Abstract

We propose a rigorous definition of the minimal set of parameters that characterize the difference between two partially polarized states of light whose electric fields vary in three dimensions with Gaussian fluctuations. Although two such states are a priori defined by eighteen parameters, we demonstrate that the performance of processing tasks such as detection, localization, or segmentation of spatial or temporal polarization variations is uniquely determined by three scalar functions of these parameters. These functions define a “polarimetric contrast” that simplifies the analysis and the specification of processing techniques on polarimetric signals and images. This result can also be used to analyze the definition of the degree of polarization of a three-dimensional state of light with Gaussian fluctuations in comparison, with respect to its polarimetric contrast parameters, with a totally depolarized light. We show that these contrast parameters are a simple function of the degrees of polarization previously proposed by Barakat [Opt. Acta 30, 1171 (1983) ] and Setälä et al. [Phys. Rev. Lett. 88, 123902 (2002) ]. Finally, we analyze the dimension of the set of contrast parameters in different particular situations.

© 2006 Optical Society of America

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  1. R. S. Cloude and E. Pottier, "Concept of polarization entropy in optical scattering," Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
    [CrossRef]
  2. M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, "Polarimetric considerations to optimize lidar detection of immersed targets," Pure Appl. Opt. 7, 1327-1340 (1998).
    [CrossRef]
  3. S. Breugnot and Ph. Clémenceau, "Modeling and performances of a polarization active imager at lambda=806nm," in Laser Radar Technology and Applications IV, G.W.Kamerman and C.H.Werner, eds., Proc. SPIE 3707, 449-460 (1999).
  4. A. Gleckler and A. Gelbart, "Multiple-slit steak tube imaging lidar MS-STIL applications," in Laser Radar Technology and Applications V, G.W.Kamerman, U.N.Singh, C.H.Werner, and V.V.Molebny, eds., Proc. SPIE 4035, 266-278 (2000).
  5. L. B. Wolff, "Polarization camera for computer vision with a beam splitter," J. Opt. Soc. Am. A 11, 2935-2945 (1994).
    [CrossRef]
  6. 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]
  7. J. E. Solomon, "Polarization imaging," Appl. Opt. 20, 1537-1544 (1981).
    [CrossRef] [PubMed]
  8. W. G. Egan, W. R. Johnson, and V. S. Whitehead, "Terrestrial polarization imagery obtained from the Space Shuttle: characterization and interpretation," Appl. Opt. 30, 435-442 (1991).
    [CrossRef] [PubMed]
  9. J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. (Bellingham) 34, 1558-1568 (1995).
    [CrossRef]
  10. Ph. Réfrégier and F. Goudail, "Invariant polarimetric contrast parameters for coherent light," J. Opt. Soc. Am. A 19, 1223-1233 (2002).
    [CrossRef]
  11. Ph. Réfrégier, F. Goudail, P. Chavel, and A. Friberg, "Entropy of partially polarized light and application to statistical processing techniques," J. Opt. Soc. Am. A 21, 2124-2134 (2004).
    [CrossRef]
  12. R. Barakat, "N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation," Opt. Acta 30, 1171-1182 (1983).
    [CrossRef]
  13. P. Pellat-Finet, "Geometrical approach to polarization optics-II: Quaternionic representation of polarized light," Optik (Stuttgart) 87, 68-76 (1991).
  14. T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
    [CrossRef] [PubMed]
  15. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
    [CrossRef]
  16. G. S. Agarwal, "Utility of 3×3 polarization matrix for partially polarized transverse electromagnetic fields," J. Mod. Opt. 32, 651-654 (2005).
    [CrossRef]
  17. T. S. Ferguson, "Invariant statistical decision problems," in Mathematical Statistics, a Decision Theoretic Approach, (Academic Press, 1967), pp. 143-197.
  18. J. W. Goodman, "Some first-order properties of light waves," in Statistical Optics (Wiley, 1985), pp. 116-156.
  19. C. Brosseau, "Stokes parameters and coherency matrix formalism," in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 105-109.
  20. J. W. Goodman, "The speckle effect in coherent imaging," in Statistical Optics (Wiley, 1985), pp. 347-356.
  21. T. M. Cover and J. A. Thomas, "Information theory and statistics," in Elements of Information Theory (Wiley, 1991), pp. 279-335.
    [CrossRef]
  22. F. Goudail, N. Roux, and Ph. Réfrégier, "Performance parameters for detection in low-flux coherent images," Opt. Lett. 28, 81-83 (2003).
    [CrossRef] [PubMed]
  23. F. Goudail, Ph. Réfrégier, and G. Delyon, "Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images," J. Opt. Soc. Am. A 21, 1231-1240 (2004).
    [CrossRef]
  24. F. Goudail and Ph. Réfrégier, "Contrast definition for optical coherent polarimetric images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 947-951 (2004).
    [CrossRef]
  25. J. C. Samson, "Descriptions of the polarization states of vector processes: applications to ULF magnetic fields," Geophys. J. R. Astron. Soc. 34, 403-419 (1973).
  26. B. R. Frieden, "Continuous random variables," in Probability, Statistical Optics and Data Testing (Springer-Verlag, 2001), p. 71.

2005

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

G. S. Agarwal, "Utility of 3×3 polarization matrix for partially polarized transverse electromagnetic fields," J. Mod. Opt. 32, 651-654 (2005).
[CrossRef]

2004

2003

2002

Ph. Réfrégier and F. Goudail, "Invariant polarimetric contrast parameters for coherent light," J. Opt. Soc. Am. A 19, 1223-1233 (2002).
[CrossRef]

T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

1998

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, "Polarimetric considerations to optimize lidar detection of immersed targets," Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

1996

1995

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

R. S. Cloude and E. Pottier, "Concept of polarization entropy in optical scattering," Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
[CrossRef]

1994

1991

W. G. Egan, W. R. Johnson, and V. S. Whitehead, "Terrestrial polarization imagery obtained from the Space Shuttle: characterization and interpretation," Appl. Opt. 30, 435-442 (1991).
[CrossRef] [PubMed]

P. Pellat-Finet, "Geometrical approach to polarization optics-II: Quaternionic representation of polarized light," Optik (Stuttgart) 87, 68-76 (1991).

1983

R. Barakat, "N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation," Opt. Acta 30, 1171-1182 (1983).
[CrossRef]

1981

1973

J. C. Samson, "Descriptions of the polarization states of vector processes: applications to ULF magnetic fields," Geophys. J. R. Astron. Soc. 34, 403-419 (1973).

Agarwal, G. S.

G. S. Agarwal, "Utility of 3×3 polarization matrix for partially polarized transverse electromagnetic fields," J. Mod. Opt. 32, 651-654 (2005).
[CrossRef]

Barakat, R.

R. Barakat, "N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation," Opt. Acta 30, 1171-1182 (1983).
[CrossRef]

Breugnot, S.

S. Breugnot and Ph. Clémenceau, "Modeling and performances of a polarization active imager at lambda=806nm," in Laser Radar Technology and Applications IV, G.W.Kamerman and C.H.Werner, eds., Proc. SPIE 3707, 449-460 (1999).

Brosseau, C.

C. Brosseau, "Stokes parameters and coherency matrix formalism," in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 105-109.

Cariou, J.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, "Polarimetric considerations to optimize lidar detection of immersed targets," Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Chavel, P.

Chipman, R. A.

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

Clémenceau, Ph.

S. Breugnot and Ph. Clémenceau, "Modeling and performances of a polarization active imager at lambda=806nm," in Laser Radar Technology and Applications IV, G.W.Kamerman and C.H.Werner, eds., Proc. SPIE 3707, 449-460 (1999).

Cloude, R. S.

R. S. Cloude and E. Pottier, "Concept of polarization entropy in optical scattering," Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
[CrossRef]

Cover, T. M.

T. M. Cover and J. A. Thomas, "Information theory and statistics," in Elements of Information Theory (Wiley, 1991), pp. 279-335.
[CrossRef]

Delyon, G.

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Egan, W. G.

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Engheta, N.

Ferguson, T. S.

T. S. Ferguson, "Invariant statistical decision problems," in Mathematical Statistics, a Decision Theoretic Approach, (Academic Press, 1967), pp. 143-197.

Floc'h, M.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, "Polarimetric considerations to optimize lidar detection of immersed targets," Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Friberg, A.

Friberg, A. T.

T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Frieden, B. R.

B. R. Frieden, "Continuous random variables," in Probability, Statistical Optics and Data Testing (Springer-Verlag, 2001), p. 71.

Gelbart, A.

A. Gleckler and A. Gelbart, "Multiple-slit steak tube imaging lidar MS-STIL applications," in Laser Radar Technology and Applications V, G.W.Kamerman, U.N.Singh, C.H.Werner, and V.V.Molebny, eds., Proc. SPIE 4035, 266-278 (2000).

Gleckler, A.

A. Gleckler and A. Gelbart, "Multiple-slit steak tube imaging lidar MS-STIL applications," in Laser Radar Technology and Applications V, G.W.Kamerman, U.N.Singh, C.H.Werner, and V.V.Molebny, eds., Proc. SPIE 4035, 266-278 (2000).

Goodman, J. W.

J. W. Goodman, "Some first-order properties of light waves," in Statistical Optics (Wiley, 1985), pp. 116-156.

J. W. Goodman, "The speckle effect in coherent imaging," in Statistical Optics (Wiley, 1985), pp. 347-356.

Goudail, F.

Johnson, W. R.

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Kieleck, C.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, "Polarimetric considerations to optimize lidar detection of immersed targets," Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Le Brun, G.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, "Polarimetric considerations to optimize lidar detection of immersed targets," Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Lotrian, J.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, "Polarimetric considerations to optimize lidar detection of immersed targets," Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Pellat-Finet, P.

P. Pellat-Finet, "Geometrical approach to polarization optics-II: Quaternionic representation of polarized light," Optik (Stuttgart) 87, 68-76 (1991).

Pezzaniti, J. L.

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

Ponomarenko, S.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Pottier, E.

R. S. Cloude and E. Pottier, "Concept of polarization entropy in optical scattering," Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
[CrossRef]

Pugh, E. N.

Réfrégier, Ph.

Roux, N.

Rowe, M. P.

Samson, J. C.

J. C. Samson, "Descriptions of the polarization states of vector processes: applications to ULF magnetic fields," Geophys. J. R. Astron. Soc. 34, 403-419 (1973).

Setälä, T.

T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Solomon, J. E.

Thomas, J. A.

T. M. Cover and J. A. Thomas, "Information theory and statistics," in Elements of Information Theory (Wiley, 1991), pp. 279-335.
[CrossRef]

Tyo, J. S.

Whitehead, V. S.

Wolf, E.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Wolff, L. B.

Appl. Opt.

Geophys. J. R. Astron. Soc.

J. C. Samson, "Descriptions of the polarization states of vector processes: applications to ULF magnetic fields," Geophys. J. R. Astron. Soc. 34, 403-419 (1973).

IEEE Trans. Pattern Anal. Mach. Intell.

F. Goudail and Ph. Réfrégier, "Contrast definition for optical coherent polarimetric images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 947-951 (2004).
[CrossRef]

J. Mod. Opt.

G. S. Agarwal, "Utility of 3×3 polarization matrix for partially polarized transverse electromagnetic fields," J. Mod. Opt. 32, 651-654 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

R. Barakat, "N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation," Opt. Acta 30, 1171-1182 (1983).
[CrossRef]

Opt. Commun.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
[CrossRef]

Opt. Eng. (Bellingham)

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

R. S. Cloude and E. Pottier, "Concept of polarization entropy in optical scattering," Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

P. Pellat-Finet, "Geometrical approach to polarization optics-II: Quaternionic representation of polarized light," Optik (Stuttgart) 87, 68-76 (1991).

Phys. Rev. Lett.

T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Pure Appl. Opt.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, "Polarimetric considerations to optimize lidar detection of immersed targets," Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Other

S. Breugnot and Ph. Clémenceau, "Modeling and performances of a polarization active imager at lambda=806nm," in Laser Radar Technology and Applications IV, G.W.Kamerman and C.H.Werner, eds., Proc. SPIE 3707, 449-460 (1999).

A. Gleckler and A. Gelbart, "Multiple-slit steak tube imaging lidar MS-STIL applications," in Laser Radar Technology and Applications V, G.W.Kamerman, U.N.Singh, C.H.Werner, and V.V.Molebny, eds., Proc. SPIE 4035, 266-278 (2000).

B. R. Frieden, "Continuous random variables," in Probability, Statistical Optics and Data Testing (Springer-Verlag, 2001), p. 71.

T. S. Ferguson, "Invariant statistical decision problems," in Mathematical Statistics, a Decision Theoretic Approach, (Academic Press, 1967), pp. 143-197.

J. W. Goodman, "Some first-order properties of light waves," in Statistical Optics (Wiley, 1985), pp. 116-156.

C. Brosseau, "Stokes parameters and coherency matrix formalism," in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 105-109.

J. W. Goodman, "The speckle effect in coherent imaging," in Statistical Optics (Wiley, 1985), pp. 347-356.

T. M. Cover and J. A. Thomas, "Information theory and statistics," in Elements of Information Theory (Wiley, 1991), pp. 279-335.
[CrossRef]

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Equations (72)

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Γ = [ E 1 2 E 1 E 2 * E 2 E 1 * E 2 2 ] = [ μ 1 ρ ρ * μ 2 ] ,
P Γ ( E ) = 1 π 2 det ( Γ ) exp ( E Γ 1 E ) ,
Γ a 1 2 Γ b Γ a 1 2 ,
Υ = Γ a 1 2 Γ b Γ a 1 2 .
Q Υ ( λ ) = det ( Υ λ I d ) ,
Q Υ ( λ ) = λ 3 + t 1 λ 2 Φ λ + Δ ,
Φ = λ 1 λ 2 + λ 2 λ 3 + λ 1 λ 3 ,
Δ = λ 1 λ 2 λ 3 ,
t n = tr ( Υ n ) .
[ λ 1 , λ 2 , λ 3 ] , [ t 1 , Φ , Δ ] , [ t 1 , t 2 , t 3 ] ,
K ( P Γ b P Γ a ) = P Γ b ( E ) log [ P Γ b ( E ) P Γ a ( E ) ] d E ,
P N ( P Γ b P Γ a ) exp [ N K ( P Γ b P Γ a ) ] .
K ( P Γ b P Γ a ) = log ( det Γ a det Γ b ) + tr ( Γ b Γ a 1 ) 3 .
K ( P Γ b P Γ a ) = log ( Δ ) + t 1 3 .
C ( s ) = log { [ P Γ a ( E ) ] 1 s [ P Γ b ( E ) ] s d E } ,
C ( s ) = log { det [ ( 1 s ) Γ b + s Γ a ] det ( Γ a ) s det ( Γ b ) 1 s } .
C ( s ) = 3 log ( 1 s ) + log [ Q Υ ( s 1 s ) ] ( 1 s ) log Q Υ ( 0 ) ,
B ( P Γ b P Γ a ) = 3 log 2 + log ( Δ + Φ + t 1 + 1 ) ( 1 2 ) log Δ ,
K ( P Γ b P Γ a ) = log ( Δ ) + t 1 3
P B 2 = 1 27 det ( Γ ) tr ( Γ ) 3 ,
S 3 = 3 [ 1 + log ( π 3 ) ] + 3 log ( I ) + log ( 1 P B 2 ) .
P S 2 = 3 2 tr ( Γ 2 ) [ tr ( Γ ) ] 2 1 2 ,
B ( P Γ P Γ i d ) = 3 log 2 + log ( 27 δ + 9 ϕ + 3 I + 1 ) ( 1 2 ) log ( 27 δ ) ,
K ( P Γ P Γ i d ) = log ( 27 δ ) + 3 I 3 .
K ( P Γ P Γ i d ) = log ( 1 P B 2 ) + 3 I 3 log ( I e ) ,
B ( P Γ P Γ i d ) = 3 log 2 + log [ U ( I , P B 2 , P S 2 ) ] 1 2 log [ V ( I , P B 2 ) ] ,
V ( I , P B 2 ) = ( 1 P B 2 ) I 3 ,
U ( I , P B 2 , P S 2 ) = ( 1 P B 2 ) I 3 + 3 ( 1 P S 2 ) I 2 + 3 I + 1 .
C ( s ) = log [ W s ( I , P B 2 , P S 2 ) ] ( 1 s ) log [ V ( I , P B 2 ) ] + 3 log ( 1 s ) ,
W s ( I , P B 2 , P S 2 ) = ( s 1 s ) 3 + 3 ( s 1 s ) 2 I + 3 s 1 s ( 1 P S 2 ) I 2 + ( 1 P B 2 ) I 3 .
I ̃ ( P Γ b P Γ a ) = t 1 3
P ̃ B 2 ( P Γ b P Γ a ) = 1 27 Δ t 1 3
P ̃ S 2 ( P Γ b P Γ a ) = 1 3 Φ t 1 2
K ( P Γ b P Γ a ) = log ( 1 P ̃ B 2 ) + 3 I ̃ 3 log ( I ̃ e ) ,
B ( P Γ b P Γ a ) = 3 log 2 + log [ U ( I ̃ , P ̃ B 2 , P ̃ S 2 ) ] 1 2 log [ V ( I ̃ , P ̃ B 2 ) ] ,
K ( P Γ P Γ I ) = log ( 1 P B 2 ) ,
B ( P Γ P Γ I ) = 3 log 2 + log ( 8 P B 2 3 P S 2 ) 1 2 log ( 1 P B 2 ) .
D S 2 = 3 i j ( μ i + μ j ) 2 ( μ 1 + μ 2 + μ 3 ) 2 μ i μ j ( μ i + μ j ) 2 .
D S 2 ( i , j ) = 4 μ i μ j ( μ i + μ j ) 2 .
α i , j = 3 ( μ i + μ j ) 2 4 ( μ 1 + μ 2 + μ 3 ) 2 ,
D S 2 = α 1 , 2 D S 2 ( 1 , 2 ) + α 2 , 3 D S 2 ( 2 , 3 ) + α 1 , 3 D S 2 ( 1 , 3 ) ,
R 18 R 3 ,
{ Γ a , Γ b } { λ 1 , λ 2 , λ 3 } ,
Q Υ ( λ ) = det ( Γ a 1 2 Γ b Γ a 1 2 λ I d ) .
Q Υ ( λ ) = det ( Γ b Γ a 1 λ I d ) ,
K ( P Γ b P Γ a ) = log ( det Γ a det Γ b ) + tr ( Γ b Γ a 1 ) 3 .
K ( P Γ b P Γ a ) = log ( det Υ ) + tr ( Υ ) 3 ,
K ( P Γ b P Γ a ) = log Δ + t 1 3 .
C ( s ) = log { det [ ( 1 s ) Γ b + s Γ a ] det ( Γ a ) s det ( Γ b ) 1 s } .
C ( s ) = log { det [ ( 1 s ) Γ a 1 2 Γ b Γ a 1 2 + s I d ] det ( Γ a 1 2 Γ b Γ a 1 2 ) 1 s } ,
C ( s ) = log [ ( 1 s ) 3 det ( Υ + s 1 s I d ) ] log det ( Υ ) 1 s ,
C ( s ) = 3 log ( 1 s ) + log [ Q Υ ( s 1 s ) ] ( 1 s ) log Q Υ ( 0 ) .
C ( s ) = 3 log ( 1 s ) + log [ Q Υ ( s 1 s ) ] ( 1 s ) log Q Υ ( 0 ) ,
Q Υ ( s 1 s ) = ( s 1 s ) 3 + 3 I ( s 1 s ) 2 + 9 ϕ s 1 s + 27 δ .
δ = I 3 27 ( 1 P B 2 )
ϕ = I 2 3 ( 1 P S 2 ) ,
Q Υ ( s 1 s ) = ( s 1 s ) 3 + 3 ( s 1 s ) 2 I + 3 s 1 s ( 1 P S 2 ) I 2 + ( 1 P B 2 ) I 3 .
Γ a = [ μ 1 a 0 0 0 μ 2 a 0 0 0 μ 3 a ] , Γ b = [ μ 1 b 0 0 0 μ 2 b 0 0 0 μ 3 b ] ,
Υ = [ μ 1 b μ 1 a 0 0 0 μ 2 b μ 2 a 0 0 0 μ 3 b μ 3 a ] .
( μ 1 b μ 1 a , μ 2 b μ 2 a , μ 3 b μ 3 a ) .
( μ 1 a , μ 2 a , μ 3 a , μ 1 b , μ 2 b , μ 3 b ) ( μ 1 b μ 1 a , μ 2 b μ 2 a , μ 3 b μ 3 a ) .
μ 1 a + μ 2 a + μ 3 a = μ 1 b + μ 2 b + μ 3 b = 1 ,
( μ 1 a , μ 2 a , μ 1 b , μ 2 b ) ( μ 1 b μ 1 a , μ 2 b μ 2 a , 1 μ 1 b μ 2 b 1 μ 1 a μ 2 a ) .
( μ 1 b μ 1 a , μ 2 b μ 2 a , 1 μ 1 b μ 2 b 1 μ 1 a μ 2 a )
V = V ( μ 1 a , μ 2 a , μ 1 b , μ 2 b ) = ( μ 1 b μ 1 a , μ 2 b μ 2 a , 1 μ 1 b μ 2 b 1 μ 1 a μ 2 a ) T ,
V μ 1 a = [ μ 1 b ( μ 1 a ) 2 , 0 , 1 μ 1 b μ 2 b ( 1 μ 1 a μ 2 a ) 2 ] T ,
V μ 2 a = [ 0 , μ 2 b ( μ 2 a ) 2 , 1 μ 1 b μ 2 b ( 1 μ 1 a μ 2 a ) 2 ] T ,
V μ 1 b = [ 1 μ 1 a , 0 , 1 1 μ 1 a μ 2 a ] T ,
V μ 2 b = [ 0 , 1 μ 2 a , 1 1 μ 1 a μ 2 a ] T .
W T V μ 1 a = μ 1 b μ 1 a + 1 μ 1 b μ 2 b 1 μ 1 a μ 2 a ,
W T V μ 2 a = μ 2 b μ 2 a + 1 μ 1 b μ 2 b 1 μ 1 a μ 2 a .
( μ 1 b , μ 2 b ) [ 3 μ 1 b , 3 μ 2 b , 3 ( 1 μ b a μ 2 b ) ] ,

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