Abstract

We show that there is an isometry between the real ambient space of all Mueller matrices and the space of all Hermitian matrices that maps the Mueller matrices onto the positive semidefinite matrices. We use this to establish an optimality result for the filtering of Mueller matrices, which roughly says that it is always enough to filter the eigenvalues of the corresponding “coherency matrix.” Then we further explain how the knowledge of the cone of Hermitian positive semidefinite matrices can be transferred to the cone of Mueller matrices with a special emphasis towards optimisation. In particular, we suggest that means of Mueller matrices should be computed within the corresponding Riemannian geometry.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. S. R. Cloude, “Conditions For The Physical Realisability Of Matrix Operators In Polarimetry,” in Polarization Considerations for Optical Systems II, vol. 1166R. A. Chipman, ed., International Society for Optics and Photonics (SPIE, 1990), pp. 177–187.
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    [Crossref]
  3. F. Le Roy-Bréhonnet, B. Le Jeune, P. Elies, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D: Appl. Phys. 29(1), 34–38 (1996).
    [Crossref]
  4. A. Aiello, G. Puentes, D. Voigt, and J. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. 31(6), 817–819 (2006).
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  6. F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
    [Crossref]
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    [Crossref]
  10. B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27(2), 188–199 (2010).
    [Crossref]
  11. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42(5), 293–297 (1982).
    [Crossref]
  12. J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2017).
  13. R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34(4), 569–575 (1987).
    [Crossref]
  14. C. Van Der Mee, “An eigenvalue criterion for matrices transforming stokes parameters,” J. Math. Phys. 34(11), 5072–5088 (1993).
    [Crossref]
  15. T. Zander, “Logikerkit/muellerconefilter (Jupyter/iPython notebooks),” Zenodo (2020) [retrieved 7 May 2020], https://doi.org/10.5281/zenodo.3813681 .
  16. C.-K. Li and N.-K. Tsing, “On the unitarily invariant norms and some related results,” Linear Multilinear Algebr. 20(2), 107–119 (1987).
    [Crossref]
  17. H. Wielandt, “An extremum property of sums of eigenvalues,” Proc. Am. Math. Soc. 6(1), 106 (1955).
    [Crossref]
  18. R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205(4), 720–727 (2008).
    [Crossref]
  19. R. D. Hill and S. R. Waters, “On the cone of positive semidefinite matrices,” Linear Algebr. its Appl. 90, 81–88 (1987).
    [Crossref]
  20. R. Bhatia, Positive definite matrices, Princeton series in applied mathematics (Princeton University Press, 2007), illustrated edition ed.
  21. J. Löfberg, “Yalmip : A toolbox for modeling and optimization in matlab,” in In Proceedings of the CACSD Conference (Taipei, Taiwan, 2004).
  22. J. C. Gilbert and C. Josz, “Plea for a semidefinite optimization solver in complex numbers,” Research report, Inria Paris (2017).
  23. N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre, “Manopt, a Matlab toolbox for optimization on manifolds,” Journal of Machine Learning Research 15, 1455–1459 (2014).
  24. J. Townsend, N. Koep, and S. Weichwald, “Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation,” Journal of Machine Learning Research 17, 1–5 (2016).
  25. J. F. Sturm, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” Optim. methods software 11(1-4), 625–653 (1999).
    [Crossref]
  26. S. Yatawatta, “Radio interferometric calibration using a riemannian manifold,” in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, (2013), pp. 3866–3870.
  27. B. Vandereycken, P.-A. Absil, and S. Vandewalle, “Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank,” 2009 IEEE/SP 15th Workshop on Statistical Signal Processing (IEEE, 2009), pp. 389–392.
  28. V. Devlaminck and P. Terrier, “Geodesic distance on non-singular coherency matrix space in polarization optics,” J. Opt. Soc. Am. A 27(8), 1756–1763 (2010).
    [Crossref]
  29. C. J. Sheppard, A. Le Gratiet, and A. Diaspro, “Factorization of the coherency matrix of polarization optics,” J. Opt. Soc. Am. A 35(4), 586–590 (2018).
    [Crossref]
  30. J. M. Lee, Introduction to Smooth Manifolds, Vol. 218 of Graduate Texts in Mathematics (Springer, 2013).
  31. N. J. Higham, “Analysis of the Cholesky decomposition of a semi-definite matrix,” in Reliable numerical computation, (Oxford Univ. Press, New York, 1990), Oxford Sci. Publ., pp. 161–185.
  32. T. Eftimov, “Müller matrix analysis of pdl components,” Fiber Integr. Opt. 23(6), 453–466 (2004).
    [Crossref]
  33. V. Devlaminck, “Mueller matrix interpolation in polarization optics,” J. Opt. Soc. Am. A 27(7), 1529–1534 (2010).
    [Crossref]
  34. W. N. Anderson Jr and R. J. Duffin, “Series and parallel addition of matrices,” J. Math. Analysis Appl. 26(3), 576–594 (1969).
    [Crossref]
  35. F. Kubo and T. Ando, “Means of positive linear operators,” Math. Ann. 246(3), 205–224 (1980).
    [Crossref]
  36. W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,” Rep. Math. Phys. 8(2), 159–170 (1975).
    [Crossref]
  37. T. Ando, C.-K. Li, and R. Mathias, “Geometric means,” Linear Multilinear Algebr. 385, 305–334 (2004).
    [Crossref]
  38. S. Bonnabel and R. Sepulchre, “Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank,” SIAM J. on Matrix Analysis Appl. 31(3), 1055–1070 (2010).
    [Crossref]
  39. V. Devlaminck and P. Terrier, “Definition of a parametric form of nonsingular Mueller matrices,” J. Opt. Soc. Am. A 25(11), 2636–2643 (2008).
    [Crossref]
  40. N. G. Parke, “Optical algebra,” J. Math. Phys. 28(1-4), 131–139 (1949).
    [Crossref]

2018 (1)

2016 (2)

J. J. Gil, “On optimal filtering of measured Mueller matrices,” Appl. Opt. 55(20), 5449–5455 (2016).
[Crossref]

J. Townsend, N. Koep, and S. Weichwald, “Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation,” Journal of Machine Learning Research 17, 1–5 (2016).

2014 (1)

N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre, “Manopt, a Matlab toolbox for optimization on manifolds,” Journal of Machine Learning Research 15, 1455–1459 (2014).

2013 (1)

2011 (2)

2010 (4)

2009 (1)

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[Crossref]

2008 (2)

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205(4), 720–727 (2008).
[Crossref]

V. Devlaminck and P. Terrier, “Definition of a parametric form of nonsingular Mueller matrices,” J. Opt. Soc. Am. A 25(11), 2636–2643 (2008).
[Crossref]

2006 (1)

2004 (2)

T. Ando, C.-K. Li, and R. Mathias, “Geometric means,” Linear Multilinear Algebr. 385, 305–334 (2004).
[Crossref]

T. Eftimov, “Müller matrix analysis of pdl components,” Fiber Integr. Opt. 23(6), 453–466 (2004).
[Crossref]

1999 (1)

J. F. Sturm, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” Optim. methods software 11(1-4), 625–653 (1999).
[Crossref]

1996 (1)

F. Le Roy-Bréhonnet, B. Le Jeune, P. Elies, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D: Appl. Phys. 29(1), 34–38 (1996).
[Crossref]

1993 (1)

C. Van Der Mee, “An eigenvalue criterion for matrices transforming stokes parameters,” J. Math. Phys. 34(11), 5072–5088 (1993).
[Crossref]

1992 (1)

1987 (3)

C.-K. Li and N.-K. Tsing, “On the unitarily invariant norms and some related results,” Linear Multilinear Algebr. 20(2), 107–119 (1987).
[Crossref]

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34(4), 569–575 (1987).
[Crossref]

R. D. Hill and S. R. Waters, “On the cone of positive semidefinite matrices,” Linear Algebr. its Appl. 90, 81–88 (1987).
[Crossref]

1982 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42(5), 293–297 (1982).
[Crossref]

1980 (1)

F. Kubo and T. Ando, “Means of positive linear operators,” Math. Ann. 246(3), 205–224 (1980).
[Crossref]

1975 (1)

W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,” Rep. Math. Phys. 8(2), 159–170 (1975).
[Crossref]

1969 (1)

W. N. Anderson Jr and R. J. Duffin, “Series and parallel addition of matrices,” J. Math. Analysis Appl. 26(3), 576–594 (1969).
[Crossref]

1955 (1)

H. Wielandt, “An extremum property of sums of eigenvalues,” Proc. Am. Math. Soc. 6(1), 106 (1955).
[Crossref]

1949 (1)

N. G. Parke, “Optical algebra,” J. Math. Phys. 28(1-4), 131–139 (1949).
[Crossref]

Absil, P.-A.

N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre, “Manopt, a Matlab toolbox for optimization on manifolds,” Journal of Machine Learning Research 15, 1455–1459 (2014).

B. Vandereycken, P.-A. Absil, and S. Vandewalle, “Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank,” 2009 IEEE/SP 15th Workshop on Statistical Signal Processing (IEEE, 2009), pp. 389–392.

Aiello, A.

Anastasiadou, M.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205(4), 720–727 (2008).
[Crossref]

Anderson Jr, W. N.

W. N. Anderson Jr and R. J. Duffin, “Series and parallel addition of matrices,” J. Math. Analysis Appl. 26(3), 576–594 (1969).
[Crossref]

Ando, T.

T. Ando, C.-K. Li, and R. Mathias, “Geometric means,” Linear Multilinear Algebr. 385, 305–334 (2004).
[Crossref]

F. Kubo and T. Ando, “Means of positive linear operators,” Math. Ann. 246(3), 205–224 (1980).
[Crossref]

Anna, G.

Ben Hatit, S.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205(4), 720–727 (2008).
[Crossref]

Bhatia, R.

R. Bhatia, Positive definite matrices, Princeton series in applied mathematics (Princeton University Press, 2007), illustrated edition ed.

Bonnabel, S.

S. Bonnabel and R. Sepulchre, “Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank,” SIAM J. on Matrix Analysis Appl. 31(3), 1055–1070 (2010).
[Crossref]

Borghi, R.

Boulvert, F.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[Crossref]

Boumal, N.

N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre, “Manopt, a Matlab toolbox for optimization on manifolds,” Journal of Machine Learning Research 15, 1455–1459 (2014).

Cariou, J.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[Crossref]

F. Le Roy-Bréhonnet, B. Le Jeune, P. Elies, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D: Appl. Phys. 29(1), 34–38 (1996).
[Crossref]

Cloude, S. R.

S. R. Cloude, “Conditions For The Physical Realisability Of Matrix Operators In Polarimetry,” in Polarization Considerations for Optical Systems II, vol. 1166R. A. Chipman, ed., International Society for Optics and Photonics (SPIE, 1990), pp. 177–187.

De Martino, A.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205(4), 720–727 (2008).
[Crossref]

Devlaminck, V.

Diaspro, A.

Dolfi, D.

Duffin, R. J.

W. N. Anderson Jr and R. J. Duffin, “Series and parallel addition of matrices,” J. Math. Analysis Appl. 26(3), 576–594 (1969).
[Crossref]

Eftimov, T.

T. Eftimov, “Müller matrix analysis of pdl components,” Fiber Integr. Opt. 23(6), 453–466 (2004).
[Crossref]

Elies, P.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Elies, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D: Appl. Phys. 29(1), 34–38 (1996).
[Crossref]

Faisan, S.

Garcia-Caurel, E.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205(4), 720–727 (2008).
[Crossref]

Gil, J. J.

J. J. Gil, “On optimal filtering of measured Mueller matrices,” Appl. Opt. 55(20), 5449–5455 (2016).
[Crossref]

J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2017).

Gilbert, J. C.

J. C. Gilbert and C. Josz, “Plea for a semidefinite optimization solver in complex numbers,” Research report, Inria Paris (2017).

Gori, F.

Goudail, F.

Heinrich, C.

Higham, N. J.

N. J. Higham, “Analysis of the Cholesky decomposition of a semi-definite matrix,” in Reliable numerical computation, (Oxford Univ. Press, New York, 1990), Oxford Sci. Publ., pp. 161–185.

Hill, R. D.

R. D. Hill and S. R. Waters, “On the cone of positive semidefinite matrices,” Linear Algebr. its Appl. 90, 81–88 (1987).
[Crossref]

Josz, C.

J. C. Gilbert and C. Josz, “Plea for a semidefinite optimization solver in complex numbers,” Research report, Inria Paris (2017).

Koep, N.

J. Townsend, N. Koep, and S. Weichwald, “Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation,” Journal of Machine Learning Research 17, 1–5 (2016).

Kostinski, A. B.

Kubo, F.

F. Kubo and T. Ando, “Means of positive linear operators,” Math. Ann. 246(3), 205–224 (1980).
[Crossref]

Le Brun, G.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[Crossref]

Le Gratiet, A.

Le Jeune, B.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[Crossref]

F. Le Roy-Bréhonnet, B. Le Jeune, P. Elies, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D: Appl. Phys. 29(1), 34–38 (1996).
[Crossref]

Le Roy-Bréhonnet, F.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Elies, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D: Appl. Phys. 29(1), 34–38 (1996).
[Crossref]

Lee, J. M.

J. M. Lee, Introduction to Smooth Manifolds, Vol. 218 of Graduate Texts in Mathematics (Springer, 2013).

Li, C.-K.

T. Ando, C.-K. Li, and R. Mathias, “Geometric means,” Linear Multilinear Algebr. 385, 305–334 (2004).
[Crossref]

C.-K. Li and N.-K. Tsing, “On the unitarily invariant norms and some related results,” Linear Multilinear Algebr. 20(2), 107–119 (1987).
[Crossref]

Löfberg, J.

J. Löfberg, “Yalmip : A toolbox for modeling and optimization in matlab,” in In Proceedings of the CACSD Conference (Taipei, Taiwan, 2004).

Lotrian, J.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Elies, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D: Appl. Phys. 29(1), 34–38 (1996).
[Crossref]

Martin, L.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[Crossref]

Mathias, R.

T. Ando, C.-K. Li, and R. Mathias, “Geometric means,” Linear Multilinear Algebr. 385, 305–334 (2004).
[Crossref]

Mishra, B.

N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre, “Manopt, a Matlab toolbox for optimization on manifolds,” Journal of Machine Learning Research 15, 1455–1459 (2014).

Mukunda, N.

Ossikovski, R.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205(4), 720–727 (2008).
[Crossref]

J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2017).

Parke, N. G.

N. G. Parke, “Optical algebra,” J. Math. Phys. 28(1-4), 131–139 (1949).
[Crossref]

Puentes, G.

Pusz, W.

W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,” Rep. Math. Phys. 8(2), 159–170 (1975).
[Crossref]

Santarsiero, M.

Sepulchre, R.

N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre, “Manopt, a Matlab toolbox for optimization on manifolds,” Journal of Machine Learning Research 15, 1455–1459 (2014).

S. Bonnabel and R. Sepulchre, “Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank,” SIAM J. on Matrix Analysis Appl. 31(3), 1055–1070 (2010).
[Crossref]

Sfikas, G.

Sheppard, C. J.

Simon, B. N.

Simon, R.

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27(2), 188–199 (2010).
[Crossref]

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34(4), 569–575 (1987).
[Crossref]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42(5), 293–297 (1982).
[Crossref]

Simon, S.

Sturm, J. F.

J. F. Sturm, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” Optim. methods software 11(1-4), 625–653 (1999).
[Crossref]

Terrier, P.

Townsend, J.

J. Townsend, N. Koep, and S. Weichwald, “Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation,” Journal of Machine Learning Research 17, 1–5 (2016).

Tsing, N.-K.

C.-K. Li and N.-K. Tsing, “On the unitarily invariant norms and some related results,” Linear Multilinear Algebr. 20(2), 107–119 (1987).
[Crossref]

Tyo, J. S.

Van Der Mee, C.

C. Van Der Mee, “An eigenvalue criterion for matrices transforming stokes parameters,” J. Math. Phys. 34(11), 5072–5088 (1993).
[Crossref]

Vandereycken, B.

B. Vandereycken, P.-A. Absil, and S. Vandewalle, “Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank,” 2009 IEEE/SP 15th Workshop on Statistical Signal Processing (IEEE, 2009), pp. 389–392.

Vandewalle, S.

B. Vandereycken, P.-A. Absil, and S. Vandewalle, “Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank,” 2009 IEEE/SP 15th Workshop on Statistical Signal Processing (IEEE, 2009), pp. 389–392.

Voigt, D.

Waters, S. R.

R. D. Hill and S. R. Waters, “On the cone of positive semidefinite matrices,” Linear Algebr. its Appl. 90, 81–88 (1987).
[Crossref]

Weichwald, S.

J. Townsend, N. Koep, and S. Weichwald, “Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation,” Journal of Machine Learning Research 17, 1–5 (2016).

Wielandt, H.

H. Wielandt, “An extremum property of sums of eigenvalues,” Proc. Am. Math. Soc. 6(1), 106 (1955).
[Crossref]

Woerdman, J.

Woronowicz, S. L.

W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,” Rep. Math. Phys. 8(2), 159–170 (1975).
[Crossref]

Yatawatta, S.

S. Yatawatta, “Radio interferometric calibration using a riemannian manifold,” in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, (2013), pp. 3866–3870.

Zallat, J.

Zander, T.

T. Zander, “Logikerkit/muellerconefilter (Jupyter/iPython notebooks),” Zenodo (2020) [retrieved 7 May 2020], https://doi.org/10.5281/zenodo.3813681 .

Appl. Opt. (2)

Fiber Integr. Opt. (1)

T. Eftimov, “Müller matrix analysis of pdl components,” Fiber Integr. Opt. 23(6), 453–466 (2004).
[Crossref]

J. Math. Analysis Appl. (1)

W. N. Anderson Jr and R. J. Duffin, “Series and parallel addition of matrices,” J. Math. Analysis Appl. 26(3), 576–594 (1969).
[Crossref]

J. Math. Phys. (2)

C. Van Der Mee, “An eigenvalue criterion for matrices transforming stokes parameters,” J. Math. Phys. 34(11), 5072–5088 (1993).
[Crossref]

N. G. Parke, “Optical algebra,” J. Math. Phys. 28(1-4), 131–139 (1949).
[Crossref]

J. Mod. Opt. (1)

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34(4), 569–575 (1987).
[Crossref]

J. Opt. Soc. Am. A (6)

J. Phys. D: Appl. Phys. (1)

F. Le Roy-Bréhonnet, B. Le Jeune, P. Elies, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D: Appl. Phys. 29(1), 34–38 (1996).
[Crossref]

Journal of Machine Learning Research (2)

N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre, “Manopt, a Matlab toolbox for optimization on manifolds,” Journal of Machine Learning Research 15, 1455–1459 (2014).

J. Townsend, N. Koep, and S. Weichwald, “Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation,” Journal of Machine Learning Research 17, 1–5 (2016).

Linear Algebr. its Appl. (1)

R. D. Hill and S. R. Waters, “On the cone of positive semidefinite matrices,” Linear Algebr. its Appl. 90, 81–88 (1987).
[Crossref]

Linear Multilinear Algebr. (2)

C.-K. Li and N.-K. Tsing, “On the unitarily invariant norms and some related results,” Linear Multilinear Algebr. 20(2), 107–119 (1987).
[Crossref]

T. Ando, C.-K. Li, and R. Mathias, “Geometric means,” Linear Multilinear Algebr. 385, 305–334 (2004).
[Crossref]

Math. Ann. (1)

F. Kubo and T. Ando, “Means of positive linear operators,” Math. Ann. 246(3), 205–224 (1980).
[Crossref]

Opt. Commun. (2)

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[Crossref]

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Supplementary Material (1)

NameDescription
» Code 1       Complementary code with calculations.

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Equations (15)

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I 2 Q 2 + U 2 + V 2 ,
M = i = 1 N c i M i .
h 00 = 1 2 ( m 11 + m 22 + m 33 + m 44 ) , h 11 = 1 2 ( m 11 + m 22 m 33 m 44 ) , h 22 = 1 2 ( m 11 m 22 + m 33 m 44 ) , h 33 = 1 2 ( m 11 m 22 m 33 + m 44 )
h 03 = 1 2 ( m 14 + m 41 i m 23 + i m 32 ) , h 30 = 1 2 ( m 14 + m 41 + i m 23 i m 32 ) , h 12 = 1 2 ( m 14 i m 41 + m 23 + m 32 ) , h 21 = 1 2 ( m 14 + i m 41 + m 23 + m 32 )
h 01 = 1 2 ( m 12 + m 21 i m 34 + i m 43 ) , h 10 = 1 2 ( m 12 + m 21 + i m 34 i m 43 ) , h 23 = 1 2 ( m 12 i m 21 + m 34 + m 43 ) , h 32 = 1 2 ( m 12 + i m 21 + m 34 + m 43 )
h 02 = 1 2 ( m 13 + m 31 i m 24 + i m 42 ) , h 20 = 1 2 ( m 13 + m 31 + i m 24 i m 42 ) , h 13 = 1 2 ( m 13 i m 31 + m 24 + m 42 ) , h 31 = 1 2 ( m 13 + i m 31 + m 24 + m 42 )
{ ( x 1 , , x n ) R n : x 1 x n c } .
{ V [ d ] V : V U n , d Y } .
| | A U [ b ] U | | | | A X | | for all X S Y .
{ ( x 1 , , x 4 ) R n : x 1 x 4 c }
M Y = { T 1 ( V [ d ] V ) : V U n , d Y } .
{ ( x 1 , , x 4 ) R 4 : x 1 x 4 0 }
{ ( x 1 , , x 4 ) R 4 : x 1 x 4 c } ,
E = { ( x 1 , x 2 , x 3 , x 4 ) R 4 : x 1 , x 2 , x 3 , x 4 0 x 1 + x 2 + x 3 + x 4 = m 11 } .
R 2 × 4 × k R 2 = C C 4 × k Y Y HPSD T 1 M

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