Abstract

This paper employs a recently developed asymptotic Bayesian multi-hypothesis testing (MHT) based error analysis to treat the problem of superresolution imaging of a pair of closely spaced, equally bright point sources. The analysis exploits the notion of the minimum probability of error (MPE) in discriminating between two competing equi-probable hypotheses, a single point source of a certain brightness at the origin vs. a pair of point sources, each of half the brightness of the single source and located symmetrically about the origin, as the distance between the source pair is changed. For a Gaussian point-spread function (PSF), the analysis makes predictions on the scaling of the minimum source strength, expressed in units of photon number, required to disambiguate the pair as a function of their separation in both the signal-dominated and background-dominated regimes. Certain logarithmic corrections to the quartic scaling of the minimum source strength with respect to the degree of superresolution characterize the signal-dominated regime, while the scaling is purely quadratic in the background-dominated regime. For the Gaussian PSF, general results for arbitrary strengths of the signal, background, and sensor noise levels are also presented.

© 2014 Optical Society of America

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References

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  1. B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annual Rev. Biochem.78, 993–1016 (2009).
    [CrossRef]
  2. S. Prasad, “Asymptotics of Bayesian error probability and source super-localization in three dimensions,” Opt. Express22, 16008–16028 (2014).
  3. G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annual Rev. Phys. Chem.61, 345–367 (2010).
    [CrossRef]
  4. T. Klein, S. Proppert, and M. Sauer, “Eight years of single-molecule localization microscopy,” Histochem. Cell Biol., Feb2014 (e-print).
    [CrossRef] [PubMed]
  5. A. Gahlmann and W. Moerner, “Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging,” Nature Rev. Microbiol.12, 9–22 (2014).
    [CrossRef]
  6. J. Harris, “Resolving power and decision theory,” J. Opt. Soc. Am.54, 606–611 (1964).
    [CrossRef]
  7. C. Helstrom, “Detection and resolution of incoherent objects by a background-limited optical system,” J. Opt. Soc. Am.59, 164–175 (1969).
    [CrossRef]
  8. A. van den Bos, “Resolution in model-based measurement,” IEEE Trans. Instrum. Meas.51, 1055–1060 (2002).
    [CrossRef]
  9. M. Shahram and P. Milanfar, “Imaging below the diffraction limit: a statistical analysis,” IEEE Trans. Image Process.13, 677–689 (2004).
    [CrossRef] [PubMed]
  10. J. Chao, S. Ram, E. S. Ward, and R. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express17, 24377–24402 (2009).
    [CrossRef]
  11. S. Ram, E. S. Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, pp. 426–435 (2005).
    [CrossRef] [PubMed]
  12. C. Rushforth and R. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am.58, 539–545 (1968).
    [CrossRef]
  13. M. Bertero and C. De Mol, “Superresolution by data inversion,” Progress in OpticsXXXVI, 129–178 (1996).
    [CrossRef]
  14. E. Kosarev, “Shannon’s superresolution limit for signal recovery,” Inverse Prob.6, 55–76 (1990).
    [CrossRef]
  15. E. Boukouvala and A. Lettington, “Restoration of astronomical images by an iterative superresolving algorithm,” Astron. Astrophys.399, 807–811 (2003).
    [CrossRef]
  16. C. Matson and D. Tyler, “Primary and secondary superresolution by data inversion,” Opt. Express14, 456–473 (2006).
    [CrossRef] [PubMed]
  17. S. Prasad and X. Luo, “Support-assisted optical superresolution of low-resolution image sequences: the one-dimensional problem,” Opt. Express17, 23213–23233 (2009).
    [CrossRef]
  18. R. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta21, 709–721 (1974).
    [CrossRef]
  19. D. Fried, “Analysis of the CLEAN algorithm and implications for superresolution,” J. Opt. Soc. Am. A12, 853–860 (1995).
    [CrossRef]
  20. P. Magain, F. Courbin, and S. Sohy, “Deconvolution with correct sampling,” Astrophys. J.494, 472–477 (1998).
    [CrossRef]
  21. J. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pacific114, 1051–1069 (2002).
    [CrossRef]
  22. R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE7094, 709405 (2008).
    [CrossRef]
  23. L. Lucy, “Statistical limits to superresolution,” Astron. Astrophys.261, 706–710 (1992).
  24. R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J.82, 2775–2783 (2002).
    [CrossRef] [PubMed]
  25. J. Enderlein, E. Toprak, and P. Selvin, “Polarization effect on position accuracy of fluorophore localization,” Opt. Express14, 8111–8120 (2006).
    [CrossRef] [PubMed]
  26. S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA103, 4457–4462 (2006).
    [CrossRef]
  27. G. Carrier, M. Krook, and C. Pearson, Functions of a Complex Variable (McGraw HillNew York, 1966), Sec. 2.8,
  28. S. Smith, “Statistical resolution limits and the complexified Cramér-Rao bounds,” IEEE Trans. Signal Process.53, 1597–1609 (2005).
    [CrossRef]

2014 (2)

S. Prasad, “Asymptotics of Bayesian error probability and source super-localization in three dimensions,” Opt. Express22, 16008–16028 (2014).

A. Gahlmann and W. Moerner, “Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging,” Nature Rev. Microbiol.12, 9–22 (2014).
[CrossRef]

2010 (1)

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annual Rev. Phys. Chem.61, 345–367 (2010).
[CrossRef]

2009 (3)

2008 (1)

R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE7094, 709405 (2008).
[CrossRef]

2006 (3)

2005 (2)

S. Ram, E. S. Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, pp. 426–435 (2005).
[CrossRef] [PubMed]

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

2004 (1)

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

2003 (1)

E. Boukouvala and A. Lettington, “Restoration of astronomical images by an iterative superresolving algorithm,” Astron. Astrophys.399, 807–811 (2003).
[CrossRef]

2002 (3)

A. van den Bos, “Resolution in model-based measurement,” IEEE Trans. Instrum. Meas.51, 1055–1060 (2002).
[CrossRef]

J. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pacific114, 1051–1069 (2002).
[CrossRef]

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

1998 (1)

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

1996 (1)

M. Bertero and C. De Mol, “Superresolution by data inversion,” Progress in OpticsXXXVI, 129–178 (1996).
[CrossRef]

1995 (1)

1992 (1)

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

1990 (1)

E. Kosarev, “Shannon’s superresolution limit for signal recovery,” Inverse Prob.6, 55–76 (1990).
[CrossRef]

1974 (1)

R. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta21, 709–721 (1974).
[CrossRef]

1969 (1)

1968 (1)

1964 (1)

Bates, M.

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annual Rev. Biochem.78, 993–1016 (2009).
[CrossRef]

Bertero, M.

M. Bertero and C. De Mol, “Superresolution by data inversion,” Progress in OpticsXXXVI, 129–178 (1996).
[CrossRef]

Boukouvala, E.

E. Boukouvala and A. Lettington, “Restoration of astronomical images by an iterative superresolving algorithm,” Astron. Astrophys.399, 807–811 (2003).
[CrossRef]

Carrier, G.

G. Carrier, M. Krook, and C. Pearson, Functions of a Complex Variable (McGraw HillNew York, 1966), Sec. 2.8,

Chao, J.

Courbin, F.

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

Davidson, M.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annual Rev. Phys. Chem.61, 345–367 (2010).
[CrossRef]

De Mol, C.

M. Bertero and C. De Mol, “Superresolution by data inversion,” Progress in OpticsXXXVI, 129–178 (1996).
[CrossRef]

Enderlein, J.

Fried, D.

Gahlmann, A.

A. Gahlmann and W. Moerner, “Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging,” Nature Rev. Microbiol.12, 9–22 (2014).
[CrossRef]

Gerchberg, R.

R. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta21, 709–721 (1974).
[CrossRef]

Harris, J.

Harris, R.

Helstrom, C.

Hier, R.

R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE7094, 709405 (2008).
[CrossRef]

Huang, B.

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annual Rev. Biochem.78, 993–1016 (2009).
[CrossRef]

Klein, T.

T. Klein, S. Proppert, and M. Sauer, “Eight years of single-molecule localization microscopy,” Histochem. Cell Biol., Feb2014 (e-print).
[CrossRef] [PubMed]

Kosarev, E.

E. Kosarev, “Shannon’s superresolution limit for signal recovery,” Inverse Prob.6, 55–76 (1990).
[CrossRef]

Krook, M.

G. Carrier, M. Krook, and C. Pearson, Functions of a Complex Variable (McGraw HillNew York, 1966), Sec. 2.8,

Larson, D.

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

Lettington, A.

E. Boukouvala and A. Lettington, “Restoration of astronomical images by an iterative superresolving algorithm,” Astron. Astrophys.399, 807–811 (2003).
[CrossRef]

Lippincott-Schwartz, J.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annual Rev. Phys. Chem.61, 345–367 (2010).
[CrossRef]

Lucy, L.

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

Luo, X.

Magain, P.

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

Manley, S.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annual Rev. Phys. Chem.61, 345–367 (2010).
[CrossRef]

Matson, C.

Milanfar, P.

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

Moerner, W.

A. Gahlmann and W. Moerner, “Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging,” Nature Rev. Microbiol.12, 9–22 (2014).
[CrossRef]

Murtagh, F.

J. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pacific114, 1051–1069 (2002).
[CrossRef]

Ober, R.

J. Chao, S. Ram, E. S. Ward, and R. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express17, 24377–24402 (2009).
[CrossRef]

S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA103, 4457–4462 (2006).
[CrossRef]

S. Ram, E. S. Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, pp. 426–435 (2005).
[CrossRef] [PubMed]

Pantin, E.

J. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pacific114, 1051–1069 (2002).
[CrossRef]

Patterson, G.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annual Rev. Phys. Chem.61, 345–367 (2010).
[CrossRef]

Pearson, C.

G. Carrier, M. Krook, and C. Pearson, Functions of a Complex Variable (McGraw HillNew York, 1966), Sec. 2.8,

Prasad, S.

Proppert, S.

T. Klein, S. Proppert, and M. Sauer, “Eight years of single-molecule localization microscopy,” Histochem. Cell Biol., Feb2014 (e-print).
[CrossRef] [PubMed]

Puetter, R.

R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE7094, 709405 (2008).
[CrossRef]

Ram, S.

J. Chao, S. Ram, E. S. Ward, and R. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express17, 24377–24402 (2009).
[CrossRef]

S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA103, 4457–4462 (2006).
[CrossRef]

S. Ram, E. S. Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, pp. 426–435 (2005).
[CrossRef] [PubMed]

Rushforth, C.

Sally Ward, E.

S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA103, 4457–4462 (2006).
[CrossRef]

Sauer, M.

T. Klein, S. Proppert, and M. Sauer, “Eight years of single-molecule localization microscopy,” Histochem. Cell Biol., Feb2014 (e-print).
[CrossRef] [PubMed]

Selvin, P.

Shahram, M.

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

Smith, S.

S. Smith, “Statistical resolution limits and the complexified Cramér-Rao bounds,” 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]

Starck, J.

J. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pacific114, 1051–1069 (2002).
[CrossRef]

Thompson, R.

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

Toprak, E.

Tyler, D.

van den Bos, A.

A. van den Bos, “Resolution in model-based measurement,” IEEE Trans. Instrum. Meas.51, 1055–1060 (2002).
[CrossRef]

Ward, E. S.

J. Chao, S. Ram, E. S. Ward, and R. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express17, 24377–24402 (2009).
[CrossRef]

S. Ram, E. S. Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, pp. 426–435 (2005).
[CrossRef] [PubMed]

Webb, W.

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

Zhuang, X.

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annual Rev. Biochem.78, 993–1016 (2009).
[CrossRef]

Annual Rev. Biochem. (1)

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annual Rev. Biochem.78, 993–1016 (2009).
[CrossRef]

Annual Rev. Phys. Chem. (1)

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annual Rev. Phys. Chem.61, 345–367 (2010).
[CrossRef]

Astron. Astrophys. (2)

E. Boukouvala and A. Lettington, “Restoration of astronomical images by an iterative superresolving algorithm,” Astron. Astrophys.399, 807–811 (2003).
[CrossRef]

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] [PubMed]

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] [PubMed]

IEEE Trans. Instrum. Meas. (1)

A. van den Bos, “Resolution in model-based measurement,” IEEE Trans. Instrum. Meas.51, 1055–1060 (2002).
[CrossRef]

IEEE Trans. Signal Process. (1)

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

Inverse Prob. (1)

E. Kosarev, “Shannon’s superresolution limit for signal recovery,” Inverse Prob.6, 55–76 (1990).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Nature Rev. Microbiol. (1)

A. Gahlmann and W. Moerner, “Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging,” Nature Rev. Microbiol.12, 9–22 (2014).
[CrossRef]

Opt. Acta (1)

R. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta21, 709–721 (1974).
[CrossRef]

Opt. Express (5)

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

S. Ram, E. Sally Ward, and R. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA103, 4457–4462 (2006).
[CrossRef]

Proc. SPIE (2)

R. Puetter and R. Hier, “Pixon sub-diffraction space imaging,” Proc. SPIE7094, 709405 (2008).
[CrossRef]

S. Ram, E. S. Ward, and R. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, pp. 426–435 (2005).
[CrossRef] [PubMed]

Progress in Optics (1)

M. Bertero and C. De Mol, “Superresolution by data inversion,” Progress in OpticsXXXVI, 129–178 (1996).
[CrossRef]

Publ. Astron. Soc. Pacific (1)

J. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: a review,” Publ. Astron. Soc. Pacific114, 1051–1069 (2002).
[CrossRef]

Other (2)

G. Carrier, M. Krook, and C. Pearson, Functions of a Complex Variable (McGraw HillNew York, 1966), Sec. 2.8,

T. Klein, S. Proppert, and M. Sauer, “Eight years of single-molecule localization microscopy,” Histochem. Cell Biol., Feb2014 (e-print).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

The image of a pair of point sources of equal brightness under the Gaussian-shaped PSF (14), for (a) d/w = 0.5; (b) d/w = 2; and (c) d/w = 3.

Fig. 2
Fig. 2

A plot of MPE vs. signal strength for three different background levels. (a) b = 102; (b) b = 106; and (c) b = 109.

Fig. 3
Fig. 3

Log-log plots of the minimum source strength, Kmin, vs. the OSR ratio, w/d, for the same three different background levels as in Figs. 2.

Fig. 4
Fig. 4

Plots of MPE vs. signal strength for two different background levels, b = 1 (solid curves) and b = 100 (dashed curves), for four different OSR factors, 4x, 8x, 12x, and 20x, and sensor noise variance, σ2 = 1.

Fig. 5
Fig. 5

A plot of MPE vs. signal strength for two different noise levels, (a) b = 1, σ = 1 and (b) b = 100, σ = 10, for three different pixel sizes, ΔA/(2πw2) equal to 1/100, 4/100, and 9/100. The dashed (red), solid (blue), and dot-dashed (black) curves correspond to the plots for the smallest to the largest pixel sizes, respectively. Five different values of the OSR factor, namely 4x, 8x, 12x, 16x, and 20x, were considered in these plots. The vertical dashed line passes through the common intersection points of the curves grouped by the OSR factor.

Equations (56)

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

P e ( min ) = 1 𝔼 [ P ( m ^ MAP | X ) ] ,
m ^ MAP ( X ) = argmax m = 1 , , M P ( m | X ) .
P e ( min ) = m = 1 M p m m m m d x P ( x | m )
m ˜ = argmax m m max x m { P ( x | m ) } .
P e ( min ) = m = 1 M p m m ˜ d x P ( x | m )
P ( x | m ) = 1 ( 2 π ) N / 2 det 1 / 2 ( Σ m ) exp [ ( 1 / 2 ) ( x T x m T ) Σ m 1 ( x x m ) ] .
Σ m = diag ( σ 2 + b + x m ) ,
P e ( min ) = 1 2 m p m erfc ( U m 2 / 2 ) ,
U m 2 = 1 2 i = 1 N ( σ 2 + b + x m i ) 1 / 2 ( σ 2 + b + x ¯ m i ) 2 ( δ x m ˜ m i ) 2 [ i = 1 N 1 ( σ 2 + b + x ¯ m i ) 2 ( δ x m ˜ m i ) 2 ] 1 / 2
U 0 2 = def 1 2 Q R 1 / 2 ,
Q = def i = 1 N 1 ( σ 2 + b + x 1 i ) 3 / 2 ( x 2 i x 1 i ) 2 ; R = def i = 1 N 1 ( σ 2 + b + x 1 i ) 2 ( x 2 i x 1 i ) 2 .
P e ( min ) = 1 2 erfc ( U 0 2 / 2 ) .
H ( ξ , η ) = 1 2 π w 2 exp [ ( ξ 2 + η 2 ) 2 w 2 ] ,
H i j = 1 2 π w 2 exp [ ( ξ i j 2 + η i j 2 ) 2 w 2 ] Δ A ,
i , j H i j = 1 ,
x 1 i j = K H 1 i j = K 2 π w 2 exp [ ξ i j 2 + η i j 2 2 w 2 ] Δ A ; x 2 i j = K H 2 i j = K 4 π w 2 { exp [ ( ξ i j d / 2 ) 2 + η i j 2 2 w 2 ] + exp [ ( ξ i j + d / 2 ) 2 + η i j 2 2 w 2 ] } Δ A ,
Q = K 2 Δ A ( 2 π w 2 ) ( σ 2 + b ) 3 / 2 n = 0 ( 1 ) n ( 2 n + 1 ) ! ! n ! ( n + 2 ) [ K Δ A 2 ( σ 2 + b ) 2 π w 2 ] n × [ 1 + 1 2 e n d 2 4 ( n + 2 ) w 2 2 e ( n + 1 ) d 2 8 ( n + 2 ) w 2 + 1 2 e d 2 4 w 2 ] ,
( d 2 8 w 2 ) 2 ( n 2 + 2 n + 3 ) ( n + 2 ) 2 ,
Q = K 2 Δ A ( d / w ) 4 ( 128 π w 2 ) ( σ 2 + b ) 3 / 2 n = 0 ( 1 ) n ( 2 n + 1 ) ! ! n ! ( n + 2 ) × ( n 2 + 2 n + 3 ) ( n + 2 ) 2 [ K Δ A 2 ( σ 2 + b ) 2 π w 2 ] n .
n 2 + 2 n + 3 ( n + 2 ) 2 = 1 2 n + 2 + 3 ( n + 2 ) 2 ,
Q = K 2 Δ A ( d / w ) 4 ( 128 π w 2 ) ( σ 2 + b ) 3 / 2 [ q 1 ( u ) 2 q 2 ( u ) + 3 q 3 ( u ) ] ,
q k ( u ) = def n = 0 ( 1 ) n ( 2 n + 1 ) ! ! 2 n n ! ( n + 2 ) k u n , u = def K Δ A 2 π w 2 ( σ 2 + b 2 ) .
n = 0 ( 1 ) n ( 2 n + 1 ) ! ! 2 n n ! ( n + 2 ) k 1 v n + 1 ,
u 2 q k ( u ) = 0 u v q k 1 ( v ) d u .
u 2 q 1 ( u ) = 0 u d v v ( 1 + v ) 3 / 2 = 0 u d v [ 1 ( 1 + v ) 1 / 2 1 ( 1 + v ) 3 / 2 ] = 2 ( 1 + u ) 1 / 2 + 2 ( 1 + u ) 1 / 2 4 ;
u 2 q 2 ( u ) = 2 0 u d v v 1 [ ( 1 + v ) 1 / 2 + ( 1 + v ) 1 / 2 2 ] = 4 ( 1 + u 1 ) 8 ln ( 1 + 1 + u 2 ) ;
u 2 q 3 ( u ) = 4 0 u d v v [ ( 1 + v ) 1 / 2 1 ] 8 0 u d v v ln ( 1 + 1 + v 2 ) .
R = K 2 Δ A ( d / w ) 4 ( 128 π w 2 ) ( σ 2 + b ) 2 n = 0 ( 1 ) n ( n + 1 ) ( n + 2 ) × ( n 2 + 2 n + 3 ) ( n + 2 ) 2 [ K Δ A 2 π w 2 ( σ 2 + b ) ] n .
n 2 + 2 n + 3 ( n + 2 ) 2 = 1 2 n + 1 + 3 ( n + 2 ) 2 ,
R = K 2 Δ A ( d / w ) 4 ( 128 π w 2 ) ( σ 2 + b ) 2 [ r 1 ( u ) 2 r 2 ( u ) + 3 r 3 ( u ) ] ,
r k ( u ) = def n = 0 ( 1 ) n ( n + 1 ) ( n + 2 ) k u n , k = 1 , 2 , 3 .
u 2 r k ( u ) = 0 u v r k 1 ( v ) d v .
r 1 ( u ) = 1 u 2 0 u v ( 1 + v ) 2 d v = 1 u 2 0 u [ 1 1 + v 1 ( 1 + v ) 2 ] d v = 1 u 2 [ ln ( 1 + u ) + 1 1 + u 1 ] = 1 u 2 ln ( 1 + u ) 1 u ( 1 + u ) .
r 2 ( u ) = 1 u 2 0 u d v [ ln ( 1 + v ) v 1 ( 1 + v ) ] = 1 u 2 [ 0 u d v ln ( 1 + v ) v ln ( 1 + u ) ] ,
r 3 ( u ) = 1 u 2 0 u d v v [ 0 v d w ln ( 1 + w ) w ln ( 1 + v ) ] = 1 u 2 0 u ( ln u ln v ) ln ( 1 + v ) v 1 u 2 0 u d v ln ( 1 + v ) ,
U 0 2 2 = K Δ A 1 / 2 ( d / w ) 2 16 [ ( 4 π w 2 ) ( σ 2 + b ) ] 1 / 2 [ q 1 ( u ) 2 q 2 ( u ) + 3 q 3 ( u ) ] [ r 1 ( u ) 2 r 2 ( u ) + 3 r 3 ( u ) ] 1 / 2 ,
q 1 ( u ) ~ 2 u 3 / 2 ; q 2 ( u ) ~ 4 u 3 / 2 ; q 3 ( u ) ~ 8 u 3 / 2 ; r 1 ( u ) ~ ln u u 2 ; r 2 ( u ) ~ 1 2 u 2 [ ( ln u ) 2 2 ln u ] ; r 3 ( u ) ~ 1 u 2 [ ( ln u ) 3 6 ( ln u ) 2 2 ] .
U 0 2 2 = K Δ A 1 / 2 ( d / w ) 2 16 [ ( 4 π w 2 ) ( σ 2 + b ) ] 1 / 2 18 2 u ln 3 / 2 ( u ) = 9 8 K d 2 / w 2 ln 3 / 2 ( u ) .
min U 0 2 2 = erfc 1 ( 2 p )
K min = [ 8 erfc 1 ( 2 p ) 9 ] 2 w 4 d 3 ln 3 [ K min Δ A / ( 2 π w 2 ) ( b + σ 2 ) ] .
U 0 2 2 = 3 8 K Δ A 1 / 2 ( d / w ) 2 16 [ ( 4 π w 2 ) ( σ 2 + b ) ] 1 / 2 ,
K min Δ A 1 / 2 ( d / w ) 2 16 [ ( 4 π w 2 ) ( σ 2 + b ) ] 1 / 2 , = 8 3 erfc 1 ( 2 p ) .
K CO Δ A 0 2 π w 2 100 ( σ 2 + b ) .
d ξ d η exp [ n ( ξ 2 + η 2 ) 2 w 2 ] { exp [ ξ 2 + η 2 2 w 2 ] 1 2 exp [ ( ξ d / 2 ) 2 + η 2 2 w 2 ] 1 2 exp [ ( ξ + d / 2 ) 2 + η 2 2 w 2 ] } 2 .
I 0 = def exp [ ( ξ 2 + η 2 ) ( n + 2 ) 2 w 2 ] d ξ d η ; I ± = def exp [ ( n + 1 ) ξ 2 + ( ξ d / 2 ) 2 + ( n + 2 ) η 2 2 w 2 ] d ξ d η ; J ± = def exp [ n ξ 2 + 2 ( ξ d / 2 ) 2 + ( n + 2 ) η 2 2 w 2 ] d ξ d η ; and J 0 = def exp [ n ξ 2 + ( ξ d / 2 ) 2 + ( ξ + d / 2 ) 2 + ( n + 2 ) η 2 2 w 2 ] d ξ d η .
I 0 = 2 π w 2 n + 2 .
( n + 1 ) ξ 2 + ( ξ d / 2 ) 2 = ( n + 2 ) [ ξ d 2 ( n + 2 ) ] 2 + ( n + 1 ) d 2 4 ( n + 2 ) ,
I ± = 2 π w 2 ( n + 2 ) exp [ d 2 8 w 2 ( n + 1 n + 2 ) ] ; J ± = 2 π w 2 ( n + 2 ) exp [ d 2 4 w 2 ( n n + 2 ) ] ; and J 0 = 2 π w 2 ( n + 2 ) exp [ d 2 4 w 2 ] .
2 0 sinh 1 u d α ( cosh α 1 ) 2 sinh α = 4 0 sinh 1 u d α sinh 3 ( α / 2 ) cosh ( α / 2 ) ,
8 1 cosh ( sinh 1 u / 2 ) d β β 2 1 β .
4 [ cosh 2 ( sinh 1 u / 2 ) 1 ] 8 lncosh ( sinh 1 u / 2 ) = 4 sinh 2 ( sinh 1 u / 2 ) 4 ln [ 1 + cosh ( sinh 1 u ) 2 ] = 2 [ cosh ( sinh 1 u ) 1 ] 4 ln [ 1 + 1 + u 2 ] = 2 [ 1 + u 1 ] 4 ln ( 1 + 1 + u 2 ) ,
q 1 ( u 1 ) ~ 2 u 3 / 2 , q 2 ( u 1 ) ~ 4 u 3 / 2 .
[ ( 1 + v ) 1 / 2 1 ] v ~ v 1 / 2 , ln ( 1 + 1 + v 2 ) ~ 1 2 ln v ,
q 2 ( u > > 1 ) ~ 8 u 3 / 2 2 u 2 ln 2 ( u ) ~ 8 u 3 / 2 ,
r 2 ( u 1 ) ~ 1 2 u 2 ( ln 2 u 2 ln u ) .
r 3 ( u ) ~ 1 u 2 ( ln 3 u 2 ln 3 u 3 ln 2 u 2 ) = 1 u 2 ( ln 3 u 6 ln 2 u 2 ) .

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