Abstract

Necessary and sufficient conditions for an optical system characterized by a Mueller matrix to be characterized by a Mueller–Jones, and thence by a Jones, matrix are considered, and the issue of measurement error is examined. It is shown that a Mueller matrix can be expressed as a linear combination of at most four trace orthonormal Mueller–Jones matrices, and an algorithm for the construction is given.

© 1994 Optical Society of America

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References

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  1. R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidths,” J. Opt. Soc. Am. 53, 317–323 (1963).
    [CrossRef]
  2. E. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  3. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).
  4. R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, New York, 1985).
    [CrossRef]
  5. R. C. Jones, “A new calculus for the treatment of optical systems. V. A more general formulation and description of another calculus,” J. Opt. Soc. Am. 37, 107–110 (1947).
    [CrossRef]
  6. P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phenomènes de fluorescence,” Ann. Phys. (N.Y.) 12, 23–97 (1929).
  7. F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
    [CrossRef]
  8. R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, New York, 1991).
    [CrossRef]
  9. R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  10. E. S. Frye, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
    [CrossRef]
  11. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
    [CrossRef]
  12. J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
    [CrossRef]
  13. B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 1809–1812 (1979).
    [CrossRef]
  14. D. A. Ramsey, “Thin film measurements on rough substrates using Mueller matrix ellipsometry,” Ph.D. dissertation (Department of Mechanical Engineering, University of Michigan, Ann Arbor, Mich., 1985).
  15. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  16. S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 177–185 (1989).
    [CrossRef]
  17. J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
    [CrossRef]
  18. K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  19. J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).
  20. W. M. Boerner, W. L. Yan, “Basic principles of radar polarimetry and its applications to target recognition with assessments of the historical development and of the current state-of-the-art,” in Electromagnetic Modelling and Measurements for Analysis and Synthesis Problems, B. de Neumann, ed. (Kluwer, Norwell, Mass., 1991), pp. 311–363.
    [CrossRef]
  21. W. M. Boerner, ed., Direct and Inverse Methods in Radar Polarimetry (Kluwer, Norwell, Mass., 1992), Parts 1 and 2.
  22. J. W. Hovenier, H. C. van der Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

1987 (2)

K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
[CrossRef]

J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).

1986 (2)

J. W. Hovenier, H. C. van der Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[CrossRef]

1985 (1)

J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

1982 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

1981 (2)

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

E. S. Frye, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
[CrossRef]

1979 (1)

B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 1809–1812 (1979).
[CrossRef]

1963 (1)

1947 (1)

1942 (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

1929 (1)

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phenomènes de fluorescence,” Ann. Phys. (N.Y.) 12, 23–97 (1929).

Barakat, R.

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidths,” J. Opt. Soc. Am. 53, 317–323 (1963).
[CrossRef]

Bernabeu, E.

J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).

J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[CrossRef]

J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Boerner, W. M.

W. M. Boerner, W. L. Yan, “Basic principles of radar polarimetry and its applications to target recognition with assessments of the historical development and of the current state-of-the-art,” in Electromagnetic Modelling and Measurements for Analysis and Synthesis Problems, B. de Neumann, ed. (Kluwer, Norwell, Mass., 1991), pp. 311–363.
[CrossRef]

Cloude, S. R.

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 177–185 (1989).
[CrossRef]

Frye, E. S.

Gil, J. J.

J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).

J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[CrossRef]

J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

Horn, R. A.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, New York, 1985).
[CrossRef]

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, New York, 1991).
[CrossRef]

Hovenier, J. W.

J. W. Hovenier, H. C. van der Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Howell, B. J.

B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 1809–1812 (1979).
[CrossRef]

Johnson, C. R.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, New York, 1991).
[CrossRef]

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, New York, 1985).
[CrossRef]

Jones, R. C.

Kattawar, G. W.

Kim, K.

Mandel, L.

O’Neill, E.

E. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Perrin, F.

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

Ramsey, D. A.

D. A. Ramsey, “Thin film measurements on rough substrates using Mueller matrix ellipsometry,” Ph.D. dissertation (Department of Mechanical Engineering, University of Michigan, Ann Arbor, Mich., 1985).

Simon, R.

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Soleillet, P.

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phenomènes de fluorescence,” Ann. Phys. (N.Y.) 12, 23–97 (1929).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

van der Hulst, H. C.

J. W. Hovenier, H. C. van der Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

van der Mee, C. V. M.

J. W. Hovenier, H. C. van der Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

Wolf, E.

Yan, W. L.

W. M. Boerner, W. L. Yan, “Basic principles of radar polarimetry and its applications to target recognition with assessments of the historical development and of the current state-of-the-art,” in Electromagnetic Modelling and Measurements for Analysis and Synthesis Problems, B. de Neumann, ed. (Kluwer, Norwell, Mass., 1991), pp. 311–363.
[CrossRef]

Ann. Phys. (N.Y.) (1)

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phenomènes de fluorescence,” Ann. Phys. (N.Y.) 12, 23–97 (1929).

Appl. Opt. (2)

B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 1809–1812 (1979).
[CrossRef]

E. S. Frye, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
[CrossRef]

Astron. Astrophys. (1)

J. W. Hovenier, H. C. van der Hulst, C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

J. Chem. Phys. (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Acta (2)

J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[CrossRef]

Opt. Commun. (2)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Optik (Stuttgart) (1)

J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).

Other (9)

W. M. Boerner, W. L. Yan, “Basic principles of radar polarimetry and its applications to target recognition with assessments of the historical development and of the current state-of-the-art,” in Electromagnetic Modelling and Measurements for Analysis and Synthesis Problems, B. de Neumann, ed. (Kluwer, Norwell, Mass., 1991), pp. 311–363.
[CrossRef]

W. M. Boerner, ed., Direct and Inverse Methods in Radar Polarimetry (Kluwer, Norwell, Mass., 1992), Parts 1 and 2.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, New York, 1991).
[CrossRef]

E. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1989).

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, New York, 1985).
[CrossRef]

D. A. Ramsey, “Thin film measurements on rough substrates using Mueller matrix ellipsometry,” Ph.D. dissertation (Department of Mechanical Engineering, University of Michigan, Ann Arbor, Mich., 1985).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 177–185 (1989).
[CrossRef]

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Tables (7)

Tables Icon

Table 1 Necessary-Conditions Test (4.8) for M1

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Table 2 Necessary-Conditions Test (4.8) for M2

Tables Icon

Table 3 Necessary-Conditions Test (4.8) for M3

Tables Icon

Table 4 Necessary-Conditions Test (4.8) for M4

Tables Icon

Table 5 Necessary-Conditions Test (4.8) for M5

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Table 6 Mueller–Jones Components

Tables Icon

Table 7 Dominant-Component Jones Matrix

Equations (167)

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σ 0 = [ 1 0 0 1 ] , σ 1 = [ 1 0 0 1 ] σ 2 = [ 0 1 1 0 ] , σ 3 = [ 0 i i 0 ] ,
tr ( σ j σ k ) = 2 δ j k
Φ o = 1 2 k = 0 3 s k σ k = 1 2 [ s 0 + s 1 s 2 i s 3 s 2 + i s 3 s 0 s 1 ] ,
Φ i = 1 2 k = 0 3 $ k σ k = 1 2 [ $ 0 + $ 1 $ 2 i $ 3 $ 2 + i $ 3 $ 0 $ 1 ]
s k = tr ( Φ o σ k ) , $ k = tr ( Φ i σ k ) .
o = s 0 2 s 1 2 s 2 2 s 3 2 , i = $ 0 2 $ 1 2 $ 2 2 $ 3 2 ,
P o = ( s 1 2 + s 2 2 + s 3 2 ) 1 / 2 / s 0 , P i = ( $ 1 2 + $ 2 2 + $ 3 2 ) 1 / 2 / $ 0 ,
Φ o = J Φ o J * .
J * J = 1 2 k = 0 3 J k σ k = 1 2 [ J 0 + J 1 J 2 i J 3 J 2 + i J 3 J 0 J 1 ] ,
J k = tr ( J * J σ k ) .
s 0 = tr ( Φ o ) = tr ( J Φ i J * ) = tr ( J * J Φ i )
s 0 = tr [ ( 1 2 k = 0 3 J k σ k ) ( 1 2 l = 0 3 $ l σ l ) ] = 1 2 k = 0 3 J k $ k .
g = 1 2 $ 0 k = 0 3 J k $ k = 1 2 ( J 0 + k = 1 3 J k $ k / $ 0 ) .
| k = 0 3 J k $ k | ( J 1 2 + J 2 2 + J 3 2 ) 1 / 2 ( $ 1 2 + $ 2 2 + $ 3 2 ) 1 / 2 ,
1 2 [ J 0 ( J 1 2 + J 2 2 + J 3 2 ) 1 / 2 P i ] g 1 2 [ J 0 + ( J 1 2 + J 2 2 + J 3 2 ) 1 / 2 P i ]
1 2 [ J 0 ( J 1 2 + J 2 2 + J 3 2 ) 1 / 2 ] g 1 2 [ J 0 + ( J 1 2 + J 2 2 + J 3 2 ) 1 / 2 ] ,
J 0 + ( J 1 2 + J 2 2 + J 3 2 ) 1 / 2 2 .
s 0 2 s 1 2 s 2 2 s 3 2 = det ( J * J ) ( $ 0 2 $ 1 2 $ 2 2 $ 3 2 )
o = det ( J * J ) i ,
1 P o 2 = [ det ( J * J ) / g 2 ] ( 1 P i 2 ) ,
det ( J * J ) = | det ( J ) | 2 = ¼ ( J 0 2 J 1 2 J 2 2 J 3 2 ) .
J 0 2 J 1 2 J 2 2 J 3 2 = [ J 0 ( J 1 2 + J 2 2 + J 3 2 ) 1 / 2 ] × [ J 0 + ( J 1 2 + J 2 2 + J 3 2 ) 1 / 2 ] ,
det ( J * J ) 1 ,
s = M $ .
M = A ( J J ¯ ) A 1
A = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] ,
A 1 = ½ A * .
1 2 k = 0 3 s k σ k = 1 2 k = 0 3 $ k ( J σ k J * ) .
s j = k = 0 3 1 2 tr ( σ j J σ k J * ) $ k
tr ( σ j J σ k J * ) ¯ = tr [ ( σ j J σ k J * ) * ] = tr ( J σ k J * σ j ) = tr ( σ j J σ k J * ) ,
m j k = ½ tr ( σ j J σ k J * )
m 0 k = ½ J k , k = 0 , 1 , 2 , 3 ,
g = 1 $ 0 k = 0 3 m 0 k $ k .
m 00 + ( m 01 2 + m 02 2 + m 03 2 ) 1 / 2 1 .
ρ ( M ) = ρ ( J J ¯ ) ,
tr ( M ) = tr ( J J ¯ ) ,
det ( M ) = det ( J J ¯ ) .
ρ ( M ) = ρ ( J ) ρ ( J ¯ ) = [ ρ ( J ) ] 2 ,
tr ( M ) = tr ( J ) tr ( J ¯ ) = | tr ( J ) | 2 ,
det ( M ) = [ det ( J ) ] 2 [ det ( J ¯ ) ] 2 = | det ( J ) | 4 .
o = [ det ( M ) ] 1 / 2 i .
s 0 2 s 1 2 s 2 2 s 3 2 = [ det ( M ) ] 1 / 2 ( $ 0 2 $ 1 2 $ 2 2 $ 3 2 ) .
G = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
s * Gs = [ det ( M ) ] 1 / 2 $ * G $ ,
$ * M * GM $ = [ det ( M ) ] 1 / 2 $ * G $ .
M * GM = [ det ( M ) ] 1 / 2 G .
α = [ det ( M ) ] 1 / 2 = ½ tr ( M * GM ) ,
Q j k = ( σ j σ 0 ) F ( σ k σ 0 ) F *
tr ( Q j k ) = 4 m 00 m j k , j , k = 0 , 1 , 2 , 3 .
F = J J ¯ ,
Q j k = ( σ j σ 0 ) ( J J ¯ ) ( σ k σ 0 ) ( J J ¯ ) * .
Q j k = ( σ j J σ 0 J * ) ( σ 0 J ¯ σ 0 J ¯ * ) ,
tr ( Q j k ) = tr ( σ j J σ k J * ) tr ( σ 0 J ¯ σ 0 J ¯ * ) .
F = ½ A * MA .
M 1 = [ 0.7599 0.0623 0.0295 0.1185 0.0573 0.4687 0.1811 0.1863 0.0384 0.1714 0.5394 0.0282 0.1240 0.2168 0.0120 0.6608 ] ,
M 2 = [ 0.8488 0.0503 0.0294 0.0617 0.0503 0.8304 0.0913 0.0920 0.0294 0.0913 0.8277 0.0000 0.0617 0.0920 0.0000 0.7947 ] ,
M 3 = [ 0.5491 0.1982 0.0096 0.0004 0.1991 0.5492 0.0389 0.0102 0.0064 0.0372 0.4285 0.2804 0.0004 0.0111 0.2814 0.4281 ] ,
M 4 = [ 1.0000 0.7476 0.0396 0.0030 0.7486 0.9992 0.0475 0.0258 0.0394 0.0467 0.0237 0.6605 0.0051 0.0261 0.6621 0.0203 ] ,
M 5 = [ 1.0000 0.5261 0.1205 0.1279 0.4131 0.5508 0.1924 0.2762 0.3908 0.5560 0.0417 0.2983 0.0498 0.0923 0.4381 0.2684 ] .
e J * M e k = tr ( Q j k ) / 4 m 0 , 0 .
F = A 1 MA = ½ A * MA
F = J J ¯ .
F = B ¯ B .
sgn ( z ) = { z / | z | , 1 , z 0 z = 0 .
z = | z | sgn ( z )
( i , j ) = n i + j .
f ( i , k ) , ( j , l ) = b ¯ i , j b k , l
f ( i , i ) , ( j , j ) = | b i , j | 2
f ( i , i ) , ( j , j ) 0
| f ( i , k ) , ( j , l ) | = | b i , j | | b k , l |
| f ( i , k ) , ( j , l ) | 2 = f ( i , i ) , ( j , j ) f ( k , k ) , ( l , l )
sgn ( f ( i , k ) , ( j , l ) ) = sgn ( b k , l ) / sgn ( b i , j )
sgn ( f ( 0 , k ) , ( 0 , l ) ) = sgn ( b k , l )
sgn ( f ( i , k ) , ( j , l ) ) = sgn ( f ( 0 , k ) , ( 0 , l ) ) / sgn ( f ( 0 , i ) , ( 0 , j ) )
| b i , j | = ( f ( i , i ) , ( j , j ) ) 1 / 2
sgn ( b k , l ) = sgn ( f ( 0 , k ) , ( 0 , l ) )
b k , l = | b k , l | sgn ( b k , l )
M ± = [ c 0 c 1 0 0 c 1 c 0 0 0 0 0 c 2 c 3 0 0 ± c 3 c 2 ] .
F + = [ c 0 + c 1 0 0 0 0 c 2 i c 3 0 0 i c 3 c 2 0 0 0 0 c 0 c 1 ] .
c 0 | c 1 | 0 ,
c 2 = ( c 0 2 c 1 2 ) 1 / 2 ,
c 3 = 0 ;
B + = [ ( c 0 + c 1 ) 1 / 2 0 0 ( c 0 c 1 ) 1 / 2 ] .
c 0 + | c 1 | 1 ,
F = [ c 0 + c 1 0 0 0 0 c 2 + i c 3 0 0 0 0 c 2 i c 3 0 0 0 0 c 0 c 1 ] .
c 0 | c 1 | 0 ,
( c 2 2 + c 3 2 ) 1 / 2 = ( c 0 2 c 1 2 ) 1 / 2 ;
B = [ ( c 0 + c 1 ) 1 / 2 0 0 ( c 2 + i c 3 ) / ( c 0 + c 1 ) 1 / 2 ] .
c 0 + | c 1 | 1 .
υ ( i , j ) = υ i , j , i , j = 0 , 1 , . . . , n 1 .
ϕ o = 1 2 [ s 0 + s 1 s 2 i s 3 s 2 + i s 3 s 0 s 1 ] ;
ϕ 0 = A 1 s .
h ( i , j ) , ( k , l ) = f ( i , k ) , ( j , l )
h ( i , j ) , ( k , l ) = b ¯ ( i , j ) b ( k , l )
H = b ¯ b .
H = uv * ,
H 2 = uv * uv * = ( v * u ) uv * = ( v * u ) H .
tr ( H ) = tr ( uv * ) = tr ( v * u ) = v * u ,
H 2 = tr ( H ) H ,
P = H / tr ( H ) ,
P 2 = P ,
H = 1 2 i = 0 3 j = o 3 m i , j σ i σ ¯ j ,
C = 1 2 A ,
M ̂ = C ( J J ¯ ) C * .
M ̂ = C ( B B ¯ ) C * ,
F = C * MC ,
F ̂ = C * M ̂ C = B ¯ B .
T F = ( i = 0 n 1 j = 0 n 1 | t i j | 2 ) 1 / 2 = [ tr ( T * T ) ] 1 / 2 .
U T F = T F = T U F
M M ̂ F = F F ̂ F = F B ¯ B F
χ ( B ) = F B ¯ B F 2 .
b k l = u k l + i υ k l
F B ¯ B F = H b ¯ b F .
H = UDV * ,
H = W Λ W * ,
Λ = | Λ | sgn ( Λ ) = sgn ( Λ ) | Λ | ,
d r 1 , r 1 δ d 0 , 0 > d r , r .
b ¯ = λ 0 , 0 w 0 .
H = k = 0 3 λ k , k w k w k * .
b ¯ k = w k , k = 0 , 1 , 2 , 3 ,
H = k = 0 3 λ k , k b ¯ k b k ,
F = k = 0 3 λ k , k , ( B ¯ k B k ) ,
M = k = 0 3 λ k , k A ( J k J ¯ k ) A 1 ,
M M ̂ F = ( d 1 , 1 2 + d 2 , 2 2 + d 3 , 3 2 ) 1 / 2
M M ̂ F = ( | λ 1 , 1 | 2 + | λ 2 , 2 | 2 + | λ 3 , 3 | 2 ) 1 / 2 .
M ̂ F = d 0 , 0 = | λ 0 , 0 | .
δ ( M , M ̂ ) = max i , j = 0 3 | e i * ( M M ̂ ) e j | .
w j * w k = δ j k = b j * b k ,
tr ( B j * B k ) = δ j k = tr ( J j * J k ) .
tr { [ A ( J j J ¯ j ) A 1 ] * [ A ( J k J ¯ k ) A 1 ] } = δ j k ,
tr { [ A ( J j J ¯ j ) A 1 ] * [ A ( J k J ¯ k ) A 1 ] } = | tr ( J j * J ¯ k ) | 2 ,
M M ̂ F < τ M ̂ F
δ ( M , M ̂ ) < τ m 0 , 0 .
ψ = ½ cos 1 ( c 1 / c 0 ) ,
Δ = tan 1 ( c 3 / c 2 ) .
tr ( Q 0 , 0 ) = tr ( FF * ) = 4 m 0 , 0 2 .
M F = F F = H F = 2 m 0 , 0 .
( i = 0 n 1 | z i | ) 2 = i = 0 n 1 | z i | 2 + 2 j > i = 0 n 1 | z i | | z j | .
( i = 0 n 1 | z i | ) 2 i = 0 n 1 | z i | 2
i = 0 n 1 | z i | ( i = 0 n 1 | z i | 2 ) 1 / 2
tr ( H ) = k = 0 3 λ k , k = 2 m 0 , 0 ,
M F = H F = ( k = 0 3 λ k , k 2 ) 1 / 2 .
M F 2 m 0 , 0
M F = F F = H F = ( k = 0 3 | λ k , k | 2 ) 1 / 2
M M ̂ F
δ ( M , M ̂ )
υ ( i , j ) = υ i , j , i , j = 0 , 1 , . . . , n 1 ,
x = ( U W ) v .
x i , j = k = 0 n 1 l = 0 n 1 u i , k υ k , l w l , j
y ( i , j ) , ( k , l ) = u i , k w l , j ;
x ( i , j ) = ( k , l ) = 0 n 2 1 y ( i , j ) , ( k , l ) υ ( k , l ) ,
ϕ o = A 1 s = ½ A * s ,
ϕ i = A 1 $ = ½ A * $ .
ϕ o = [ J ( J * ) ] ϕ i
A 1 s = ( J J ¯ ) A 1 $
s = A ( J J ¯ ) S 1 $ = M $
M = A ( J J ¯ ) A 1 .
Φ o = k = 0 3 λ k , k J k Φ i J k *
μ i , 1 y * Φ i y μ i , 2
x * Φ o x = k = 0 3 λ k , k x * J k Φ i J k * x .
μ i , 1 x * J k J k * x x * J k Φ i J k * x μ i , 2 x * J k J k * x .
ν k , 1 x * J k J k * x ν k , 2 , k = 0 , 1 , 2 , 3 ,
k = 0 3 λ k , k ν k , 1 k = 0 3 λ k , k x * J k J k * x k = 0 3 λ k , k ν k , 2 ,
μ i , 1 k = 0 3 λ k , k ν k , 1 x * Φ o x μ i , 2 k = 0 3 λ k , k ν k , 2 .
k = 0 3 λ k , k ν k , 1 0 .
g = k = 0 3 λ k , k g k .
0 < tr ( H ) = k = 0 3 λ k , k = 2 m 0 , 0 1 .

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