Abstract

A general formulation of the additive composition and decomposition of Mueller matrices is presented, which is expressed in adequate terms for properly performing the “polarimetric subtraction,” from a given depolarizing Mueller matrix M, of the Mueller matrix of a given nondepolarizing component that is incoherently embedded in the whole system represented by M. A general and comprehensive procedure for the polarimetric subtraction of depolarizing Mueller matrices is also developed.

© 2013 Optical Society of America

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  1. J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” VII Jornadas Zaragoza-Pau de Matemática Aplicada y estadística, Jaca, Spain, 17–18 September, 2001. Monografías del Semin. Matem. Garc a de Galdeano. 27, 233–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .
  2. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  3. J. J. Gil, I. San José, and R. Ossikovski, “Serial-parallel decompositions of Mueller matrices,” J. Opt. Soc. Am. A. 30, 32–50 (2013).
    [CrossRef]
  4. Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod Opt. 39, 461–484 (1992).
    [CrossRef]
  5. C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
    [CrossRef]
  6. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
    [CrossRef]
  7. J. J. Gil and E. Bernabéu, “Depolarization and polarization indices of an optical system,” Optica Acta 33, 185–189 (1986).
    [CrossRef]
  8. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 2012).
  9. N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 182–195 (1930).
    [CrossRef]
  10. N. G. Parke, “Matrix optics,” Ph.D. thesis (Massachusetts Institute of Technology, 1948).
  11. U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1954).
    [CrossRef]
  12. E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
    [CrossRef]
  13. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
    [CrossRef]
  14. G. B. Parrent and P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
    [CrossRef]
  15. R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
    [CrossRef]
  16. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
  17. J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
    [CrossRef]
  18. J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Semin. Matem. García de Galdeano. 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .
  19. J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010).
    [CrossRef]
  20. I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
    [CrossRef]
  21. J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28, 1578–1585 (2011).
    [CrossRef]
  22. J. J. Gil, “Determination of polarization parameters in matricial representation. Theoretical contribution and development of an automatic measurement device,” Ph.D. thesis (Facultad de Ciencias, Univ. Zaragoza, Spain, 1983). Available from http://www.pepegil.es/PhD-Thesis-JJ-Gil-English.pdf .
  23. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
    [CrossRef]
  24. R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010).
    [CrossRef]
  25. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [CrossRef]
  26. 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]
  27. S. S. Girgel, “Structure of Mueller matrices of depolarized optical systems,” Sov. Phys. Crystallogr. 36, 890 (1991).
  28. J. J. Gil and J. Correas, “Polarimetric subtraction for obtaining the Mueller matrices of components which appear combined in a whole material sample under measurement,” Proceedings of ICO Topical Meeting on Polarization Optics (ICOPO), Polvijärvi, Finland, July 2, 2003. Available at http://www.pepegil.es/Polarimetric-subtraction-ICOPO-2003-.pdf .
  29. M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices,” Opt. Express 17, 12794–12806 (2009).
    [CrossRef]
  30. A. Freeman and S. L. Durden, “A three component model for polarimetric SAR data,” IEEE Trans. Geosci. Remote Sens. 36, 963–973 (1998).
    [CrossRef]
  31. Y. Yamaguchi, T. Moriyama, M. Ishido, and H. Yamada, “Four-component scattering model for polarimetric SAR image decomposition,” IEEE Trans. Geosci. Remote Sens. 43, 1699–1706 (2005).
    [CrossRef]
  32. J. J. van Zyl, “Application of Cloude’s target decomposition theorem to polarimeric imaging radar,” Proc. SPIE 127, 184–212 (1992).
    [CrossRef]
  33. J. J. van Zyl, M. Arii, and Y. Kim, “Model-based decomposition of polarimetric SAR covariance matrices constrained for nonnegative eigenvalues,” IEEE Trans. Geosci. Remote Sens. 49, 3452–3459 (2011).
    [CrossRef]
  34. Y. Cui, Y. Yamaguchi, J. Yang, H. Kobayashi, S.-E. Park, and G. Singh, “On complete model based three component decomposition,” Proceedings of International Niigata Polarimetric SAR Workshop, University of Niigata, Niigata, Japan, 2012).
  35. J. H. M. Wedderburn, Lectures on Matrices (Dover, 1964).
  36. M. T. Chu, R. E. Funderlic, and G. H. Golub, “A rank-one reduction formula and its applications to matrix factorizations,” SIAM Rev. 37, 512–530 (1995).
    [CrossRef]
  37. J. J. Gil and E. Bernabéu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from its Mueller matrix,” Optik 76, 67–71 (1987).
  38. S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).
  39. N. Gosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications and outlook,” J. Biomed. Opt. 16, 110801 (2011).
    [CrossRef]
  40. T. Novikova, A. Pierangelo, A. De Martino, A. Benali, and P. Validire, “Polarimetric imaging for cancer diagnosis and staging,” Opt. Photon. News 23(10), 26–33 (2012).
    [CrossRef]

2013 (1)

J. J. Gil, I. San José, and R. Ossikovski, “Serial-parallel decompositions of Mueller matrices,” J. Opt. Soc. Am. A. 30, 32–50 (2013).
[CrossRef]

2012 (1)

T. Novikova, A. Pierangelo, A. De Martino, A. Benali, and P. Validire, “Polarimetric imaging for cancer diagnosis and staging,” Opt. Photon. News 23(10), 26–33 (2012).
[CrossRef]

2011 (4)

J. J. van Zyl, M. Arii, and Y. Kim, “Model-based decomposition of polarimetric SAR covariance matrices constrained for nonnegative eigenvalues,” IEEE Trans. Geosci. Remote Sens. 49, 3452–3459 (2011).
[CrossRef]

N. Gosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications and outlook,” J. Biomed. Opt. 16, 110801 (2011).
[CrossRef]

J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28, 1578–1585 (2011).
[CrossRef]

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

2010 (2)

J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010).
[CrossRef]

R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010).
[CrossRef]

2009 (2)

2007 (1)

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

2005 (1)

Y. Yamaguchi, T. Moriyama, M. Ishido, and H. Yamada, “Four-component scattering model for polarimetric SAR image decomposition,” IEEE Trans. Geosci. Remote Sens. 43, 1699–1706 (2005).
[CrossRef]

2004 (1)

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Semin. Matem. García de Galdeano. 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

2003 (1)

J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” VII Jornadas Zaragoza-Pau de Matemática Aplicada y estadística, Jaca, Spain, 17–18 September, 2001. Monografías del Semin. Matem. Garc a de Galdeano. 27, 233–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .

2000 (1)

1998 (1)

A. Freeman and S. L. Durden, “A three component model for polarimetric SAR data,” IEEE Trans. Geosci. Remote Sens. 36, 963–973 (1998).
[CrossRef]

1996 (1)

1995 (1)

M. T. Chu, R. E. Funderlic, and G. H. Golub, “A rank-one reduction formula and its applications to matrix factorizations,” SIAM Rev. 37, 512–530 (1995).
[CrossRef]

1993 (1)

C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

1992 (2)

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod Opt. 39, 461–484 (1992).
[CrossRef]

J. J. van Zyl, “Application of Cloude’s target decomposition theorem to polarimeric imaging radar,” Proc. SPIE 127, 184–212 (1992).
[CrossRef]

1991 (1)

S. S. Girgel, “Structure of Mueller matrices of depolarized optical systems,” Sov. Phys. Crystallogr. 36, 890 (1991).

1987 (3)

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

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]

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

1986 (2)

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

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

1963 (1)

1960 (1)

G. B. Parrent and P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
[CrossRef]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

1954 (2)

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1954).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

1930 (1)

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 182–195 (1930).
[CrossRef]

Arii, M.

J. J. van Zyl, M. Arii, and Y. Kim, “Model-based decomposition of polarimetric SAR covariance matrices constrained for nonnegative eigenvalues,” IEEE Trans. Geosci. Remote Sens. 49, 3452–3459 (2011).
[CrossRef]

Barakat, R.

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

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

Benali, A.

T. Novikova, A. Pierangelo, A. De Martino, A. Benali, and P. Validire, “Polarimetric imaging for cancer diagnosis and staging,” Opt. Photon. News 23(10), 26–33 (2012).
[CrossRef]

Bernabéu, E.

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

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

Chipman, R. A.

Chu, M. T.

M. T. Chu, R. E. Funderlic, and G. H. Golub, “A rank-one reduction formula and its applications to matrix factorizations,” SIAM Rev. 37, 512–530 (1995).
[CrossRef]

Cloude, S. R.

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

S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).

Correas, J.

J. J. Gil and J. Correas, “Polarimetric subtraction for obtaining the Mueller matrices of components which appear combined in a whole material sample under measurement,” Proceedings of ICO Topical Meeting on Polarization Optics (ICOPO), Polvijärvi, Finland, July 2, 2003. Available at http://www.pepegil.es/Polarimetric-subtraction-ICOPO-2003-.pdf .

Correas, J. M.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Semin. Matem. García de Galdeano. 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” VII Jornadas Zaragoza-Pau de Matemática Aplicada y estadística, Jaca, Spain, 17–18 September, 2001. Monografías del Semin. Matem. Garc a de Galdeano. 27, 233–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .

Cui, Y.

Y. Cui, Y. Yamaguchi, J. Yang, H. Kobayashi, S.-E. Park, and G. Singh, “On complete model based three component decomposition,” Proceedings of International Niigata Polarimetric SAR Workshop, University of Niigata, Niigata, Japan, 2012).

De Martino, A.

T. Novikova, A. Pierangelo, A. De Martino, A. Benali, and P. Validire, “Polarimetric imaging for cancer diagnosis and staging,” Opt. Photon. News 23(10), 26–33 (2012).
[CrossRef]

M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices,” Opt. Express 17, 12794–12806 (2009).
[CrossRef]

Durden, S. L.

A. Freeman and S. L. Durden, “A three component model for polarimetric SAR data,” IEEE Trans. Geosci. Remote Sens. 36, 963–973 (1998).
[CrossRef]

Fano, U.

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1954).
[CrossRef]

Ferreira, C.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Semin. Matem. García de Galdeano. 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

Foldyna, M.

Freeman, A.

A. Freeman and S. L. Durden, “A three component model for polarimetric SAR data,” IEEE Trans. Geosci. Remote Sens. 36, 963–973 (1998).
[CrossRef]

Funderlic, R. E.

M. T. Chu, R. E. Funderlic, and G. H. Golub, “A rank-one reduction formula and its applications to matrix factorizations,” SIAM Rev. 37, 512–530 (1995).
[CrossRef]

Garcia-Caurel, E.

Gil, J. J.

J. J. Gil, I. San José, and R. Ossikovski, “Serial-parallel decompositions of Mueller matrices,” J. Opt. Soc. Am. A. 30, 32–50 (2013).
[CrossRef]

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28, 1578–1585 (2011).
[CrossRef]

J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010).
[CrossRef]

M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices,” Opt. Express 17, 12794–12806 (2009).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Semin. Matem. García de Galdeano. 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” VII Jornadas Zaragoza-Pau de Matemática Aplicada y estadística, Jaca, Spain, 17–18 September, 2001. Monografías del Semin. Matem. Garc a de Galdeano. 27, 233–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .

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

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

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

J. J. Gil and J. Correas, “Polarimetric subtraction for obtaining the Mueller matrices of components which appear combined in a whole material sample under measurement,” Proceedings of ICO Topical Meeting on Polarization Optics (ICOPO), Polvijärvi, Finland, July 2, 2003. Available at http://www.pepegil.es/Polarimetric-subtraction-ICOPO-2003-.pdf .

J. J. Gil, “Determination of polarization parameters in matricial representation. Theoretical contribution and development of an automatic measurement device,” Ph.D. thesis (Facultad de Ciencias, Univ. Zaragoza, Spain, 1983). Available from http://www.pepegil.es/PhD-Thesis-JJ-Gil-English.pdf .

Girgel, S. S.

S. S. Girgel, “Structure of Mueller matrices of depolarized optical systems,” Sov. Phys. Crystallogr. 36, 890 (1991).

Givens, C. R.

C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

Golub, G. H.

M. T. Chu, R. E. Funderlic, and G. H. Golub, “A rank-one reduction formula and its applications to matrix factorizations,” SIAM Rev. 37, 512–530 (1995).
[CrossRef]

Gosh, N.

N. Gosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications and outlook,” J. Biomed. Opt. 16, 110801 (2011).
[CrossRef]

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 2012).

Ishido, M.

Y. Yamaguchi, T. Moriyama, M. Ishido, and H. Yamada, “Four-component scattering model for polarimetric SAR image decomposition,” IEEE Trans. Geosci. Remote Sens. 43, 1699–1706 (2005).
[CrossRef]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 2012).

Jose, I. San

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

José, I. San

J. J. Gil, I. San José, and R. Ossikovski, “Serial-parallel decompositions of Mueller matrices,” J. Opt. Soc. Am. A. 30, 32–50 (2013).
[CrossRef]

J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010).
[CrossRef]

Kim, K.

Kim, Y.

J. J. van Zyl, M. Arii, and Y. Kim, “Model-based decomposition of polarimetric SAR covariance matrices constrained for nonnegative eigenvalues,” IEEE Trans. Geosci. Remote Sens. 49, 3452–3459 (2011).
[CrossRef]

Kobayashi, H.

Y. Cui, Y. Yamaguchi, J. Yang, H. Kobayashi, S.-E. Park, and G. Singh, “On complete model based three component decomposition,” Proceedings of International Niigata Polarimetric SAR Workshop, University of Niigata, Niigata, Japan, 2012).

Kostinski, A. B.

C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

Lu, S.-Y.

Mandel, L.

Melero, P. A.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Semin. Matem. García de Galdeano. 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” VII Jornadas Zaragoza-Pau de Matemática Aplicada y estadística, Jaca, Spain, 17–18 September, 2001. Monografías del Semin. Matem. Garc a de Galdeano. 27, 233–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .

Moriyama, T.

Y. Yamaguchi, T. Moriyama, M. Ishido, and H. Yamada, “Four-component scattering model for polarimetric SAR image decomposition,” IEEE Trans. Geosci. Remote Sens. 43, 1699–1706 (2005).
[CrossRef]

Novikova, T.

T. Novikova, A. Pierangelo, A. De Martino, A. Benali, and P. Validire, “Polarimetric imaging for cancer diagnosis and staging,” Opt. Photon. News 23(10), 26–33 (2012).
[CrossRef]

Ossikovski, R.

Park, S.-E.

Y. Cui, Y. Yamaguchi, J. Yang, H. Kobayashi, S.-E. Park, and G. Singh, “On complete model based three component decomposition,” Proceedings of International Niigata Polarimetric SAR Workshop, University of Niigata, Niigata, Japan, 2012).

Parke, N. G.

N. G. Parke, “Matrix optics,” Ph.D. thesis (Massachusetts Institute of Technology, 1948).

Parrent, G. B.

G. B. Parrent and P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
[CrossRef]

Pierangelo, A.

T. Novikova, A. Pierangelo, A. De Martino, A. Benali, and P. Validire, “Polarimetric imaging for cancer diagnosis and staging,” Opt. Photon. News 23(10), 26–33 (2012).
[CrossRef]

Roman, P.

G. B. Parrent and P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
[CrossRef]

Singh, G.

Y. Cui, Y. Yamaguchi, J. Yang, H. Kobayashi, S.-E. Park, and G. Singh, “On complete model based three component decomposition,” Proceedings of International Niigata Polarimetric SAR Workshop, University of Niigata, Niigata, Japan, 2012).

Validire, P.

T. Novikova, A. Pierangelo, A. De Martino, A. Benali, and P. Validire, “Polarimetric imaging for cancer diagnosis and staging,” Opt. Photon. News 23(10), 26–33 (2012).
[CrossRef]

van Zyl, J. J.

J. J. van Zyl, M. Arii, and Y. Kim, “Model-based decomposition of polarimetric SAR covariance matrices constrained for nonnegative eigenvalues,” IEEE Trans. Geosci. Remote Sens. 49, 3452–3459 (2011).
[CrossRef]

J. J. van Zyl, “Application of Cloude’s target decomposition theorem to polarimeric imaging radar,” Proc. SPIE 127, 184–212 (1992).
[CrossRef]

Vitkin, I. A.

N. Gosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications and outlook,” J. Biomed. Opt. 16, 110801 (2011).
[CrossRef]

Wedderburn, J. H. M.

J. H. M. Wedderburn, Lectures on Matrices (Dover, 1964).

Wiener, N.

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 182–195 (1930).
[CrossRef]

Wolf, E.

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]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

Xing, Z.-F.

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod Opt. 39, 461–484 (1992).
[CrossRef]

Yamada, H.

Y. Yamaguchi, T. Moriyama, M. Ishido, and H. Yamada, “Four-component scattering model for polarimetric SAR image decomposition,” IEEE Trans. Geosci. Remote Sens. 43, 1699–1706 (2005).
[CrossRef]

Yamaguchi, Y.

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Figures (4)

Fig. 1.
Fig. 1.

Two identical beams fall on two different material samples characterized by respective Mueller matrices M1 and M2.

Fig. 2.
Fig. 2.

Matrices M1 and M2 represent respective incoherent components of the optical system. Each component is partially illuminated by the beam represented by p1s and p2s, respectively. The complete polarimetric interaction is given by s=Ms=M1(p1s)+M2(p2s).

Fig. 3.
Fig. 3.

Matrices p1M1 and p2M2 represent respective incoherent components of the optical system. p1 and p2 are the ratios between the respective cross sections and the illuminated areas. The complete polarimetric interaction is given by s=Ms=(p1M1)s+(p2M2)s.

Fig. 4.
Fig. 4.

Component M0 to be subtracted is included in the parallel decomposition of the minuend matrix M. The coefficient p0 of M0 is calculated and then the subtraction is performed as MX=(Mp0M0)/(1p0); (MX)00=(M0)00=(M)00m00.

Equations (71)

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M=m00(1DTPm);D1m00(m01,m02,m03,)T,P1m00(m10,m20,m30,)T,m1m00(m11m12m13m21m22m23m31m32m33).
PΔ=(D2+P2+m22)/3,
Φ(s)=12k=03skσk=12[s0+s1s2is3s2+is3s0s1],
σ0=[1001],σ1=[1001],σ2=[0110],σ3=[0ii0].
λ0=12I(1+P),λ1=12I(1P);I=s0=trΦ,P=λ0λ1trΦ.
s=I[1Pu]=p0s0+(1p0)s1;s0I[1v],s1I[1w];0<p0<1;|u|=|v|=|w|=1.
s0=s0=I.
ΦtrtrΦ=s0=s0=I.
s1=11p0(sp0s0)=I{11p0[1p0Pup0v]}.
p0=1P22(1PuTv),0<p0<1;s1=I[1w],wPup0v1p0.
s=I[1Pu]=λ0I(I[1u])+λ1I(I[1u]).
H(M)=14i,j=03mij(σiσj),
mij=tr[(σiσj)H].
H=Udiag(λ0,λ1,λ2,λ3)U,
H=i=03λitrH[(trH)UDiU];D0diag(1,0,0,0),D1diag(0,1,0,0),D2diag(0,0,1,0),D3diag(0,0,0,1),
H=i=03λitrHHi,Hi(trH)(uiui).
M=i=03λim00MJi,
M0=m00.
HtrtrH=m00=M0,
s=M1(p1s)+M2(p2s)=(p1M1)s+(p2M2)s,s0=p1(M1s)0+p2(M2s)0.
s0(su)=m00=p1(M1)00+p2(M2)00;p1+p2=1,
s=isi=i(piMJis)=Ms,M(ipiMJi);(MJi)00=m00;pi0,ipi=1,
Ddiag(λ0,λ1,λ2,λ3),
V=(Vk000),
H=UDU=UDDU=UDIkDU=UDVVDU=UD(i=0k1vivi)DU=i=0k1(UDvi)(UDvi),
H=i=0k1piHJi;HJitrH(wiwi),wiUDvi|Dvi|,pi|Dvi|2trH(i=0,,k1),i=0k1pi=1,
vitrHpi(D)Uwi(i=0,,k1);(D)(1/λ0,,1/λk1,0,,0),
H=UDU=UDDU=UDIkDU,
Ik=(D)UHU(D)=(D)U[trHi=0k1pi(wiwi)]U(D)=i=0k1[trHpi(D)Uwi][trHpi(D)Uwi]=i=0k1vivi.
|Dvi|2=1j=0k11λj|(Uwi)j|2,
pi=1trHj=0k11λj|(Uwi)j|2.
M=i=0k1piMJi;(MJi)qr=m00tr[(σqσr)(wiwi)]=m00|Dvi|2tr[(σqσr)(UDviviDU)],pi1m00j=0k11λj|(Uwi)j|2=|Dvi|2m00(i=0,,k1),
H=p0HJ0+p1HJ1=p0m00(w0w0)+p1m00(w1w1);|w0|=|w1|=1;p0+p1=1;trH=trHJ1=trHJ0m00.
H=i=0n1piHJi=m00i=0n1pi(wiwi);|wi|=1;i=0n1pi=1;trH=trHJim00.
mij=12trj=03(σiTσjT)
MJ=L(TT*)L1;L[1001100101100ii0],L1=12L.
T[t0t1t2t3],
HJ=12(tt);t(t0,t1,t2,t3)T,
t=(2trH)w
T=(2trH)W;W[w0w1w2w3].
w=l=0k1clul;l=0k1cl2=1.
T0=c[w00w01w02w03];c>0,
w0ul=0,lk,
Hrk=p0HJ0+i=1k1piHJi;trHJi=trHr3(i=0,,k1);i=0k1pi=1.
Hrkp0HJ0=(1p0)HX
HX11p0i=1k1piHJi;trHX=trHJ0=trHrk,
p0=1trHj=0k11λj|(w0U)j|2;HX=1(1p0)(Hrkp0HJ0).
rankHX=rankHrkrankHJ0=k1.
MX=1(1p0)(Mrkp0MJ0);(MX)00=(MJ0)00=(Mrk)00m00.
Hrk=p0HJ0+(1p0)HXr(k1);trHXrk1=trHJ0=trHrk.
Hrk=aHJ0+[(p0a)HJ0+(1p0)HXr(k1)],
HXrk=11a(HrkaHJ0)=11a[(p0a)HJ0+(1p0)HXr(k1)];0<a<p0;rankHXrk=rankHrkk;trHXrk=trHXrk1=trHJ0=trHrk.
RangeHrmRangeHrk
Hrk=UDU;Ddiag(λ0,λ1,λ2,λ3);λ0λ1λ2λ3,
HX=11p(HrkpHrm);0<p<1;trHX=trHrk=trHrm,
Ker(Hrk)={uk,,u3},
(i=k3uiui)Hrm(i=k3uiui)=0,
Hrm=(i=03uiui)Hrm(i=03uiui)=(UIkU)Hrm(UIkU).
Hrk=UDU=UDDU=UDIkDU,Ik=(D)IkD=DIk(D),
Hrm=trHrki=0m1qizizi,
HX=HrkpHrm=UD[Ikp(D)UHrmU(D)]DU=UD[Ikp(D)U(trHrki=0m1qizizi)U(D)]DU=UD[Ikpi=0m1([trHrkqi(D)Uzi][trHrkqi(D)Uzi])]DU.
rmi=0m1yiyi;yitrHrkqi(D)Uzi,
rm=rm,
(m00I),
HX=UD[Ikprm]DU=UD[Ikprm]DU=UD[Ikprm]DU.
(Dkξ)i=1pλ͡i(i=0,,m1);(Dkξ)i=1(i=m,,k1);(Dkξ)i=1(ik),DkξIkpDrm.
rankHX=rank{UD(Dkξ)DU}=rank{D(Dkξ)D},
rankHX=rank(Dkξ).
pmax=1/λ͡0,
rankHX=kξ,
m00(1+D)1,m00(1+P)1.

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