Abstract

We analyze the estimation precision of the parameter of the orthogonal state contrast image (OSCI) under coherent illumination. This parameter represents the degree of polarization of the light if the materials that compose the scene are purely depolarizing. Two different estimation modes are considered, depending on the uniformity of the illumination of the scene. We first determine lower bounds on the estimation precision in both cases by computing the Cramer–Rao lower bounds (CRLBs) for unbiased estimation. This allows us to compare the potential precision that can be reached in each mode. We then consider the estimators based on empirical averaging of the data, and we show that there are cases where they are strongly biased. We thus propose and characterize another estimator based on the natural representation of the OSCI, which is asymptotically unbiased and whose variance is close to the unbiased CRLB.

© 2005 Optical Society of America

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References

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  1. R. B. Holmes, “Applications of lasers to imaging of distant objects,” in Intense Laser Beams and Applications, W. E. McDermott, ed., Proc. SPIE1871, 306–315 (1993).
  2. G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
    [CrossRef]
  3. D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman and Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
  4. R. C. Hardie, M. Vadyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
    [CrossRef]
  5. L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
    [CrossRef]
  6. B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
  7. J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
    [CrossRef]
  8. R. A. Chipman, “Polarization diversity active imaging,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 68–73 (1997).
  9. S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., Proc. SPIE3707, 449–460 (1999).
  10. P. Terrier, V. DeVlaminck, “Robust and accurate estimate of the orientation of partially polarized light from a camera sensor,” Appl. Opt. 40, 5233–5239 (2001).
    [CrossRef]
  11. P. Gerligand, M. H. Smith, R. A. Chipman, “Polarimetric images of a cone,” Opt. Express 4, 420–430 (1999).
    [CrossRef] [PubMed]
  12. S. Breugnot, P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806nm,” Opt. Eng. 39, 2681–2688 (2000).
    [CrossRef]
  13. J. W. Goodman, “Laser speckle and related phenomena,” in Statistical Properties of Laser Speckle Patterns, Vol. 9 of Topics in Applied Physics (Springer-Verlag, 1975), pp. 9–75.
  14. W. G. Egan, W. R. Johnson, V. S. Whitehead, “Terrestrial polarization imagery obtained from the Space Shuttle: characterization and interpretation,” Appl. Opt. 30, 435–442 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
  16. F. Goudail, Ph. Réfrégier, “Statistical algorithms for target detection in coherent active polarimetric images,” J. Opt. Soc. Am. A 18, 3049–3060 (2001).
    [CrossRef]
  17. F. Goudail, Ph. Réfrégier, “Target segmentation in active polarimetric images by use of statistical active contours,” Appl. Opt. 41, 874–883 (2002).
    [CrossRef] [PubMed]
  18. S. L. Jacques, J. C. Ramella-Roman, K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
    [CrossRef] [PubMed]
  19. T. S. Ferguson, “Exponential families of distributions,” in Mathematical Statistics, a Decision Theoretic Approach (Academic, 1967), pp. 125–132.
  20. J. W. Goodman, “Some problems involving high-order coherence,” in Statistical Optics (Wiley, 1985), pp. 237–285.
  21. J. W. Goodman, “The speckle effect in coherent imaging,” in Statistical Optics (Wiley, 1985), pp. 347–356.
  22. H. L. Van Trees, Detection, Estimation and Modulation Theory. Part I: Detection, Estimation and Linear Modulation Theory (Wiley, 1968).
  23. Ph. Réfrégier, F. Goudail, N. Roux, “Estimation of the degree of polarization in active coherent imagery using the natural representation,” J. Opt. Soc. Am. A 21, 2292–2300 (2004).
    [CrossRef]
  24. Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004).
    [CrossRef]
  25. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

2004 (1)

2002 (2)

F. Goudail, Ph. Réfrégier, “Target segmentation in active polarimetric images by use of statistical active contours,” Appl. Opt. 41, 874–883 (2002).
[CrossRef] [PubMed]

S. L. Jacques, J. C. Ramella-Roman, K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

2001 (3)

2000 (1)

S. Breugnot, P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806nm,” Opt. Eng. 39, 2681–2688 (2000).
[CrossRef]

1999 (1)

1998 (1)

R. C. Hardie, M. Vadyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

1996 (1)

G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
[CrossRef]

1995 (1)

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

1991 (1)

1990 (1)

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[CrossRef]

Barclay, H. T.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

Beraldin, J. A.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman and Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).

Blais, F.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman and Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).

Breugnot, S.

S. Breugnot, P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806nm,” Opt. Eng. 39, 2681–2688 (2000).
[CrossRef]

S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., Proc. SPIE3707, 449–460 (1999).

Chipman, R. A.

P. Gerligand, M. H. Smith, R. A. Chipman, “Polarimetric images of a cone,” Opt. Express 4, 420–430 (1999).
[CrossRef] [PubMed]

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

R. A. Chipman, “Polarization diversity active imaging,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 68–73 (1997).

Clémenceau, P.

S. Breugnot, P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806nm,” Opt. Eng. 39, 2681–2688 (2000).
[CrossRef]

Clémenceau, Ph.

S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., Proc. SPIE3707, 449–460 (1999).

Cournoyer, L.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman and Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).

DeVlaminck, V.

Egan, W. G.

Ferguson, T. S.

T. S. Ferguson, “Exponential families of distributions,” in Mathematical Statistics, a Decision Theoretic Approach (Academic, 1967), pp. 125–132.

Gerligand, P.

Goodman, J. W.

J. W. Goodman, “Laser speckle and related phenomena,” in Statistical Properties of Laser Speckle Patterns, Vol. 9 of Topics in Applied Physics (Springer-Verlag, 1975), pp. 9–75.

J. W. Goodman, “Some problems involving high-order coherence,” in Statistical Optics (Wiley, 1985), pp. 237–285.

J. W. Goodman, “The speckle effect in coherent imaging,” in Statistical Optics (Wiley, 1985), pp. 347–356.

Goudail, F.

Hardie, R. C.

R. C. Hardie, M. Vadyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

Holmes, R. B.

R. B. Holmes, “Applications of lasers to imaging of distant objects,” in Intense Laser Beams and Applications, W. E. McDermott, ed., Proc. SPIE1871, 306–315 (1993).

Jacques, S. L.

S. L. Jacques, J. C. Ramella-Roman, K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

Johnson, B.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

Johnson, W. R.

Joseph, R.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

Kerekes, J. P.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

Laurin, D. G.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman and Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).

Lee, K.

S. L. Jacques, J. C. Ramella-Roman, K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

McManamon, P. F.

R. C. Hardie, M. Vadyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

Newbury, A.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

Nischan, M. L.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

Osche, G. R.

G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

Pezzaniti, J. L.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Ramella-Roman, J. C.

S. L. Jacques, J. C. Ramella-Roman, K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

Réfrégier, Ph.

Rioux, M.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman and Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).

Roux, N.

Smith, M. H.

Terrier, P.

Vadyanathan, M.

R. C. Hardie, M. Vadyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory. Part I: Detection, Estimation and Linear Modulation Theory (Wiley, 1968).

Whitehead, V. S.

Willard, B. C.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

Wolff, L. B.

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[CrossRef]

Young, D. S.

G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
[CrossRef]

Zayhowski, J. J.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

Appl. Opt. (3)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[CrossRef]

J. Biomed. Opt. (1)

S. L. Jacques, J. C. Ramella-Roman, K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

Opt. Eng. (3)

S. Breugnot, P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806nm,” Opt. Eng. 39, 2681–2688 (2000).
[CrossRef]

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

R. C. Hardie, M. Vadyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. IEEE (1)

G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
[CrossRef]

Other (12)

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman and Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).

R. B. Holmes, “Applications of lasers to imaging of distant objects,” in Intense Laser Beams and Applications, W. E. McDermott, ed., Proc. SPIE1871, 306–315 (1993).

R. A. Chipman, “Polarization diversity active imaging,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 68–73 (1997).

S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., Proc. SPIE3707, 449–460 (1999).

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, and R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).

J. W. Goodman, “Laser speckle and related phenomena,” in Statistical Properties of Laser Speckle Patterns, Vol. 9 of Topics in Applied Physics (Springer-Verlag, 1975), pp. 9–75.

T. S. Ferguson, “Exponential families of distributions,” in Mathematical Statistics, a Decision Theoretic Approach (Academic, 1967), pp. 125–132.

J. W. Goodman, “Some problems involving high-order coherence,” in Statistical Optics (Wiley, 1985), pp. 237–285.

J. W. Goodman, “The speckle effect in coherent imaging,” in Statistical Optics (Wiley, 1985), pp. 347–356.

H. L. Van Trees, Detection, Estimation and Modulation Theory. Part I: Detection, Estimation and Linear Modulation Theory (Wiley, 1968).

Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004).
[CrossRef]

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

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Figures (9)

Fig. 1
Fig. 1

Variation of the product P n × CRLB ( u 0 ) in the scalar and vectorial cases as a function of u 0 for different values of the speckle order n.

Fig. 2
Fig. 2

(a) ρ u ( u 0 ) as a function of u 0 for three values of n, (b) ρ u ( u 0 ) as a function of n for u 0 = 0 (unpolarized light) and u 0 = 0.9 (almost purely polarized light).

Fig. 3
Fig. 3

Product P n CRLB ( γ 0 ) in the vectorial case and in the scalar case for different values of n as a function of γ 0 .

Fig. 4
Fig. 4

Bias of estimator u ̂ V for different values of P n as a function of u 0 . The solid curves represent bias computed with numerical integration, and the dotted curves represent bias computed with approximation (24).

Fig. 5
Fig. 5

In (a) and (b), the solid curves represent variance of estimator u ̂ V estimated by numerical integration for different values of P n . In (a) the dotted curves represent the approximation of the variance defined in relation (25), and in (b) the dotted curves represent CRLB V .

Fig. 6
Fig. 6

Bias of estimator u ̂ S for different values of n as a function of u 0 . The solid curves represent bias computed with numerical integration, and the dotted curves represent bias computed with approximation (27).

Fig. 7
Fig. 7

Ratio ξ between the variance of γ ̂ and CRLB S ( γ 0 ) as a function of n.

Fig. 8
Fig. 8

Bias of u ̂ γ for different values of n and P. The solid curves represent bias estimated with Monte Carlo simulations on 10 4 samples, and the dotted curves represent bias computed with approximation (32).

Fig. 9
Fig. 9

The solid curves represent variance of u ̂ γ for different values of n and P estimated with Monte Carlo simulations on 10 4 samples, and the dotted curves represent CRLB S for the same values of n and P.

Equations (50)

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P X i ( x ) = n n x n 1 Γ [ n ] m X n exp ( n x m X ) .
u = m X m Y m X + m Y ,
R i = X i Y i X i + Y i = X i Y i X i + Y i .
P β ( β ) = ( 2 n 1 ) ! [ ( n 1 ) ! ] 2 1 { exp [ ( β γ ) 2 ] + exp [ ( β γ ) 2 ] } 2 n ,
u = exp ( γ ) 1 exp ( γ ) + 1 .
CRLB ( u 0 ) = 1 2 ln L ( χ u ) u 2 u = u 0 ,
L ( χ u ) = i = 1 P [ n n x i n 1 m X n Γ ( n ) exp ( n x i m X ) ] [ n n y i n 1 m Y n Γ ( n ) exp ( n y i m Y ) ] .
ln L ( χ u ) = i = 1 P { n ln n + ( n 1 ) ln x i 2 n x i I ( 1 + u ) n ln [ I 2 ( 1 + u ) ] ln Γ ( n ) } + { n ln n + ( n 1 ) ln y i 2 n y i I ( 1 u ) n ln [ I 2 ( 1 u ) ] ln Γ ( n ) } .
2 u 2 ln L ( χ u ) = [ 4 n I ( 1 + u ) 3 i = 1 P x i P n ( 1 + u ) 2 ] [ 4 n I ( 1 u ) 3 i = 1 P y i P n ( 1 u ) 2 ] .
CRLB V ( u 0 ) = 1 2 P n ( 1 u 0 2 ) 2 ( 1 + u 0 2 ) .
P R i ( r ) = B n ( 1 u 2 ) n ( 1 r 2 ) n 1 ( 1 u r ) 2 n ,
2 ln L ( χ u ) u 2 = 2 P n [ 1 + u 2 ( 1 u 2 ) 2 r 2 ( 1 u r ) 2 ] ,
r 2 ( 1 u r ) 2 = B n ( 1 u 2 ) n 1 + 1 r 2 ( 1 r 2 ) n 1 ( 1 u r ) 2 n + 2 d r ,
1 + 1 r 2 ( 1 r 2 ) n 1 ( 1 u r ) 2 n + 2 d r = 1 2 n ( 2 n + 1 ) d 2 d u 2 [ 1 + 1 ( 1 r 2 ) n 1 ( 1 u r ) 2 n d r ] ,
1 + 1 ( 1 r 2 ) n 1 ( 1 u r ) 2 n d r = 1 B n 1 ( 1 u 2 ) n .
r 2 ( 1 u r ) 2 = 1 2 n ( 2 n + 1 ) ( 1 u 2 ) n d 2 d u 2 [ 1 ( 1 u 2 ) n ] = 1 2 n + 1 [ 1 1 u 2 + 2 u 2 ( n + 1 ) ( 1 u 2 ) 2 ] .
CRLB S ( u 0 ) = 1 2 P n 2 n + 1 2 n ( 1 u 0 2 ) 2 .
ρ u ( u 0 ) = CRLB S ( u 0 ) CRLB V ( u 0 ) = 2 n + 1 2 n ( 1 + u 0 2 ) .
CRLB ( γ ) = ( γ u ) 2 CRLB ( u ) .
CRLB V ( γ 0 ) = 1 P n [ exp ( γ 0 ) + 1 ] exp ( 2 γ 0 ) + 1 ,
CRLB S ( γ 0 ) = 1 P n 2 n + 1 n .
ρ γ ( γ 0 ) = CRLB S ( γ 0 ) CRLB V ( γ 0 ) = CRLB S ( u 0 ) CRLB V ( u 0 ) = 2 n + 1 2 n ( 1 + u 0 2 ) = 2 n + 1 2 n exp ( 2 γ 0 ) + 1 [ exp ( γ 0 + 1 ) ] 2 .
u ̂ V = i = 1 P X i i = 1 P Y i i = 1 P X i + i = 1 P Y i .
b V u 0 3 u 0 2 P n + 1 ,
σ V 2 1 2 P n + 1 ( 1 4 P n + 3 2 P n + 1 u 0 2 + 2 P n + 3 2 P n + 1 u 0 4 ) .
u ̂ S = 1 P i = 1 P R i = 1 P i = 1 P Y i X i Y i + X i .
b S u 0 3 u 0 2 n + 1 ,
σ S 2 1 ( 2 n + 1 ) P ( 1 4 n + 3 2 n + 1 u 0 2 + 2 n + 3 2 n + 1 u 0 4 ) .
γ ̂ = 1 P i = 1 P β i .
σ ̃ γ 2 ( 1 ) = π 2 3 , σ ̃ γ 2 ( n + 1 ) = σ ̃ γ 2 ( n ) 2 n 2 for n 2 .
u ̂ γ = exp ( γ ̂ ) 1 exp ( γ ̂ ) + 1 .
b γ ( u 0 ) = u ̂ u 0 = ( u 0 3 u 0 ) σ ̃ γ 2 ( n ) 4 P ,
σ ̃ γ 2 ( n ) 2 n , n + ,
ln L ( χ u ) u = ln L ( χ γ ) γ γ u
2 ln L ( χ u ) u 2 = ln L ( χ γ ) γ 2 γ u 2 + 2 ln L ( χ γ ) γ 2 ( γ u ) 2 .
ln L ( χ γ ) γ = ln L ( χ γ ) γ L ( χ γ ) d χ = L ( χ γ ) γ d χ = γ [ L ( χ γ ) d χ ] = 0 .
2 ln L ( χ u ) u 2 = 2 ln L ( χ γ ) γ 2 ( γ u ) 2 .
CRLB ( γ ) = CRLB ( u ) ( γ u ) 2 .
P η ( η ) = B n ( 1 η 2 ) n 1 with B n = ( 2 n 1 ) ! 2 2 n 1 [ ( n 1 ) ! ] 2 .
R = 1 + 1 r P R ( r ) d r = 1 + 1 u + η 1 + u η P η ( η ) d η .
u + η 1 + u η = u + η ( 1 u 2 ) + η 2 u ( u 2 1 ) + o ( η 2 ) .
R u + ( 1 u 2 ) 1 + 1 η P ( η ) d η + ( u 3 u ) 1 + 1 η 2 P η ( η ) d η .
1 B n + 1 = 1 + 1 ( 1 η 2 ) ( 1 η 2 ) n 1 d η = 1 + 1 ( 1 η 2 ) n 1 d η σ η 2 ( n ) B n = 1 B n σ η 2 ( n ) B n ,
σ η 2 ( n ) = 1 B n B n + 1 = 1 2 n + 1 .
R u + u 3 u 2 n + 1 .
R 2 = 1 + 1 ( u + η 1 + u η ) 2 P η ( η ) d η .
( u + η 1 + u η ) 2 = u 2 + 2 u η ( 1 u 2 ) + η 2 ( 1 4 u 2 + 3 u 4 ) + o ( η 2 ) .
R 2 u 2 + 1 4 u 2 + 3 u 4 2 n + 1 = 1 2 n + 1 + 1 4 2 n + 1 u 2 + 3 u 4 2 n + 1 .
σ R 2 = R 2 R 2 1 2 n + 1 ( 1 4 n + 3 2 n + 1 u 2 + 2 n + 3 2 n + 1 u 4 ) ,
u ̂ γ g ( γ ̂ ) + g ( γ ̂ ) σ γ 2 2 ,

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