Abstract

We address the problem of estimating the polarization degree of polarimetric images in coherent illumination. It has been recently shown that the degree of polarization associated with polarimetric images can be estimated by the method of moments applied to two or four images assuming fully developed speckle. We show that the estimation can also be conducted by using maximum likelihood methods. The maximum likelihood estimators of the polarization degree are derived from the joint distribution of the image intensities. We show that the joint distribution of polarimetric images is a multivariate gamma distribution whose marginals are univariate, bivariate, or trivariate gamma distributions. This property is used to derive maximum likelihood estimators of the polarization degree using two, three, or four images. The proposed estimators provide better performance than the estimators of moments. These results are illustrated by estimations conducted on synthetic and real images.

© 2009 Optical Society of America

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    [CrossRef]
  25. D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, “Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A 19, 687-694 (2002).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008

G. Letac and J. Wesolowski, “Laplace transforms which are negative powers of quadratic polynomials,” Trans. Am. Math. Soc. 360, 6475-6496 (2008).
[CrossRef]

J. Fade, M. Roche, and P. Réfrégier, “Precision of moment-based estimation of the degree of polarization in coherent imagery without polarization device,” J. Opt. Soc. Am. A 25, 483-492 (2008).
[CrossRef]

2007

2006

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453-5469 (2006).
[CrossRef] [PubMed]

P. Bernardoff, “Which multivariate gamma distributions are infinitely divisible?” Bernoulli 12, 169-189 (2006).

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

2005

T. Shibata, T. Takahashi, D. Miyazaki, Y. Sato, and K. Ikeuchi, “Creating photorealistic virtual model with polarization-based vision system,” Proc. SPIE 5888, 25-35 (2005).

2004

I. K. Miyazaki D. and M. Kagesawa, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
[CrossRef] [PubMed]

S. Umeyama and G. Godin, “Separation of diffuse and specular components of surface reflection by use of polarization and statistical analysis of images,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 639-647 (2004).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

2003

2002

2001

J. W. Williams, J. S. Tee, and M. A. Poulter, “Image processing and classification for the UK remote minefield detection system infrared polarimetric camera,” Proc. SPIE 4394, 139-152 (2001).
[CrossRef]

N. Kikuchi, “Analysis of signal degree of polarization degradation used as control signal for optical polarization mode dispersion compensation,” J. Lightwave Technol. 19, 480-486 (2001).
[CrossRef]

2000

1999

1997

1996

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, and R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439-442 (1996).
[CrossRef]

J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optical scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855-1870 (1996).
[CrossRef] [PubMed]

1990

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

1989

B. Porat and B. Friedlander, “Performance analysis of parameter estimation algorithms based on high-order moments,” Int. J. Adapt. Control Signal Process. 3, 191-229 (1989).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Alfano, R. R.

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, and R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439-442 (1996).
[CrossRef]

Artal, P.

Baum, S. A.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Bernardoff, P.

P. Bernardoff, “Which multivariate gamma distributions are infinitely divisible?” Bernoulli 12, 169-189 (2006).

Biretta, J. A.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Birkinshaw, M.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Breugnot, S.

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

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998).

Bueno, J. M.

Capetti, A.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Chang, P. C. Y.

Chatelain, F.

F. Chatelain, J.-Y. Tourneret, A. Ferrari, and J. Inglada, “Bivariate gamma distributions for image registration and change detection,” IEEE Trans. Image Process. 16, 1796-1806 (2007).
[CrossRef] [PubMed]

F. Chatelain, G. Letac, and J.-Y. Tourneret, “Estimating the correlation coefficient of bivariate gamma distributions using the maximum likelihood principle and the inference for margins,” Tech. rep., IRIT/ENSEEIHT/TéSA (2007).

Chen, P. C. Y.

Chenault, D. B.

Chun, C. S. L.

Clémenceau, P.

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

de Boer, J. F.

Demos, S. G.

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, and R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439-442 (1996).
[CrossRef]

Dulwich, F.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Engheta, N.

Everett, M. J.

Fade, J.

Ferrari, A.

F. Chatelain, J.-Y. Tourneret, A. Ferrari, and J. Inglada, “Bivariate gamma distributions for image registration and change detection,” IEEE Trans. Image Process. 16, 1796-1806 (2007).
[CrossRef] [PubMed]

Flitton, J. C.

Friedlander, B.

B. Porat and B. Friedlander, “Performance analysis of parameter estimation algorithms based on high-order moments,” Int. J. Adapt. Control Signal Process. 3, 191-229 (1989).
[CrossRef]

Georganopoulos, M.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Godin, G.

S. Umeyama and G. Godin, “Separation of diffuse and specular components of surface reflection by use of polarization and statistical analysis of images,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 639-647 (2004).
[CrossRef] [PubMed]

Goldstein, D. L.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Goudail, F.

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

F. Goudail and P. Réfrégier, Statistical Image Processing Techniques for Noisy Images: An Application Oriented Approach (Kluwer, 2004).

Graczyk, P.

P. Graczyk, G. Letac, and H. Massam, “The complex Wishart distribution and the symmetric group,” Ann. Stat. 31, 287-309 (2003).
[CrossRef]

Heerdt, A. S.

S. G. Demos, H. Savage, A. S. Heerdt, S. Schantz, and R. R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439-442 (1996).
[CrossRef]

Hopcraft, K. I.

Huard, S.

S. Huard, Polarization of Light (Wiley, 1997).

Ikeuchi, K.

T. Shibata, T. Takahashi, D. Miyazaki, Y. Sato, and K. Ikeuchi, “Creating photorealistic virtual model with polarization-based vision system,” Proc. SPIE 5888, 25-35 (2005).

D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, “Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A 19, 687-694 (2002).
[CrossRef]

Inglada, J.

F. Chatelain, J.-Y. Tourneret, A. Ferrari, and J. Inglada, “Bivariate gamma distributions for image registration and change detection,” IEEE Trans. Image Process. 16, 1796-1806 (2007).
[CrossRef] [PubMed]

Jacques, S. L.

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

Jakeman, E.

Jester, S.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Jordan, D. L.

Kagesawa, M.

I. K. Miyazaki D. and M. Kagesawa, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
[CrossRef] [PubMed]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

Kendall, M. G.

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 2 (Griffin, 1961).

Kikuchi, N.

Kiryati, N.

Korotkova, O.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Koshikawa, K.

K. Koshikawa, “A polarimetric approach to shape understanding of glossy objects,” in Proceedings of the 6th International Joint Conference on Artificial Intelligence, Tokyo, Japan, August 20-23, 1979, pp. 493-495.

Leahy, J. P.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Lee, K.

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

Lensch, H.

C. F. Tongbo Chen, H. Lensch, and H.-P. Seidel, “Polarization and phase-shifting for 3d scanning of translucent objects,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2007), pp. 1-8.
[CrossRef]

Letac, G.

G. Letac and J. Wesolowski, “Laplace transforms which are negative powers of quadratic polynomials,” Trans. Am. Math. Soc. 360, 6475-6496 (2008).
[CrossRef]

P. Graczyk, G. Letac, and H. Massam, “The complex Wishart distribution and the symmetric group,” Ann. Stat. 31, 287-309 (2003).
[CrossRef]

F. Chatelain, G. Letac, and J.-Y. Tourneret, “Estimating the correlation coefficient of bivariate gamma distributions using the maximum likelihood principle and the inference for margins,” Tech. rep., IRIT/ENSEEIHT/TéSA (2007).

Luis, A.

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806-1-013806-8 (2002).
[CrossRef]

Maitland, D. J.

Martel, A.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Massam, H.

P. Graczyk, G. Letac, and H. Massam, “The complex Wishart distribution and the symmetric group,” Ann. Stat. 31, 287-309 (2003).
[CrossRef]

Milner, T. E.

Miyazaki, D.

T. Shibata, T. Takahashi, D. Miyazaki, Y. Sato, and K. Ikeuchi, “Creating photorealistic virtual model with polarization-based vision system,” Proc. SPIE 5888, 25-35 (2005).

D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, “Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A 19, 687-694 (2002).
[CrossRef]

Miyazaki D., I. K.

I. K. Miyazaki D. and M. Kagesawa, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
[CrossRef] [PubMed]

Narasimhan, S. G.

Nayar, S. K.

Nelson, J. S.

O'Dea, C. P.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Padgett, C. A.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Perlman, E. S.

E. S. Perlman, C. A. Padgett, M. Georganopoulos, W. B. Sparks, J. A. Biretta, C. P. O'Dea, S. A. Baum, M. Birkinshaw, D. M. Worrall, F. Dulwich, S. Jester, A. Martel, A. Capetti, and J. P. Leahy, “Optical polarimetry of the jets of nearby radio galaxies: I. the data,” Astrophys. J. 651, 735-748 (2006).
[CrossRef]

Porat, B.

B. Porat and B. Friedlander, “Performance analysis of parameter estimation algorithms based on high-order moments,” Int. J. Adapt. Control Signal Process. 3, 191-229 (1989).
[CrossRef]

Poulter, M. A.

J. W. Williams, J. S. Tee, and M. A. Poulter, “Image processing and classification for the UK remote minefield detection system infrared polarimetric camera,” Proc. SPIE 4394, 139-152 (2001).
[CrossRef]

Pugh, E. N.

Ramella-Roman, J. C.

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

Réfrégier, P.

Roche, M.

Rowe, M. P.

Sadjadi, A. F.

Saito, M.

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Sankaran, V.

Sato, Y.

T. Shibata, T. Takahashi, D. Miyazaki, Y. Sato, and K. Ikeuchi, “Creating photorealistic virtual model with polarization-based vision system,” Proc. SPIE 5888, 25-35 (2005).

D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, “Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A 19, 687-694 (2002).
[CrossRef]

Savage, H.

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Figures (9)

Fig. 1
Fig. 1

Log MSEs of the square DoP estimates using two and four images versus P 2 for the set of polarization matrices defined in Table 2 ( n = 15 × 15 , MoM method of moments estimators, MLE maximum likelihood estimators, Asympt. theoretical asymptotic value of the log MSE for a given estimator).

Fig. 2
Fig. 2

Log MSEs of the square DoP estimates using three and four images versus P 2 for the set of polarization matrices defined in Table 2; parameters as in Fig. 1. ( n = 15 × 15 , for a given estimator).

Fig. 3
Fig. 3

Log MSE of the estimated square DoP P 2 using two or three intensity images versus the logarithm of the sample size for the matrix Γ 2 ; labels as in Fig. 2.

Fig. 4
Fig. 4

Log MSE of the estimated square DoP P 2 using two or three intensity images versus the logarithm of the sample size for the matrix Γ 7 ; labels as in Fig. 2.

Fig. 5
Fig. 5

Composition of the scene used to generate synthetic polarimetric intensity images.

Fig. 6
Fig. 6

Synthetic intensity images and polarimetric contrast image C = I 2 α I 1 [20] (with a typical value of α = 12 % ) for the scene depicted in Fig. 5 and described in Table 3.

Fig. 7
Fig. 7

Estimates of P 2 using two, three, or four intensity images for the synthetic polarimetric images (numbers appearing in each region of the image are given for identification and represent the means of the estimates for each object assuming that the region constituting each object is perfectly known) for an estimation window of size n = 15 × 15 ; MoM method of moments estimators, MLE maximum likelihood estimators.

Fig. 8
Fig. 8

Real polarimetric intensity images of a scene composed of a plastic disk (left) and a steel disk (right).

Fig. 9
Fig. 9

Estimates of P 2 using two, three, or four intensity images for the real polarimetric images; size of the estimation window n = 9 × 9 , MoM method of moments estimators, MLE maximum likelihood estimators.

Tables (3)

Tables Icon

Table 1 Polarimetric Image DoPs

Tables Icon

Table 2 Covariance Matrices of the Jones Vector

Tables Icon

Table 3 Polarimetric Properties of Image DoPs

Equations (73)

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E ( r , t ) = [ A X ( r , t ) e X + A Y ( r , t ) e Y ] e i 2 π ν t ,
Γ = [ E [ A X A X * ] E [ A X A Y * ] E [ A Y A X * ] E [ A Y A Y * ] ] [ a 1 a 3 + i a 4 a 3 i a 4 a 2 ] ,
p A ( A ) = 1 π 2 Γ e A Γ 1 A ,
I 1 = A X 2 , I 2 = A Y 2 ,
I 3 = 1 2 A X 2 + 1 2 A Y 2 + Re ( A X A Y * ) ,
I 4 = 1 2 A X 2 + 1 2 A Y 2 + Im ( A X A Y * ) .
P 2 = 1 4 Γ [ trace ( Γ ) ] 2 = 1 4 [ a 1 a 2 ( a 3 2 + a 4 2 ) ] ( a 1 + a 2 ) 2 ,
L I ( θ ) = E [ exp ( j = 1 4 θ j I j ) ] ,
S = [ s 1 s 3 + i s 4 s 3 i s 4 s 2 ] = [ A X 2 A X A Y * A Y A X * A Y 2 ] .
s = M I = [ 1 0 0 0 0 1 0 0 1 2 1 2 1 0 1 2 1 2 0 1 ] I .
L I ( θ ) = E [ exp ( θ T I ) ] = E [ exp ( θ T M 1 s ) ] = L s ( M T θ ) .
L S ( Θ ) = E [ exp [ trace ( S Θ ) ] ] ,
Θ = [ θ 1 θ 3 + i θ 4 θ 3 i θ 4 θ 2 ] .
trace ( S Θ ) = θ 1 s 1 + θ 2 s 2 + 2 θ 3 s 3 + 2 θ 4 s 4 ;
L S ( Θ ) = L s ( θ 1 , θ 2 , 2 θ 3 , 2 θ 4 ) ,
L S ( Θ ) = E { exp [ trace ( S Θ ) ] } = I 2 + Γ Θ 1 ,
L I ( θ ) = 1 P ( θ ) ,
P ( θ ) = 1 + α T θ + k [ 2 θ 1 θ 2 + θ 3 θ 4 + ( θ 1 + θ 2 ) ( θ 3 + θ 4 ) ] ,
α ̂ ML = 1 n j = 1 n I j .
α 1 = E [ I 1 ] = a 1 , α 3 = E [ I 3 ] = ( a 1 + a 2 + 2 a 3 ) 2 ,
α 2 = E [ I 2 ] = a 2 , α 4 = E [ I 4 ] = ( a 1 + a 2 + 2 a 4 ) 2 ,
a ̂ ML = M α ̂ ML .
P ̂ 4 2 = 1 4 [ a ̂ 1 a ̂ 2 ( a ̂ 3 2 + a ̂ 4 2 ) ] ( a ̂ 1 + a ̂ 2 ) 2 .
cov ( a ̂ ML ) = ( 1 n ) M cov ( I ) M T ,
cov ( a ̂ ML ) = 1 n [ a 1 2 a 3 2 + a 4 2 a 1 a 3 a 1 a 4 a 3 2 + a 4 2 a 2 2 a 2 a 3 a 2 a 4 a 1 a 3 a 2 a 3 c 3 , 3 a 3 a 4 a 1 a 4 a 2 a 4 a 3 a 4 c 4 , 4 ] ,
var A ( P ̂ 4 2 ) = G 4 T cov ( a ̂ ML ) G 4 = 2 ( 1 P 2 ) 2 P 2 n ,
G 4 ( a ) = [ 4 ( a 1 a 2 a 2 2 2 a 3 2 2 a 4 2 ) ( a 1 + a 2 ) 3 , 4 ( a 1 a 2 a 1 2 2 a 3 2 2 a 4 2 ) ( a 1 + a 2 ) 3 , 8 a 3 ( a 1 + a 2 ) 2 , 8 a 4 ( a 1 + a 2 ) 2 ] T
[ I 3 ( I 1 + I 2 ) 2 ] 2 + [ I 4 ( I 1 + I 2 ) 2 ] 2 = I 1 I 2 .
L I ̃ ( θ ̃ ) = E [ exp ( j = 1 3 θ j I j ) ] = 1 P ̃ ( θ ̃ ) ,
L I ̃ ( θ ̃ ) = [ 1 2 c T θ ̃ + v ( θ ̃ ) ] p ,
p ( I ̃ ) = 1 k π v ( I ̃ ) exp [ ( a 2 + a 3 ) I 1 + ( a 1 + a 3 ) I 2 2 a 3 I 3 2 k ] f 1 2 ( a 4 2 v ( I ̃ ) 16 k 2 ) I Ω ( I ̃ ) ,
f q ( z ) = m = 0 z m Γ ( m + q ) m ! , q > 0 ,
( a ̂ 1 a ̂ 2 a ̂ 3 ) = [ 1 0 0 0 1 0 1 2 1 1 2 ] ( α ̂ 1 α ̂ 2 α ̂ 3 ) ,
1 2 n j = 1 n v ( I ̃ j ) a 4 2 ˜ tanh ( a 4 2 ˜ v ( I ̃ j ) d ̂ a 4 2 ˜ ) = 1 ,
P ̃ 3 2 = 1 4 [ a ̂ 1 a ̂ 2 ( a ̂ 3 2 + a 4 2 ˜ ) ] ( a ̂ 1 + a ̂ 2 ) 2 .
F 3 ( η ) = E [ 2 log p ( I ̃ ; η ) η η T ] .
[ F 3 ( η ) ] i j 1 N k = 1 N 2 log p ( x k ) η i η j ,
var ( P ̃ 3 2 ) = G 3 T F 3 1 G 3 ,
G 3 = [ 4 ( a 1 a 2 a 2 2 2 a 3 2 2 a 4 2 ) ( a 1 + a 2 ) 3 , 4 ( a 1 a 2 a 1 2 2 a 3 2 2 a 4 2 ) ( a 1 + a 2 ) 3 , 8 a 3 ( a 1 + a 4 ) 2 , 4 ( a 1 + a 2 ) 2 ] T .
L I ̱ ( θ ̱ ) = E [ exp ( j = 1 2 θ j I j ) ] = 1 P ̱ ( θ ̱ ) ,
p ( I ̱ ) = 1 2 k exp ( a 2 I 1 + a 1 I 2 2 k ) f 1 ( c I 1 I 2 ) I R + 2 ( I ̱ ) ,
a ̂ 1 = α ̂ 1 , a ̂ 2 = α ̂ 2 ,
a ̂ 1 a ̂ 2 r 1 n j = 1 n I 1 j I 2 j f 2 [ r I 1 j I 2 j ( a ̂ 1 a ̂ 2 r ) 2 ] f 1 [ r I 1 j I 2 j ( a ̂ 1 a ̂ 2 r ) 2 ] = 0 .
P ̱ 2 2 = 1 4 ( a ̂ 1 a ̂ 2 r ̱ ) ( a ̂ 1 + a ̂ 2 ) 2 .
var A ( P ̱ 2 2 ) = G 2 T F 2 1 G 2 ,
G 2 = [ 4 ( a 1 a 2 a 2 2 2 r ) ( a 1 + a 2 ) 3 , 4 ( a 1 a 2 a 1 2 2 r ) ( a 1 + a 2 ) 3 , 4 ( a 1 + a 2 ) 2 ] T ,
a ̂ Mo = M α ̂ Mo ,
E [ I 1 ] = a 1 , E [ I 2 ] = a 2 , E [ I 3 ] = 1 2 ( a 1 + a 2 ) + a 3 ,
( a ̃ 1 , a ̃ 2 , a ̃ 3 ) T = ( a ̂ 1 , a ̂ 2 , a ̂ 3 ) T .
E [ I 1 I 2 ] = a 1 a 2 + a 3 2 + a 4 2 ,
E [ I 1 I 3 ] = a 1 2 + ( a 2 2 + 2 a 3 ) a 1 + ( a 3 2 + a 4 2 ) 2 ,
E [ I 2 I 3 ] = a 2 2 + ( a 1 2 + 2 a 3 ) a 2 + ( a 3 2 + a 4 2 ) 2 .
a 4 Mo 2 ˜ = arg min x > 0 [ f ̃ ( x ) s n ] T C ̃ ( x ) 1 [ f ̃ ( x ) s n ] ,
P ̃ 3 Mo 2 = 1 4 [ a ̂ 1 a ̂ 2 ( a ̂ 3 2 + a 4 Mo 2 ˜ ) ] ( a ̂ 1 + a ̂ 2 ) 2 .
s n = 1 n j = 1 n h ( I j ) ,
E [ s n ] = f ( η ) = E [ h ( I 1 ) ] ,
n cov [ s n ] = C ( η ) = cov [ h ( I 1 ) ] .
n cov [ s n ] B ( η ) = ( H ( η ) C ( η ) 1 H ( η ) T ) 1 ,
H ( η ) = [ 1 0 1 2 a 2 2 a 1 + a 2 2 + 2 a 3 a 2 2 0 1 1 2 a 1 a 1 2 2 a 2 + a 1 2 + 2 a 3 0 0 1 2 a 3 2 a 1 + a 3 2 a 2 + a 3 0 0 0 1 1 2 1 2 ] .
var A ( P ̃ 3 Mo 2 ) G 3 T B ( η ) G 3 .
E [ I 1 ] = a 1 , E [ I 2 ] = a 2 ,
E [ I 1 I 2 ] = a 1 a 2 + r .
( a ̱ 1 , a ̱ 2 ) T = ( a ̂ 1 , a ̂ 2 ) T ,
r ̱ Mo = 1 n j = 1 n I 1 j I 2 j a ̂ 1 a ̂ 2 .
P ̱ 2 Mo 2 = 1 4 [ a ̂ 1 a ̂ 2 r ̱ Mo ] ( a ̂ 1 + a ̂ 2 ) 2 .
var A ( η ̱ 2 , Mo ) = 1 n [ a 1 2 r 2 a 1 r r a 2 2 2 a 2 r 2 a 1 2 a 2 r a 1 2 a 2 2 + 4 a 1 a 2 r + 3 r 2 ] .
var A ( P ̱ 2 Mo 2 ) = G 2 T var A ( η ̱ 2 , Mo ) G 2 ,
var A ( P ̱ 2 Mo 2 ) = 2 ( 1 P 2 ) 2 ( P 2 + 1 2 ) n + 64 a 1 a 2 r n ( a 1 + a 2 ) 4 .
l 3 ( I ̃ ( n ) ; a ) = j = 1 n [ ( a 2 + a 3 ) I 1 + ( a 1 + a 3 ) I 2 2 a 3 I 3 2 k log f 1 2 ( a 4 2 v ( I ̃ j ) 16 k 2 ) ] n log ( k ) ,
( a ̂ 1 a ̂ 2 a ̂ 3 ) = [ 1 0 0 0 1 0 1 2 1 1 2 ] ( α ̂ 1 α ̂ 2 α ̂ 3 ) ,
g 3 ( I ̃ ( n ) ; a ) = n 4 k 2 [ 2 k ( a 2 + a 3 ) m ̂ 1 ( a 1 + a 3 ) m ̂ 2 + 2 a 3 m ̂ 3 ] + ( a 1 a 2 a 3 2 + a 4 2 ) 32 k 3 j = 1 n v ( I ̃ j ) f 3 2 [ a 4 2 v ( I ̃ j ) 16 k 2 ] f 1 2 [ a 4 2 v ( I ̃ j ) 16 k 2 ] .
1 1 4 n 1 d ̂ a 4 2 j = 1 n v ( I ̃ j ) f 3 2 [ a 4 2 v ( I ̃ j ) 4 ( d ̂ a 4 2 ) 2 ] f 1 2 [ a 4 2 v ( I ̃ j ) 4 ( d ̂ a 4 2 ) 2 ] = 0 ,
1 1 2 n j = 1 n v ( I ̃ j ) a 4 2 tanh ( a 4 2 v ( I ̃ j ) d ̂ a 4 2 ) = 0 ,

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