Abstract

Channeled spectropolarimetry measures the spectrally resolved Stokes parameters. A key aspect of this technique is to accurately reconstruct the Stokes parameters from a modulated measurement of the channeled spectropolarimeter. The state-of-the-art reconstruction algorithm uses the Fourier transform to extract the Stokes parameters from channels in the Fourier domain. While this approach is straightforward, it can be sensitive to noise and channel cross-talk, and it imposes bandwidth limitations that cut off high frequency details. To overcome these drawbacks, we present a reconstruction method called compressed channeled spectropolarimetry. In our proposed framework, reconstruction in channeled spectropolarimetry is an underdetermined problem, where we take N measurements and solve for 3N unknown Stokes parameters. We formulate an optimization problem by creating a mathematical model of the channeled spectropolarimeter with inspiration from compressed sensing. We show that our approach offers greater noise robustness and reconstruction accuracy compared with the Fourier transform technique in simulations and experimental measurements. By demonstrating more accurate reconstructions, we push performance to the native resolution of the sensor, allowing more information to be recovered from a single measurement of a channeled spectropolarimeter.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
  7. M. Lowenstern and M. W. Kudenov, “Field deployable pushbroom hyperspectral imagining polarimeter,” Proc. SPIE 9853, 98530 (2016).
  8. E. R. Woodard and M. W. Kudenov, “Spectrally resolved longitudinal spatial coherence inteferometry,” Proc. SPIE 10198, 101980 (2016).
  9. C. F. LaCasse, O. G. Rodríguez-Herrera, R. A. Chipman, and J. S. Tyo, “Spectral density response functions for modulated polarimeters,” Appl. Opt. 54, 9490–9499 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  11. D. E. Aspnes, “Analysis of Semiconductor Materials and Structures by Spectroellipsometry,” Proc. SPIE 0946, 84 (1988).
    [Crossref]
  12. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475–1477 (1999).
    [Crossref]
  13. R. A. Chipman, “Handbook of Optics,” Polarimetry, 2nd ed. M. Bass, ed. McGraw Hill, New York2 (1995).
  14. M. W. Kudenov, N. A. Hagen, E. L. Dereniak, and G. R. Gerhart, “Fourier transform channeled spectropolarimetry in the MWIR,” Opt. Express 15, 12792–12805 (2007).
    [Crossref] [PubMed]
  15. D. J. Lee and A. M. Weiner, “Optical phase imaging using a synthetic aperture phase retrieval technique,” Opt. Express 22(8), 9380–9394 (2014).
    [Crossref] [PubMed]
  16. D. J. Lee, K. Han, H. J. Lee, and A. M. Weiner, “Synthetic aperture microscopy based on referenceless phase retrieval with an electrically tunable lens,” Appl. Opt. 54(17), 5346–5352 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  20. D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single Shot Digital Holography Using Iterative Reconstruction with Alternating Updates of Amplitude and Phase,” http://www.arxiv.org/abs/1609.02978 (2016).
  21. D. J. Lee, C. F. LaCasse, and J. M. Craven, “Compressed channeled linear imaging polarimetry,” Proc. SPIE 10407, 104070D (2017).
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    [Crossref] [PubMed]
  23. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
    [Crossref]
  24. S. Wright, Primal-Dual Interior-Point Methods (Society for Industrial and Applied Mathematics, 1997).
    [Crossref]
  25. A. Domahidi, E. Chu, and S. Boyd, “ECOS: An SOCP solver for embedded systems,” in “European Control Conference (ECC),” (2013), pp. 3071–3076.

2017 (1)

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Compressed channeled linear imaging polarimetry,” Proc. SPIE 10407, 104070D (2017).

2016 (4)

J. Boyer, J. C. Keresztes, W. Saeys, and J. Koshel, “An automated imaging BRDF polarimeter for fruit quality inspection,” Proc. SPIE 9948, 99480 (2016).

M. Lowenstern and M. W. Kudenov, “Field deployable pushbroom hyperspectral imagining polarimeter,” Proc. SPIE 9853, 98530 (2016).

E. R. Woodard and M. W. Kudenov, “Spectrally resolved longitudinal spatial coherence inteferometry,” Proc. SPIE 10198, 101980 (2016).

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Channeled spectropolarimetry using iterative reconstruction,” Proc. SPIE 9853, 98530V (2016).
[Crossref]

2015 (2)

2014 (2)

2012 (2)

2011 (2)

2007 (3)

2006 (2)

1999 (1)

1988 (1)

D. E. Aspnes, “Analysis of Semiconductor Materials and Structures by Spectroellipsometry,” Proc. SPIE 0946, 84 (1988).
[Crossref]

Alenin, A. S.

A. S. Alenin and J. S. Tyo, “Generalized channeled polarimetry,” JOSA A 31, 1013–1022 (2014).
[Crossref] [PubMed]

Aspnes, D. E.

D. E. Aspnes, “Analysis of Semiconductor Materials and Structures by Spectroellipsometry,” Proc. SPIE 0946, 84 (1988).
[Crossref]

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

A. Domahidi, E. Chu, and S. Boyd, “ECOS: An SOCP solver for embedded systems,” in “European Control Conference (ECC),” (2013), pp. 3071–3076.

Boyer, J.

J. Boyer, J. C. Keresztes, W. Saeys, and J. Koshel, “An automated imaging BRDF polarimeter for fruit quality inspection,” Proc. SPIE 9948, 99480 (2016).

Cairns, B.

Chenault, D. B.

Chipman, R. A.

Chu, E.

A. Domahidi, E. Chu, and S. Boyd, “ECOS: An SOCP solver for embedded systems,” in “European Control Conference (ECC),” (2013), pp. 3071–3076.

Craven, J. M.

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Compressed channeled linear imaging polarimetry,” Proc. SPIE 10407, 104070D (2017).

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Channeled spectropolarimetry using iterative reconstruction,” Proc. SPIE 9853, 98530V (2016).
[Crossref]

Davis, A.

Dereniak, E.

Dereniak, E. L.

Diner, D. J.

Ding, T.

Domahidi, A.

A. Domahidi, E. Chu, and S. Boyd, “ECOS: An SOCP solver for embedded systems,” in “European Control Conference (ECC),” (2013), pp. 3071–3076.

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52, 1289–1306 (2006).
[Crossref]

Escuti, M. J.

Gerhart, G. R.

Goldstein, D. L.

Gutt, G.

Hagen, N.

Hagen, N. A.

Han, K.

Hancock, B.

Kato, T.

Keresztes, J. C.

J. Boyer, J. C. Keresztes, W. Saeys, and J. Koshel, “An automated imaging BRDF polarimeter for fruit quality inspection,” Proc. SPIE 9948, 99480 (2016).

Koshel, J.

J. Boyer, J. C. Keresztes, W. Saeys, and J. Koshel, “An automated imaging BRDF polarimeter for fruit quality inspection,” Proc. SPIE 9948, 99480 (2016).

Kudenov, M. W.

LaCasse, C. F.

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Compressed channeled linear imaging polarimetry,” Proc. SPIE 10407, 104070D (2017).

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Channeled spectropolarimetry using iterative reconstruction,” Proc. SPIE 9853, 98530V (2016).
[Crossref]

C. F. LaCasse, O. G. Rodríguez-Herrera, R. A. Chipman, and J. S. Tyo, “Spectral density response functions for modulated polarimeters,” Appl. Opt. 54, 9490–9499 (2015).
[Crossref] [PubMed]

C. F. LaCasse, R. A. Chipman, and J. S. Tyo, “Band limited data reconstruction in modulated polarimeters,” Opt. Express 19(16), 14976–14989 (2011).
[Crossref] [PubMed]

Lee, D. J.

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Compressed channeled linear imaging polarimetry,” Proc. SPIE 10407, 104070D (2017).

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Channeled spectropolarimetry using iterative reconstruction,” Proc. SPIE 9853, 98530V (2016).
[Crossref]

D. J. Lee, K. Han, H. J. Lee, and A. M. Weiner, “Synthetic aperture microscopy based on referenceless phase retrieval with an electrically tunable lens,” Appl. Opt. 54(17), 5346–5352 (2015).
[Crossref] [PubMed]

D. J. Lee and A. M. Weiner, “Optical phase imaging using a synthetic aperture phase retrieval technique,” Opt. Express 22(8), 9380–9394 (2014).
[Crossref] [PubMed]

Lee, H. J.

Lowenstern, M.

M. Lowenstern and M. W. Kudenov, “Field deployable pushbroom hyperspectral imagining polarimeter,” Proc. SPIE 9853, 98530 (2016).

Oka, K.

Peng, B.

Rodríguez-Herrera, O. G.

Saeys, W.

J. Boyer, J. C. Keresztes, W. Saeys, and J. Koshel, “An automated imaging BRDF polarimeter for fruit quality inspection,” Proc. SPIE 9948, 99480 (2016).

Shaw, J. A.

Tyo, J. S.

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

Wang, P.

Weiner, A. M.

Woodard, E. R.

E. R. Woodard and M. W. Kudenov, “Spectrally resolved longitudinal spatial coherence inteferometry,” Proc. SPIE 10198, 101980 (2016).

Wright, S.

S. Wright, Primal-Dual Interior-Point Methods (Society for Industrial and Applied Mathematics, 1997).
[Crossref]

Appl. Opt. (6)

IEEE Transactions on Information Theory (1)

D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52, 1289–1306 (2006).
[Crossref]

JOSA A (1)

A. S. Alenin and J. S. Tyo, “Generalized channeled polarimetry,” JOSA A 31, 1013–1022 (2014).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (2)

Proc. SPIE (6)

J. Boyer, J. C. Keresztes, W. Saeys, and J. Koshel, “An automated imaging BRDF polarimeter for fruit quality inspection,” Proc. SPIE 9948, 99480 (2016).

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Channeled spectropolarimetry using iterative reconstruction,” Proc. SPIE 9853, 98530V (2016).
[Crossref]

M. Lowenstern and M. W. Kudenov, “Field deployable pushbroom hyperspectral imagining polarimeter,” Proc. SPIE 9853, 98530 (2016).

E. R. Woodard and M. W. Kudenov, “Spectrally resolved longitudinal spatial coherence inteferometry,” Proc. SPIE 10198, 101980 (2016).

D. E. Aspnes, “Analysis of Semiconductor Materials and Structures by Spectroellipsometry,” Proc. SPIE 0946, 84 (1988).
[Crossref]

D. J. Lee, C. F. LaCasse, and J. M. Craven, “Compressed channeled linear imaging polarimetry,” Proc. SPIE 10407, 104070D (2017).

Other (5)

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

S. Wright, Primal-Dual Interior-Point Methods (Society for Industrial and Applied Mathematics, 1997).
[Crossref]

A. Domahidi, E. Chu, and S. Boyd, “ECOS: An SOCP solver for embedded systems,” in “European Control Conference (ECC),” (2013), pp. 3071–3076.

R. A. Chipman, “Handbook of Optics,” Polarimetry, 2nd ed. M. Bass, ed. McGraw Hill, New York2 (1995).

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single Shot Digital Holography Using Iterative Reconstruction with Alternating Updates of Amplitude and Phase,” http://www.arxiv.org/abs/1609.02978 (2016).

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Figures (8)

Fig. 1
Fig. 1 Reconstructions at two SNRs from a simulated measurement with added white Gaussian noise. The input Stokes parameters are defined in Eqs. (46)(48). The fit metrics for Fourier reconstruction and CCSP and the percent change Δ% from Fourier reconstruction to CCSP are defined in Eqs. (58)(60). GT: Ground truth; FR: Fourier reconstruction; CCSP: Compressed channeled spectropolarimetry.
Fig. 2
Fig. 2 Fit values at different SNRs, showing how well each reconstruction matches ground truth. We add noise to a simulated measurement, where the input Stokes parameters are defined in Eqs. (46)(48). As SNR decreases, CCSP outperforms Fourier reconstruction. The fit metrics for Fourier reconstruction and CCSP are defined in Eqs. (58)(60). The lines show polynomial fits to the data points. FR: Fourier reconstruction; CCSP: Compressed channeled spectropolarimetry.
Fig. 3
Fig. 3 Optimal regularizer weights β at different SNRs that resulted in the most accurate reconstructions. We add varying noise to a simulated measurement, where the input Stokes parameters are defined in Eqs. (46)(48). β is the regularizer weight from Problem (45). At lower SNR, increasing the regularizer weight improves robustness to noise and results in more accurate reconstructions. The line is a polynomial fit to the data points.
Fig. 4
Fig. 4 Reconstructions at two simulated values of fS0/fc. The frequency of S0(σ) varies relative to the carrier frequency of the measurement. The input Stokes parameters are defined in Eqs. (46)(48). The fit metrics for Fourier reconstruction and CCSP and the percent change Δ% from Fourier reconstruction to CCSP are defined in Eqs. (58)(60). GT: Ground truth; FR: Fourier reconstruction; CCSP: Compressed channeled spectropolarimetry.
Fig. 5
Fig. 5 Fit at different values of fS0/fc, showing how well each reconstruction matches ground truth. The frequency of S0(σ) varies relative to the carrier frequency of the measurement. The input Stokes parameters are defined in Eqs. (46)(48). As fS0 increases, the Fourier reconstruction degrades because it imposes a windowing constraint that cuts off high frequency details, whereas CCSP does not impose this constraint and can recover higher frequencies. The fit metrics for Fourier reconstruction and CCSP and the percent change Δ% from Fourier reconstruction to CCSP are defined in Eqs. (58)(60). FR: Fourier reconstruction; CCSP: Compressed channeled spectropolarimetry.
Fig. 6
Fig. 6 Experimental setup of the channeled spectropolarimeter. OPD: Optical path difference. QWP: Quarter wave plate; R: Retarder; LP: Linear polarizer.
Fig. 7
Fig. 7 Reconstructions of four measured samples with the sample polarizer oriented at 67.5°. We compare ground truth from a rotating polarizer spectropolarimeter with reconstructions from a channeled spectropolarimeter. The fit metrics for Fourier reconstruction and CCSP and the percent change Δ% from Fourier reconstruction to CCSP are defined in Eqs. (58)(60). The samples are summarized in Table 1. GT: Ground truth; FR: Fourier reconstruction; CCSP: Compressed channeled spectropolarimetry.
Fig. 8
Fig. 8 Fit values for four measured samples over varying sample polarizer angles, showing how well the reconstructions match ground truth. The fit metrics for Fourier reconstruction and CCSP are defined in Eqs. (58)(60). The average fits over all angles are shown on the righthand side of the plots for each reconstruction. Here Δ% denotes the percent change of the averaged fits from Fourier reconstruction to CCSP. Note that Fig. 7 shows reconstructions with the sample polarizer angles oriented at 67.5°, and this figure plots the fit metrics over all measured polarizer angles. The samples are summarized in Table 1. FR: Fourier reconstruction; CCSP: Compressed channeled spectropolarimetry.

Tables (3)

Tables Icon

Algorithm 1 Proposed algorithm for compressed channeled spectropolarimetry

Tables Icon

Table 1 Samples used for experimental measurements. Each sample has a linear polarizer, and we rotate this polarizer from 0° to 180° in 22.5° increments. We measure a total of 36 test cases, corresponding to the 4 samples and 9 polarizer angles per sample. LP: Linear polarizer.

Tables Icon

Table 2 Fit values for each sample, averaged over all sample polarizer angles. The fit values quantify how well each reconstruction matches ground truth. This table summarizes the fit values that appear in Fig. 8. All quantites are percentages. The columns indicate which sample is measured. Each row shows statistics on reconstruction fit. FR: Fourier reconstruction; CCSP: Compressed channeled spectropolarimetry.

Equations (71)

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[ S 0 S 1 S 2 S 3 ] = [ I H + I V I H I V I 45 I 135 I L I R ]
s 0 = [ S 0 ( σ 1 ) S 0 ( σ N ) ] T N ,
s 1 = [ S 1 ( σ 1 ) S 1 ( σ N ) ] T N ,
s 2 = [ S 2 ( σ 1 ) S 2 ( σ N ) ] T N ,
S = [ s 0 T s 1 T s 2 T ] 3 × N .
Θ = [ 1 cos ( 2 θ 1 ) sin ( 2 θ 1 )     1 cos ( 2 θ A ) sin ( 2 θ A ) ] A × 3
Y = [ y 1     y A ] = Θ S A × N
S = Θ 1 Y
S = ( Θ T Θ ) 1 Θ T Y .
M QWP = [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] ,
M R ( σ ) = [ 1 0 0 0 0 cos [ ϕ ( σ ) ] 0 sin [ ϕ ( σ ) ] 0 0 1 0 0 sin [ ϕ ( σ ) ] 0 cos [ ϕ ( σ ) ] ] ,
M LP = 1 2 [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] .
s in ( σ ) = [ S 0 ( σ ) S 1 ( σ ) S 2 ( σ ) S 3 ( σ ) ] T ,
s out ( σ ) = M L P M R ( σ ) M QWP s in ( σ ) .
y ( σ ) = 1 2 { S 0 ( σ ) + S 1 ( σ ) cos [ ϕ ( σ ) ] + S 2 ( σ ) sin [ ϕ ( σ ) ] } = 1 2 { S 0 ( σ ) + S 1 ( σ ) cos [ 2 π B ( σ ) t σ ] + S 2 ( σ ) sin [ 2 π B ( σ ) t σ ] } = 1 2 { S 0 ( σ ) + S 1 ( σ ) cos ( 2 π f c σ ) + S 2 ( σ ) sin ( 2 π f c σ ) }
ϕ ( σ ) = 2 π B ( σ ) t σ
f c = B ( σ ) t .
y ^ ( d ) = 1 [ y ( σ ) ] = 1 2 S ^ 0 ( d ) + 1 4 [ S ^ 1 ( d B t ) + S ^ 1 ( d + B t ) ] + 1 4 i [ S ^ 2 ( d + B t ) S ^ 2 ( d B t ) ]
C ^ 0 ( d ) = 2 H LPF ( d ) y ^ ( d ) = S ^ 0 ( d ) ,
C ^ 1 ( d ) = H LPF ( d ) y ^ ( d ) = 1 4 [ S ^ 1 ( d + B t ) + i S ^ 2 ( d + B t ) ]
C 0 ( σ ) = [ C ^ 0 ( d ) ] = S 0 ( σ )
C 1 ( σ ) = [ C ^ 1 ( d ) ] = 1 4 [ S 1 ( σ ) e i 2 π σ B t + i S 2 ( σ ) e i 2 π σ B t ] .
C 0 R ( σ ) = [ C ^ 0 R ( d ) ] = S 0 R ( σ )
C 1 R ( σ ) = 1 4 S 1 R ( σ ) e i 2 π σ B t .
ϕ ^ ( σ ) = arg [ C 1 R ( σ ) ] .
C ¯ 1 ( σ ) = C 1 ( σ ) C 1 R ( σ ) S 0 R ( σ ) S 0 ( σ ) = S 1 ( σ ) / S 0 ( σ ) S 1 R ( σ ) / S 0 R ( σ ) + i S 2 ( σ ) / S 0 ( σ ) S 1 R ( σ ) / S 0 R ( σ ) = S 1 ( σ ) S 0 ( σ ) + i S 2 ( σ ) S 0 ( σ )
S 1 ( σ ) S 0 ( σ ) = Re [ C ¯ 1 ( σ ) ]
S 2 ( σ ) S 0 ( σ ) = Im [ C ¯ 1 ( σ ) ] .
s = [ s 0 s 1 s 2 ] 3 N ,
M cos = diag [ cos ( ϕ 1 ) , , cos ( ϕ N ) ] N × N
M sin = diag [ sin ( ϕ 1 ) , , sin ( ϕ N ) ] N × N .
ϕ = [ ϕ 1 ϕ N ] T
y model = 1 2 ( s 0 + M cos s 1 + M sin s 2 ) ,
y model , i = 1 2 [ S 0 ( σ i ) + S 1 ( σ i ) cos ( ϕ i ) + S 2 ( σ i ) sin ( ϕ i ) ] .
p n = [ P n ( x 1 ) P n ( x N ) ] T N
P n ( x ) = 2 n k = 0 n x k ( n k ) ( n + k 1 2 n ) .
M dct ( m , n ) = { 2 N cos ( π 2 N ( 2 n 1 ) ( m 1 ) ) , for m = 2 , , N 1 N , for m = 1 , M dct N × N .
M support = [ |   | p 1 p L |   | | M dct ] N × ( N + L )
s i = M support s ^ i ,
s ^ i = [ s ^ i poly s ^ i DCT ]
y model = 1 2 ( M support s ^ 0 + M cos M support s ^ 1 + M sin M support s ^ 2 ) .
L ( s ^ 0 , s ^ 1 , s ^ 2 ) = y model y measured 2 2 = 1 2 ( M support s ^ 0 + M cos M support s ^ 1 + M sin M support s ^ 2 ) y measured 2 2 ,
R ( s ^ 0 , s ^ 1 , s ^ 2 ) = s ^ 0 1 + s ^ 1 1 + s ^ 2 1 .
c ( s ^ 0 , s ^ 1 , s ^ 2 ) = L ( s ^ 0 , s ^ 1 , s ^ 2 ) + β R ( s ^ 0 , s ^ 1 , s ^ 2 ) = 1 2 ( M support s ^ 0 + M cos M support s ^ 1 + M sin M support s ^ 2 ) y measured 2 2 + β ( s ^ 0 1 + s ^ 1 1 + s ^ 2 1 ) .
minimize s ^ 0 , s ^ 1 , s ^ 2 L ( s ^ 0 , s ^ 1 , s ^ 2 ) + β R ( s ^ 0 , s ^ 1 , s ^ 2 ) subject to s ^ i , j DCT = 0 , j τ ( f τ ) , i = 0 , 1 , 2
S 0 ( σ ) = a 1 cos ( 2 π f S 0 σ ) + b
S 1 ( σ ) = a 2 S 0 ( σ )
S 2 ( σ ) = 0 ,
H LPF ( d ) = rect ( d Δ )
H BPF ( d ) = rect ( d d 0 Δ )
rect ( d ) = { 0 , if | d | 1 2 1 , if | d | < 1 2 .
y ( σ ) = 1 2 { S 0 ( σ ) + S 1 ( σ ) + S 1 ( σ ) cos ( 2 π f c σ ) + S 2 ( σ ) sin ( 2 π f c σ ) } + n
n ~ N ( μ = 0 , σ n 2 ) ,
y i = 1 2 { S 0 ( σ i ) + S 1 ( σ i ) cos ( 2 π f c σ i ) + S 2 ( σ i ) sin ( 2 π f c σ i ) } + n i
P n = 1 N i = 1 N n i 2 .
P s = 1 N i = 1 N y i 2 .
SNR = 10 log 10 P s P n P n .
FR Fit ( s i FR , s i GT ) = 1 s i FR s i GT s i GT
CCSP Fit ( s i CCSP , s i GT ) = 1 s i CCSP s i GT s i GT .
Δ % ( FR Fit , CCSP Fit ) = CCSP Fit FR Fit FR Fit 100 .
DOLP ( σ ) = S 1 2 ( σ ) + S 2 2 ( σ ) S 0 ( σ ) ,
f ( x ) = | | Ax | | 1
f ( λ x + ( 1 λ ) y ) = | | A ( λ x + ( 1 λ ) y | | 1 = | | λ A x + ( 1 λ ) A y | | 1 λ | | A x | | 1 + ( 1 λ ) | | A y | | 1 = λ f ( x ) + ( 1 λ ) f ( y )
s ^ = [ s ^ 0 s ^ 1 s ^ 2 ] 3 ( N + L ) .
M model = 1 2 [ I | M cos | M sin ] N × 3 N
A = M model M support 3 N N × 3 ( N + L ) .
y model = A s ^ ,
L ( s ^ ) = | | y model y measured | | 2 2 = | | A s ^ y measured | | 2 2 .
R ( s ^ ) = | | D 0 s ^ | | 1 + | | D 1 s ^ | | 1 + | | D 2 s ^ | | 1 .
g i j ( s ^ ) = δ i j T s ^ , j > τ ( f τ ) , i = 0 , 1 , 2 .
minimize s ^ L ( s ^ ) + β R ( s ^ ) subject to g i j ( s ^ ) = 0 , j > τ ( f τ ) , i = 0 , 1 , 2 .

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