Abstract

In the field of imaging polarimetry Stokes parameters are sought and must be inferred from noisy and blurred intensity measurements. Using a penalized-likelihood estimation framework we investigate reconstruction quality when estimating intensity images and then transforming to Stokes parameters, and when estimating Stokes parameters directly. We define our cost function for reconstruction by a weighted least-squares data fit term and a regularization penalty. We show that for quadratic regularization the estimators of Stokes and intensity images can be made equal by appropriate choice of regularization parameters. It is empirically shown that, when using edge preserving regularization, estimating the Stokes parameters directly leads to lower RMS error. Also, the addition of a cross channel regularization term further lowers the RMS error for both methods, especially in the case of low SNR.

© 2009 Optical Society of America

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References

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  1. 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]
  2. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  3. W. G. Egan, “Polarization and surface roughness,” Proc. SPIE 3426, 144-152 (1998).
    [Crossref]
  4. R. A. Chipman, Polarimetry, Vol. 2 (McGraw-Hill, 1995).
  5. E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).
  6. J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. 45, 5497-5503 (2006).
    [Crossref] [PubMed]
  7. J. Zallat and C. Heinrich, “Polarimetric data reduction: a bayesian approach,” Opt. Express 15, 83-96 (2007).
    [Crossref] [PubMed]
  8. T. F. Chan and C. Wong, “Multichannel image deconvolution by total variation regularization,” Proc. SPIE 3162, 358-366 (1997).
    [Crossref]
  9. L. Bar, A. Brook, and K. Nahum, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
    [Crossref] [PubMed]
  10. J. R. Valenzuela and J. A. Fessler, “Regularized estimation of Stokes images from polarimetric measurements,” Proc. SPIE 6814, 681403-681403-10 (2008).
    [Crossref]
  11. D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503-528 (1989).
    [Crossref]
  12. J. Fessler and W. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction methods: Space-invariant tomographs,” IEEE Trans. Image Process. 5, 1346-1358 (1996).
    [Crossref] [PubMed]
  13. J. Fessler, Department of Electrical and Computer Engineering, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109-2122, USA, is preparing a book to be titled Image Reconstruction: Algorithms and Analysis.
  14. P. Blomgren and T. Chan, “Color tv: Total variation methods for restoration of vector-valued images,” IEEE Trans. Image Process. 7, 304-309 (1998).
    [Crossref]

2008 (1)

J. R. Valenzuela and J. A. Fessler, “Regularized estimation of Stokes images from polarimetric measurements,” Proc. SPIE 6814, 681403-681403-10 (2008).
[Crossref]

2007 (2)

L. Bar, A. Brook, and K. Nahum, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[Crossref] [PubMed]

J. Zallat and C. Heinrich, “Polarimetric data reduction: a bayesian approach,” Opt. Express 15, 83-96 (2007).
[Crossref] [PubMed]

2006 (2)

1998 (2)

W. G. Egan, “Polarization and surface roughness,” Proc. SPIE 3426, 144-152 (1998).
[Crossref]

P. Blomgren and T. Chan, “Color tv: Total variation methods for restoration of vector-valued images,” IEEE Trans. Image Process. 7, 304-309 (1998).
[Crossref]

1997 (1)

T. F. Chan and C. Wong, “Multichannel image deconvolution by total variation regularization,” Proc. SPIE 3162, 358-366 (1997).
[Crossref]

1996 (1)

J. Fessler and W. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction methods: Space-invariant tomographs,” IEEE Trans. Image Process. 5, 1346-1358 (1996).
[Crossref] [PubMed]

1989 (1)

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503-528 (1989).
[Crossref]

Bar, L.

L. Bar, A. Brook, and K. Nahum, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[Crossref] [PubMed]

Barrett, H.

H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Blomgren, P.

P. Blomgren and T. Chan, “Color tv: Total variation methods for restoration of vector-valued images,” IEEE Trans. Image Process. 7, 304-309 (1998).
[Crossref]

Brook, A.

L. Bar, A. Brook, and K. Nahum, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[Crossref] [PubMed]

Chan, T.

P. Blomgren and T. Chan, “Color tv: Total variation methods for restoration of vector-valued images,” IEEE Trans. Image Process. 7, 304-309 (1998).
[Crossref]

Chan, T. F.

T. F. Chan and C. Wong, “Multichannel image deconvolution by total variation regularization,” Proc. SPIE 3162, 358-366 (1997).
[Crossref]

Chenault, D. B.

Chipman, R. A.

R. A. Chipman, Polarimetry, Vol. 2 (McGraw-Hill, 1995).

Egan, W. G.

W. G. Egan, “Polarization and surface roughness,” Proc. SPIE 3426, 144-152 (1998).
[Crossref]

Fessler, J.

J. Fessler and W. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction methods: Space-invariant tomographs,” IEEE Trans. Image Process. 5, 1346-1358 (1996).
[Crossref] [PubMed]

J. Fessler, Department of Electrical and Computer Engineering, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109-2122, USA, is preparing a book to be titled Image Reconstruction: Algorithms and Analysis.

Fessler, J. A.

J. R. Valenzuela and J. A. Fessler, “Regularized estimation of Stokes images from polarimetric measurements,” Proc. SPIE 6814, 681403-681403-10 (2008).
[Crossref]

Goldstein, D. L.

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).

Heinrich, C.

Liu, D.

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503-528 (1989).
[Crossref]

Myers, K. J.

H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Nahum, K.

L. Bar, A. Brook, and K. Nahum, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[Crossref] [PubMed]

Nocedal, J.

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503-528 (1989).
[Crossref]

Rogers, W.

J. Fessler and W. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction methods: Space-invariant tomographs,” IEEE Trans. Image Process. 5, 1346-1358 (1996).
[Crossref] [PubMed]

Shaw, J. A.

Tyo, J. S.

Valenzuela, J. R.

J. R. Valenzuela and J. A. Fessler, “Regularized estimation of Stokes images from polarimetric measurements,” Proc. SPIE 6814, 681403-681403-10 (2008).
[Crossref]

Wei, H.

Wong, C.

T. F. Chan and C. Wong, “Multichannel image deconvolution by total variation regularization,” Proc. SPIE 3162, 358-366 (1997).
[Crossref]

Zallat, J.

Appl. Opt. (2)

IEEE Trans. Image Process. (3)

L. Bar, A. Brook, and K. Nahum, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[Crossref] [PubMed]

J. Fessler and W. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction methods: Space-invariant tomographs,” IEEE Trans. Image Process. 5, 1346-1358 (1996).
[Crossref] [PubMed]

P. Blomgren and T. Chan, “Color tv: Total variation methods for restoration of vector-valued images,” IEEE Trans. Image Process. 7, 304-309 (1998).
[Crossref]

Math. Program. (1)

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503-528 (1989).
[Crossref]

Opt. Express (1)

Proc. SPIE (3)

T. F. Chan and C. Wong, “Multichannel image deconvolution by total variation regularization,” Proc. SPIE 3162, 358-366 (1997).
[Crossref]

J. R. Valenzuela and J. A. Fessler, “Regularized estimation of Stokes images from polarimetric measurements,” Proc. SPIE 6814, 681403-681403-10 (2008).
[Crossref]

W. G. Egan, “Polarization and surface roughness,” Proc. SPIE 3426, 144-152 (1998).
[Crossref]

Other (4)

R. A. Chipman, Polarimetry, Vol. 2 (McGraw-Hill, 1995).

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).

H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

J. Fessler, Department of Electrical and Computer Engineering, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109-2122, USA, is preparing a book to be titled Image Reconstruction: Algorithms and Analysis.

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Figures (7)

Fig. 1
Fig. 1

Noisy and blurred polarimetric imagery. The first row has an SNR of 25 dB and the second row has an SNR of 15 dB. From left to right the angle of the polarizer is {0°, 45°, 90°, 135°}.

Fig. 2
Fig. 2

Estimates of Stokes images for SNR = 25 dB . All rows read from left to right: pristine, proposed method, traditional method. First row, S 0 ; second row, S 1 ; third row, S 2 .

Fig. 3
Fig. 3

Estimates of Stokes images for SNR = 15 dB . All rows read from left to right: pristine, proposed method, traditional method. First row, S 0 ; second row, S 1 ; third row, S 2 .

Fig. 4
Fig. 4

Estimates of the DOLP for SNR = 25 dB ; from left to right: pristine, proposed method, traditional method.

Fig. 5
Fig. 5

Estimates of the DOLP for SNR = 15 dB ; from left to right: pristine, proposed method, traditional method.

Fig. 6
Fig. 6

Estimates of the DOLP for SNR = 25 dB ; from left to right: pristine, proposed method with cross-channel regularization, traditional method with cross-channel regularization.

Fig. 7
Fig. 7

Estimates of the DOLP for SNR = 15 dB ; from left to right: pristine, proposed method with cross-channel regularization, traditional method with cross-channel regularization.

Tables (2)

Tables Icon

Table 1 Simulation Results ( SNR = 25 dB ) : RMS Error Percentages

Tables Icon

Table 2 Simulation Results ( SNR = 15 dB ) : RMS Error Percentages

Equations (47)

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E ( t ) = E x ( t ) i ̂ + E y ( t ) j ̂ .
Γ ( θ ) = ( E ( t ) p θ ) 2 = E x 2 ( t ) cos 2 ( θ ) + E y 2 ( t ) sin 2 ( θ ) + 2 E x ( t ) E y ( t ) sin ( θ ) cos ( θ ) ,
Γ ( θ ) = 1 2 ( E x 2 ( t ) + E y 2 ( t ) ) + 1 2 cos ( 2 θ ) ( E x 2 ( t ) E y 2 ( t ) ) + E x ( t ) E y ( t ) sin ( 2 θ ) .
S 0 = Γ ( 0 ° ) + Γ ( 90 ° ) = E x 2 ( t ) + E y 2 ( t ) ,
S 1 = Γ ( 0 ° ) Γ ( 90 ° ) = E x 2 ( t ) E y 2 ( t ) ,
S 2 = Γ ( 45 ° ) Γ ( 135 ° ) = 2 E x ( t ) E y ( t ) .
Γ ( θ ) = 1 2 [ S 0 + S 1 cos ( 2 θ ) + S 2 sin ( 2 θ ) ] .
Γ = T J × 3 S .
S ̂ = T J × 3 Γ .
y j = A j Γ j + ϵ j .
ϵ j N ( 0 , σ 2 I ) .
Γ ̂ = argmin Γ { log p ( y Γ ) + R Γ ( Γ ) } ,
R Γ ( Γ ) = j = 1 J R j ( Γ j ) ,
Γ ̂ j = argmin Γ j { log p ( y j Γ j ) + R j ( Γ j ) } ,
log p ( y j Γ j ) = 1 2 σ 2 y j A j Γ j 2 .
S ̂ 1 = T ̌ Γ ̂ = ( T J × 3 I n p ) Γ ̂ ,
S ̂ = argmin S { log p ( y S ) + R S ( S ) } ,
R S ( S ) = j = 0 2 R j ( S j ) .
T 4 = [ 1 2 1 2 0 1 2 0 1 2 1 2 1 2 0 1 2 0 1 2 ] , T 4 = [ 1 2 1 2 1 2 1 2 1 0 1 0 0 1 0 1 ] .
y = ( I 4 B ) Γ + ϵ .
R Γ ( Γ ) = 1 2 β j = 1 J k = 2 n p ( Γ j k Γ j k 1 ) 2 = 1 2 β j = 1 J C Γ j 2 ,
R Γ ( Γ ) = 1 2 β C Γ 2 .
Γ ̂ = argmin Γ { 1 2 σ 2 y ( I 4 B ) Γ 2 + 1 2 β C Γ 2 } .
Γ [ 1 2 σ 2 y ( I 4 B ) Γ 2 + 1 2 β C Γ 2 ] = 0 .
S ̂ Γ = T ̌ { I 4 [ ( B B + σ 2 β R ) 1 B ] } y ,
y = ( T 4 B ) S + ϵ .
S ̂ = argmin S { 1 2 σ 2 y ( T 4 B ) S 2 + 1 2 ( β 0 C S 0 2 + β 1 C S 1 2 + β 2 C S 2 2 ) } .
S ̂ = ( T 4 T 4 B B + σ 2 β 3 R ) 1 ( T 4 B ) y ,
E [ S ̂ S ] = ( P 1 Q ϴ ϴ Q + σ 2 β 3 Q Ω Ω Q ) 1 ( P 1 Q ϴ ϴ Q ) S = Q [ ( P 1 ϴ ϴ + σ 2 β 3 Ω Ω ) 1 ( P 1 ϴ ϴ ) ] Q S .
Filter for S 0 L k = B k 2 B k 2 + β 0 σ 2 F k 2 ,
Filter for S 1 L k = 1 2 B k 2 1 2 B k 2 + β 1 σ 2 F k 2 ,
Filter for S 2 L k = 1 2 B k 2 1 2 B k 2 + β 2 σ 2 F k 2 ,
β 3 = β diag ( 1 , 1 2 , 1 2 ) ,
S ̂ = ( T 4 T 4 B B + σ 2 β 3 R ) 1 ( T 4 B ) y = [ diag ( 1 , 1 2 , 1 2 ) ( B B + σ 2 β R ) ] 1 ( T 4 B ) y = diag ( 1 , 2 , 2 ) ( B B + σ 2 β R ) 1 ( T 4 B ) y = T 4 [ B B + σ 2 β R ] 1 B y = S ̂ Γ .
Cov ( S ̂ S ) = Cov ( L y S ) = σ 2 ( P 1 B B + σ 2 β 3 R ) 1 ( P 1 B B ) ( P 1 B B + σ 2 β 3 R ) 1 .
Cov ( S ̂ S ) = σ 2 ( P 1 Q ϴ ϴ Q + σ 2 β 3 Q Ω Ω Q ) 1 ( P 1 Q ϴ ϴ Q ) ( P 1 Q ϴ ϴ Q + σ 2 β 3 Q Ω Ω Q ) 1 = σ 2 Q ( P 1 ϴ ϴ + σ 2 β 3 Ω Ω ) 1 ( P 1 ϴ ϴ ) ( P 1 ϴ ϴ + σ 2 β 3 Ω Ω ) 1 Q .
Var { S 0 i S } = σ 2 n p k B k 2 ( B k 2 + β 0 σ 2 F k 2 ) 2 ,
Var { S 1 i S } = σ 2 n p k 2 B k 2 ( B k 2 + β 1 σ 2 F k 2 ) 2 ,
Var { S 2 i S } = σ 2 n p k 2 B k 2 ( B k 2 + β 2 σ 2 F k 2 ) 2 .
R Γ ( Γ ) = j k ψ ( [ C Γ j ] ; δ ) ,
R S ( S ) = = 0 2 k ψ ( [ C S ] ; δ ) .
l j ( S ) = lim ϵ 0 S ̂ [ y ¯ ( S + ϵ e j ) ] S ̂ [ y ¯ ( S ) ] ϵ = S ̂ [ y ¯ ( S ) ] y ¯ ( S ) e j .
l k j = ( B B + σ 2 β k R ) 1 B B e k j ,
R cross ( Γ ) = β cross k = 1 n p j = 1 J [ C Γ j ] k 2 ,
R cross ( S ) = β cross k = 1 n p [ C S 0 ] k 2 + [ C S 1 ] k 2 + [ C S 2 ] k 2 .
Γ ̂ = argmin Γ { 1 2 σ 2 y ( I 4 B ) Γ 2 + β j ψ ( C Γ j ; δ ) } + β cross k = 1 n p j = 1 4 [ C Γ j ] k 2 ,
S ̂ = argmin S { 1 2 σ 2 y ( T 4 × 3 B ) S 2 + β 0 ψ ( C S 0 ; δ 0 ) + β 1 ψ ( C S 1 ; δ 1 ) + β 2 ψ ( C S 2 ; δ 2 ) + β cross k = 1 n p [ C S 0 ] k 2 + [ C S 1 ] k 2 + [ C S 2 ] k 2 } .

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