Abstract

A maximum likelihood blind deconvolution algorithm is derived for incoherent polarimetric imagery using expectation maximization. In this approach, the unpolarized and fully polarized components of the scene are estimated along with the corresponding angles of polarization and channel point spread functions. The scene state of linear polarization is determined unambiguously using this parameterization. Results are demonstrated using laboratory data.

© 2008 Optical Society of America

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References

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  1. J. S. Tyo, M. P. Rowe, E. N. Pugh, Jr., and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855-1870 (1996).
    [CrossRef] [PubMed]
  2. M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).
  3. W. G. Egan, “Polarization in remote sensing,” Proc. SPIE 1747, 2-48 (1992).
    [CrossRef]
  4. S. Lin, K. Yemelyanov, E. Pugh, Jr., and N. Engheta, “Separation and contrast enhancement of overlapping cast shadow components using polarization,” Opt. Express 14, 7099-7108 (2006).
    [CrossRef] [PubMed]
  5. W. Victor and K. Coulson, “Remote sensing in polarized light,” in NASA Conference Publication 3014 (NASA, 1987).
  6. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1992).
  7. W. Egan, Photometry and Polarization in Remote Sensing (Elsevier, 1985).
  8. W. A. Shurcliff, Polarized Light: Production and Use (Harvard U. Press, 1962).
  9. D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13, 43-64 (1996).
    [CrossRef]
  10. D. Kundur and D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13, 61-63 (1996).
    [CrossRef]
  11. T. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064-1073 (1993).
    [CrossRef]
  12. T. Schulz, B. Stribling, and J. Miller, “Multiframe blind deconvolution with real data: imagery of the Hubble Space Telescope,” Opt. Express 1, 355-362 (1997).
    [CrossRef] [PubMed]
  13. J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83-96 (2007).
    [CrossRef] [PubMed]
  14. D. M. Strong, “Polarimeter blind deconvolution using image diversity,” Ph.D. dissertation (Air Force Institute of Technology, 2007).
  15. N. R. D. Dempster and A. P. Laird, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B (Methodol.) 39, 1-37 (1977).
  16. T. Moon, “The expectation-maximization algorithm,” IEEE Signal Process. Mag. 13, 47-60 (1996).
    [CrossRef]
  17. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 2000).
    [PubMed]
  18. E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642-644 (2008).
    [CrossRef] [PubMed]
  19. L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113-122 (1982).
    [CrossRef] [PubMed]
  20. R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

2008 (1)

2007 (1)

2006 (1)

2002 (1)

M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).

1997 (1)

1996 (4)

D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13, 43-64 (1996).
[CrossRef]

D. Kundur and D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13, 61-63 (1996).
[CrossRef]

T. Moon, “The expectation-maximization algorithm,” IEEE Signal Process. Mag. 13, 47-60 (1996).
[CrossRef]

J. S. Tyo, M. P. Rowe, E. N. Pugh, Jr., and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855-1870 (1996).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

W. G. Egan, “Polarization in remote sensing,” Proc. SPIE 1747, 2-48 (1992).
[CrossRef]

1982 (1)

L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

1977 (1)

N. R. D. Dempster and A. P. Laird, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B (Methodol.) 39, 1-37 (1977).

1972 (1)

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Antonucci, R.

M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 2000).
[PubMed]

Cohen, R. D.

M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1992).

Coulson, K.

W. Victor and K. Coulson, “Remote sensing in polarized light,” in NASA Conference Publication 3014 (NASA, 1987).

Dempster, N. R. D.

N. R. D. Dempster and A. P. Laird, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B (Methodol.) 39, 1-37 (1977).

Egan, W.

W. Egan, Photometry and Polarization in Remote Sensing (Elsevier, 1985).

Egan, W. G.

W. G. Egan, “Polarization in remote sensing,” Proc. SPIE 1747, 2-48 (1992).
[CrossRef]

Engheta, N.

Gerchberg, R.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Hatzinakos, D.

D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13, 43-64 (1996).
[CrossRef]

D. Kundur and D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13, 61-63 (1996).
[CrossRef]

Heinrich, C.

Hurt, T. W.

M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).

Kay, L. E.

M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).

Kishimoto, M.

M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).

Krolik, J. H.

M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).

Kundur, D.

D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13, 43-64 (1996).
[CrossRef]

D. Kundur and D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13, 61-63 (1996).
[CrossRef]

Laird, A. P.

N. R. D. Dempster and A. P. Laird, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B (Methodol.) 39, 1-37 (1977).

Lin, S.

Miller, J.

Moon, T.

T. Moon, “The expectation-maximization algorithm,” IEEE Signal Process. Mag. 13, 47-60 (1996).
[CrossRef]

Pugh, E.

Pugh, E. N.

Rowe, M. P.

Saxton, W.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Schulz, T.

Shepp, L.

L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light: Production and Use (Harvard U. Press, 1962).

Stribling, B.

Strong, D. M.

D. M. Strong, “Polarimeter blind deconvolution using image diversity,” Ph.D. dissertation (Air Force Institute of Technology, 2007).

Tyo, J. S.

Vardi, Y.

L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

Victor, W.

W. Victor and K. Coulson, “Remote sensing in polarized light,” in NASA Conference Publication 3014 (NASA, 1987).

Wolf, E.

E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642-644 (2008).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 2000).
[PubMed]

Yemelyanov, K.

Zallat, J.

Appl. Opt. (1)

Astrophys. J., Suppl. (1)

M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet imaging polarimetry of the Seyfert 2 galaxy Markarian 3,” Astrophys. J., Suppl. 565, 155-162 (2002).

IEEE Signal Process. Mag. (3)

D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13, 43-64 (1996).
[CrossRef]

D. Kundur and D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Process. Mag. 13, 61-63 (1996).
[CrossRef]

T. Moon, “The expectation-maximization algorithm,” IEEE Signal Process. Mag. 13, 47-60 (1996).
[CrossRef]

IEEE Trans. Med. Imaging (1)

L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

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

J. R. Stat. Soc. Ser. B (Methodol.) (1)

N. R. D. Dempster and A. P. Laird, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B (Methodol.) 39, 1-37 (1977).

Opt. Express (3)

Opt. Lett. (1)

Optik (Jena) (1)

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Proc. SPIE (1)

W. G. Egan, “Polarization in remote sensing,” Proc. SPIE 1747, 2-48 (1992).
[CrossRef]

Other (6)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 2000).
[PubMed]

D. M. Strong, “Polarimeter blind deconvolution using image diversity,” Ph.D. dissertation (Air Force Institute of Technology, 2007).

W. Victor and K. Coulson, “Remote sensing in polarized light,” in NASA Conference Publication 3014 (NASA, 1987).

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1992).

W. Egan, Photometry and Polarization in Remote Sensing (Elsevier, 1985).

W. A. Shurcliff, Polarized Light: Production and Use (Harvard U. Press, 1962).

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Figures (5)

Fig. 1
Fig. 1

Laboratory test imagery. The 0° (left), 60° (middle), and 60 ° (right) channels.

Fig. 2
Fig. 2

Unaberrated 60° (left), and 60 ° (right) channels.

Fig. 3
Fig. 3

Top row: initial λ u (left) and λ p (right). Bottom row: λ u (left) and λ p (right) after restoration.

Fig. 4
Fig. 4

Recovered target angle of polarization (on target pixels only).

Fig. 5
Fig. 5

Close-up of the estimated PSFs. Top row: initial guess for all channels (left) and final 0° channel estimate (right). Bottom row: final estimates for the 60° (left) and 60 ° (right) channels.

Equations (37)

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S 1 = P S 0 cos 2 α ,
S 2 = P S 0 sin 2 α ,
I = 1 2 ( S 0 + S 1 cos 2 θ + S 2 sin 2 θ ) .
I c = 1 2 ( 1 P ) S 0 + 1 2 P S 0 ( 1 + cos 2 α cos 2 θ c + sin 2 α sin 2 θ c ) = 1 2 λ u + λ p cos 2 ( α θ c ) ,
i c ( y ) = x o c ( x ) h c ( y x ) .
o c ( x ) = 1 2 λ u ( x ) + λ p ( x ) cos 2 ( α ( x ) θ c ) .
d c ( y ) = x d ̃ u c ( y , x ) + x d ̃ p c ( y , x ) ,
E [ d ̃ u c ( y , x ) ] = 1 2 λ u ( x ) h c ( y x ) ,
E [ d ̃ p c ( y , x ) ] = λ p ( x ) cos 2 ( α ( x ) θ c ) h c ( y x ) .
L CD ( λ u , λ p , α , h ) = c y x { d ̃ u c ( y , x ) ln [ 1 2 λ u ( x ) h c ( y x ) ] 1 2 λ u ( x ) h c ( y x ) } + c y x { d ̃ p c ( y , x ) ln [ λ p ( x ) cos 2 ( α ( x ) θ c ) h c ( y x ) ] λ p ( x ) cos 2 ( α ( x ) θ c ) h c ( y x ) } ,
Q n + 1 ( λ u , λ p , α , h ) = E [ L CD ( λ u , λ p , α , h ) d , λ u n , λ p n , α n , h n ] ,
ψ k c n + 1 ( y , x ) = E [ d ̃ k c ( y , x ) d c , λ k n , α n , h c n ] ,
ψ p c n + 1 ( y , x ) = d c ( y ) i c n ( y ) λ p n ( x ) cos 2 ( α n ( x ) θ c ) h c n ( y x ) ,
ψ u c n + 1 ( y , x ) = 1 2 d c ( y ) i c n ( y ) λ u n ( x ) h c n ( y x ) .
Q n + 1 λ p ( x 0 ) = c y ψ p c n + 1 ( y , x 0 ) λ p ( x 0 ) c cos 2 ( α ( x 0 ) θ c ) = 0 ,
Q n + 1 λ u ( x 0 ) = c y ψ u c n + 1 ( y , x 0 ) λ u ( x 0 ) C 2 = 0 ,
Q n + 1 α ( x 0 ) = 2 c y ψ p c n + 1 ( y , x 0 ) tan ( α ( x 0 ) θ c ) + c λ p ( x 0 ) sin [ 2 ( α ( x 0 ) θ c ) ] = 0 ,
c cos 2 ( α n + 1 ( x 0 ) θ c ) = C 2 ,
c sin [ 2 ( α n + 1 ( x 0 ) θ c ) ] = 0 .
λ k n + 1 ( x 0 ) = 2 C c y ψ k c n + 1 ( y , x 0 ) ,
S n + 1 = M 1 Ψ p ,
α n + 1 ( x 0 ) = 1 2 tan 1 S 2 n + 1 ( x 0 ) S 1 n + 1 ( x 0 ) ,
Q n + 1 h c ( z ) = y ψ p c n + 1 ( y , y z ) + ψ u c n + 1 ( y , y z ) h c ( z ) y o c n + 1 ( y z ) ,
h c ( x , φ c n + 1 ) = u A ( u ) exp [ i φ c n + 1 ( u ) ] e i 2 π k u x 2 ,
φ c n + 1 = { φ ̃ c if x ξ ( x ) ln h c ( x , φ ̃ c ) x ξ ( x ) ln h c ( x , φ c n ) φ c n otherwise } ,
ξ ( x ) = h c ( x , φ c n ) D c y d c ( y ) i c n ( y ) o c n ( y x ) ,
D c = x o c n + 1 ( x ) = y d c ( y ) ,
φ ̃ c = ph { F 1 [ ξ ( x , φ c n ) exp ( i ph ( h c ( x , φ c n ) ) ) ] } ,
tan ( α θ c ) = tan α tan θ c 1 + tan α tan θ c ,
tan 2 α = ( 1 cos 2 α + 1 ) tan α ,
sin 2 α = 2 tan α tan 2 α + 1 .
3 ( Ψ 3 Ψ 2 ) [ tan 2 α + 1 ] + 4 ( Ψ 1 + Ψ 2 + Ψ 3 ) tan α 3 Ψ 1 tan α [ tan 2 α + 1 ] = 0 ,
S 0 n + 1 = 2 3 ( Ψ 1 + Ψ 2 + Ψ 3 ) ,
S 1 n + 1 = 2 3 ( Ψ 1 Ψ 2 Ψ 3 ) ,
S 2 n + 1 = 2 3 ( Ψ 2 Ψ 3 ) ,
3 2 S 2 n + 1 + 3 S 0 n + 1 2 tan α tan 2 α + 1 + 3 2 ( S 0 n + 1 + S 1 n + 1 ) tan α = 0 .
3 2 S 2 n + 1 + 3 S 2 n + 1 + 3 S 1 n + 1 2 tan 2 α = 0 ,

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