Abstract

We address constrained estimation of the Mueller matrices from noisy measurements, taking into account the physical realizability. Physical realizability is enforced based on the positive semi-definite Hermitian coherency matrix, and the statistics of the noise is taken into account by employing Maximum Likelihood (ML) method. We consider two types of noise sources frequently encountered in optical imaging systems: additive Gaussian noise and Poisson shot noise. In both cases, we demonstrate reduction of estimation error by enforcing the physical realizability constraint, and superiority of the ML constrained solutions compared to empirically constrained ones. The ML constrained estimation method proposed in this paper provides a justified and effective way to exploit experimental measurements of Mueller matrices.

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  1. J. E. Solomon, “Polarization imaging,” Appl. Opt.20, 1537–1544 (1981).
    [CrossRef] [PubMed]
  2. 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]
  3. S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ = 806 nm,” Opt. Eng.39, 2681–2688 (2000).
    [CrossRef]
  4. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt.7, 329–340 (2002).
    [CrossRef] [PubMed]
  5. Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” Appl. Opt.42, 511–525 (2003).
    [CrossRef] [PubMed]
  6. J. M. Bueno, J. Hunter, C. Cookson, M. Kisilak, and M. Campbell, “Improved scanning laser fundus imaging using polarimetry,” J. Opt. Soc. Am. A24, 1337–1348 (2007).
    [CrossRef]
  7. A. Pierangelo, B. Abdelali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by mueller polarimetric imaging,” Opt. Express19, 1582–1593 (2011).
    [CrossRef] [PubMed]
  8. J. Cariou, B. L. Jeune, J. Lotrian, and Y. Guern, “Polarization effects of seawater and underwater targets,” Appl. Opt.29, 1689–1695 (1990).
    [CrossRef] [PubMed]
  9. B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt.18, 809–812 (1979).
    [CrossRef] [PubMed]
  10. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a mueller matrix to be derivable from a jones matrix,” J. Opt. Soc. Am. A11, 2305–2319 (1994).
    [CrossRef]
  11. R. S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng.34, 1599–1610 (1995).
    [CrossRef]
  12. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of mueller matrices,” Opt. Lett.31, 817–819 (2006).
    [CrossRef] [PubMed]
  13. J. Zallat, C. Heinrich, and M. Petremand, “A bayesian approach for polarimetric data reduction: the mueller imaging case,” Opt. Express16, 7119–7133 (2008).
    [CrossRef] [PubMed]
  14. J. E. Ahmad and Y. Takakura, “Estimation of physically realizable mueller matrices from experimentsusing global constrained optimization,” Opt. Express16, 14274–14287 (2008).
    [CrossRef] [PubMed]
  15. S. M. Kay, Fundamentals of statistical signal processing - Volume I : Estimation Theory (Prentice-Hall, Englewood Cliffs, 1993).
  16. R. Ossikovski, “Retrieval of a nondepolarizing estimate from an experimental mueller matrix through virtual experiment,” Opt. Lett.37, 578–580 (2012).
    [CrossRef] [PubMed]
  17. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).
  18. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part ii,” Opt. Eng.34, 1656–1658 (1995).
    [CrossRef]
  19. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete stokes polarimeter,” Opt. Lett.25, 802–804 (2000).
    [CrossRef]
  20. J. S. Tyo, “Noise equalization in stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett.25, 1198–1200 (2000).
    [CrossRef]
  21. F. Goudail, “Noise minimization and equalization for stokes polarimeters in the presence of signal-dependent poisson shot noise,” Opt. Lett.34, 647–649 (2009).
    [CrossRef] [PubMed]
  22. G. Anna and F. Goudail, “Optimal mueller matrix estimation in the presence of poisson shot noise,” Opt. Express20, 21331–21340 (2012).
    [CrossRef] [PubMed]
  23. Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl.76, 501–521 (1993).
    [CrossRef]
  24. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Optica Acta: International Journal of Optics33, 185–189 (1986).
    [CrossRef]

2012 (2)

2011 (1)

2009 (1)

2008 (2)

2007 (1)

2006 (1)

2003 (1)

2002 (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt.7, 329–340 (2002).
[CrossRef] [PubMed]

2000 (3)

1996 (1)

1995 (2)

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

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part ii,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

1994 (1)

1993 (1)

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl.76, 501–521 (1993).
[CrossRef]

1990 (1)

1986 (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Optica Acta: International Journal of Optics33, 185–189 (1986).
[CrossRef]

1981 (1)

1979 (1)

Abdelali, B.

Ahmad, J. E.

Aiello, A.

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part ii,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

Anderson, D. G. M.

Anna, G.

Antonelli, M.-R.

Barakat, R.

Barrett, H. H.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Optica Acta: International Journal of Optics33, 185–189 (1986).
[CrossRef]

Breugnot, S.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ = 806 nm,” Opt. Eng.39, 2681–2688 (2000).
[CrossRef]

Bueno, J. M.

Campbell, M.

Cariou, J.

Clémenceau, P.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ = 806 nm,” Opt. Eng.39, 2681–2688 (2000).
[CrossRef]

Cloude, R. S.

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

Cookson, C.

De Martino, A.

Dereniak, E. L.

Descour, M. R.

Duan, Q. Y.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl.76, 501–521 (1993).
[CrossRef]

Engheta, N.

Gayet, B.

Gil, J. J.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Optica Acta: International Journal of Optics33, 185–189 (1986).
[CrossRef]

Goudail, F.

Guern, Y.

Gupta, V. K.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl.76, 501–521 (1993).
[CrossRef]

Heinrich, C.

Howell, B. J.

Hunter, J.

Jacques, S. L.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt.7, 329–340 (2002).
[CrossRef] [PubMed]

Jeune, B. L.

Kay, S. M.

S. M. Kay, Fundamentals of statistical signal processing - Volume I : Estimation Theory (Prentice-Hall, Englewood Cliffs, 1993).

Kemme, S. A.

Kisilak, M.

Lee, K.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt.7, 329–340 (2002).
[CrossRef] [PubMed]

Look, D. C.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part ii,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

Lotrian, J.

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).

Narasimhan, S. G.

Nayar, S. K.

Novikova, T.

Ossikovski, R.

Petremand, M.

Phipps, G. S.

Pierangelo, A.

Pottier, E.

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

Puentes, G.

Pugh, E. N.

Ramella-Roman, J. C.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt.7, 329–340 (2002).
[CrossRef] [PubMed]

Rowe, M. P.

Sabatke, D. S.

Schechner, Y. Y.

Solomon, J. E.

Sorooshian, S.

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl.76, 501–521 (1993).
[CrossRef]

Sweatt, W. C.

Takakura, Y.

Tyo, J. S.

Validire, P.

Voigt, D.

Woerdman, J. P.

Zallat, J.

Appl. Opt. (5)

J. Biomed. Opt. (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt.7, 329–340 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

J. Optim. Theory Appl. (1)

Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl.76, 501–521 (1993).
[CrossRef]

Opt. Eng. (3)

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ = 806 nm,” Opt. Eng.39, 2681–2688 (2000).
[CrossRef]

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

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part ii,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

Opt. Express (4)

Opt. Lett. (5)

Optica Acta: International Journal of Optics (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Optica Acta: International Journal of Optics33, 185–189 (1986).
[CrossRef]

Other (2)

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).

S. M. Kay, Fundamentals of statistical signal processing - Volume I : Estimation Theory (Prentice-Hall, Englewood Cliffs, 1993).

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Figures (5)

Fig. 1
Fig. 1

Total RMSE as a function of SNR for the UC, EC and GMLC estimators in the presence of the additive Gaussian noise. The RMSE is estimated from 2000 noise realizations and the error bars correspond to the standard deviation of this estimated RMSE.

Fig. 2
Fig. 2

Total RMSE as a function of SNR for the GMLC and MJC estimators in the presence of the additive Gaussian noise. The RMSE is estimated from 2000 noise realizations

Fig. 3
Fig. 3

Mean of DI of EC and GMLC estimators as a function of SNR. The average DI is estimated from 2000 noise realizations and the error bars correspond to the standard deviation of this estimated mean.

Fig. 4
Fig. 4

Total RMSE as a function of SNR for the UC, EC and PMLC estimators in the presence of Poisson shot noise. The number of realizations is 2000.

Fig. 5
Fig. 5

Mean of DI as a function of SNR for the EC and PMLC estimators in the presence of Poisson shot noise. The number of realizations is 2000.

Tables (4)

Tables Icon

Table 1 UC, EC and GMLC estimates of one realization as well as the corresponding eigenvalues of the coherency matrix. The measurements are perturbed with additive Gaussian noise.

Tables Icon

Table 2 The norms between the estimates and the UC estimate or the true value M0.

Tables Icon

Table 3 UC, EC and PMLC estimates of one realization as well as the corresponding eigenvalues of the coherency matrix. The measurements are perturbed with Poisson shot noise.

Tables Icon

Table 4 The norm between the EC or PMLC estimate and the UC estimate or the true value M0.

Equations (31)

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i k = I 0 t k T M s k , k [ 1 , N ]
i k = I 0 [ t k s k ] T m = q k T m , k [ 1 , N ]
q k = I 0 [ t k s k ] , k [ 1 , N ]
i = Q m .
m ^ u = Q 1 i .
C [ m ] = i , j = 0 3 M i j σ i σ j ,
C [ m ^ u ] = V diag ( λ ) V T ,
λ k = { λ k if λ k 0 0 otherwise
C [ m ^ e ] = V diag ( λ ) V T .
m ^ e = arg min m { m m ^ u 2 } ,
m ^ opt = arg min m { m m 0 2 } ,
m ^ mlc = arg max m { log [ P i m ( i ) ] } .
A = [ a 1 0 0 0 a 5 + i a 6 a 2 0 0 a 11 + i a 12 a 7 + i a 8 a 3 0 a 15 + i a 16 a 13 + i a 14 a 9 + i a 10 a 4 ] ,
m ^ mlc = arg max a 16 { log [ P i m ( a ) ( i ) ] } ,
B = [ b 1 0 0 0 b 2 + i b 3 0 0 0 b 4 + i b 5 0 0 0 b 6 + i b 7 0 0 0 ] ,
M 00 + M 01 2 + M 02 2 + M 03 2 1.
P i m 0 ( i ) = k = 1 N P i k m 0 ( i k ) .
G = 1 2 σ 2 i Q m 2 ln ( 2 π σ ) ,
m ^ gmlc = arg min m { i Q m 2 } .
m ^ gmlc = arg min m { Q ( m ^ u m ) 2 } .
P ( i k ) = e [ Q m ] k [ Q m ] k i k i k ! , k [ 1 , N ] .
P ( m ) = k = 1 16 [ [ Q m ] k + i k ln [ Q m ] k + ln ( i k ! ) ] .
m ^ pmlc = arg min m { k = 1 16 [ Q m ] k i k ln ( [ Q m ] k ) } .
m ^ G virtual = arg min m { i Q m 2 } ,
m ^ P virtual = arg min m { k = 1 16 [ Q m ] k i k ln ( [ Q m ] k ) } .
S = T = 1 2 [ 1 1 1 1 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 ] .
M 0 = [ 0.5 0.14 0 0 0.14 0.5 0 0 0 0 0.48 0 0 0 0 0.48 ] ,
RMSE = i , j = 0 3 ( M ^ i j M 0 i j ) 2 ,
SNR G = 10 log 10 [ I 0 2 σ 2 ]
DI = i , j = 0 3 M i j 2 M 00 2 3 M 00 ,
SNR P = 10 log 10 [ I 0 ]

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