)dx.
MMSE=mσmθm2mθm2RmP2(m|x)P(x)dx2mmmθmθmRmP(m|x)P(m|x)P(x)dx+O(ϵ2).
P(x)=P(x|m)σm+mmP(x|m)σm,
MMSE=mσmθm2[1RmP(x|m)dx]+mθm2RmmmP(x|m)σmdx2mmmθmθmRmσmP(x|m)P(m|x)dx+O(ϵ2).
RmP(x|m)dx+R¯mP(x|m)dx=1,
MMSE=mσmθm2R¯mP(x|m)dx+mθm2RmmmP(x|m)σmdx2mmmθmθmRmP(x|m)σmdx+O(ϵ2).
Imm=RmσmP(x|m)dx,
MMSE=mθm2mm(Imm+Imm)2mmmθmθmImm+O(ϵ2).
MMSE=mmm(θm2θmθm)(Imm+Imm)+O(ϵ2)=12mm(θmθm)2(Imm+Imm)+O(ϵ2).
Pe(min)=1mRmσmP(x|m)dx=mmmRmσmP(x|m)dx=mmmImm=12mmm(Imm+Imm),
θm=(m1/2)Mθ0,m=1,,M.
h(θθm)=1πθ02exp[(θθm)2/θ02].
Km(p)=ηK(p0.5)L(p+0.5)Lh(θθm)dθ=ηKπ(p0.5)L/θ0(p+0.5)L/θ0exp[(uθm/θ0)2]du=ηK2[erfc((p0.5)Lθ0m1/2M)erfc((p+0.5)Lθ0m1/2M)],
erfc(t)=2πtexp(u2)du.
ddterfc(t)=2πexp(t2),
dKm(p)dm=KMπ{exp[((p0.5)lm1/2M)2]exp[((p+0.5)lm1/2M)2]}
P(k|m)=12πσ2exp[(kKm)2/(2σ2)],
Pe(min)=11MP(k|m*(k))dk,
m*(k)={1forK1+K22<k<2forK2+K32<kK1+K22Mfor<kKM1+KM2.
Pe(min)=11Mm=1MAmBm12πσ2dkexp[(kKm)2/(2σ2)],
Am=Km+Km+12,Bm=Km+Km12,m=1,,M,
Pe(min)=11M12Mm=1M[erfc(am)erfc(bm)],
am=(kmkm+1)22SNR,bm=(km1km)22SNR,
p(k|m)=Kmkexp(Km)k!,k=0,1,,.
kln(Km/Km±1)KmKm±1,
ξm+1kξm1,ξmfloor[Kkmkm+1ln(km/km+1)],
Pe(min)=11Mm=1Mk=ξm+1ξm1Kmkexp(Km)k!,
k=ξ1+1p(k|1)=k=0p(k|1)k=0ξ1p(k|1)=1k=0ξ1p(k|1),
Pe(min)=11M1Mm=2M1k=ξm+1ξm1p(k|m)+1M[k=0ξ1p(k|1)k=0ξM1p(k|M)].
σP2=1Mm=1M[m(M+1)/2]2θ02M2=θ0212(11M2),
E[θ^2(k)]=θ02M4m=1Mm=1M(m1/2)(m1/2)kp(k|m)p(k|m)p(k),
CRB(θ)=1/J(θ),J(θ)=defE[2θ2lnp(k|θ)].
J(θ)=(Mθ0)21Mm=1M1Km(Kmm)2.
MMSEE[δθ^(k)δθ]2,
MMSEθ02M2m=1M(ηm+1Imm+1+ηm1Imm1)+O(ϵ2);Pe(min)m=1M(ηm+1Imm+1+ηm1Imm1),

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