Abstract

The question of the physical significance of the Mueller matrix average is addressed by means of an analysis of interpolation processes. We draw a comparison between two interpolation processes. The first one is related to the classical Euclidean metrics and the second one is based on the log-Euclidean metrics. Both the associated interpolation procedures are depicted with their underlying physical models. Addressing the question of the physical meaning of the log-Euclidean process of interpolation is founded on a very similar approach to the layered-medium interpretation proposed by Jones [J. Opt. Soc. Am. 38, 671 (1948)] in the seventh paper of his series. Based on the analysis of their respective properties, we eventually show that the choice between both these interpolation processes may depend on what statistical situation is considered or what underlying physical model is assumed.

© 2010 Optical Society of America

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  1. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824–2832 (2004).
    [CrossRef] [PubMed]
  2. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38, 3490–3502 (1999).
    [CrossRef]
  3. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. 31, 817–819 (2006).
    [CrossRef] [PubMed]
  4. J. Zallat, C. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express 16, 7119–7133 (2008).
    [CrossRef] [PubMed]
  5. F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
    [CrossRef]
  6. A. Hielscher, A. Eick, J. Mourant, D. Shen, J. Freyer, and I. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express 1, 441–453 (1997).
    [CrossRef] [PubMed]
  7. M. H. Smith, P. Burke, A. Lompado, E. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000).
    [CrossRef]
  8. J. M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Lett. 24, 64–66 (1999).
    [CrossRef]
  9. M. H. Smith, E. A. Sornsin, R. A. Chipman, and T. J. Tayag, “Mueller matrix imaging of GaAs/AlGaAs self-imaging beamsplitting waveguides,” Proc. SPIE 3121, 47–54 (1997).
  10. L. Jin, T. Hamada, Y. Otani, and N. Umeda, “Measurement of characteristics of magnetic fluid by the Mueller matrix imaging polarimeter,” Opt. Eng. (Bellingham) 43, 181–185 (2004).
    [CrossRef]
  11. W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).
  12. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  13. S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).
  14. J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
    [CrossRef]
  15. A. Aiello and J. P. Woerdman, arXiv:math-ph/0412061.
  16. J. J. Gill and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
    [CrossRef]
  17. J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, Monograph Seminario Matemático García de Galdeano (Edificio de Matemáticas, Universidad de Zaragoza, 2004); available at http://www.unizar.es/galdeano/actas pau/PDFVIII/pp161-167.pdf.
  18. V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. 29, 328–347 (2007).
    [CrossRef]
  19. R. C. Jones, “A new calculus for the treatment of optical systems. VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  20. R. Barakat, “Exponential versions of the Jones and Mueller–Jones polarization matrices,” J. Opt. Soc. Am. A 13, 158–163 (1996).
    [CrossRef]

2008

2007

V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. 29, 328–347 (2007).
[CrossRef]

2006

2005

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

2004

L. Jin, T. Hamada, Y. Otani, and N. Umeda, “Measurement of characteristics of magnetic fluid by the Mueller matrix imaging polarimeter,” Opt. Eng. (Bellingham) 43, 181–185 (2004).
[CrossRef]

B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824–2832 (2004).
[CrossRef] [PubMed]

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, Monograph Seminario Matemático García de Galdeano (Edificio de Matemáticas, Universidad de Zaragoza, 2004); available at http://www.unizar.es/galdeano/actas pau/PDFVIII/pp161-167.pdf.

2000

M. H. Smith, P. Burke, A. Lompado, E. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000).
[CrossRef]

J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
[CrossRef]

1999

1997

1996

1987

1986

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).

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

1948

Aiello, A.

Arsigny, V.

V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. 29, 328–347 (2007).
[CrossRef]

Artal, P.

Ayache, N.

V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. 29, 328–347 (2007).
[CrossRef]

Barakat, R.

Bernabeu, E.

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

Bigio, I.

Boothby, W. M.

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).

Boulbry, B.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Boulvert, F.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Bueno, J. M.

Burke, P.

M. H. Smith, P. Burke, A. Lompado, E. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000).
[CrossRef]

Cariou, J.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Chipman, R. A.

M. H. Smith, E. A. Sornsin, R. A. Chipman, and T. J. Tayag, “Mueller matrix imaging of GaAs/AlGaAs self-imaging beamsplitting waveguides,” Proc. SPIE 3121, 47–54 (1997).

Cloude, S. R.

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).

Compain, E.

Correas, J. M.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, Monograph Seminario Matemático García de Galdeano (Edificio de Matemáticas, Universidad de Zaragoza, 2004); available at http://www.unizar.es/galdeano/actas pau/PDFVIII/pp161-167.pdf.

De Martino, A.

Drevillon, B.

Drévillon, B.

Eick, A.

Ferreira, C.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, Monograph Seminario Matemático García de Galdeano (Edificio de Matemáticas, Universidad de Zaragoza, 2004); available at http://www.unizar.es/galdeano/actas pau/PDFVIII/pp161-167.pdf.

Fillard, P.

V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. 29, 328–347 (2007).
[CrossRef]

Freyer, J.

Gil, J. J.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, Monograph Seminario Matemático García de Galdeano (Edificio de Matemáticas, Universidad de Zaragoza, 2004); available at http://www.unizar.es/galdeano/actas pau/PDFVIII/pp161-167.pdf.

J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
[CrossRef]

Gill, J. J.

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

Hamada, T.

L. Jin, T. Hamada, Y. Otani, and N. Umeda, “Measurement of characteristics of magnetic fluid by the Mueller matrix imaging polarimeter,” Opt. Eng. (Bellingham) 43, 181–185 (2004).
[CrossRef]

Heinrich, C.

Hielscher, A.

Hillman, L. W.

M. H. Smith, P. Burke, A. Lompado, E. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000).
[CrossRef]

Jin, L.

L. Jin, T. Hamada, Y. Otani, and N. Umeda, “Measurement of characteristics of magnetic fluid by the Mueller matrix imaging polarimeter,” Opt. Eng. (Bellingham) 43, 181–185 (2004).
[CrossRef]

Jones, R. C.

Kim, K.

Laude-Boulesteix, B.

Le Brun, G.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Le Jeune, B.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Lompado, A.

M. H. Smith, P. Burke, A. Lompado, E. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000).
[CrossRef]

Mandel, L.

Melero, P. A.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, Monograph Seminario Matemático García de Galdeano (Edificio de Matemáticas, Universidad de Zaragoza, 2004); available at http://www.unizar.es/galdeano/actas pau/PDFVIII/pp161-167.pdf.

Mourant, J.

Otani, Y.

L. Jin, T. Hamada, Y. Otani, and N. Umeda, “Measurement of characteristics of magnetic fluid by the Mueller matrix imaging polarimeter,” Opt. Eng. (Bellingham) 43, 181–185 (2004).
[CrossRef]

Pennec, X.

V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. 29, 328–347 (2007).
[CrossRef]

Petremand, M.

Poirier, S.

Puentes, G.

Rivet, S.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

Schwartz, L.

Shen, D.

Smith, M. H.

M. H. Smith, P. Burke, A. Lompado, E. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000).
[CrossRef]

M. H. Smith, E. A. Sornsin, R. A. Chipman, and T. J. Tayag, “Mueller matrix imaging of GaAs/AlGaAs self-imaging beamsplitting waveguides,” Proc. SPIE 3121, 47–54 (1997).

Sornsin, E. A.

M. H. Smith, E. A. Sornsin, R. A. Chipman, and T. J. Tayag, “Mueller matrix imaging of GaAs/AlGaAs self-imaging beamsplitting waveguides,” Proc. SPIE 3121, 47–54 (1997).

Tanner, E.

M. H. Smith, P. Burke, A. Lompado, E. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000).
[CrossRef]

Tayag, T. J.

M. H. Smith, E. A. Sornsin, R. A. Chipman, and T. J. Tayag, “Mueller matrix imaging of GaAs/AlGaAs self-imaging beamsplitting waveguides,” Proc. SPIE 3121, 47–54 (1997).

Umeda, N.

L. Jin, T. Hamada, Y. Otani, and N. Umeda, “Measurement of characteristics of magnetic fluid by the Mueller matrix imaging polarimeter,” Opt. Eng. (Bellingham) 43, 181–185 (2004).
[CrossRef]

Voigt, D.

Woerdman, J. P.

Wolf, E.

Zallat, J.

Appl. Opt.

J. Opt. A, Pure Appl. Opt.

F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A, Pure Appl. Opt. 7, 21–28 (2005).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

Opt. Eng. (Bellingham)

L. Jin, T. Hamada, Y. Otani, and N. Umeda, “Measurement of characteristics of magnetic fluid by the Mueller matrix imaging polarimeter,” Opt. Eng. (Bellingham) 43, 181–185 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttgart)

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).

Proc. SPIE

M. H. Smith, P. Burke, A. Lompado, E. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000).
[CrossRef]

SIAM J. Matrix Anal. Appl.

V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. 29, 328–347 (2007).
[CrossRef]

SPIE

M. H. Smith, E. A. Sornsin, R. A. Chipman, and T. J. Tayag, “Mueller matrix imaging of GaAs/AlGaAs self-imaging beamsplitting waveguides,” Proc. SPIE 3121, 47–54 (1997).

Other

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, Monograph Seminario Matemático García de Galdeano (Edificio de Matemáticas, Universidad de Zaragoza, 2004); available at http://www.unizar.es/galdeano/actas pau/PDFVIII/pp161-167.pdf.

A. Aiello and J. P. Woerdman, arXiv:math-ph/0412061.

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Figures (2)

Fig. 1
Fig. 1

Interpolation process for three values of the t parameter. (a) Euclidean distance model. (b) LE distance model.

Fig. 2
Fig. 2

Medium and the corresponding Mueller matrix for different values of the z parameter.

Equations (26)

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

M = ( m i j ) = Λ F Λ ,     F = Λ M Λ ,
Λ = 1 2 [ 1 1 0 0 0 0 1 i 0 0 1 i 1 1 0 0 ] ,     F = J J e = [ j 0 j 0 e j 0 j 1 e j 1 j 0 e j 1 j 1 e j 0 j 2 e j 0 j 3 e j 1 j 2 e j 1 j 3 e j 2 j 0 e j 2 j 1 e j 3 j 0 e j 3 j 1 e j 2 j 2 e j 2 j 3 e j 3 j 2 e j 3 j 3 e ] ,     J = [ j 0 j 1 j 2 j 3 ] .
H = i j m i j ( σ i σ j ) = Per ( F ) = Per ( Λ M Λ ) .
C = Λ H Λ .
d E ( M 1 , M 2 ) = M 1 M 2 = [ Tr [ ( M 1 M 2 ) ( M 1 M 2 ) ] ] 1 / 2 ,
M ( t ) = ( 1 t ) M 1 + t M 2 ,     with   0 t 1.
H ( t ) = Per ( Λ M ( t ) Λ ) = Per [ Λ ( ( 1 t ) M 1 + t M 2 ) Λ ] = ( 1 t ) Per [ Λ M 1 Λ ] + t Per [ Λ M 2 Λ ] = ( 1 t ) H 1 + t H 2 .
d LE ( H 1 , H 2 ) = log ( H 1 ) log ( H 2 ) = [ Tr [ ( log ( H 1 ) log ( H 2 ) ) 2 ] ] 1 / 2 .
H ( t ) = exp ( [ ( 1 t ) log ( H 1 ) + t   log ( H 2 ) ] ) .
H ( τ ) = exp ( τ D ) = I + τ D + 1 2 ( τ D ) 2 + 1 6 ( τ D ) 3 + ,
M 0 = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] .
H 1 e = exp ( τ 1 D 1 ) = I + τ 1 D 1 + O ( τ 1 2 ) ,
H 2 e = exp ( τ 2 D 2 ) = I + τ 2 D 2 + O ( τ 2 2 ) ,
H s = I + τ 1 D 1 + τ 2 D 2 + O [ ( τ 1 + τ 2 ) 2 ]     or
H s = I + τ [ ( 1 t ) D 1 + t D 2 ] + O ( τ 2 ) ,     with   t = τ 2 τ ,
τ = τ 1 + τ 2 .
log ( H s ) = log ( H 1 e H 2 e ) = log ( I + τ [ ( 1 t ) D 1 + t D 2 ] + O ( τ 2 ) ) = τ [ ( 1 t ) D 1 + t D 2 ] + O ( τ 2 ( 1 t ) D 1 + t D 2 2 ) ,
H s = exp ( τ [ ( 1 t ) D 1 + t D 2 ] + O ( τ 2 ( 1 t ) D 1 + t D 2 2 ) ) .
( H s ) q = ( H 1 e H 2 e ) q = exp ( q τ [ ( 1 t ) D 1 + t D 2 ] + O ( q τ 2 [ ( 1 t ) D 1 + t D 2 ] 2 ) ) = exp ( z [ ( 1 t ) D 1 + t D 2 ] + O ( z τ [ ( 1 t ) D 1 + t D 2 ] 2 ) ) .
H ( z , t ) = lim q + ( H s ) q = lim τ 0 ( H s ) z / τ = exp ( z [ ( 1 t ) D 1 + t D 2 ] ) .
ψ [ ( H 1 e H 2 e ) q ] [ ψ ( H 1 e H 2 e ) ] q .
M ( L z , t ) = ψ [ H ( z , t ) ] ,
M ( L z 2 , t ) M rs ( z 2 z 1 , t ) = M ( L z 1 , t ) ,
H = Per ( F ) = [ j 0 j 0 e j 0 j 1 e j 0 j 2 e j 0 j 3 e j 1 j 0 e j 1 j 1 e j 1 j 2 e j 1 j 3 e j 2 j 0 e j 2 j 1 e j 2 j 2 e j 2 j 3 e j 3 j 0 e j 3 j 1 e j 3 j 2 e j 3 j 3 e ] .
det [ exp ( X ) ] = exp [ Tr ( X ) ] ,
det [ H ( t ) ] = det [ exp ( [ ( 1 t ) D 1 + t D 2 ] ) ] = exp ( [ ( 1 t ) Tr ( D 1 ) + t Tr ( D 2 ) ] ) = exp ( Tr ( D 1 ) ) ( 1 t ) exp ( Tr ( D 2 ) ) t = ( det [ H ( 0 ) ] ) ( 1 t ) ( det [ H ( 1 ) ] ) t .

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