Abstract

Precise channel-to-channel registration is a prerequisite for effective exploitation of passive polarimetric imagery. In this paper, the Cramer-Rao bound is employed to determine the limits of registration precision in the presence of scene polarization diversity, channel noise, and random translational registration errors between channels. The effects of misregistration on Stokes image estimation are also explored in depth. Algorithm bias is discussed in the context of the bound, without being estimator specific. Finally, case studies are presented for polarization insensitive imagery (a special case) and linear polarization imaging systems with three and four channels. An optimum polarization channel arrangement is proposed in the context of the bound.

© 2008 Optical Society of America

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  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," apj 565, 155-162 (2002).
  2. W. G. Egan, "Polarization in remote sensing," in Polarization and Remote Sensing, W. G. Egan, ed., Proc. SPIE 1747, 2-48 (1992).
  3. 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]
  4. C. M. Persons, D. B. Chenault, M. W. Jones, K. D. Spradley, M. G. Gulley, and C. A. Farlow, "Automated registration of polarimetric imagery using Fourier transform techniques," Proc. SPIE 4819, 107-117 (2002).
    [CrossRef]
  5. X. Wang, S. Yang, J. Ma, and Y. Qiao, "Automated registration of polarimetric image using wavelet transform techniques," (SPIE, 2005) Vol. 5832, pp. 695-702
  6. D. A. LeMaster, "A Comparison of Template Matching Registration Methods for Polarimetric Imagery," in Aerospace Conference, 2008 IEEE, Vol. 1, pp. 1-9 (2008).
  7. Guyot, S.  and Anastasiadou, M.  and Del�??echelle, E.  and De Martino, A. , "Registration scheme suitable to Mueller matrix imaging for biomedical applications," Opt. Express 15, 7393-7400 (2007).
    [CrossRef] [PubMed]
  8. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewood Cliffs, New Jersey, 1993).
  9. D. Robinson and P. Milanfar, "Fundamental performance limits in image registration." IEEE Trans. Image Process 13, 1185-1199 (2004).
    [CrossRef] [PubMed]
  10. A. Yetik, and I. S. Nehorai, "Performance bounds on image registration," IEEE Trans. Signal Process 54, 1737-1749 (May 2006).
    [CrossRef]
  11. B. Zitova and J. Flusser, "Image registration methods: a survey," Image and Vision Computing 21, 977-1000 (2003).
    [CrossRef]
  12. L. L. Scharf, Statistical Signal Processing: detection, estimation, and time series analysis (Addison-Wesley, Reading, Massachusetts, 1991).
  13. L. L. Scharf and L. T. McWhorter, "Geometry of the Cramer-Rao bound," Signal Processing 31, 301-311 (1993).
    [CrossRef]
  14. H. Van Trees, Detection, estimation, and modulation theory. Part 1: detection, estimation, and linear modulation theory (Wiley, New York, 2001).
    [CrossRef]
  15. J. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt 41, 619-630 (2002).
    [CrossRef] [PubMed]
  16. J. Tyo, "Optimum linear combination strategy for an N-channel polarization-sensitive imaging or vision system," J. Opt. Soc. Am. A 15, 359-366 (1998).
    [CrossRef]
  17. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, Inc., New York, 1992).
  18. M. Healy, Matrices for Statistics (Oxford University Press, USA, 1986).
  19. A. Graham, Kronecker Products and Matrix Calculus With Applications. (Wiley, New York, 1982).

2007

2006

2004

D. Robinson and P. Milanfar, "Fundamental performance limits in image registration." IEEE Trans. Image Process 13, 1185-1199 (2004).
[CrossRef] [PubMed]

2003

B. Zitova and J. Flusser, "Image registration methods: a survey," Image and Vision Computing 21, 977-1000 (2003).
[CrossRef]

2002

J. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt 41, 619-630 (2002).
[CrossRef] [PubMed]

1998

1993

L. L. Scharf and L. T. McWhorter, "Geometry of the Cramer-Rao bound," Signal Processing 31, 301-311 (1993).
[CrossRef]

Anastasiadou, S.

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," apj 565, 155-162 (2002).

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," apj 565, 155-162 (2002).

De Martino, E.

Del???echelle, M.

Engheta, N.

Flusser, J.

B. Zitova and J. Flusser, "Image registration methods: a survey," Image and Vision Computing 21, 977-1000 (2003).
[CrossRef]

Guyot,

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," apj 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," apj 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," apj 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," apj 565, 155-162 (2002).

Lin, S.

Ma, J.

X. Wang, S. Yang, J. Ma, and Y. Qiao, "Automated registration of polarimetric image using wavelet transform techniques," (SPIE, 2005) Vol. 5832, pp. 695-702

McWhorter, L. T.

L. L. Scharf and L. T. McWhorter, "Geometry of the Cramer-Rao bound," Signal Processing 31, 301-311 (1993).
[CrossRef]

Milanfar, P.

D. Robinson and P. Milanfar, "Fundamental performance limits in image registration." IEEE Trans. Image Process 13, 1185-1199 (2004).
[CrossRef] [PubMed]

Pugh, E.

Qiao, Y.

X. Wang, S. Yang, J. Ma, and Y. Qiao, "Automated registration of polarimetric image using wavelet transform techniques," (SPIE, 2005) Vol. 5832, pp. 695-702

Robinson, D.

D. Robinson and P. Milanfar, "Fundamental performance limits in image registration." IEEE Trans. Image Process 13, 1185-1199 (2004).
[CrossRef] [PubMed]

Scharf, L. L.

L. L. Scharf and L. T. McWhorter, "Geometry of the Cramer-Rao bound," Signal Processing 31, 301-311 (1993).
[CrossRef]

Tyo, J.

J. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt 41, 619-630 (2002).
[CrossRef] [PubMed]

J. Tyo, "Optimum linear combination strategy for an N-channel polarization-sensitive imaging or vision system," J. Opt. Soc. Am. A 15, 359-366 (1998).
[CrossRef]

Wang, X.

X. Wang, S. Yang, J. Ma, and Y. Qiao, "Automated registration of polarimetric image using wavelet transform techniques," (SPIE, 2005) Vol. 5832, pp. 695-702

Yang, S.

X. Wang, S. Yang, J. Ma, and Y. Qiao, "Automated registration of polarimetric image using wavelet transform techniques," (SPIE, 2005) Vol. 5832, pp. 695-702

Yemelyanov, K.

Yetik, A.

A. Yetik, and I. S. Nehorai, "Performance bounds on image registration," IEEE Trans. Signal Process 54, 1737-1749 (May 2006).
[CrossRef]

Zitova, B.

B. Zitova and J. Flusser, "Image registration methods: a survey," Image and Vision Computing 21, 977-1000 (2003).
[CrossRef]

Appl. Opt

J. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt 41, 619-630 (2002).
[CrossRef] [PubMed]

IEEE Trans. Image Process

D. Robinson and P. Milanfar, "Fundamental performance limits in image registration." IEEE Trans. Image Process 13, 1185-1199 (2004).
[CrossRef] [PubMed]

IEEE Trans. Signal Process

A. Yetik, and I. S. Nehorai, "Performance bounds on image registration," IEEE Trans. Signal Process 54, 1737-1749 (May 2006).
[CrossRef]

Image and Vision Computing

B. Zitova and J. Flusser, "Image registration methods: a survey," Image and Vision Computing 21, 977-1000 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Signal Processing

L. L. Scharf and L. T. McWhorter, "Geometry of the Cramer-Rao bound," Signal Processing 31, 301-311 (1993).
[CrossRef]

Other

H. Van Trees, Detection, estimation, and modulation theory. Part 1: detection, estimation, and linear modulation theory (Wiley, New York, 2001).
[CrossRef]

L. L. Scharf, Statistical Signal Processing: detection, estimation, and time series analysis (Addison-Wesley, Reading, Massachusetts, 1991).

C. M. Persons, D. B. Chenault, M. W. Jones, K. D. Spradley, M. G. Gulley, and C. A. Farlow, "Automated registration of polarimetric imagery using Fourier transform techniques," Proc. SPIE 4819, 107-117 (2002).
[CrossRef]

X. Wang, S. Yang, J. Ma, and Y. Qiao, "Automated registration of polarimetric image using wavelet transform techniques," (SPIE, 2005) Vol. 5832, pp. 695-702

D. A. LeMaster, "A Comparison of Template Matching Registration Methods for Polarimetric Imagery," in Aerospace Conference, 2008 IEEE, Vol. 1, pp. 1-9 (2008).

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," apj 565, 155-162 (2002).

W. G. Egan, "Polarization in remote sensing," in Polarization and Remote Sensing, W. G. Egan, ed., Proc. SPIE 1747, 2-48 (1992).

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

M. Healy, Matrices for Statistics (Oxford University Press, USA, 1986).

A. Graham, Kronecker Products and Matrix Calculus With Applications. (Wiley, New York, 1982).

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewood Cliffs, New Jersey, 1993).

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Figures (6)

Fig. 1.
Fig. 1.

The misregistered polarimetric images

Fig. 2.
Fig. 2.

The Stokes parameter images of the bar target

Fig. 3.
Fig. 3.

Polarization insensitive imagery and its estimation bounds

Fig. 4.
Fig. 4.

The Markarian 3 test scene

Fig. 5.
Fig. 5.

The printed circuit board test scene

Fig. 6.
Fig. 6.

Bound results for the four channel case

Tables (1)

Tables Icon

Table 1. Average bounds on Stokes parameter estimates in the S −1 dominant regime

Equations (73)

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

I = 1 2 ( S 0 + S 1 cos 2 θ + S 2 cos ϕ sin 2 θ + S 3 sin ϕ sin 2 θ ) .
I x = a x 0 S 0 + a x 1 S 1 + a x 2 S 2
A = [ a 10 a 12 a N 0 a N 2 ]
Cov [ θ ̂ ] J 1 ,
J = E [ θ ( θ L θ z ) T ]
L θ z = ln p θ ( z ) .
z i ( m n ) = f i ( m n ) + ε i ( m n ) ,
f 1 ( m n ) = j = 0 2 a 1 j S j ( m n )
f i ( m n ) = j = 0 2 a ij S j ( m n v i ) ,
z i = [ z i ( m 1 ) z i ( m p 2 ) ] T
f i = [ f i ( m 1 ) f i ( m p 2 ) ] T ,
L θ z = 1 2 σ 2 i = 1 N ( z i f i ) T ( z i f i ) + ξ
θ = [ v 2 T v N T S 0 T S 2 T ] T ,
S i = [ S i ( m 1 ) S i ( m p 2 ) ] T .
J ij = m f T θ i R 1 f θ j
J ij = 1 σ 2 n = 1 N f n T θ i f n θ j
J = V H T H S
V ij = { 1 σ 2 ( v i f i T ) ( v j f j T ) T if i = j , 0 2 × 2 if i j
V ij ~ = { 0 2 × 2 if i = j , 1 σ 2 ( v i f i T ) ( v j f j T ) T if i j
S jk = 1 σ 2 ( I p 2 × p 2 ) i = 1 N a ij a ik
C = [ c 00 c 02 c 20 c 22 ] , c jk = i = 1 N a ij a ik
S = 1 σ 2 C I p 2 × p 2
S 1 = σ 2 C 1 I p 2 × p 2
H ij = a ji σ 2 ( v j f j T ) T .
B v = ( V H T S 1 H ) 1
B S = S 1 + S 1 ( H B v H T ) S 1
H T S 1 H = σ 2 H T ( C 1 I p 2 × p 2 ) H .
D hk = σ 2 i = 0 2 j = 0 2 C ij 1 H ih T H jk .
D hk = { i = 0 2 j = 0 2 C ij 1 a hi a hj V hh if h = k , i = 0 2 j = 0 2 C ij 1 a hi a kj V hk ~ if h k
D = W v ( V + V ~ ) + V
W v = ( I ( N 1 ) × ( N 1 ) M C T M T ) 1 2 × 2
B v = ( V D ) 1 = [ W v ( V + V ~ ) ] 1
Γ i 1 = σ 2 C ii 1 I p 2 × p 2
Φ i 1 = σ 2 C i 1 I p 2 × p 2
C i 1 = [ C i 0 1 C i 2 1 ]
B Si = 1 p 2 tr ( B Si )
tr ( B Si ) = tr ( Γ i 1 ) + tr [ Φ i 1 ( H B v H T ) Φ i T ]
tr [ Φ i 1 ( H B v H T ) Φ i T ] = σ 2 vec [ W Si ( V + V ~ ) ] T vec ( B v )
= σ 2 tr [ W Si ( V + V ~ ) B v ]
W Si = M ( C i T C i 1 ) M T 1 2 × 2
B Si = σ 2 C ii 1 + σ 2 p 2 tr [ W Si ( V + V ~ ) B v ]
Cov [ θ ̂ ] Δ T J 1 Δ
Δ = θ E [ θ ̂ ] T
B ~ v = Δ v T ( W v ( V + V ~ ) ) 1 Δ v
tr ( B ~ Si ) = tr ( Δ Si T Γ i 1 Δ Si + Δ Si T Φ i 1 ( H B v H T ) Φ i T Δ Si )
Φ i T Δ Si Δ Si T Φ i 1 = σ 4 C i T C i 1 Δ Si Δ Si T
tr ( B Si ~ ) = tr ( Δ Si T Γ i 1 Δ Si ) + σ 4 tr ( H T ( C i T C i 1 Δ Si Δ Si T ) HB v )
( W v ) ij = { 1 1 N if i = j , 1 N if i j
W S 0 = σ 2 N 2 1 2 ( N 1 ) × 2 ( N 1 ) .
B S 0 N = 2 = σ 2 p 2 + σ 2 2
B S 0 N large = σ 2 p 2 2 ( N 1 ) N 2 + σ 2 N
W v = 0 4 × 4
( H ij ) T H kj = a ji a jk σ 4 ( v j f j T ) ( v j f j T ) T = a ji a jk σ 2 V jj
( H lj ) T H ki = a jl a ik σ 4 ( v j f j T ) ( v i f i T ) T = a jl a ik σ 2 V ~ ji
v i f i T = x f i T .
tr ( A + G ) = tr ( A ) + tr ( G )
tr ( CGD ) = tr ( DCG )
tr ( EF ) = vec ( E T ) T vec ( F )
vec ( AGC ) = ( C T A ) vec ( G )
( A C ) ( G D ) = ( AG CD )
( A G ) T = A T G T
( A G ) T = A T G T
tr [ Φ i 1 ( HB v H T ) Φ i T ] = tr [ Φ i T Φ i 1 ( H B v H T ) ]
= vec ( Φ i T Φ i 1 ) T vec ( H B v H T )
= vec ( Φ i T Φ i 1 ) T ( H H ) vec ( B v )
= vec [ H T ( Φ i T Φ i 1 ) H ] T vec ( B v )
H T ( Φ i T Φ i 1 ) H = σ 4 H T ( C i T C i 1 I p 2 × p 2 ) H
= σ 2 W Si ( V + V ~ )
W Si = M ( C i T C i 1 ) M T 1 2 × 2
tr [ Φ i 1 ( H B v H T ) Φ i T ] = σ 2 vec [ W Si ( V + V ~ ) ] T vec ( B v )
= σ 2 tr [ W Si ( V + V ~ ) B v ]
B Si = 1 p 2 [ tr ( Γ i 1 ) ] + σ 2 p 2 tr [ W Si ( V + V ~ ) B v ]
= σ 2 C ii 1 + σ 2 p 2 tr [ W Si ( V + V ~ ) B v ]

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