Abstract

We consider the degree of linear polarization (DOLP) polarimetry system, which performs two intensity measurements at orthogonal polarization states to estimate DOLP. We show that if the total integration time of intensity measurements is fixed, the variance of the DOLP estimator depends on the distribution of integration time for two intensity measurements. Therefore, by optimizing the distribution of integration time, the variance of the DOLP estimator can be decreased. In this paper, we obtain the closed-form solution of the optimal distribution of integration time in an approximate way by employing Delta method and Lagrange multiplier method. According to the theoretical analyses and real-world experiments, it is shown that the variance of the DOLP estimator can be decreased for any value of DOLP. The method proposed in this paper can effectively decrease the measurement variance and thus statistically improve the measurement accuracy of the polarimetry system.

© 2016 Optical Society of America

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References

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  1. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
    [Crossref] [PubMed]
  2. S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ= 806 nm,” Opt. Eng. 39(10), 2681–2688 (2000).
    [Crossref]
  3. M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
    [Crossref]
  4. N. J. Brock, C. Crandall, and J. E. Millerd, “Snap-shot imaging polarimeter: performance and applications,” Proc. SPIE 9099, 909903 (2014).
    [Crossref]
  5. G. A. Kafidova, E. T. Aksenov, and V. M. Petrov, “Measurement of glucose concentration in turbid media by the polarization state of backscattered laser light,” Proc. SPIE 8803, 880306 (2013).
    [Crossref]
  6. G. Anna, F. Goudail, and D. Dolfi, “Polarimetric target detection in the presence of spatially fluctuating Mueller matrices,” Opt. Lett. 36(23), 4590–4592 (2011).
    [Crossref] [PubMed]
  7. F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt. 49(4), 683–693 (2010).
    [Crossref] [PubMed]
  8. A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278(2), 264–269 (2007).
    [Crossref]
  9. A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rap. Public. 3, 08002 (2008).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  17. D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic, 1982).
  18. A. Lizana, I. Estévez, F. A. Torres-Ruiz, A. Peinado, C. Ramirez, and J. Campos, “Arbitrary state of polarization with customized degree of polarization generator,” Opt. Lett. 40(16), 3790–3793 (2015).
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    [Crossref] [PubMed]

2015 (2)

2014 (1)

N. J. Brock, C. Crandall, and J. E. Millerd, “Snap-shot imaging polarimeter: performance and applications,” Proc. SPIE 9099, 909903 (2014).
[Crossref]

2013 (2)

G. A. Kafidova, E. T. Aksenov, and V. M. Petrov, “Measurement of glucose concentration in turbid media by the polarization state of backscattered laser light,” Proc. SPIE 8803, 880306 (2013).
[Crossref]

H. Hu, G. Anna, and F. Goudail, “On the performance of the physicality-constrained maximum-likelihood estimation of Stokes vector,” Appl. Opt. 52(27), 6636–6644 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (1)

2010 (1)

2009 (1)

2008 (2)

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Minimization of the influence of passive-light contribution in active imaging of the degree of polarization,” Opt. Lett. 33(20), 2335–2337 (2008).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rap. Public. 3, 08002 (2008).
[Crossref]

2007 (2)

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278(2), 264–269 (2007).
[Crossref]

P. Shukla, R. Sumathi, S. Gupta, and A. Pradhan, “Influence of size parameter and refractive index of the scatterer on polarization-gated optical imaging through turbid media,” J. Opt. Soc. Am. A 24(6), 1704–1713 (2007).
[Crossref] [PubMed]

2004 (1)

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[Crossref]

2002 (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

2000 (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ= 806 nm,” Opt. Eng. 39(10), 2681–2688 (2000).
[Crossref]

1999 (1)

Aksenov, E. T.

G. A. Kafidova, E. T. Aksenov, and V. M. Petrov, “Measurement of glucose concentration in turbid media by the polarization state of backscattered laser light,” Proc. SPIE 8803, 880306 (2013).
[Crossref]

Alouini, M.

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, A. Bénière, I. Baarstad, T. Løke, P. Kaspersen, X. Normandin, and G. Berginc, “Near-infrared active polarimetric and multispectral laboratory demonstrator for target detection,” Appl. Opt. 48(8), 1610–1618 (2009).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Minimization of the influence of passive-light contribution in active imaging of the degree of polarization,” Opt. Lett. 33(20), 2335–2337 (2008).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rap. Public. 3, 08002 (2008).
[Crossref]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278(2), 264–269 (2007).
[Crossref]

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[Crossref]

Anna, G.

Baarstad, I.

Bénière, A.

Berginc, G.

Bourderionnet, J.

Breugnot, S.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ= 806 nm,” Opt. Eng. 39(10), 2681–2688 (2000).
[Crossref]

Brock, N. J.

N. J. Brock, C. Crandall, and J. E. Millerd, “Snap-shot imaging polarimeter: performance and applications,” Proc. SPIE 9099, 909903 (2014).
[Crossref]

Campos, J.

Chavel, P.

Clémenceau, P.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ= 806 nm,” Opt. Eng. 39(10), 2681–2688 (2000).
[Crossref]

Crandall, C.

N. J. Brock, C. Crandall, and J. E. Millerd, “Snap-shot imaging polarimeter: performance and applications,” Proc. SPIE 9099, 909903 (2014).
[Crossref]

Dolfi, D.

G. Anna, F. Goudail, P. Chavel, and D. Dolfi, “On the influence of noise statistics on polarimetric contrast optimization,” Appl. Opt. 51(8), 1178–1187 (2012).
[Crossref] [PubMed]

G. Anna, F. Goudail, and D. Dolfi, “Polarimetric target detection in the presence of spatially fluctuating Mueller matrices,” Opt. Lett. 36(23), 4590–4592 (2011).
[Crossref] [PubMed]

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, A. Bénière, I. Baarstad, T. Løke, P. Kaspersen, X. Normandin, and G. Berginc, “Near-infrared active polarimetric and multispectral laboratory demonstrator for target detection,” Appl. Opt. 48(8), 1610–1618 (2009).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Minimization of the influence of passive-light contribution in active imaging of the degree of polarization,” Opt. Lett. 33(20), 2335–2337 (2008).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rap. Public. 3, 08002 (2008).
[Crossref]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278(2), 264–269 (2007).
[Crossref]

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[Crossref]

Estévez, I.

Goudail, F.

H. Hu, G. Anna, and F. Goudail, “On the performance of the physicality-constrained maximum-likelihood estimation of Stokes vector,” Appl. Opt. 52(27), 6636–6644 (2013).
[Crossref] [PubMed]

G. Anna, F. Goudail, P. Chavel, and D. Dolfi, “On the influence of noise statistics on polarimetric contrast optimization,” Appl. Opt. 51(8), 1178–1187 (2012).
[Crossref] [PubMed]

G. Anna, F. Goudail, and D. Dolfi, “Polarimetric target detection in the presence of spatially fluctuating Mueller matrices,” Opt. Lett. 36(23), 4590–4592 (2011).
[Crossref] [PubMed]

F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt. 49(4), 683–693 (2010).
[Crossref] [PubMed]

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, A. Bénière, I. Baarstad, T. Løke, P. Kaspersen, X. Normandin, and G. Berginc, “Near-infrared active polarimetric and multispectral laboratory demonstrator for target detection,” Appl. Opt. 48(8), 1610–1618 (2009).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Minimization of the influence of passive-light contribution in active imaging of the degree of polarization,” Opt. Lett. 33(20), 2335–2337 (2008).
[Crossref] [PubMed]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rap. Public. 3, 08002 (2008).
[Crossref]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278(2), 264–269 (2007).
[Crossref]

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[Crossref]

Grisard, A.

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, A. Bénière, I. Baarstad, T. Løke, P. Kaspersen, X. Normandin, and G. Berginc, “Near-infrared active polarimetric and multispectral laboratory demonstrator for target detection,” Appl. Opt. 48(8), 1610–1618 (2009).
[Crossref] [PubMed]

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[Crossref]

Gupta, S.

Hu, H.

Huang, B.

Jacques, S. L.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

Kafidova, G. A.

G. A. Kafidova, E. T. Aksenov, and V. M. Petrov, “Measurement of glucose concentration in turbid media by the polarization state of backscattered laser light,” Proc. SPIE 8803, 880306 (2013).
[Crossref]

Kaspersen, P.

Lallier, E.

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[Crossref]

Lee, K.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

Li, X.

Liu, T.

Lizana, A.

Løke, T.

Millerd, J. E.

N. J. Brock, C. Crandall, and J. E. Millerd, “Snap-shot imaging polarimeter: performance and applications,” Proc. SPIE 9099, 909903 (2014).
[Crossref]

Normandin, X.

Peinado, A.

Petrov, V. M.

G. A. Kafidova, E. T. Aksenov, and V. M. Petrov, “Measurement of glucose concentration in turbid media by the polarization state of backscattered laser light,” Proc. SPIE 8803, 880306 (2013).
[Crossref]

Pradhan, A.

Ramella-Roman, J. C.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

Ramirez, C.

Réfrégier, P.

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[Crossref]

Shaw, J. A.

Shukla, P.

Song, Z.

Sumathi, R.

Torres-Ruiz, F. A.

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(3), 329–340 (2002).
[Crossref] [PubMed]

J. Eur. Opt. Soc. Rap. Public. (1)

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rap. Public. 3, 08002 (2008).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278(2), 264–269 (2007).
[Crossref]

Opt. Eng. (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ= 806 nm,” Opt. Eng. 39(10), 2681–2688 (2000).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Proc. SPIE (3)

M. Alouini, F. Goudail, P. Réfrégier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133–144 (2004).
[Crossref]

N. J. Brock, C. Crandall, and J. E. Millerd, “Snap-shot imaging polarimeter: performance and applications,” Proc. SPIE 9099, 909903 (2014).
[Crossref]

G. A. Kafidova, E. T. Aksenov, and V. M. Petrov, “Measurement of glucose concentration in turbid media by the polarization state of backscattered laser light,” Proc. SPIE 8803, 880306 (2013).
[Crossref]

Other (2)

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic, 1982).

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Figures (4)

Fig. 1
Fig. 1

The optimal integration times of intensity measurements at different values ofP.

Fig. 2
Fig. 2

(a) The normalized variance of P ^ with equalized integration times and with optimal integration times as a function of P; (b) The degree of optimization Ψ at optimal distribution of integration time as a function of P.

Fig. 3
Fig. 3

Schematic of experimental setup of DOLP polarimetry. QWP: quarter-wave plate; LP: linear polarizer; P-BS: polarizing beam-splitter.

Fig. 4
Fig. 4

The experimental and theoretical results of the degree of optimization Ψat optimal distribution of integration time as a function of P.

Tables (1)

Tables Icon

Table 1 The experimental variances of estimator at different distributions of integration time. The experimental and theoretical results of degree of optimization Ψ are also presented. The true value of P is 0.6.

Equations (19)

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

P= I // I I // + I .
{ I // m = t // I // + n // I m = t I + n ,
{ I ^ // = I // m t // = I // + n // t // I ^ = I m t = I + n t .
Γ X =[ σ 2 t // 2 0 0 σ 2 t 2 ].
I // = I( 1+P ) 2 , I = I( 1P ) 2 .
P ^ = I ^ // I ^ I ^ // + I ^ = ( I // + n //   t // )( I + n t ) ( I // + n //   t // )+( I + n t ) .
y f( X ) , VAR[y] [ f( X ) ] T Γ X [ f( X ) ],
P ^ ( X )= 2 ( I // + I ) 2 [ I , I // ] T = 1 I [1P,(1+P)] T .
VAR[ P ^ ]= σ 2 I 2 [ (1P) 2 t // 2 + (1+P) 2 t 2 ].
min t VAR[ P ^ ]= min t { σ 2 I 2 [ (1P) 2 t // 2 + (1+P) 2 t 2 ] },
h(t)= t // + t 2=0.
L(λ,t)=VAR[ P ^ ]+λh(t)= σ 2 I 2 [ (1P) 2 t // 2 + (1+P) 2 t 2 ]+λ( t // + t 2 ),
{ L t i = 2 C i t i 3 +λ=0 L λ = t // + t 2=0 i{ //, } ,
2 C // t // 3 = 2 C t 3 =λ.
t // : t = (1P) 2 3 : (1+P) 2 3 .
t // = 2 (1P) 2 3 (1+P) 2 3 + (1P) 2 3 , t = 2 (1+P) 2 3 (1+P) 2 3 + (1P) 2 3 .
VAR [ P ^ ] opt = σ 2 4 I 2 [ ( 1+P ) 2 3 + ( 1P ) 2 3 ] 3 ,
VAR [ P ^ ] equ = σ 2 I 2 [ ( 1+P ) 2 + ( 1P ) 2 ].
Ψ=1 VAR [ P ^ ] opt VAR [ P ^ ] equ =1 [ ( 1+P ) 2 3 + ( 1P ) 2 3 ] 3 4[ ( 1+P ) 2 + ( 1P ) 2 ] ,

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