Abstract

Within the general framework of active imaging we address the degree of polarization (DOP) estimation in the presence of additive Gaussian detector noise. We first study the performance of standard DOP estimators and propose a method to increase estimation precision using physically relevant a priori information. We then consider the realistic case of nonuniform illumination distribution. We derive the Cramer–Rao lower bound and determine a profile likelihood-based estimator. We demonstrate the efficiency of this new estimator and compare its performance with other standard estimators as a function of the degree of nonuniformity of the illumination.

© 2008 Optical Society of America

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References

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  1. L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059-1071 (1990).
    [CrossRef]
  2. 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]
  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. S. Breugnot and P. Clémenceau, “Modeling and performances of a Polarization Active Imager at λ=806 nm,” Opt. Eng. (Bellingham) 39, 2681-2688 (2000).
    [CrossRef]
  5. F. Goudail and P. Réfrégier, “Statistical techniques for target detection in polarisation diversity images,” Opt. Lett. 26, 644-646 (2001).
    [CrossRef]
  6. V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973-980 (2002).
    [CrossRef]
  7. P. Réfrégier, F. Goudail, and N. Roux, “Estimation of the degree of polarization in active coherent imagery using the natural representation,” J. Opt. Soc. Am. A 21, 2292-2300 (2004).
    [CrossRef]
  8. P. Réfrégier, M. Roche, and F. Goudail, “Cramer-Rao lower bound for the estimation of the degree of polarization in active coherent imagery at low photon level,” Opt. Lett. 31, 3565-3567 (2006).
    [CrossRef] [PubMed]
  9. 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, 264-269 (2007).
    [CrossRef]
  10. M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
    [CrossRef]
  11. S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806 nm,” Proc. SPIE 3707, 449-460 (1999).
    [CrossRef]
  12. P. Terrier and V. DeVlaminck, “Robust and accurate estimate of the orientation of partially polarized light from a camera sensor,” Appl. Opt. 40, 5233-5239 (2001).
    [CrossRef]
  13. O. Germain and P. Réfrégier, “Snake-based method for the segmentation of objects in multichannel images degraded by speckle,” Opt. Lett. 24, 814-816 (1999).
    [CrossRef]
  14. S. M. Kay, Fundamentals of Statistical Signal Processing Vol. I, Estimation Theory (Prentice-Hall, 1993).
  15. D. V. Hinkley, “On the ratio of two correlated normal random variables,” Biometrika 56, 635 (1969).
    [CrossRef]
  16. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

2007 (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, 264-269 (2007).
[CrossRef]

2006 (2)

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

P. Réfrégier, M. Roche, and F. Goudail, “Cramer-Rao lower bound for the estimation of the degree of polarization in active coherent imagery at low photon level,” Opt. Lett. 31, 3565-3567 (2006).
[CrossRef] [PubMed]

2004 (2)

P. Réfrégier, F. Goudail, and N. Roux, “Estimation of the degree of polarization in active coherent imagery using the natural representation,” J. Opt. Soc. Am. A 21, 2292-2300 (2004).
[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]

2002 (2)

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]

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973-980 (2002).
[CrossRef]

2001 (2)

2000 (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a Polarization Active Imager at λ=806 nm,” Opt. Eng. (Bellingham) 39, 2681-2688 (2000).
[CrossRef]

1999 (2)

O. Germain and P. Réfrégier, “Snake-based method for the segmentation of objects in multichannel images degraded by speckle,” Opt. Lett. 24, 814-816 (1999).
[CrossRef]

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806 nm,” Proc. SPIE 3707, 449-460 (1999).
[CrossRef]

1990 (1)

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059-1071 (1990).
[CrossRef]

1969 (1)

D. V. Hinkley, “On the ratio of two correlated normal random variables,” Biometrika 56, 635 (1969).
[CrossRef]

Appl. Opt. (1)

Biometrika (1)

D. V. Hinkley, “On the ratio of two correlated normal random variables,” Biometrika 56, 635 (1969).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059-1071 (1990).
[CrossRef]

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 (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, 264-269 (2007).
[CrossRef]

Opt. Eng. (1)

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973-980 (2002).
[CrossRef]

Opt. Eng. (Bellingham) (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a Polarization Active Imager at λ=806 nm,” Opt. Eng. (Bellingham) 39, 2681-2688 (2000).
[CrossRef]

Opt. Lett. (3)

Proc. SPIE (3)

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806 nm,” Proc. SPIE 3707, 449-460 (1999).
[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]

Other (2)

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

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

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Figures (9)

Fig. 1
Fig. 1

PDF P [see Eq. (7)] for two values of the SNR and P = 0.5 ; Gaussian PDF with variance ( 1 + P 2 ) ( SNR 2 ) for SNR = 9 , P = 0.5 .

Fig. 2
Fig. 2

Estimated means; standard deviations of estimators P ̂ em , P ̂ med , P ̂ ml ; and square root of CRLB for N = 9 samples and P = 0.5 . The estimations are made with 10 4 realizations.

Fig. 3
Fig. 3

Histograms of P ̂ em , P ̂ med , P ̂ ml , and P ̂ mlt [see Eq. (8), Eq. (9), Eq. (5), and Eq. (10)], obtained with N = 9 and 2500 realizations. The conditions of acquisition with a Basler A312f camera were such that SNR 6.1 and the real value of the DOP was P = 0.15 (this value has been estimated at high SNR ).

Fig. 4
Fig. 4

Estimated bias, standard deviation, and root mean square deviation of estimators P ̂ ml and P ̂ mlt with 25 samples as a function of the SNR and square root of the CRLB. We performed Monte Carlo simulations with 10 4 realizations for three different values of the DOP: first row, P = 0.1 ; second row, P = 0.5 ; third row: P = 0.9 .

Fig. 5
Fig. 5

Standard deviations σ median , σ ml , and σ pl of, respectively, the median, ML, and PL estimators for SNR F = 12 with the square roots of CRLB corresponding to the use of the uniform model [ CRLB U ; see Eq. (3)] and nonuniform model [ CRLB F ; see Eq. (16)] as a function of Q [see Eq. (19)], which characterizes the nonuniformity of the illumination. The estimations were made by Monte Carlo simulations on 10 4 realizations and nine samples with P = 0.5 . For Q = 0 , we estimate σ ml = 0.114 and σ pl = 0.116 .

Fig. 6
Fig. 6

Standard deviation of estimators P ̂ pl and P ̂ ml , F and the square root of the CRLB F [see Eq. (16)] as a function of Q [see Eq. (19)], for SNR F = 12 , P = 0.5 , and N = 9 . The estimations were made by Monte Carlo simulations on 10 4 realizations.

Fig. 7
Fig. 7

Standard deviation of the estimators P ̂ pl and P ̂ ml , F and the square root of the CRLB F as a function of N for SNR F = 12 , P = 0.5 and Q = 1 . The estimations were made by Monte Carlo simulations on 10 5 realizations.

Fig. 8
Fig. 8

Estimated means and standard deviations of estimators P ̂ pl , P ̂ plt , P ̂ med , and P ̂ mlt and the square root of CRLB as a function of SNR F for N = 9 , P = 0.5 and Q = 1 . The estimations are made with 10 5 realizations.

Fig. 9
Fig. 9

Images of DOP estimated on nine successive images of the same scene with estimators P ̂ med (a), P ̂ mlt (b), and P ̂ plt (c). The nonuniformity parameter Q is close to 1 and SNR F 23 . The scene is composed of pieces of transparent plastic on a diffusive white background obtained with a Basler A312f camera with a 70 ms exposure time. The DOP of the plastic and the background have been estimated, respectively, at 0.18 and 0.10.

Tables (2)

Tables Icon

Table 1 Estimated Mean and Standard Deviation of the Empirical Mean ( P ̂ em ) , the Median ( P ̂ med ) , the Maximum Likelihood ( P ̂ ml ) , and the Truncated Maximum Likelihood ( P ̂ mlt ) Estimators a

Tables Icon

Table 2 Values of Q a and Gain in Precision Using the Nonuniform Model for Different Types of Illumination

Equations (52)

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X i = m X + n i x ,
Y i = m Y + n i y ,
P = m X m Y m X + m Y .
CRLB = ( 1 + P 2 ) SNR 2 ,
SNR = I N 2 σ
P ̂ ml = i = 1 N X i i = 1 N Y i i = 1 N X i + i = 1 N Y i .
P ̂ ml = SNR × P + b 1 SNR + b 2 ,
P ( ρ ) = α P 1 ( ρ ) + ( 1 α ) P 2 ( ρ ) ,
P 1 ( ρ ) = 1 π ( 1 + ρ 2 ) ,
P 2 ( ρ ) = SNR 2 π ( 1 α ) 1 + ρ P ( 1 + ρ 2 ) 3 2 F [ SNR ( 1 + ρ P ) 1 + ρ 2 ] exp [ SNR 2 ( ρ P ) 2 2 ( 1 + ρ 2 ) ] ,
P ̂ em = 1 N i = 1 N X i Y i X i + Y i ,
P ̂ med = median ( X i Y i X i + Y i ) .
P ̂ mlt = P ̂ ml if P ̂ ml [ 0 , 1 ] ;
= 0 if P ̂ ml < 0 ;
= 1 if P ̂ ml > 1 .
X i = F i m X + n i x ,
Y i = F i m Y + n i y ,
I = [ 2 l I 2 2 l I P 2 l P I 2 l P 2 ] ,
l ( χ F , m X , m Y ) = 2 N ln ( 2 π σ ) 1 2 σ 2 i = 1 N ( X i F i m X ) 2 1 2 σ 2 i = 1 N ( Y i F i m Y ) 2 .
l ( χ F , I , P ) = 2 N ln ( 2 π σ ) 1 2 σ 2 i = 1 N ( X i F i I ( 1 + P ) 2 ) 2 1 2 σ 2 i = 1 N ( Y i F i I ( 1 P ) 2 ) 2 .
J = I 1 = 2 σ 2 i = 1 N F i 2 [ 1 P I P I ( 1 + P 2 ) I 2 ] .
CRLB F = 2 σ 2 i = 1 N F i 2 ( 1 + P 2 ) I 2 = ( 1 + P 2 ) SNR F 2 ,
SNR F = I i = 1 N F i 2 2 σ .
SNR F min SNR F SNR F max
CRLB U = 1 + P 2 ( SNR F min ) 2 .
CRLB U = CRLB F ( SNR F SNR F min ) 2 = CRLB F ( 1 + σ F 2 m F 2 ) ,
Q = 1 N 1 σ F 2 m F 2 .
CRLB U = CRLB F [ 1 + ( N 1 ) Q ] .
I ̂ ml , F = i = 1 N F i ( X i + Y i ) i = 1 N F i ,
P ̂ ml , F = i = 1 N ( F i X i F i Y i ) i = 1 N ( F i X i + F i Y i ) .
P ̂ ml , F = SNR F × P + b 1 SNR F + b 2 ,
l p ( χ P ) = 2 N ln ( 2 π σ ) 1 2 σ 2 1 N [ ( P 1 ) X i ( P + 1 ) Y i ] 2 1 + P 2 .
P 2 2 R P 1 = 0 ,
P ̂ p l = R + sign [ i = 1 N ( X i 2 Y i 2 ) ] 1 + R 2 .
R = SNR F 2 ( 1 P 2 2 ) + SNR F ( b s P b d ) + N b 3 SNR F 2 P + SNR F ( b s + P b d ) + N b 4 ,
P ̂ plt = P ̂ pl if P ̂ pl [ 0 , 1 ] ;
= 0 if P ̂ pl < 0 ;
= 1 if P ̂ pl > 1 .
P mlt ( ρ ) = P ml ( ρ ) if ρ ( 0 , 1 ) ;
= 0 if ρ < 0 ;
= 0 if ρ > 1 ;
= 1 2 if ρ = 0 ;
= 1 2 tan 1 ( 1 ) π if ρ = 1 .
P ̂ mlt = 0 1 ρ P mlt ( ρ ) d ρ = 0 + 0 1 ρ P ml ( ρ ) d ρ + P mlt ( 1 ) 0.36 .
l ( χ I , P ) = 2 N ln ( 2 π σ ) 1 2 σ 2 1 N [ X i I ( 1 + P ) 2 ] 2 1 2 σ 2 1 N [ Y i I ( 1 P ) 2 ] 2 ,
f ( S X , S Y I , P ) = 1 2 σ 2 [ S X I ( 1 + P ) 2 + S Y I ( 1 P ) 2 ] + I 2 ( 1 + P 2 ) 2 ,
SNR = I N 2 σ = I F 0 N 2 σ = SNR F min .
CRLB U = 1 + P 2 [ SNR F min ] 2 .
i = 1 N X i Y i = ( i = 1 N F i 2 ) m x m y + ( i = 1 N F i 2 ) σ ( m x b y + m y b x ) + N σ 2 b 3 ,
i = 1 N X i 2 Y i 2 = i = 1 N F i 2 ( m x 2 m y 2 ) + 2 ( i = 1 N F i 2 ) σ ( m x b x m y b y ) + 2 N σ 2 b 4 ,
R = ( SNR F ) 2 ( 1 P 2 2 ) + SNR F ( b s P b d ) + N b 3 ( SNR F ) 2 P + SNR F ( b s + P b d ) + N b 4 ,
sign ( i = 1 N X i 2 Y i 2 ) = sign { σ 2 [ ( SNR F ) 2 P + SNR F ( b s + P b d ) + N b 4 ] } ,

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