Abstract

We consider imaging systems that measure the three first elements of the Stokes vector and deduce from them the degree of linear polarization and the angle of polarization. They require the acquisition of at least three intensity measurements, but performing more measurements is often thought to improve the estimation precision. We show that if the total acquisition time is fixed, the optimal number of measurements depends on the type of noise that affects the image: the estimation variance increases with the number of measurements N when the noise is additive; it is independent of N in the presence of Poisson shot noise and decreases with N when the angles of the analyzers fluctuate. In general, the optimal number of measurements results from a compromise on the robustness of these different types of perturbations.

© 2010 Optical Society of America

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References

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    [CrossRef]
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2009 (1)

2008 (4)

2007 (1)

2006 (4)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453-5469 (2006).
[CrossRef] [PubMed]

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to 3D reconstruction of specular metallic surfaces by polarization imaging,” Appl. Opt. 45, 4062-4068 (2006).
[CrossRef] [PubMed]

J. Hough, “Polarimetry: a powerful diagnostic tool in astronomy,” Astron. Geophys. 47, 3.31-3.35 (2006).
[CrossRef]

2004 (1)

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. Pattern Anal. Machine Intell. 26, 73-82 (2004).
[CrossRef]

2002 (4)

M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. 41, 2488-2493 (2002).
[CrossRef] [PubMed]

J. S. Tyo, “Design of optimal polarimeters: maximization of the signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41, 619-630 (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]

A. G. Andreau and Z. K. Kalyjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566-576 (2002).
[CrossRef]

2000 (1)

1998 (1)

1995 (1)

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1658 (1995).
[CrossRef]

1991 (1)

L. B. Wolff and T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Machine Intell. 13, 635-657 (1991).
[CrossRef]

1990 (1)

1988 (1)

M. Unser and M. Eden, “Maximum likelihood estimation of linear signal parameters for Poisson processes,” IEEE Trans. Acoust. Speech Signal Processing 36, 942-945 (1988).
[CrossRef]

Ahmad, J. Elsayed

Ainouz, S.

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1658 (1995).
[CrossRef]

Andreau, A. G.

A. G. Andreau and Z. K. Kalyjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566-576 (2002).
[CrossRef]

Belsher, J. F.

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]

Black, B. M.

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Boger, J. K.

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Boult, T. E.

L. B. Wolff and T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Machine Intell. 13, 635-657 (1991).
[CrossRef]

Bowers, D. L.

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Chenault, D. B.

Chipman, R.

Chipman, R. A.

Collados, M.

Dereniak, E. L.

Descour, M. R.

Eden, M.

M. Unser and M. Eden, “Maximum likelihood estimation of linear signal parameters for Poisson processes,” IEEE Trans. Acoust. Speech Signal Processing 36, 942-945 (1988).
[CrossRef]

Fetrow, M. P.

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Gamiz, V. L.

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]

Goldstein, D. H.

Goldstein, D. L.

Gorria, P.

Goudail, F.

Holst, G. C.

G. C. Holst, CCD Arrays, Cameras, and Displays, 2nd ed. (JCD, 1998).

Hoover, B. G.

Hough, J.

J. Hough, “Polarimetry: a powerful diagnostic tool in astronomy,” Astron. Geophys. 47, 3.31-3.35 (2006).
[CrossRef]

Huard,, S.

S. Huard, “Polarized optical wave,” in Polarization of Light (Wiley, 1997), pp. 1-35.

Hubbs, J. E.

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Ikeuchi, K.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. Pattern Anal. Machine Intell. 26, 73-82 (2004).
[CrossRef]

Kagesawa, M.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. Pattern Anal. Machine Intell. 26, 73-82 (2004).
[CrossRef]

Kalyjian, Z. K.

A. G. Andreau and Z. K. Kalyjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566-576 (2002).
[CrossRef]

Kay, S. M.

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

Kemme, S. A.

Look, D. C.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1658 (1995).
[CrossRef]

Meriaudeau, F.

Miyazaki, D.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. Pattern Anal. Machine Intell. 26, 73-82 (2004).
[CrossRef]

Morel, O.

Ortega, S. E.

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Papoulis, A.

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

Phipps, G. S.

Ramos, A.

Ratliff, B. M.

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Sabatke, D. S.

Shaw, J. A.

Smith, M. H.

Stoll, M. P.

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Stolz, C.

Sweatt, W. C.

Takakura, Y.

Twietmeyer, K. M.

Tyo, J. S.

Unser, M.

M. Unser and M. Eden, “Maximum likelihood estimation of linear signal parameters for Poisson processes,” IEEE Trans. Acoust. Speech Signal Processing 36, 942-945 (1988).
[CrossRef]

Vaughn, I. J.

Wellems, L. D.

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Wolff, L. B.

L. B. Wolff and T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Machine Intell. 13, 635-657 (1991).
[CrossRef]

Zallat, J.

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Appl. Opt. (6)

Astron. Geophys. (1)

J. Hough, “Polarimetry: a powerful diagnostic tool in astronomy,” Astron. Geophys. 47, 3.31-3.35 (2006).
[CrossRef]

IEEE Sens. J. (1)

A. G. Andreau and Z. K. Kalyjian, “Polarization imaging: principles and integrated polarimeters,” IEEE Sens. J. 2, 566-576 (2002).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Processing (1)

M. Unser and M. Eden, “Maximum likelihood estimation of linear signal parameters for Poisson processes,” IEEE Trans. Acoust. Speech Signal Processing 36, 942-945 (1988).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (2)

L. B. Wolff and T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Machine Intell. 13, 635-657 (1991).
[CrossRef]

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. Pattern Anal. Machine Intell. 26, 73-82 (2004).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

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

Opt. Eng. (3)

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]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1658 (1995).
[CrossRef]

D. L. Bowers, J. K. Boger, L. D. Wellems, S. E. Ortega, M. P. Fetrow, J. E. Hubbs, B. M. Black, B. M. Ratliff, and J. S. Tyo, “Unpolarized calibration and nonuniformity correction for LWIR microgrid imaging polarimeters,” Opt. Eng. 47, 046403 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Other (4)

G. C. Holst, CCD Arrays, Cameras, and Displays, 2nd ed. (JCD, 1998).

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

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

S. Huard, “Polarized optical wave,” in Polarization of Light (Wiley, 1997), pp. 1-35.

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Figures (2)

Fig. 1
Fig. 1

Theoretical and estimated variances of (a) the DOLP and (b) the AOP α as a function of the true value of the AOP in the presence of Poisson noise. S 0 = 2000 , P = 0.7 , α [ 45 ° , + 45 ° ] , and three values of N = 3 , 4, and 8 are considered.

Fig. 2
Fig. 2

Theoretical and estimated variances of (a) the DOLP and (b) the AOP α as a function of the true value of the AOP in the presence of random fluctuations of the analyzer angles. S 0 = 2000 , P = 0.7 , α [ 45 ° , + 45 ° ] , σ θ = 3 ° , and three values of N = 3 , 4, and 8 are considered.

Equations (63)

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

S = S 0 ( 1 , P cos 2 α , P sin 2 α ) T ,
I i = 1 2 N [ S 0 + S 1 cos 2 θ i + S 2 sin 2 θ i ] ,
I i = S 0 2 N [ 1 + P cos ( 2 θ i 2 α ) ] .
I = 1 N W S
W = 1 2 [ 1 cos 2 θ i sin 2 θ i ] .
S ^ = N W I
W = ( W T W ) 1 W T .
P ^ = S ^ 1 2 + S ^ 2 2 S ^ 0 ,
α ^ = 1 2 arg [ S ^ 1 + i S ^ 2 ] ,
if S 1 > 0 : α ^ = 1 2 arctan [ S ^ 2 S ^ 1 ] , if S 1 < 0 : α ^ = π 2 × sgn ( S 2 ) + 1 2 arctan [ S ^ 2 S ^ 1 ] ,
I = 1 N W S + b ,
Γ S ^ = N 2 W Γ I ( W ) T ,
Γ S ^ = N 2 σ 2 ( W T W ) 1 ,
θ i = θ 0 + ( i 1 ) × 180 ° N ,
( W T W ) 1 = 4 N [ 1 0 0 0 2 0 0 0 2 ] ,
i = 1 N cos 2 θ i = i = 1 N sin 2 θ i = i = 1 N cos 2 θ i sin 2 θ i = 0 , i = 1 N cos 2 2 θ i = i = 1 N sin 2 2 θ i = N 2
W = 2 N [ 1 1 2 cos 2 θ 1 2 cos 2 θ N 2 sin 2 θ 1 2 sin 2 θ N ] .
Γ S ^ = 4 N σ 2 [ 1 0 0 0 2 0 0 0 2 ] .
y f ( X ) and VAR [ y ] [ f ( X ) ] T Γ X f ( X ) ,
P ^ = 1 P S 0 2 [ P 2 S 0 , S 1 , S 2 ] T .
VAR [ P ^ ] = ( σ S 0 ) 2 4 N ( 2 + P 2 ) .
α ^ = 1 2 P 2 S 0 2 [ 0 , S 2 , S 1 ] T .
VAR [ α ] = ( σ S 0 ) 2 2 N P 2 .
I = VAR [ I ] = 1 N W S ,
Γ i j S ^ = N k = 0 2 S k n = 1 N W i n W j n W n k .
Γ S ^ = 2 [ S 0 S 1 S 2 S 1 2 S 0 + S 1 S 2 S 2 S 2 2 S 0 S 1 ] ,
Γ S ^ = 2 [ S 0 S 1 S 2 S 1 2 S 0 0 S 2 0 2 S 0 ] .
N 3 : i = 1 N cos 2 2 θ i sin 2 θ i = i = 1 N sin 3 2 θ i = 0 ,
N = 3 : i = 1 N cos 3 2 θ i = 3 4 , i = 1 N sin 2 2 θ i cos 2 θ i = 3 4 ,
N 4 : i = 1 N cos 3 2 θ i = 0 , i = 1 N sin 2 2 θ i cos 2 θ i = 0.
{ γ 0 = 2 S 0 γ 1 = 4 S 0 + 2 S 1 = 2 S 0 ( 2 + P cos 2 α ) γ 2 = 4 S 0 2 S 1 = 2 S 0 ( 2 P cos 2 α ) .
γ 0 = 2 S 0 , γ 1 = 4 S 0 , γ 2 = 4 S 0 ,
if N = 3 , VAR [ P ^ ] = 2 S 0 [ 2 P 2 + P cos ( 6 α ) ] ,
if N 4 , VAR [ P ^ ] = 2 S 0 [ 2 P 2 ] .
if N = 3 , VAR [ α ^ ] = 1 P 2 S 0 [ 1 P 2 cos ( 6 α ) ] ,
if N 4 , VAR [ α ^ ] = 1 P 2 S 0 .
I = 1 N W S and S ^ = N W 0 I .
S ^ = S + d M S ,
d M = W 0 ( W W 0 ) .
d M = i = 1 N D i d θ i with     D i = W 0 W θ i .
Γ S ^ = d M d M T = i = 1 N j = 1 N D i S S T D j T d θ i d θ j .
Γ S ^ = σ θ 2 i = 1 N D i S S T D i T .
if N = 3 , Γ S ^ = 2 σ θ 2 3 [ P 2 S 0 2 S 2 2 S 1 2 2 S 1 S 2 S 2 2 S 1 2 P 2 S 0 2 + 2 S 2 2 2 S 1 S 2 2 S 1 S 2 2 S 1 S 2 P 2 S 0 2 + 2 S 1 2 ] ,
if N = 4 , Γ S ^ = σ θ 2 2 [ P 2 S 0 2 0 0 0 4 S 2 2 0 0 0 4 S 1 2 ] ,
if N > 4 , Γ S ^ = 2 σ θ 2 N [ P 2 S 0 2 0 0 0 P 2 S 0 2 + 2 S 2 2 2 S 1 S 2 0 2 S 1 S 2 P 2 S 0 2 + 2 S 1 2 ] .
N 3 : i = 1 N cos 3 2 θ i sin 2 θ i = i = 1 N cos 2 θ i sin 3 2 θ i = 0 ,
N = 4 : i = 1 N cos 4 2 θ i = 2 , i = 1 N sin 4 2 θ i = 2 , i = 1 N cos 2 2 θ i sin 2 2 θ i = 0 ,
N 4 : i = 1 N cos 4 2 θ i = 3 N 8 , i = 1 N sin 4 2 θ i = 3 N 8 , i = 1 N cos 2 2 θ i sin 2 2 θ i = N 8 .
trace [ Γ S ^ ] = 10 N σ θ 2 P 2 S 0 2 .
if N = 3 , VAR [ P ^ ] = 2 σ θ 2 3 P 2 [ 1 + P 2 + 2 P cos ( 6 α ) ] ,
if N = 4 , VAR [ P ^ ] = σ θ 2 2 P 2 [ 1 + P 2 cos ( 8 α ) ] ,
if N > 4 , VAR [ P ^ ] = 2 σ θ 2 N P 2 [ 1 + P 2 ] .
if N = 4 , VAR [ α ^ ] = σ θ 2 8 [ 3 + cos ( 8 α ) ] ,
if N 4 , VAR [ α ^ ] = 3 2 σ θ 2 N .
VAR [ P ^ ] ( N ) = 4 ( 2 + P 2 ) SNR 2 N + 2 S 0 ( 2 P 2 ) + 2 σ θ 2 P 2 ( 1 + P 2 ) N ,
N min P = σ θ SNR P 2 ( 1 + P 2 ) 2 ( 2 + P 2 ) .
N min α = σ θ SNR 3 2 P .
I = ( 1 N ε ) W S ,
S ^ = N 1 N ε W I .
Γ S ^ = ( N 1 N ε ) 2 W Γ I ( W ) T = N 2 ( 1 N ε ) 2 σ 2 ( W T W ) 1 .
Γ i j I = ( 1 N ε ) k = 0 2 W i k S k if     i = j , 0 , otherwise,
Γ i j S ^ = N 1 N ε k = 0 2 S k n = 1 N W i n W j n W n k .
VAR [ P ^ ] ( N ) = 4 ( 2 + P 2 ) N SNR 2 ( 1 N ε ) 2 + 2 ( 2 P 2 ) S 0 ( 1 N ε ) + 2 σ θ 2 P 2 ( 1 + P 2 ) N .

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