Abstract

Two methods used to retrieve Mueller matrices from intensity measurements are revisited. It is shown that with symmetry or orthogonality considerations, numerical inversions of polarimetric equations can be avoided. With the obtained analytical formulas, noise propagation can be analyzed. If the intensity noise is a Gaussian white noise, the noise of Mueller matrices features remarkable properties. Mueller components are mutually correlated according to a scheme that involves decomposition into four blocks of 2×2 matrices. Variances are unequally distributed: the middle 2×2 block has the highest variance, the element on the bottom right has the lowest. These characteristics have been validated on experimental Mueller images of the free space.

© 2007 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  2. C. Brosseau, Fundamentals of Polarized Light--A Statistical Optics Approach (Wiley, 1998).
  3. H. Mueller, "The foundations of optics," J. Opt. Soc. Am. 38, 661-661 (1948).
  4. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).
  5. D. H. Goldstein, ed., "Polarization measurement, analysis, and applications V," Proc. SPIE 4819, 68-74 (2002).
  6. J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
    [CrossRef]
  7. D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
    [CrossRef]
  8. J. S. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt. 41, 619-630 (2002).
    [CrossRef] [PubMed]
  9. S.-M. Nee, "Error analysis for Mueller matrix measurement," J. Opt. Soc. Am. A 20, 1651-1657 (2003).
    [CrossRef]
  10. R. A. Chipman, "Polarimetry" in Handbook of Optics (McGraw-Hill, 1994), Chap. 22.
  11. F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
    [CrossRef]
  12. A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part I," Opt. Eng. 34, 1651-1655 (1995).
    [CrossRef]
  13. C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, 1971).
  14. A. Ruszczynski, Nonlinear Optimization (Princeton U. Press, 2006).
  15. J. S. Walker, Fast Fourier Transforms, 2nd ed. (CRC Press, 1996).
  16. A. J. Bouwens, Digital Instrumentation (McGraw-Hill, 1984).
  17. M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
    [CrossRef]
  18. G. L. Liu, Y. Li, and B. D. Cameron, "Polarization-based optical imaging and processing techniques with application to cancer diagnostics," Proc. SPIE 4617, 208-220 (2002).
    [CrossRef]
  19. J. W. Goodman, Statistical Optics (Wiley, 1985).
  20. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1984).
  21. H. Neudecker, "Some theorems on matrix differentiation with special reference to Kronecker matrix products," J. Am. Stat. Assoc. 64, 953-963 (1969).
    [CrossRef]
  22. S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
    [CrossRef]
  23. W. A. Shurcliff, Polarized Light: Production and Use (van Nostrand, 1964).
  24. A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," http://www.arxiv.org/abs/math-ph/0412061.
  25. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, "Optimization of retardance for a complete Stokes polarimeter," Opt. Lett. 25, 802-804 (2000).
    [CrossRef]
  26. C. S. Won and R. M. Gray, Stochastic Image Processing (Springer/Kluwer/Plenum Academic, 2004).

2004 (1)

M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

2003 (2)

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

S.-M. Nee, "Error analysis for Mueller matrix measurement," J. Opt. Soc. Am. A 20, 1651-1657 (2003).
[CrossRef]

2002 (3)

J. S. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt. 41, 619-630 (2002).
[CrossRef] [PubMed]

G. L. Liu, Y. Li, and B. D. Cameron, "Polarization-based optical imaging and processing techniques with application to cancer diagnostics," Proc. SPIE 4617, 208-220 (2002).
[CrossRef]

D. H. Goldstein, ed., "Polarization measurement, analysis, and applications V," Proc. SPIE 4819, 68-74 (2002).

2000 (1)

1997 (1)

F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

1995 (2)

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part I," Opt. Eng. 34, 1651-1655 (1995).
[CrossRef]

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

1990 (1)

1969 (1)

H. Neudecker, "Some theorems on matrix differentiation with special reference to Kronecker matrix products," J. Am. Stat. Assoc. 64, 953-963 (1969).
[CrossRef]

1948 (1)

H. Mueller, "The foundations of optics," J. Opt. Soc. Am. 38, 661-661 (1948).

Aiello, A.

A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," http://www.arxiv.org/abs/math-ph/0412061.

Alouini, M.

M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Ambirajan, A.

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part I," Opt. Eng. 34, 1651-1655 (1995).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bouwens, A. J.

A. J. Bouwens, Digital Instrumentation (McGraw-Hill, 1984).

Brosseau, C.

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

Cameron, B. D.

G. L. Liu, Y. Li, and B. D. Cameron, "Polarization-based optical imaging and processing techniques with application to cancer diagnostics," Proc. SPIE 4617, 208-220 (2002).
[CrossRef]

Chipman, R. A.

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
[CrossRef]

R. A. Chipman, "Polarimetry" in Handbook of Optics (McGraw-Hill, 1994), Chap. 22.

Dereniak, E. L.

Descour, M. R.

Dolfi, D.

M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Goldstein, D. H.

D. H. Goldstein, ed., "Polarization measurement, analysis, and applications V," Proc. SPIE 4819, 68-74 (2002).

D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
[CrossRef]

Goodman, J. W.

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

Goudail, F.

M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Gray, R. M.

C. S. Won and R. M. Gray, Stochastic Image Processing (Springer/Kluwer/Plenum Academic, 2004).

Grisard, A.

M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Kemme, S. A.

Lallier, E.

M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Le Jeune, B.

F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Le Roy-Brehonnet, F.

F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Li, Y.

G. L. Liu, Y. Li, and B. D. Cameron, "Polarization-based optical imaging and processing techniques with application to cancer diagnostics," Proc. SPIE 4617, 208-220 (2002).
[CrossRef]

Liu, G. L.

G. L. Liu, Y. Li, and B. D. Cameron, "Polarization-based optical imaging and processing techniques with application to cancer diagnostics," Proc. SPIE 4617, 208-220 (2002).
[CrossRef]

Look, D. C.

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part I," Opt. Eng. 34, 1651-1655 (1995).
[CrossRef]

Mitra, S. K.

C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, 1971).

Mueller, H.

H. Mueller, "The foundations of optics," J. Opt. Soc. Am. 38, 661-661 (1948).

Nee, S.-M.

Neudecker, H.

H. Neudecker, "Some theorems on matrix differentiation with special reference to Kronecker matrix products," J. Am. Stat. Assoc. 64, 953-963 (1969).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1984).

Pezzaniti, J. L.

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

Phipps, G. S.

Rao, C. R.

C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, 1971).

Réfrégier, Ph.

M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Ruszczynski, A.

A. Ruszczynski, Nonlinear Optimization (Princeton U. Press, 2006).

Sabatke, D. S.

Savenkov, S. N.

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light: Production and Use (van Nostrand, 1964).

Sweatt, W. C.

Tyo, J. S.

Walker, J. S.

J. S. Walker, Fast Fourier Transforms, 2nd ed. (CRC Press, 1996).

Woerdman, J. P.

A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," http://www.arxiv.org/abs/math-ph/0412061.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Won, C. S.

C. S. Won and R. M. Gray, Stochastic Image Processing (Springer/Kluwer/Plenum Academic, 2004).

Yushtin, K. E.

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

Appl. Opt. (1)

J. Am. Stat. Assoc. (1)

H. Neudecker, "Some theorems on matrix differentiation with special reference to Kronecker matrix products," J. Am. Stat. Assoc. 64, 953-963 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

H. Mueller, "The foundations of optics," J. Opt. Soc. Am. 38, 661-661 (1948).

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

Opt. Eng. (2)

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter: part I," Opt. Eng. 34, 1651-1655 (1995).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (4)

D. H. Goldstein, ed., "Polarization measurement, analysis, and applications V," Proc. SPIE 4819, 68-74 (2002).

M. Alouini, F. Goudail, Ph. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, "Multispectral polarimetric imaging with coherent illumination: towards higher image contrast," Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

G. L. Liu, Y. Li, and B. D. Cameron, "Polarization-based optical imaging and processing techniques with application to cancer diagnostics," Proc. SPIE 4617, 208-220 (2002).
[CrossRef]

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

Prog. Quantum Electron. (1)

F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Other (13)

R. A. Chipman, "Polarimetry" in Handbook of Optics (McGraw-Hill, 1994), Chap. 22.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).

W. A. Shurcliff, Polarized Light: Production and Use (van Nostrand, 1964).

A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," http://www.arxiv.org/abs/math-ph/0412061.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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

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

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1984).

C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, 1971).

A. Ruszczynski, Nonlinear Optimization (Princeton U. Press, 2006).

J. S. Walker, Fast Fourier Transforms, 2nd ed. (CRC Press, 1996).

A. J. Bouwens, Digital Instrumentation (McGraw-Hill, 1984).

C. S. Won and R. M. Gray, Stochastic Image Processing (Springer/Kluwer/Plenum Academic, 2004).

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Figures (6)

Fig. 1
Fig. 1

Principal schematics of active rotating polarimeters. Polarization properties of a light source is controlled thanks to a polarizer followed by a rotating quarter-wave plate constituting a PSG. Light scattered by the sample is analyzed by means of a rotating quarter-wave plate and a polarizer, which form the PSA.

Fig. 2
Fig. 2

Normalized histogram of the 12-bit camera (Hamamatsu C4742-95) used in the polarimeter of Fig. 1. Bars correspond to the normalized histogram of gray levels and the continuous curve represents the best fit by a Gaussian function. Dark is at level 177. Variance is ∼11.

Fig. 3
Fig. 3

Covariance matrix of Mueller elements. The matrix is decomposed into four blocks of 8 × 8 matrices. Blocks in gray correspond to null matrices and the two main blocks on the diagonal feature alternation between 2 × 2 nonnull and null matrices (checkerboard layout).

Fig. 4
Fig. 4

Elements of a Mueller matrix that are mutually correlated form four distinct blocks of 2 × 2 matrices.

Fig. 5
Fig. 5

Effect of sign inversion of the two middle rows of a Mueller matrix (see text). Elements of the covariance matrix that have their sign inverted are marked with the minus sign.

Fig. 6
Fig. 6

Mueller images of the free space obtained with the MI method and the polarimeter of Fig. 1 (see text). The bar on the right indicates the relative intensity scale for all the 4 × 4 images.

Equations (60)

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

S i ( θ ) = [ 1 cos 2 ( 2 θ ) 1 2  sin ( 4 θ ) sin ( 2 θ ) ] .
S o T ( θ ) = [ 1 cos 2 ( 2 θ ) 1 2  sin ( 4 θ ) sin ( 2 θ ) ] .
I ( θ , θ ) = S o T ( θ ) M S i ( θ ) .
I = [ S o T ( θ 1 ) S o T ( θ 2 ) S o T ( θ 3 ) S o T ( θ 4 ) ] A M [ S i ( θ 1 ) S i ( θ 2 ) S i ( θ 3 ) S i ( θ 4 ) ] G .
θ 1 = θ 4 ,
θ 2 = θ 3 .
[ 1 1 1 1 cos 2 ( 2 θ 1 ) cos 2 ( 2 θ 2 ) cos 2 ( 2 θ 2 ) cos 2 ( 2 θ 1 ) 1 2  sin ( 4 θ 1 ) 1 2  sin ( 4 θ 2 ) + 1 2  sin ( 4 θ 2 ) + 1 2  sin ( 4 θ 1 ) sin ( 2 θ 1 ) sin ( 2 θ 2 ) + sin ( 2 θ 2 ) + sin ( 2 θ 1 ) ] × [ u 1 v 1 x 1 y 1 u 2 v 2 x 2 y 2 u 2 v 2 x 2 y 2 u 1 v 1 x 1 y 1 ] G 1 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] I d .
u 1 = + cos 2 ( 2 θ 2 ) / ( 2   sin ( 2 ( θ 1 θ 2 ) ) sin ( 2 ( θ 1 + θ 2 ) ) ) ,
v 1 = + 1 / ( 2   sin ( 2 ( θ 1 θ 2 ) ) sin ( 2 ( θ 1 + θ 2 ) ) ) ,
u 2 = cos 2 ( 2 θ 1 ) / ( 2   sin ( 2 ( θ 1 θ 2 ) ) sin ( 2 ( θ 1 + θ 2 ) ) ) ,
v 2 = 1 / ( 2   sin ( 2 ( θ 1 θ 2 ) ) sin ( 2 ( θ 1 + θ 2 ) ) ) ,
x 1 = + 1 / ( 4   sin ( θ 1 θ 2 ) sin ( θ 1 + θ 2 ) sin ( 2 θ 1 ) ) ,
y 1 = cos ( 2 θ 2 ) / ( 4   sin ( θ 1 θ 2 ) sin ( θ 1 + θ 2 ) sin ( 2 θ 1 ) ) ,
x 2 = 1 / ( 4   sin ( θ 1 θ 2 ) sin ( θ 1 + θ 2 ) sin ( 2 θ 2 ) ) ,
y 2 = + cos ( 2 θ 1 ) / ( 4   sin ( θ 1 θ 2 ) sin ( θ 1 + θ 2 ) sin ( 2 θ 2 ) ) .
A 1 = [ + 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 + 1 ] R · [ G 1 ] T .
M = R · [ G 1 ] T I G 1 .
k = 0 N 1 sin ( 4 θ k ) sin ( 2 θ k ) = 0 ,
k = 0 N 1 sin ( 4 θ k ) = k = 0 N 1 sin ( 4 θ k ) cos 2 ( 2 θ k ) = 0 ,
k = 0 N 1 sin ( 2 θ k ) = k = 0 N 1 sin ( 2 θ k ) cos 2 ( 2 θ k ) = 0.
k = 0 N 1 1 = N ,
k = 0 N 1 cos 2 ( 2 θ k ) cos ( 4 θ k ) = N / 4 ,
k = 0 N 1 sin ( 4 θ k ) sin ( 4 θ k ) = k = 0 N 1 sin ( 2 θ k ) sin ( 2 θ k ) = N / 2 .
G · [ 1 cos ( 4 θ 0 ) sin ( 4 θ 0 ) sin ( 2 θ 0 ) 1 cos ( 4 θ N 1 ) sin ( 4 θ N 1 ) sin ( 2 θ N 1 ) ] W i = N 2 [ 2 0 0 0 1 1 / 2 0 0 0 0 1 / 2 0 0 0 0 1 ] .
P i = 2 N [ 1 / 2 0 0 0 1 2 0 0 0 0 2 0 0 0 0 1 ] .
G 1 = W i P i .
W i T · A = N 2 [ 2 1 0 0 0 1 / 2 0 0 0 0 1 / 2 0 0 0 0 1 ] .
P o = R · P i T .
A 1 = R · P i T W i T = P o W i T .
M = R · [ W i P i ] T I W i P i .
Δ M = R · [ G 1 ] T Δ I G 1 .
Δ M = R · [ G 1 ] T Δ I G 1 = 0.
Δ N = R · Δ M = [ G 1 ] T Δ I G 1 .
vec { ( a i j ) i = 0 , , N 1 ; j = 0 , , N 1 ) = ( b k ) k = 0 , , N N 1
b j N + i = a i j .
vec { U V W } = ( [ W ] T U ) vec { V } .
C N = vec { Δ N } [ vec { Δ N } ] T = ( [ G 1 ] T [ G 1 ] T ) vec { Δ I } [ vec { Δ I } ] T × ( G 1 G 1 ) .
C N = σ 2 ( [ G 1 ] T G 1 ) N D ( [ G 1 ] T G 1 ) .
N D ( M I ) = [ 0.5918 0.8455 0 0 0.8455 2.0915 0 0 0 0 2.0619 0.0195 0 0 0.0195 0.4198 ] .
N D ( F T ) = 2 N [ 3 / 2 2 0 0 2 4 0 0 0 0 4 0 0 0 0 1 ] .
N D = [ n 00 n 01 0 0 n 10 n 11 0 0 0 0 n 22 n 23 0 0 n 32 n 33 ] = [ N u O O N d ] .
C N = σ 2 · [ n 00 N u O n 01 N u O O O O O O n 00 N d O n 01 N d O O O O n 10 N u O n 11 N u O O O O O O n 10 N d O n 11 N d O O O O O O O O n 22 N u O n 23 N u O O O O O O n 22 N d O n 23 N d O O O O n 32 N u O n 33 N u O O O O O O n 32 N d O n 33 N d ] .
var { M } = σ 2 [ 0.3502 1.2377 1.2202 0.2485 1.2377 4.3742 4.3124 0.8781 1.2202 4.3124 4.2514 0.8657 0.2485 0.8781 0.8657 0.1763 ] M I = σ 2 4 N 2 [ 9 / 4 6 6 3 / 2 6 16 16 4 6 16 16 4 3 / 2 4 4 1 ] F T .
Cov ( m i j , m p q ) = 1 L l = 0 L 1 ( m i j ) l ( m p q ) l .
C 11 = [ 0.3502 0.5004 0.5004 0.7149 0.5004 1.2377 0.7149 1.7683 0.5004 0.7149 1.2377 1.7683 0.7149 1.7683 1.7683 4.3742 ] M I ,
4 N 2 [ 9 / 4 3 3 4 3 6 4 8 3 4 6 8 4 8 8 16 ] F T ,
C ^ 11 = [ 0.3519 0.5013 0.5009 0.7149 0.5013 1.2364 0.7127 1.7556 0.5009 0.7127 1.2379 1.7670 0.7149 1.7556 1.7670 4.3389 ] M I ,
4 N 2 [ 2.3275 3.0018 2.9999 4.0034 3.0018 6.0115 4.0015 8.0027 2.9999 4.0015 6.0069 7.9982 4.0034 8.0027 7.9982 15.9333 ] F T ,
C 12 = [ 1.2202 0.0115 1.7433 0.0165 0.0115 0.2485 0.0165 0.3550 1.7433 0.0165 4.3124 0.0407 0.0165 0.3550 0.0407 0.8781 ] M I ,
4 N 2 [ 6 0 8 0 0 3 / 2 0 2 8 0 16 0 0 2 0 4 ] F T ,
C ^ 12 = [ 1.2328 0.0145 1.7629 0.0175 0.0145 0.2497 0.0219 0.3568 1.7629 0.0219 4.3181 0.0368 0.0175 0.3568 0.0368 0.8812 ] M I ,
4 N 2 [ 5.9948 0.0135 7.9585 0.0128 0.0135 1.5071 0.0162 2.0091 7.9585 0.0162 15.8227 0.0036 0.0128 2.0091 0.0036 4.0121 ] F T ,
C 21 = [ 1.2202 1.7433 0.0115 0.0165 1.7433 4.3124 0.0165 0.0407 0.0115 0.0165 0.2485 0.3550 0.0165 0.0407 0.3550 0.8781 ] M I ,
4 N 2 [ 6 8 0 0 8 16 0 0 0 0 3 / 2 2 0 0 2 4 ] F T ,
C ^ 21 = [ 1.2283 1.7571 0.0086 0.0137 1.7571 4.3154 0.0122 0.0291 0.0086 0.0122 0.2508 0.3571 0.0137 0.0291 0.3571 0.8791 ] M I ,
4 N 2 [ 5.9435 7.9344 0.0207 0.0395 7.9344 15.8873 0.0015 0.0124 0.0207 0.0015 1.5092 2.0073 0.0395 0.0124 2.0073 3.9984 ] F T ,
C 22 = [ 4.2514 0.0402 0.0402 0.0004 0.0402 0.8657 0.0004 0.0082 0.0402 0.0004 0.8657 0.0082 0.0004 0.0082 0.0082 0.1763 ] M I ,
4 N 2 [ 16 0 0 0 0 4 0 0 0 0 4 0 0 0 0 1 ] F T ,
C ^ 22 = [ 4.1819 0.0225 0.0155 0.0004 0.0225 0.8644 0.0077 0.0058 0.0155 0.0077 0.8690 0.0087 0.0004 0.0058 0.0087 0.1777 ] M I ,
4 N 2 [ 15.8699 0.0219 0.0478 0.0138 0.0219 3.9856 0.0149 0.0094 0.0478 0.0149 3.9939 0.0015 0.0138 0.0094 0.0015 1.0094 ] F T .

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