Abstract

Polarimetric cameras based on micropolarizer grids make it possible to design division of focal plane (DoFP) polarimeters. However, the polarimetric estimation precision reached by these devices depends on their realization quality, which is estimated by calibration. We derive the theoretical expressions of the estimation variance of such polarimetric parameters as an angle of linear polarization and degree of linear polarization as a function of the calibrated micropolarizer characteristics. These values can be compared with the variances that would be obtained with ideal micropolarizers in order to quantitatively assess the effect of manufacturing defects on polarimetric performance. These results are validated by experimental measurements on a real-world camera.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (3)

2016 (1)

2015 (1)

Z. Chen, X. Wang, and R. Liang, “Calibration method of microgrid polarimeters with image interpolation,” Opt. Soc. Am. 54, 995–1001 (2015).

2014 (2)

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

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22, 3063 (2014).
[Crossref] [PubMed]

2013 (2)

2011 (1)

N. J. Brock, B. T. Kimbrough, and J. E. Millerd, “A pixelated micropolarizer-based camera for instantaneous interferometric measurements,” Proc. SPIE 8160, 81600W (2011).
[Crossref]

2010 (4)

2009 (1)

2006 (1)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. a. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. optics 45, 5453–5469 (2006).
[Crossref]

2000 (1)

1995 (1)

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

1988 (1)

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

Ambirajan, A.

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

Balakrishnan, K.

Bénière,

Beresna, M.

Bermak, A.

Boffety, M.

J. Dupont, M. Boffety, and F. Goudail, “Precision of polarimetric orthogonal state contrast estimation in coherent images corrupted by speckle, Poisson, and additive noise,” J. Opt. Soc. Am. A. 35(6), 977–984 (2018).
[Crossref]

Boussaid, F.

Brock, N.

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]

N. J. Brock, B. T. Kimbrough, and J. E. Millerd, “A pixelated micropolarizer-based camera for instantaneous interferometric measurements,” Proc. SPIE 8160, 81600W (2011).
[Crossref]

Chen, C.-S.

Chen, W.-C.

Chen, Z.

Z. Chen, X. Wang, and R. Liang, “Calibration method of microgrid polarimeters with image interpolation,” Opt. Soc. Am. 54, 995–1001 (2015).

Chenault, D. B.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. a. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. optics 45, 5453–5469 (2006).
[Crossref]

Chigrinov, V. G.

Crandall, C.

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

Demars, C.

Dereniak, E. L.

Descour, M. R.

Dupont, J.

J. Dupont, M. Boffety, and F. Goudail, “Precision of polarimetric orthogonal state contrast estimation in coherent images corrupted by speckle, Poisson, and additive noise,” J. Opt. Soc. Am. A. 35(6), 977–984 (2018).
[Crossref]

Eden, M.

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

Fei, H.

Feng, B.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. (United Kingdom)20 (2018).

Gecevicius, M.

Goldstein, D. L.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. a. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. optics 45, 5453–5469 (2006).
[Crossref]

Goudail, F.

Gruev, V.

Holst, G. C.

G. C. Holst, CCD arrays, cameras, and displays, Second Edtion (JCD Publishing, 1998).

Hsu, W.-L.

Ibn-Elhaj, M.

Kay, S. M.

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

Kazansky, P. G.

Kemme, S. A.

Kimbrough, B. T.

N. J. Brock, B. T. Kimbrough, and J. E. Millerd, “A pixelated micropolarizer-based camera for instantaneous interferometric measurements,” Proc. SPIE 8160, 81600W (2011).
[Crossref]

Li, F.-M.

Liang, R.

Z. Chen, X. Wang, and R. Liang, “Calibration method of microgrid polarimeters with image interpolation,” Opt. Soc. Am. 54, 995–1001 (2015).

Liu, H.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. (United Kingdom)20 (2018).

Liu, L.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. (United Kingdom)20 (2018).

Look, D. C.

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

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]

N. J. Brock, B. T. Kimbrough, and J. E. Millerd, “A pixelated micropolarizer-based camera for instantaneous interferometric measurements,” Proc. SPIE 8160, 81600W (2011).
[Crossref]

Mudge, J. D.

Myhre, G.

Papoulis, A.

A. Papoulis, Probability, random variables and stochastic processes (Mc Graw-Hill, 1991).

Pau, S.

Perkins, R.

Phipps, G. S.

Powell, S. B.

Sabatke, D. S.

Shaw, J. a.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. a. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. optics 45, 5453–5469 (2006).
[Crossref]

Shi, Z.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. (United Kingdom)20 (2018).

Sweatt, W. C.

Tyler, D. W.

Tyo, J. S.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. a. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. optics 45, 5453–5469 (2006).
[Crossref]

Unser, M.

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

Wang, X.

Z. Chen, X. Wang, and R. Liang, “Calibration method of microgrid polarimeters with image interpolation,” Opt. Soc. Am. 54, 995–1001 (2015).

York, T.

Zhang, J.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. (United Kingdom)20 (2018).

Zhang, R.

Zhao, X.

Zhao, Y.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. (United Kingdom)20 (2018).

Appl. Opt. (3)

Appl. optics (1)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. a. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. optics 45, 5453–5469 (2006).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (1)

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

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

J. Dupont, M. Boffety, and F. Goudail, “Precision of polarimetric orthogonal state contrast estimation in coherent images corrupted by speckle, Poisson, and additive noise,” J. Opt. Soc. Am. A. 35(6), 977–984 (2018).
[Crossref]

Opt. Eng. (1)

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

Opt. Express (5)

Opt. Lett. (4)

Opt. Soc. Am. (1)

Z. Chen, X. Wang, and R. Liang, “Calibration method of microgrid polarimeters with image interpolation,” Opt. Soc. Am. 54, 995–1001 (2015).

Proc. SPIE (2)

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

N. J. Brock, B. T. Kimbrough, and J. E. Millerd, “A pixelated micropolarizer-based camera for instantaneous interferometric measurements,” Proc. SPIE 8160, 81600W (2011).
[Crossref]

Other (4)

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. (United Kingdom)20 (2018).

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

G. C. Holst, CCD arrays, cameras, and displays, Second Edtion (JCD Publishing, 1998).

A. Papoulis, Probability, random variables and stochastic processes (Mc Graw-Hill, 1991).

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Figures (6)

Fig. 1
Fig. 1 Schematic representation of the micropolarizers on the camera sensor. A superpixel is composed of 4 pixels with micropolarizers oriented at 0°, 45°, 90° and 135°.
Fig. 2
Fig. 2 Maps of the extinction ratio of the micro-polarizers on the sensor. a) 0°, b) 45°, c) 90°, d) 135°.
Fig. 3
Fig. 3 Maps of the orientations of the micro-polarizers on the sensor. a) 0°, b) 45°, c) 90°, d) 135°.
Fig. 4
Fig. 4 a) Average raw intensity captured captured by the camera during the measurement. b) EWV ¯ / EWV ideal. c) VAR [ α ] ¯ / VAR [ α ] ideal. d) VAR [ P ] ¯ / VAR [ P ] ideal.
Fig. 5
Fig. 5 a) EWV divided by the ideal EWV. b) Variance of AOP divided by ideal variance of AOP. c) Variance of the DOLP divided by the ideal variance of the DOLP. Experimental (solid lines) and theoretical (dotted lines) values are represented in blue and their mean values in red.
Fig. 6
Fig. 6 a) VAR [ α ^ ] ¯ as a function of S0 for different values of σa. b) VAR[α̂]max/VAR[α̂]ideal as a function of micropixel angle fluctuation amplitude Δϕ.

Tables (1)

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Table 1 Average orientations and extinction ratio of the micro-polarizers.

Equations (44)

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d = g η t I 0 v T S + b
t = t + t
v T = 1 2 [ 1 , q cos ( 2 ϕ ) , q sin ( 2 ϕ ) ]
q = t t t + t
ζ = t t = 1 + q 1 q
d n m = g I 0 η n m t n m v n m T S + b n m
A = [ A 1 A 2 A K ] T
d n m = ( g I 0 η n m t n m ) A v n m + b n m 1
A ( d n m b n m 1 ) = g I 0 η n m t n m v n m
Ad 0 ¯ = 1 4 N n = 1 N m = 1 4 [ A ( d n m b n m 1 ) ] 0 = 1 2 g I 0 η t ¯
η t ¯ = 1 4 N n = 1 N m = 1 4 η n m t n m
w n m = A ( d n m b n m 1 ) 2 A d 0 ¯ = η n m t n m η t ¯ v n m
S = S 0 [ 1 , P cos ( 2 α ) , P sin ( 2 α ) ]
d n m = g η t ¯ w n m T S + b n m
W n = g [ w n 1 , w n 2 , w n 3 , w n 4 ] T
d n = η t ¯ W n S + b n
η t ¯ S = W n ( d n b n )
d ¯ n = d n b n
d ¯ n = W n S and S = W n d ¯ n
P = S 1 2 + S 2 2 S 0
α = 1 2 arctan [ S 2 S 1 ]
S ^ = W d ¯
VAR [ d ¯ ] = g W S + g 2 σ a 2
EWV = trace [ Γ S ^ ]
Γ S ^ = Γ add + Γ poi
Γ i j add = σ a 2 δ i j and Γ i j poi = k = 0 2 S k γ i j k
δ i j = g 2 [ ( W T W ) 1 ] i j and γ i j k = g l = 1 4 W i l W j l W l k , ( k , i , j ) [ 0 , 2 ] 3
Γ add = σ a 2 [ 1 0 0 0 2 0 0 0 2 ] and Γ poi = 1 2 [ S 0 S 1 S 2 S 1 2 S 0 0 S 2 0 2 S 0 ]
EWV ideal = 5 ( σ a 2 + S 0 2 )
EWV = σ a 2 i = 0 2 δ i i + k = 0 2 S k β k with β k = i = 0 2 γ i i k
EWV = σ a 2 i = 0 2 δ i i + S 0 β 0 { 1 + C cos [ 2 ( α θ ) ] }
θ = 1 2 arctan [ β 2 β 1 ] and C = P β 1 2 + β 2 2 β 0
EWV ¯ = 1 π 0 π EWV ( α ) d α = σ a 2 i = 0 2 δ i i + S 0 β 0
< y > f ( < X > ) and VAR [ y ] [ f ( < X > ) ] T Γ X f ( < X > )
α ^ = 1 2 P 2 S 0 2 [ 0 , S 2 , S 1 ] T
VAR [ α ^ ] ideal = 1 2 P 2 ( σ a 2 S 0 2 + 1 2 S 0 )
VAR [ α ^ ] = σ a 2 4 P 2 S 0 2 { δ 11 s 2 + δ 22 c 2 2 δ 12 c s } + 1 4 P 2 S 0 { γ 11 0 s 2 + γ 22 0 c 2 2 γ 22 2 c s + Pc 2 [ ( γ 22 2 2 γ 12 1 ) s + γ 22 1 c ] + Ps 2 [ ( γ 11 1 2 γ 12 2 ) c + γ 11 2 s ] }
c = cos ( 2 α ) and s = sin ( 2 α )
VAR [ α ^ ] ¯ = 1 π 0 π VAR [ α ^ ] ( α ) d α = 1 8 P 2 [ σ a 2 S 0 2 ( δ 11 + δ 22 ) + 1 S 0 ( γ 11 0 + γ 22 0 ) ]
P ^ = 1 P S 0 2 [ P 2 S 0 , S 1 , S 2 ] T
VAR [ P ^ ] ideal = σ a 2 S 0 2 [ 2 + P 2 ] + 1 2 S 0 [ 2 P 2 ]
VAR [ P ^ ] = σ a 2 S 0 2 { P 2 δ 00 2 P ( δ 01 c + δ 02 s ) + 2 δ 12 c s + δ 11 c 2 + δ 22 s 2 } + 1 S 0 { P 3 ( γ 00 1 c + γ 00 2 s ) + P 2 [ γ 00 0 2 γ 01 1 c 2 2 γ 02 2 s 2 2 ( γ 01 2 + γ 02 1 ) c s ] + P [ ( γ 11 2 + 2 γ 12 1 ) c 2 s + ( γ 22 1 + 2 γ 12 2 ) c s 2 + γ 11 1 c 3 + γ 22 2 s 3 2 γ 01 0 c 2 γ 02 0 s ] + [ γ 11 0 c 2 + γ 22 0 s 2 + 2 γ 12 0 c s ] }
VAR [ P ^ ] ¯ = 1 π 0 π VAR [ P ^ ] ( α ) d α = σ a 2 S 0 2 [ P 2 δ 00 + δ 11 + δ 22 2 ] + 1 2 S 0 [ ( γ 11 0 + γ 22 0 ) + 2 P 2 ( γ 00 0 γ 01 1 γ 02 2 ) ]
VAR [ α ^ ] max = max k [ max α ( VAR [ α ^ ] k ) ] .

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