Abstract

We consider the Mueller matrix ellipsometry (MME) measuring the ellipsometric parameters of the isotropic sample and the anisotropic sample under certain conditions in the presence of either Gaussian additive noise or Poisson shot noise. In this case, the ellipsometric parameters only relate to partial elements in Mueller matrix, and we optimize the instrument matrices of polarization state generator (PSG) and analyzer (PSA) to minimize the total measurement variance for these elements, in order to decrease the variance of the estimator of ellipsometric parameters. Compared with the previous instrument matrices, the optimal instrument matrices in this paper can effectively decrease the measurement variance and thus statistically improve the measurement precision of the ellipsometric parameters. In addition, it is found that the optimal instrument matrices for Poisson shot noise are same to those for Gaussian additive noise, and furthermore, the optimal instrument matrices do not depend on the ellipsometric parameters to be measured, which means that the optimal instrument matrices of MME proposed in this paper can be widely applied in various cases.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2016 (3)

2015 (2)

X. Li, T. Liu, B. Huang, Z. Song, and H. Hu, “Optimal distribution of integration time for intensity measurements in Stokes polarimetry,” Opt. Express 23(21), 27690–27699 (2015).
[Crossref] [PubMed]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

2014 (3)

2013 (6)

2012 (3)

2011 (1)

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

2010 (1)

2008 (1)

2007 (2)

T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, “Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction,” Opt. Express 15(5), 2033–2046 (2007).
[Crossref] [PubMed]

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

2004 (1)

2003 (1)

2002 (1)

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41(5), 965–972 (2002).
[Crossref]

1997 (1)

R. M. A. Azzam, “Mueller-matrix ellipsometry: a review in Polarization: Measurement, Analysis, and Remote Sensing,” Proc. SPIE 3121, 396–399 (1997).
[Crossref]

1994 (2)

D. A. Ramsey and K. C. Luderna, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65(9), 2874–2881 (1994).
[Crossref]

S. Krishnan and P. C. Nordine, “Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects,” Appl. Opt. 33(19), 4184–4192 (1994).
[Crossref] [PubMed]

1993 (1)

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

1979 (1)

Anna, G.

Azzam, R. M. A.

R. M. A. Azzam, “Mueller-matrix ellipsometry: a review in Polarization: Measurement, Analysis, and Remote Sensing,” Proc. SPIE 3121, 396–399 (1997).
[Crossref]

Bijkerk, F.

Boller, K. J.

Boyer, G. R.

Bulkin, P.

Chen, J.

Chen, X.

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

Chipman, R. A.

Chumakov, A.

De Martino, A.

Dolfi, D.

Drévillon, B.

Du, E.

Du, W.

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

Duan, Q. Y.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

Garcia-Caurel, E.

Gaston, J. P.

Goudail, F.

Gu, H.

Guo, Y.

Gupta, V. K.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

He, H.

He, Y.

Hoover, B. G.

Hu, H.

Huang, B.

Jiang, H.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

Johnson, S. J.

Kim, Y. K.

Klimov, A.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

Krishnan, S.

Lamouroux, B. F.

Laude, B.

Lee, C. J.

Li, W.

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

Li, X.

Liao, R.

Liu, F.

Liu, S.

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
[Crossref] [PubMed]

M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
[Crossref] [PubMed]

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

Liu, T.

Louis, E.

Luderna, K. C.

D. A. Ramsey and K. C. Luderna, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65(9), 2874–2881 (1994).
[Crossref]

Ma, H.

Muttiah, R.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

Muttiah, R. S.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

Nguyen, Q.

Nordine, P. C.

Novikova, T.

Oberemok, E.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

Ossikovski, R.

Popov, V.

Prade, B. S.

Ramsey, D. A.

D. A. Ramsey and K. C. Luderna, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65(9), 2874–2881 (1994).
[Crossref]

Sauer, H.

Savenkov, S.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

Savenkov, S. N.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41(5), 965–972 (2002).
[Crossref]

Song, Z.

Sorooshian, S.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

Sun, M.

E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
[Crossref] [PubMed]

M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
[Crossref] [PubMed]

Twietmeyer, K. M.

Tyo, J. S.

van der Slot, P. J. M.

Volchkov, S. A.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

Wang, S. X.

Wang, Z.

Weiner, A. M.

Wu, J.

E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
[Crossref] [PubMed]

M. Sun, H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma, “Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters,” Biomed. Opt. Express 5(12), 4223–4234 (2014).
[Crossref] [PubMed]

Yan, L.

Yun, T.

Yushtin, K. E.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

Zeng, N.

Zhang, C.

H. Gu, X. Chen, H. Jiang, C. Zhang, W. Li, and S. Liu, “Accurate alignment of optical axes of a biplate using a spectroscopic Mueller matrix ellipsometer,” Appl. Opt. 55(15), 3935–3941 (2016).
[Crossref] [PubMed]

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

Appl. Opt. (7)

Appl. Spectrosc. (1)

Biomed. Opt. Express (1)

J. Appl. Phys. (1)

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

J. Biomed. Opt. (1)

E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19(7), 076013 (2014).
[Crossref] [PubMed]

J. Opt. (1)

W. Li, C. Zhang, H. Jiang, X. Chen, and S. Liu, “Depolarization artifacts in dual rotating-compensator Mueller matrix ellipsometry,” J. Opt. 18(5), 055701 (2016).
[Crossref]

J. Optim. Theory Appl. (1)

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “Shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. 76(3), 501–521 (1993).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (2)

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete active polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1796–1802 (2011).
[Crossref]

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transf. 106(1), 475–486 (2007).
[Crossref]

Opt. Eng. (1)

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41(5), 965–972 (2002).
[Crossref]

Opt. Express (8)

H. Hu, R. Ossikovski, and F. Goudail, “Performance of Maximum Likelihood estimation of Mueller matrices taking into account physical realizability and Gaussian or Poisson noise statistics,” Opt. Express 21(4), 5117–5129 (2013).
[Crossref] [PubMed]

F. Liu, C. J. Lee, J. Chen, E. Louis, P. J. M. van der Slot, K. J. Boller, and F. Bijkerk, “Ellipsometry with randomly varying polarization states,” Opt. Express 20(2), 870–878 (2012).
[Crossref] [PubMed]

X. Li, T. Liu, B. Huang, Z. Song, and H. Hu, “Optimal distribution of integration time for intensity measurements in Stokes polarimetry,” Opt. Express 23(21), 27690–27699 (2015).
[Crossref] [PubMed]

G. Anna and F. Goudail, “Optimal Mueller matrix estimation in the presence of Poisson shot noise,” Opt. Express 20(19), 21331–21340 (2012).
[Crossref] [PubMed]

K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express 16(15), 11589–11603 (2008).
[Crossref] [PubMed]

X. Li, H. Hu, T. Liu, B. Huang, and Z. Song, “Optimal distribution of integration time for intensity measurements in degree of linear polarization polarimetry,” Opt. Express 24(7), 7191–7200 (2016).
[Crossref] [PubMed]

T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, “Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction,” Opt. Express 15(5), 2033–2046 (2007).
[Crossref] [PubMed]

Y. Guo, N. Zeng, H. He, T. Yun, E. Du, R. Liao, Y. He, and H. Ma, “A study on forward scattering Mueller matrix decomposition in anisotropic medium,” Opt. Express 21(15), 18361–18370 (2013).
[Crossref] [PubMed]

Opt. Lett. (3)

Proc. SPIE (2)

W. Du, S. Liu, C. Zhang, and X. Chen, “Optimal configurations for the dual rotating-compensator Mueller matrix ellipsometer,” Proc. SPIE 8759, 875925 (2013).
[Crossref]

R. M. A. Azzam, “Mueller-matrix ellipsometry: a review in Polarization: Measurement, Analysis, and Remote Sensing,” Proc. SPIE 3121, 396–399 (1997).
[Crossref]

Rev. Sci. Instrum. (1)

D. A. Ramsey and K. C. Luderna, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65(9), 2874–2881 (1994).
[Crossref]

Other (5)

K. Hinrichs and K. J. Eichhorn, eds., Ellipsometry of Functional Organic Surfaces and Films (Springer, 2014).

R. M. Azzam and N. M. Bashara, Ellipsometry and polarized light (Elsevier Science Publishing, 1987).

D. Goldstein, Polarized Light (Dekker, 2003).

H. Tompkins and E. Irene, Handbook of Ellipsometry (William Andrew, 2005).

F. Hiai and D. Petz, Introduction to matrix analysis and applications (Springer Science & Business Media, 2014).

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Figures (1)

Fig. 1
Fig. 1 With the instrument matrix A s i m p , evolution of the total variance of the interested partial Mueller elements (8 elements or 4 elements) in function of parameters ψ and Δ .

Tables (3)

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Table 1 Variance of each element in Mueller matrix for different instrument matrices in the presence of Gaussian additive noise. The total variance of the eight Mueller elements and the ratio of optimization are also presented.

Tables Icon

Table 2 Variance of each element in Mueller matrix for different instrument matrices in the presence of Gaussian additive noise. The total variance of the four Mueller elements and the ratio of optimization are also presented.

Tables Icon

Table 3 The total variances of the eight and four Mueller elements and the ratio of optimization for different instrument matrices in the presence of Poisson shot noise.

Equations (28)

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ρ = r p r s = tan   ψ e i Δ ,
M = r [ 1 cos 2 ψ 0 0 cos 2 ψ 1 0 0 0 0 sin 2 ψ cos Δ sin 2 ψ sin Δ 0 0 sin 2 ψ sin Δ sin 2 ψ cos Δ ] ,
M = [ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ]
I = A T M W ,
V I = [ A A ] T V M ,
V ^ M = { [ A A ] T } 1 V I = [ ( A T ) 1 ( A T ) 1 ] V I = [ A 1 A 1 ] T V I .
Γ V ^ M = [ A 1 A 1 ] T Γ V I [ A 1 A 1 ] ,
M = [ m 11 m 12 0 0 m 21 m 22 0 0 0 0 m 33 m 34 0 0 m 43 m 44 ] .
Var [ M ] = [ σ 1 2 σ 2 2 σ 5 2 σ 6 2 σ 11 2 σ 12 2 σ 15 2 σ 16 2 ] ,
A 8 - e l e m s = arg min A i Ω 1 σ i 2 , Ω 1 = { 1 , 2 , 5 , 6 , 11 , 12 , 15 , 16 } .
M = [ m 11 m 12 m 33 m 34 ] .
ψ = 1 2 cos 1 [ m 12 m 11 ] , Δ = tan 1 [ m 34 m 33 ] .
A 4 - e l e m s = arg min A i Ω 2 σ i 2 , Ω 2 = { 1 , 2 , 11 , 12 } .
σ i 2 = σ 2 [ [ A A T ] 1 [ A A T ] 1 ] i i , i [ 1 , 16 ] .
A 8 e l e m s G a u = arg min A i Ω 1 σ 2 [ [ A A T ] 1 [ A A T ] 1 ] i i , A 4 e l e m s G a u = arg min A i Ω 2 σ 2 [ [ A A T ] 1 [ A A T ] 1 ] i i ,
[ Γ V I ] i i = [ V I ] i = k = 1 16 [ A A ] i k T [ V M ] k , i [ 1 , 16 ] .
σ i 2 = k = 1 16 [ V M ] k [ n = 1 16 ( [ A 1 A 1 ] n i ) 2 [ A A ] n k T ] .
σ i 2 = [ V M ] 1 4 n = 1 16 ( [ A 1 A 1 ] n i ) 2 + k = 2 16 [ V M ] k [ n = 1 16 ( [ A 1 A 1 ] n i ) 2 [ A A ] n k T ] .
A 8 e l e m s G a u = 1 2 [ 1 1 1 1 0 .550 -0 .550 0 .550 -0 .550 -0 .653 0 .521 0 .653 -0 .521 -0 .521 -0 .653 0 .521 0 .653 ] .
Var [ M ] = σ 2 [ 1.00 3.31 3.31 10.94 2.87 2.87 9.48 9.48 2.87 9.48 2.87 9.48 8.22 8.22 8.22 8.22 ] .
A t e t r a = 1 2 [ 1 1 1 1 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 ] .
A s i m p = 1 2 [ 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 ] .
A 4 e l e m s G a u = 1 2 [ 1 1 1 1 0 .443 0 .443 -0 .443 -0 .443 0 .732 -0 .732 0 .732 -0 .732 0 .518 -0 .518 -0 .518 0 .518 ]
Var [ M ] = σ 2 [ 1.00 5 .10 5 .10 25 .99 1 .87 3 .73 9 .51 19 .03 1 .87 9 .51 3 .73 19 .03 3 .48 6 .96 6 .96 13 .93 ] .
k = 2 16 [ V M ] k [ n = 1 16 ( [ A 1 A 1 ] n i ) 2 [ A A ] n k T ] = 0.
A 8 e l e m s P o i = arg min A i Ω 1 [ V M ] 1 4 n = 1 16 ( [ A 1 A 1 ] n i ) 2 , A 4 e l e m s P o i = arg min A i Ω 2 [ V M ] 1 4 n = 1 16 ( [ A 1 A 1 ] n i ) 2 ,
A 8 e l e m s P o i = A 8 e l e m s G a u = 1 2 [ 1 1 1 1 0 .550 -0 .550 0 .550 -0 .550 -0 .653 0 .521 0 .653 -0 .521 -0 .521 -0 .653 0 .521 0 .653 ] , A 4 e l e m s P o i = A 4 e l e m s G a u = 1 2 [ 1 1 1 1 0 .443 0 .443 -0 .443 -0 .443 0 .732 -0 .732 0 .732 -0 .732 0 .518 -0 .518 -0 .518 0 .518 ] .
Eight elements : Var [ M ] = [ V M ] 1 4 [ 1.00 3.31 3.31 10.94 2.87 2.87 9.48 9.48 2.87 9.48 2.87 9.48 8.22 8.22 8.22 8.22 ] , Four elements : Var [ M ] = [ V M ] 1 4 [ 1.00 5 .10 5 .10 25 .99 1 .87 3 .73 9 .51 19 .03 1 .87 9 .51 3 .73 19 .03 3 .48 6 .96 6 .96 13 .93 ] .

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