Abstract

We address the estimation of the Stokes vectors taking into account the physical realizability constraint. We propose a fast method for computing the constrained maximum-likelihood (CML) estimator for any measurement matrix, and we compare its performance with the classical empirical physicality-constrained estimator. We show that when the measurement matrix is based on four polarization states spanning a regular tetrahedron on the Poincaré sphere, the two estimators are very similar, but the CML provides a better estimation of the intensity. For an arbitrary measurement matrix, the CML estimator does not always yield better estimation performance than the empirical one: their comparative performances depend on the measurement matrix, the actual Stokes vector and the signal-to-noise ratio.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. J. Wu and J. T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
    [CrossRef]
  2. J. A. North and M. J. Duggin, “Stokes vector imaging of the polarized sky-dome,” Appl. Opt. 36, 723–730 (1997).
    [CrossRef]
  3. D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
    [CrossRef]
  4. J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83–96 (2007).
    [CrossRef]
  5. M. R. Foreman, C. M. Romero, and P. Torok, “A priori information and optimisation in polarimetry,” Opt. Express 16, 15212–15227 (2008).
    [CrossRef]
  6. J. R. Valenzuela and J. A. Fessler, “Joint reconstruction of Stokes images from polarimetric measurements,” J. Opt. Soc. Am. A 26, 962–968 (2009).
    [CrossRef]
  7. J. R. Valenzuela, J. A. Fessler, and R. G. Paxman, “Joint estimation of Stokes images and aberrations from phase-diverse polarimetric measurements,” J. Opt. Soc. Am. A 27, 1185–1193 (2010).
    [CrossRef]
  8. S. Faisan, C. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering estimation of Stokes vector images based on a nonlocal means approach,” J. Opt. Soc. Am. A 29, 2028–2037 (2012).
    [CrossRef]
  9. J. E. Ahmad and Y. Takakura, “Estimation of physically realizable Mueller matrices from experiments using global constrained optimization,” Opt. Express 16, 14274–14287 (2008).
    [CrossRef]
  10. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. 31, 817–819 (2006).
    [CrossRef]
  11. H. Hu, R. Ossikovski, and F. Goudail, “Performance of maximum likelihood estimation of Mueller matrices taking into account physical realizability and Gaussian or Poisson noise statistics,” Opt. Express 21, 5117–5129 (2013).
    [CrossRef]
  12. 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]
  13. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25, 1198–1200 (2000).
    [CrossRef]
  14. S. M. Kay, Fundamentals of Statistical Signal Processing—Volume I: Estimation Theory (Prentice-Hall, 1993).
  15. F. Goudail, “Optimization of the contrast in active Stokes images,” Opt. Lett. 34, 121–123 (2009).
    [CrossRef]
  16. 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]
  17. G. R. Boyer, B. F. Lamouroux, and B. S. Prade, “Automatic measurement of the Stokes vector of light,” Appl. Opt. 18, 1217–1219 (1979).
    [CrossRef]
  18. S. X. Wang and A. M. Weiner, “Fast wavelength-parallel polarimeter for broadband optical networks,” Opt. Lett. 29, 923–925 (2004).
    [CrossRef]

2013 (1)

2012 (1)

2010 (1)

2009 (3)

2008 (2)

2007 (1)

2006 (2)

P. J. Wu and J. T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef]

A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. 31, 817–819 (2006).
[CrossRef]

2004 (1)

2000 (2)

1997 (1)

1992 (1)

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

1979 (1)

Ahmad, J. E.

Aiello, A.

Boyer, G. R.

Chambellan, C. W.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Dereniak, E. L.

Descour, M. R.

Duggin, M. J.

Dunn, R. B.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Elmore, D. F.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Faisan, S.

Fessler, J. A.

Foreman, M. R.

Goudail, F.

Heinrich, C.

Hu, H.

Hull, H. K.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing—Volume I: Estimation Theory (Prentice-Hall, 1993).

Kemme, S. A.

Lacey, L. B.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Lallement, A.

Lamouroux, B. F.

Leach, T. W.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Lites, B. W.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

North, J. A.

Ossikovski, R.

Paxman, R. G.

Phipps, G. S.

Prade, B. S.

Puentes, G.

Romero, C. M.

Rousseau, F.

Sabatke, D. S.

Schuenke, J. A.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Skumanich, A.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Streander, K. V.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Sweatt, W. C.

Takakura, Y.

Tomczyk, S.

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Torok, P.

Tyo, J. S.

Valenzuela, J. R.

Voigt, D.

Walsh, J. T.

P. J. Wu and J. T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef]

Wang, S. X.

Weiner, A. M.

Woerdman, J. P.

Wu, P. J.

P. J. Wu and J. T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef]

Zallat, J.

Appl. Opt. (2)

J. Biomed. Opt. (1)

P. J. Wu and J. T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Express (4)

Opt. Lett. (6)

Proc. SPIE (1)

D. F. Elmore, B. W. Lites, S. Tomczyk, A. Skumanich, R. B. Dunn, J. A. Schuenke, K. V. Streander, T. W. Leach, C. W. Chambellan, H. K. Hull, and L. B. Lacey, “Advanced Stokes polarimeter: a new instrument for solar magnetic field research,” Proc. SPIE 1746, 22–33 (1992).
[CrossRef]

Other (1)

S. M. Kay, Fundamentals of Statistical Signal Processing—Volume I: Estimation Theory (Prentice-Hall, 1993).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

RMSEs of (a) DOP and (b) intensity of unconstrained, empirical, and CML estimators as a function of SNR in the presence of the additive Gaussian noise. The number of realizations is 20,000.

Fig. 2.
Fig. 2.

(a) Ratio between DOP RMSE of CML and unconstrained estimators as a function of actual DOP. (b) Ratio between intensity RMSE of CML and empirical estimators as a function of actual DOP. Results are shown for two different values of the SNR and RMSE is estimated on 50,000 realizations.

Fig. 3.
Fig. 3.

Schematic of the distributions of the unconstrained estimator in Poincaré space in two extreme cases for arbitrary measurement matrix. The pink ellipses in the schematic refer to the ellipsoids representing the statistical distribution of the unconstrained estimator.

Fig. 4.
Fig. 4.

RMSEs of (a) intensity, (b) azimuth, and (c) ellipticity of CML and empirical estimators as a function of SNR. The number of realizations is 20,000, the azimuth and ellipticity are expressed in degrees.

Fig. 5.
Fig. 5.

Ratio of RMSE of (a) intensity, (b) azimuth, and (c) ellipticity between CML and empirical estimators as a function of true value of DOP. The number of realizations is 50,000.

Fig. 6.
Fig. 6.

RMSEs of (a) intensity, (b) azimuth, and (c) ellipticity between empirical and CML estimator as a function of SNR. The number of realizations is 20,000.

Fig. 7.
Fig. 7.

Ratio of RMSE of (a) intensity, (b) azimuth, and (c) ellipticity between CML and empirical estimator as a function of true value of DOP. The number of realizations is 50,000.

Fig. 8.
Fig. 8.

(a) Locus of the investigated Stokes vectors. RMSE of (b) intensity, (c) azimuth, and (d) ellipticity of empirical and CML estimator as a function of the angle between the S1 axis and the Stokes vector located in the meridian. SNR is 100 and the number of realizations is 100,000.

Fig. 9.
Fig. 9.

Global RMSE of the estimators corresponding to the actual Stokes vectors located at the meridians of the Poincaré sphere with (a) DOP=1 and (b) DOP=0.8. SNR is 100 and the number of realizations is 100,000.

Equations (23)

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

I=WS,
S^u=(WTW)1WTI,
S00,
S02S12+S22+S32.
P^u=[S^1u]2+[S^2u]2+[S^3u]2S^0u,
S^e=diag[1,1/Pu,1/Pu,1/Pu]S^u.
F(S)=IWS2λSTGS,
2WTI+2WTWS2λGS=0,
[Iλ(WTW)1G]S=W1I=S^u,
Sc(λ)=[Iλ(WTW)1G]1S^u.
[Sc(λ*)]TGSc(λ*)=0.
Sc(λ)=113λdiag[13λ1+λ,1,1,1]S^u.
[Sc(λ*)]TGSc(λ*)=0λ*=(1Pu)/(3+Pu).
S^c=3+Pu4diag[1,1/Pu,1/Pu,1/Pu]S^u.
Sc(λ)=[IλPDP1]1S^u=[PP1λPDP1]1S^u=P(IλD)1P1S^u.
(P1S^u)T(IλD)1PTGP(IλD)1P1S^u=0.
vT(IλD)1M(IλD)1P1v=0i,j=14viMijvj(1λdi)(1λdj)=0.
1k=14(1λdk)2i,j=14aijk=14(1λdk)2(1λdi)(1λdj)=0.
i,j=14aijk=14(1λdk)2(1λdi)(1λdj)=0,
SNR=S02σ2,
RMSE(X)=(X^X0)2,
ρ=(P^cP)2(P^uP)2,
W=12[1100110010101001].

Metrics