Abstract

We address the problem of degree of polarization estimation in polarization diversity images. We consider active imaging techniques with laser illumination, which have the appealing feature of revealing contrasts that do not appear in conventional intensity images. These techniques provide two images of the same scene that are perturbed with speckle noise. Because of the presence of nonhomogeneity in the reflected intensity, it can be preferable to perform image analysis of the orthogonal-state contrast image, which is a measure of the degree of polarization of the reflected light when the coherency matrix is diagonal. It has been shown that a simple nonlinear transformation of this orthogonal-state contrast image leads to an image perturbed with additive symmetrical noise on which simple and efficient estimation and detection techniques can be applied. We propose to precisely analyze estimation properties of the degree of polarization using this natural representation. In particular, we determine the Cramer–Rao bound of the polarization degree estimation and the variance of the proposed estimator, and we study the estimator’s efficiency as a function of the speckle order for different measurement strategies.

© 2004 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).
    [CrossRef]
  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, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
    [CrossRef]
  4. R. C. Hardie, M. Vadyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. (Bellingham) 37, 752–762 (1998).
    [CrossRef]
  5. 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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
    [CrossRef]
  6. 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]
  7. J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. (Bellingham) 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).
    [CrossRef]
  9. S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806 nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman, Ch. Werner, eds., Proc. SPIE3707, 449–460 (1999).
    [CrossRef]
  10. J. W. Goodman, “Laser speckle and related phenomena,” in Statistical Properties of Laser Speckle Patterns, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Heidelberg, Germany, 1975), pp. 9–75.
    [CrossRef]
  11. F. Goudail, Ph. Réfrégier, “Statistical techniques for target detection in polarization diversity images,” Opt. Lett. 26, 644–646 (2001).
    [CrossRef]
  12. F. Goudail, Ph. Réfrégier, “Statistical algorithms for target detection in coherent active polarimetric images,” J. Opt. Soc. Am. A 18, 3049–3060 (2001).
    [CrossRef]
  13. F. Goudail, Ph. Réfrégier, “Target segmentation in active polarimetric images by use of statistical active contours,” Appl. Opt. 41, 874–883 (2002).
    [CrossRef] [PubMed]
  14. F. Goudail, Ph. Réfrégier, “Improving target detection in active polarimetric images,” in Optical Pattern Recognition XII, T. H. Chao, D. P. Casasent, eds., Proc. SPIE4387, 140–151 (2001).
    [CrossRef]
  15. T. S. Ferguson, “Exponential families of distributions,” in Mathematical Statistics: A Decision Theoretic Approach (Academic, New York, 1967), pp. 125–132.
  16. S. Huard, “Polarized optical wave,” in Polarization of Light (Wiley, Paris, 1997), pp. 1–35.
  17. P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).
  18. J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, New York, 1985), pp. 116–156.
  19. T. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach (Academic, New York, 1967).
  20. P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Printice-Hall Europe, London, 1995).
  21. J. Garcia, J. Campos, C. Ferreira, “Multichannel pattern recognition using a minimum average correlation energy filter,” Pure Appl. Opt. 3, 221–224 (1994).
    [CrossRef]
  22. F. Goudail, Ph. Réfrégier, “Algorithmes statistiques pour le traitement des images polarimétriques en lumière cohérente,” Traitement Signal 18, 297–319 (2001).
  23. H. Stocker, J. W. Harris, Handbook of Mathematics and Computational Science (Springer-Verlag, New York, 1998).

2002 (1)

2001 (3)

1998 (1)

R. C. Hardie, M. Vadyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. (Bellingham) 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. (Bellingham) 34, 1558–1568 (1995).
[CrossRef]

1994 (1)

J. Garcia, J. Campos, C. Ferreira, “Multichannel pattern recognition using a minimum average correlation energy filter,” Pure Appl. Opt. 3, 221–224 (1994).
[CrossRef]

1991 (1)

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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

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, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

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, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

Breugnot, S.

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

P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).

Campos, J.

J. Garcia, J. Campos, C. Ferreira, “Multichannel pattern recognition using a minimum average correlation energy filter,” Pure Appl. Opt. 3, 221–224 (1994).
[CrossRef]

Chipman, R. A.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. (Bellingham) 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).
[CrossRef]

Clémenceau, P.

P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).

Clémenceau, Ph.

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

Collot, L.

P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).

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, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

Egan, W. G.

Ferguson, T. S.

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

T. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach (Academic, New York, 1967).

Ferreira, C.

J. Garcia, J. Campos, C. Ferreira, “Multichannel pattern recognition using a minimum average correlation energy filter,” Pure Appl. Opt. 3, 221–224 (1994).
[CrossRef]

Garcia, J.

J. Garcia, J. Campos, C. Ferreira, “Multichannel pattern recognition using a minimum average correlation energy filter,” Pure Appl. Opt. 3, 221–224 (1994).
[CrossRef]

Garthwaite, P. H.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Printice-Hall Europe, London, 1995).

Goodman, J. W.

J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, New York, 1985), pp. 116–156.

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

Goudail, F.

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

F. Goudail, Ph. Réfrégier, “Statistical techniques for target detection in polarization diversity images,” Opt. Lett. 26, 644–646 (2001).
[CrossRef]

F. Goudail, Ph. Réfrégier, “Statistical algorithms for target detection in coherent active polarimetric images,” J. Opt. Soc. Am. A 18, 3049–3060 (2001).
[CrossRef]

F. Goudail, Ph. Réfrégier, “Algorithmes statistiques pour le traitement des images polarimétriques en lumière cohérente,” Traitement Signal 18, 297–319 (2001).

F. Goudail, Ph. Réfrégier, “Improving target detection in active polarimetric images,” in Optical Pattern Recognition XII, T. H. Chao, D. P. Casasent, eds., Proc. SPIE4387, 140–151 (2001).
[CrossRef]

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. (Bellingham) 37, 752–762 (1998).
[CrossRef]

Harris, J. W.

H. Stocker, J. W. Harris, Handbook of Mathematics and Computational Science (Springer-Verlag, New York, 1998).

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).
[CrossRef]

Huard, S.

S. Huard, “Polarized optical wave,” in Polarization of Light (Wiley, Paris, 1997), pp. 1–35.

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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Johnson, W. R.

Jolliffe, I. T.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Printice-Hall Europe, London, 1995).

Jones, B.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Printice-Hall Europe, London, 1995).

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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

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, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

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. (Bellingham) 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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

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]

Pezzaniti, J. L.

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

Réfrégier, Ph.

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

F. Goudail, Ph. Réfrégier, “Statistical algorithms for target detection in coherent active polarimetric images,” J. Opt. Soc. Am. A 18, 3049–3060 (2001).
[CrossRef]

F. Goudail, Ph. Réfrégier, “Statistical techniques for target detection in polarization diversity images,” Opt. Lett. 26, 644–646 (2001).
[CrossRef]

F. Goudail, Ph. Réfrégier, “Algorithmes statistiques pour le traitement des images polarimétriques en lumière cohérente,” Traitement Signal 18, 297–319 (2001).

F. Goudail, Ph. Réfrégier, “Improving target detection in active polarimetric images,” in Optical Pattern Recognition XII, T. H. Chao, D. P. Casasent, eds., Proc. SPIE4387, 140–151 (2001).
[CrossRef]

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, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

Stocker, H.

H. Stocker, J. W. Harris, Handbook of Mathematics and Computational Science (Springer-Verlag, New York, 1998).

Vadyanathan, M.

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

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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Appl. Opt. (2)

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

Opt. Eng. (Bellingham) (2)

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. (Bellingham) 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. (Bellingham) 37, 752–762 (1998).
[CrossRef]

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]

Pure Appl. Opt. (1)

J. Garcia, J. Campos, C. Ferreira, “Multichannel pattern recognition using a minimum average correlation energy filter,” Pure Appl. Opt. 3, 221–224 (1994).
[CrossRef]

Traitement Signal (1)

F. Goudail, Ph. Réfrégier, “Algorithmes statistiques pour le traitement des images polarimétriques en lumière cohérente,” Traitement Signal 18, 297–319 (2001).

Other (14)

H. Stocker, J. W. Harris, Handbook of Mathematics and Computational Science (Springer-Verlag, New York, 1998).

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).
[CrossRef]

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, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

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, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

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

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

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

F. Goudail, Ph. Réfrégier, “Improving target detection in active polarimetric images,” in Optical Pattern Recognition XII, T. H. Chao, D. P. Casasent, eds., Proc. SPIE4387, 140–151 (2001).
[CrossRef]

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

S. Huard, “Polarized optical wave,” in Polarization of Light (Wiley, Paris, 1997), pp. 1–35.

P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).

J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, New York, 1985), pp. 116–156.

T. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach (Academic, New York, 1967).

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Printice-Hall Europe, London, 1995).

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Figures (3)

Fig. 1
Fig. 1

Circles, CRBn(γ) as a function of n [see Eq. (11)]; dashed line, curve y=log10(2/n). We can clearly see that CRBn(γ) varies as 2/n for large values of n. Log10() is the decimal logarithm.

Fig. 2
Fig. 2

Efficiency [i.e., CRBn(γ)/σn2(β)] of the empirical mean (i.e., of γ^P) as a function of n. One can see that the variance of the estimator γ^P reaches approximately (with a precision of 1%) the Cramer–Rao bound when n4.

Fig. 3
Fig. 3

mσm2(β) as a function of m. One can see that averaging couples of variables {(xi, yi)} before determining the γi can allow one to reduce the variance of the estimator of γ by a factor of 1.55.

Tables (1)

Tables Icon

Table 1 Comparison between the Variance of the Estimator γ^P and the Cramer–Rao Bound

Equations (81)

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

ρi=Xi-YiXi+Yi.
PX(x)=nnxn-1Γ(n)mXnexp-nxmX,
Pρ(u,n)(ρ)=122n-11Bn (1-u2)n(1-ρ2)n-1(1-uρ)2n,
Bn=[(n-1)!]2(2n-1)!,B1=1,
u=mX-mYmX+mY.
Pβ(γ,n)(β)=1Bn1{exp[(β-γ)/2]+exp[-(β-γ)/2]}2n,
u=exp(γ)-1exp(γ)+1.
CRBn,P(γ)=1/In,P,
In,P=-22γln[Pβ(γ,n)(β1)Pβ(γ,n)(βP)]×Pβ(γ,n)(β1)Pβ(γ,n)(βP)dβ1dβP.
CRBn,P(γ)=CRBn(γ)P,
CRBn(γ)=2n+1n2.
xj=1mi=(j-1)m+1jmxi,yj=1mi=(j-1)m+1jmyi,
CRBmn,Q(γ)=2mn+1Q(mn)2,
CRBmn,Q(γ)=1P2mn+1n(mn).
CRBmn,Q(γ)CRBn,P(γ)=2mn+1m(2n+1).
CRBm,Q(γ)CRB1,P(γ)=2m+13m,
γ^P=1Pi=1Pβi.
σn2(β)=-(β-γ)2Pβ(γ,n)(β)dβ.
σn+12(β)=σn2(β)-2/n2,
σ12(β)=π2/3.
σn2(β)=2qn1q2.
1-12(n+1)ηn(γ^P)1,
limn ηn(γ^P)=1.
(uˆ-u)22r(1+r)22CRBn(γ)P.
(uˆ-u)22r(1+r)22CRBn(γ)P.
u^β=exp(γ^P)-1exp(γ^P)+1.
u^P=1Pi=1Pρi
ln[Pβ(γ,n)(β)]=-ln[Bn]-2n ln{exp[(β-γ)/2]+exp[-(β-γ)/2]}.
ddγln[Pβ(γ,n)(β)]=n exp[(β-γ)/2]-exp[-(β-γ)/2]exp[(β-γ)/2]+exp[-(β-γ)/2].
d2dγ2ln[Pβ(γ,n)(β)]
=-2n{exp[(β-γ)/2]+exp[-(β-γ)/2]}2.
d2dγ2ln[Pβ(γ,n)(β)]
 =-2nBn-dβ{exp[(β-γ)/2]+exp[-(β-γ)/2]}2n+2,
d2dγ2ln[Pβ(γ,n)(β)]=-2nBn+1Bn=-n2(2n+1),
CRBn(γ)=(2n+1)n2.
-βPβ(n)(β)dβ=γ,
σn2(β)=-β2Pβ(n)(β)dβ,
I(ξ)=-1[exp(ξβ/2)+exp(-ξβ/2)]2ndβ;
dI(ξ)dξ=-2n-β2[exp(ξβ/2)-exp(-ξβ/2)][exp(ξβ/2)+exp(-ξβ/2)]2n+1dβ,
-12nd2I(ξ)dξ2
=-(2n+1)-β22×[exp(ξβ/2)-exp(-ξβ/2)]2[exp(ξβ/2)+exp(-ξβ/2)]2n+2dβ+-β22[exp(ξβ/2)+exp(-ξβ/2)][exp(ξβ/2)+exp(-ξβ/2)]2n+1dβ.
-12nd2I(ξ)dξ2ξ=1
=-(2n+1)-β22×[exp(β/2)-exp(-β/2)]2[exp(β/2)+exp(-β/2)]2n+2dβ+-β221[exp(β/2)+exp(-β/2)]2ndβ.
σn2(β)=1Bn-β2[exp(β/2)+exp(-β/2)]2ndβ,
-2nd2I(ξ)dξ2ξ=1=Bnσn2(β)-(2n+1)Jn,
Jn=-β2[exp(β/2)-exp(-β/2)]2[exp(β/2)+exp(-β/2)]2n+2dβ.
Jn-Bnσn2(β)
=-β2F(β) 1[exp(β/2)+exp(-β/2)]2ndβ,
F(β)=[exp(β/2)-exp(-β/2)]2[exp(β/2)+exp(-β/2)]2-1,
F(β)=-4[exp(β/2)+exp(-β/2)]2,
Jn-Bnσn2(β)=-4Bn+1σn+12(β).
-2nd2I(ξ)dξ2ξ=1=-2nBnσn2(β)+4(2n+1)×Bn+1σn+12(β).
-4n Bn=-2nBnσn2(β)+4(2n+1)Bn+1σn+12(β)
(2n+1)σn+12(β)=BnBn+1n2 σn2(β)-1n.
σn+12(β)=σn2(β)-2n2,
σN2(β)=σ12(β)-2n=1N-11n2.
n>01n2=π26.
σ12(β)=π23.
σN2(β)CRBN(γ)=2N2(2N+1)nN1n2
1N2+N1x2dxANN1x2dx,
2N+22N+1σN2(β)CRBN(γ)2N(2N+1).
1-12N+2CRBN(γ)σN2(β)1,
limNσN2(β)CRBN(γ)=1.
j=0L1(N+j)2+N+L1x2dxAN,
2j=0L1(N+j)2+1N+LσN2(β),
2N22N+1j=0L1(N+j)2+1N+LσN2(β)CRBN(γ),
γˆ=γ+δγˆ,
u^β-u2r(1+r)2 δγˆ,
(u^β-u)22r(1+r)22δγ^2.
(u^β-u)22r(1+r)22CRBn(γ)P.
(uˆ-u)22r(1+r)22CRBn(γ)P.
(u^-u)2<2r(1+r)22CRBn(γ)P.
eγ=1+u1-u,
γ^ρ=ln1+u^1-u^.
u^=u+δu^
γ^ρ=ln1+u+δu^1-u-δu^,
γ^ρ=ln(1+u+δu^)-ln(1-u-δu^).
γ^ρln1+u1-u+21-u2 δu^,
(δγ^ρ)221-u22(δu^)2.
(δγ^ρ)2(1+r)22r2(δu^)2,
(δγ^ρ)2<CRBn(γ)P,

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