Abstract

Performance bounds (Cramer–Rao bounds on root-mean-square errors) are computed for the estimation of the degree of polarization for reflected fields with active laser illumination [Proc. SPIE 5888, 58880N (2005)]. The bounds are computed from various sensing modalities, which involves the measurement and processing of (1) the four intensities outputs of a four-channel polarimeter, (2) the intensities of two orthogonal field components, and (3) the total intensity of the field. Each modality includes detector noise models and utilizes realistic data-collection models.

© 2010 Optical Society of America

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  1. T. J. Schulz, “Performance bounds for the estimation of the degree of polarization from active laser illumination,” Proc. SPIE 5888, 58880N (2005).
    [Crossref]
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    [Crossref]
  7. J. Zallat, P. Grabbling, and Y. Takakura, “Using polarimetric imaging for material classification,” in Proceedings of the 2003 International Conference on Image Processing (ICIP, 2003) Vol. 2, pp. II-827-30.
  8. P. Réfrégier, J. Fade, and M. Roche, “Estimation precision of the degree of polarization from a single speckle intensity image,” Opt. Lett. 32, 739–741 (2007).
    [Crossref] [PubMed]
  9. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29, 685–689(5) (1982).
    [Crossref]
  10. K. S. Miller, Complex Stochastic Processes: An Introduction to Theory and Application (Addison-Wesley, 1974).
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    [Crossref]
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  14. V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
    [Crossref]
  15. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  16. A. Jeffrey and D. Zwillinger, Table Of Integrals, Series, and Products, 6th ed.(Academic, 2000).
  17. J. Fade, P. Réfrégier, and M. Roche, “Estimation of the degree of polarization from a single speckle intensity image with photon noise,” J. Opt. A, Pure Appl. Opt. 10, 115301 (2008).
    [Crossref]
  18. F. Goudail and P. Réfrégier, “Statistical algorithms for target detection in coherent active polarimetric images,” J. Opt. Soc. Am. A 18, 3049–3060 (2001).
    [Crossref]

2008 (1)

J. Fade, P. Réfrégier, and M. Roche, “Estimation of the degree of polarization from a single speckle intensity image with photon noise,” J. Opt. A, Pure Appl. Opt. 10, 115301 (2008).
[Crossref]

2007 (1)

2005 (1)

T. J. Schulz, “Performance bounds for the estimation of the degree of polarization from active laser illumination,” Proc. SPIE 5888, 58880N (2005).
[Crossref]

2002 (1)

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
[Crossref]

2001 (1)

2000 (1)

A. Jeffrey and D. Zwillinger, Table Of Integrals, Series, and Products, 6th ed.(Academic, 2000).

1998 (1)

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

1997 (2)

B. Jähne, Practical Handbook on Image Processing for Scientific Applications (CRC, 1997).

S. G. Demos and R. R. Alfano, “Optical polarization imaging,” Appl. Opt. 36, 150–155 (1997).
[Crossref] [PubMed]

1996 (1)

1994 (1)

G. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[Crossref]

1993 (1)

S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. 1, Estimation Theory (Prentice Hall, 1993).

1990 (1)

L. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[Crossref]

1985 (1)

J. W. Goodman, Statistical Optics (Wiley, 1985).

1982 (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29, 685–689(5) (1982).
[Crossref]

1981 (1)

1974 (1)

K. S. Miller, Complex Stochastic Processes: An Introduction to Theory and Application (Addison-Wesley, 1974).

Alfano, R. R.

Azzam, R. M. A.

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29, 685–689(5) (1982).
[Crossref]

Belsher, J. F.

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
[Crossref]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Demos, S. G.

Engheta, N.

Fade, J.

J. Fade, P. Réfrégier, and M. Roche, “Estimation of the degree of polarization from a single speckle intensity image with photon noise,” J. Opt. A, Pure Appl. Opt. 10, 115301 (2008).
[Crossref]

P. Réfrégier, J. Fade, and M. Roche, “Estimation precision of the degree of polarization from a single speckle intensity image,” Opt. Lett. 32, 739–741 (2007).
[Crossref] [PubMed]

Gamiz, V. L.

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Goudail, F.

Grabbling, P.

J. Zallat, P. Grabbling, and Y. Takakura, “Using polarimetric imaging for material classification,” in Proceedings of the 2003 International Conference on Image Processing (ICIP, 2003) Vol. 2, pp. II-827-30.

Healey, G.

G. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[Crossref]

Jähne, B.

B. Jähne, Practical Handbook on Image Processing for Scientific Applications (CRC, 1997).

Jeffrey, A.

A. Jeffrey and D. Zwillinger, Table Of Integrals, Series, and Products, 6th ed.(Academic, 2000).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. 1, Estimation Theory (Prentice Hall, 1993).

Kondepudy, R.

G. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[Crossref]

Miller, K. S.

K. S. Miller, Complex Stochastic Processes: An Introduction to Theory and Application (Addison-Wesley, 1974).

Pugh, J. E. N.

Réfrégier, P.

Roche, M.

J. Fade, P. Réfrégier, and M. Roche, “Estimation of the degree of polarization from a single speckle intensity image with photon noise,” J. Opt. A, Pure Appl. Opt. 10, 115301 (2008).
[Crossref]

P. Réfrégier, J. Fade, and M. Roche, “Estimation precision of the degree of polarization from a single speckle intensity image,” Opt. Lett. 32, 739–741 (2007).
[Crossref] [PubMed]

Rowe, M. P.

Schulz, T. J.

T. J. Schulz, “Performance bounds for the estimation of the degree of polarization from active laser illumination,” Proc. SPIE 5888, 58880N (2005).
[Crossref]

Solomon, J. E.

Takakura, Y.

J. Zallat, P. Grabbling, and Y. Takakura, “Using polarimetric imaging for material classification,” in Proceedings of the 2003 International Conference on Image Processing (ICIP, 2003) Vol. 2, pp. II-827-30.

Tyo, J. S.

Wolff, L.

L. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[Crossref]

Zallat, J.

J. Zallat, P. Grabbling, and Y. Takakura, “Using polarimetric imaging for material classification,” in Proceedings of the 2003 International Conference on Image Processing (ICIP, 2003) Vol. 2, pp. II-827-30.

Zwillinger, D.

A. Jeffrey and D. Zwillinger, Table Of Integrals, Series, and Products, 6th ed.(Academic, 2000).

Appl. Opt. (3)

IEEE Trans. Pattern Anal. Mach. Intell. (2)

L. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059–1071 (1990).
[Crossref]

G. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[Crossref]

J. Mod. Opt. (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29, 685–689(5) (1982).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

J. Fade, P. Réfrégier, and M. Roche, “Estimation of the degree of polarization from a single speckle intensity image with photon noise,” J. Opt. A, Pure Appl. Opt. 10, 115301 (2008).
[Crossref]

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

Opt. Eng. (1)

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973–980 (2002).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (1)

T. J. Schulz, “Performance bounds for the estimation of the degree of polarization from active laser illumination,” Proc. SPIE 5888, 58880N (2005).
[Crossref]

Other (7)

J. W. Goodman, Statistical Optics (Wiley, 1985).

K. S. Miller, Complex Stochastic Processes: An Introduction to Theory and Application (Addison-Wesley, 1974).

B. Jähne, Practical Handbook on Image Processing for Scientific Applications (CRC, 1997).

J. Zallat, P. Grabbling, and Y. Takakura, “Using polarimetric imaging for material classification,” in Proceedings of the 2003 International Conference on Image Processing (ICIP, 2003) Vol. 2, pp. II-827-30.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

A. Jeffrey and D. Zwillinger, Table Of Integrals, Series, and Products, 6th ed.(Academic, 2000).

S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. 1, Estimation Theory (Prentice Hall, 1993).

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Figures (7)

Fig. 1
Fig. 1

Diagram of the four-channel polarimeter. The incident beam is split by an 80 20 polarizing splitter into two beams of orthogonal polarization, s-polarized and p-polarized. The 80% p-polarized and 20% s-polarized go to one direction, while the rest go to the other direction, followed by a 1 4 wave plate at 45° and a 1 2 wave plate at 22.5° in each leg. The two beams are split again by two polarizing splitters into four beams, which go to four intensity detectors to register the intensity data.

Fig. 2
Fig. 2

Cramer–Rao bounds on the RMS estimation error for unbiased estimators of the DOP from four-channel polarimeter intensity measurements of various noise models: for the Gaussian noise case I ¯ = 100 , σ = 30 ; for the Poisson noise case, I ¯ = 15 /photon counts, a = 1 , b = 0 ; for the Gaussian and Poisson noise case, I ¯ = 50 /photon counts, σ = 10 , a = 1 , b = 0 .

Fig. 3
Fig. 3

Cramer–Rao bounds on the RMS estimation error for unbiased estimators of the DOP from two orthogonal intensity measurements of various noise models: for Gaussian noise case I ¯ = 100 , σ = 30 ; for Poisson noise case, I ¯ = 15 /photon counts, a = 1 , b = 0 ; for Gaussian and Poisson noise case, I ¯ = 50 /photon counts, σ = 10 , a = 1 , b = 0 .

Fig. 4
Fig. 4

Cramer–Rao bounds on the RMS estimation error for unbiased estimators of the DOP from total intensity measurements of various noise models: for Gaussian noise case I ¯ = 100 , σ = 30 ; for Poisson noise case, I ¯ = 15 /photon counts, a = 1 , b = 0 ; for Gaussian and Poisson noise case, I ¯ = 50 /photon counts, σ = 10 , a = 1 , b = 0 .

Fig. 5
Fig. 5

Comparison of normalized Cramer–Rao bounds on RMS estimation error for unbiased estimators of the DOP. (a) Detector noise free; (b) Gaussian read-out noise I ¯ = 100 , σ = 30 ; (c) Poisson shot noise I ¯ = 15 ; (d) combined Gaussian and Poisson noise I ¯ = 50 , σ = 10 .

Fig. 6
Fig. 6

Normalized Cramer–Rao bounds on RMS error for estimators of the degree of polarization from measurements corrupted by Gaussian and Poisson noise with the same SNR: I ¯ = 15 , σ 2 = 15 .

Fig. 7
Fig. 7

Performance bounds versus SNR for total intensity measurements with Gaussian noise.

Equations (116)

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U = [ U x U y ] ,
J = E [ U U ] = [ J x x J x y J x y * J y y ] ,
P J P = [ λ 1 0 0 λ 2 ] ,
λ 1 = J x x + J y y 2 [ 1 + 1 4 det ( J ) [ tr ( J ) ] 2 ] ,
λ 2 = J x x + J y y 2 [ 1 1 4 det ( J ) [ tr ( J ) ] 2 ] .
P = [ 1 0 0 exp ( j φ x y ) ] [ cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ] ,
DOP = | λ 1 λ 2 | λ 1 + λ 2 = 1 4 det ( J ) [ tr ( J ) ] 2 ,
p U ( u | J ) = 1 π 2 det ( J ) exp ( u J 1 u ) ,
I x = | U x | 2 ,
I y = | U y | 2 ,
p I x , I y ( i x , i y ) = 1 det ( J ) exp ( J y y i x + J x x i y det ( J ) ) I 0 ( 2 | J x y | det ( J ) i x i y ) ,
I = U 2 = I x + I y ,
p I ( i ) = 1 λ 1 λ 2 [ exp ( i λ 1 ) exp ( i λ 2 ) ] .
D = Poisson ( a I + b ) + N ,
var ( θ ̂ ) 1 F ( θ ) ,
F ( θ ) = E [ ( l ( x | θ ) θ ) 2 ] = E [ 2 l ( x | θ ) 2 θ ] ,
θ = [ θ 1 , θ 2 , , θ d ] T ,
F m , k = E [ θ m ln f ( x | θ ) θ k ln f ( x | θ ) ] .
cov θ ( T ( x ) ) ψ ( θ ) θ [ F ( θ ) ] 1 ( ψ ( θ ) θ ) T ,
E [ θ m ln f ( x | θ ) θ k ln f ( x | θ ) ] 1 N n = 1 N [ θ m ln f ( x n | θ ) θ k ln f ( x n | θ ) ] ,
θ m ln f ( x | θ ) ln f ( x | θ 1 , , θ m + Δ , , θ d ) ln f ( x | θ 1 , , θ m , , θ d ) Δ ,
I = A S ,
[ I 0 I 1 I 2 I 3 ] = [ 0.25 0.15 0.2 0 0.25 0.15 0.2 0 0.25 0.15 0 0.2 0.25 0.15 0 0.2 ] [ S 0 S 1 S 2 S 3 ] ,
S 0 = | U x | 2 + | U y | 2 = a x 2 + a y 2 ,
S 1 = | U x | 2 | U y | 2 = a x 2 a y 2 ,
S 2 = U x U y * + U x * U y = 2 a x a y cos ( Δ ) ,
S 3 = j ( U x U y * U x * U y ) = 2 a x a y sin ( Δ ) ,
S = A 1 I ,
[ S 0 S 1 S 2 S 3 ] = [ 1 1 1 1 5 3 5 3 5 3 5 3 2.5 2.5 0 0 0 0 2.5 2.5 ] [ I 0 I 1 I 2 I 3 ] .
S 0 = J x x + J y y ,
S 1 = J x x J y y ,
S 2 = 2 Re ( J x y ) ,
S 3 = 2 Im ( J x y ) .
p I 0 , I 1 , I 2 , I 3 ( i 0 , i 1 , i 2 , i 3 ) = 125 δ ( s 0 s 1 2 + s 2 2 + s 3 2 ) 12 π d s 0 exp [ 1 2 d ( S 0 s 0 S 1 s 1 S 2 s 2 S 3 s 3 ) ] ,
s 0 = i 0 + i 1 + i 2 + i 3 ,
s 1 = 5 3 ( i 0 + i 1 i 2 i 3 ) ,
s 2 = 2.5 ( i 0 i 1 ) ,
s 3 = 2.5 ( i 2 i 3 ) .
S = [ S 0 S 1 S 2 S 3 ] T .
F m , k = E [ S m ln p D 0 , D 1 , D 2 , D 3 ( d 0 , d 1 , d 2 , d 3 ) S k ln p D 0 , D 1 , D 2 , D 3 ( d 0 , d 1 , d 2 , d 3 ) ] ,
DOP = ψ ( S ) = S 1 2 + S 2 2 + S 3 2 S 0 .
var ( DOP ˆ ) ψ ( S ) S F 1 ( ψ ( S ) S ) T ,
ψ ( S ) S = [ ψ ( S ) S 0 ψ ( S ) S 1 ψ ( S ) S 2 ψ ( S ) S 3 ] .
p D 0 , D 1 , D 2 , D 3 ( d 0 , d 1 , d 2 , d 3 ) = p I 0 , I 1 , I 2 , I 3 ( d 0 , d 1 , d 2 , d 3 ) ,
D = [ D 0 D 1 D 2 D 3 ] = [ I 0 + N 0 I 1 + N 1 I 2 + N 2 I 3 + N 3 ] ,
p D 0 , D 1 , D 2 , D 3 ( d 0 , d 1 , d 2 , d 3 ) = d 3 d 2 d 1 d 0 p I 0 , I 1 , I 2 , I 3 ( d 0 n 0 , d 1 n 1 , d 2 n 2 , d 3 n 3 ) p N 0 ( n 0 ) p N 1 ( n 1 ) p N 2 ( n 2 ) p N 3 ( n 3 ) d n 0 d n 1 d n 2 d n 3 ,
D = [ D 0 D 1 D 2 D 3 ] = [ Poisson ( a I 0 + b ) Poisson ( a I 1 + b ) Poisson ( a I 2 + b ) Poisson ( a I 3 + b ) ] ,
p D | I ( d 0 , d 1 , d 2 , d 3 | i 0 , i 1 , i 2 , i 3 ) = k = 0 3 exp [ ( a i k + b ) ] ( a i k + b ) d k d k ! .
p D , I ( d 0 , d 1 , d 2 , d 3 , i 0 , i 1 , i 2 , i 3 ) = p D | I ( d 0 , d 1 , d 2 , d 3 | i 0 , i 1 , i 2 , i 3 ) p I ( i 0 , i 1 , i 2 , i 3 ) .
p D 0 , D 1 , D 2 , D 3 ( d 0 , d 1 , d 2 , d 3 ) = 0 0 0 0 p D , I ( d 0 , d 1 , d 2 , d 3 , i 0 , i 1 , i 2 , i 3 ) d i 0 d i 1 d i 2 d i 3 = 0 0 0 0 k = 0 3 exp [ ( a i k + b ) ] ( a i k + b ) d k d k ! p I 0 , I 1 , I 2 , I 3 ( i 0 , i 1 , i 2 , i 3 ) d i 0 d i 1 d i 2 d i 3 ,
D = [ D 0 D 1 D 2 D 3 ] = [ Poisson ( a I 0 + b ) + N 0 Poisson ( a I 1 + b ) + N 1 Poisson ( a I 2 + b ) + N 2 Poisson ( a I 3 + b ) + N 3 ] = [ L 0 + N 0 L 1 + N 1 L 2 + N 2 L 3 + N 3 ] ,
p L 0 , L 1 , L 2 , L 3 | I 0 , I 1 , I 2 , I 3 ( l 0 , l 1 , l 2 , l 3 | i 0 , i 1 , i 2 , i 3 ) = m 0 = 0 m 1 = 0 m 2 = 0 m 3 = 0 j = 0 3 exp [ ( a i j + b ) ] ( a i j + b ) l j l j ! δ ( l 0 m 0 , l 1 m 1 , l 2 m 2 , l 3 m 3 ) .
p N 0 , N 1 , N 2 , N 3 | I 0 , I 1 , I 2 , I 3 ( n 0 , n 1 , n 2 , n 3 | i 0 , i 1 , i 2 , i 3 ) = j = 0 3 p N j ( n j ) = j = 0 3 1 2 π σ exp ( n j 2 2 σ 2 ) .
p D 0 , D 1 , D 2 , D 3 | I 0 , I 1 , I 2 , I 3 ( d 0 , d 1 , d 2 , d 3 | i 0 , i 1 , i 2 , i 3 ) = m 0 = 0 m 1 = 0 m 2 = 0 m 3 = 0 P ( l 0 = m 0 , l 1 = m 1 , l 2 = m 2 , l 3 = m 3 n 0 = d 0 m 0 , n 1 = d 1 m 1 , n 2 = d 2 m 2 , n 3 = d 3 m 3 | I ) = m 0 = 0 m 1 = 0 m 2 = 0 m 3 = 0 j = 0 3 exp [ ( a i j + b ) ] ( a i j + b ) m j m j ! 1 2 π σ exp ( ( d j m j ) 2 2 σ 2 ) .
p D 0 , D 1 , D 2 , D 3 ( d 0 , d 1 , d 2 , d 3 ) = 0 0 0 0 p D | I ( d 0 , d 1 , d 2 , d 3 | i 0 , i 1 , i 2 , i 3 ) p I ( i 0 , i 1 , i 2 , i 3 ) d i 0 d i 1 d i 2 d i 3 ,
θ = [ θ 1 θ 2 θ 3 ] = [ J x x J y y | J x y | ] .
F m , k = E [ θ m ln p D x , D y ( d x , d y ) θ k ln p D x , D y ( d x , d y ) ] .
DOP = 1 4 det ( J ) [ tr ( J ) ] 2 = ψ ( θ ) = 1 4 θ 1 θ 2 θ 3 2 ( θ 1 + θ 2 ) 2 .
var ( DOP ˆ ) ( ψ ( θ ) θ ) T F 1 ψ ( θ ) θ ,
ψ ( θ ) θ = [ ψ ( θ ) θ 1 ψ ( θ ) θ 2 ψ ( θ ) θ 3 ] .
p D x , D y ( d x , d y ) = 1 det ( J ) exp ( J y y d x + J x x d y det ( J ) ) I 0 ( 2 | J x y | det ( J ) d x d y ) ,
D = [ D x D y ] = [ I x + N x I y + N y ] ,
p D x , D y ( d x , d y ) = d y d x p I x , I y ( d x n x , d y n y ) p N x ( n x ) p N y ( n y ) d n x d n y ,
D = [ D x D y ] = [ Poisson ( a I x + b ) Poisson ( a I y + b ) ] ,
p D x , D y | I x , I y ( d x , d y | i x , i y ) = exp [ ( a i x + b ) ] ( a i x + b ) d x d x ! exp [ ( a i y + b ) ] ( a i y + b ) d y d y ! .
p D x , D y , I x , I y ( d x , d y , i x , i y ) = p D x , D y | I x , I y ( d x , d y | i x , i y ) P I x , I y ( i x , i y ) .
p D x , D y ( d x , d y ) = 0 0 p D x , D y , I x , I y ( d x , d y , i x , i y ) d i x d i y = 0 0 exp [ ( a i x + b ) ] ( a i x + b ) d x d x ! exp [ ( a i y + b ) ] ( a i y + b ) d y d y ! 1 det ( J ) exp ( J y y i x + J x x i y det ( J ) ) I 0 ( 2 | J x y | det ( J ) i x i y ) d i x d i y .
D = [ D x D y ] = [ Poisson ( a I x + b ) + N x Poisson ( a I y + b ) + N y ] = [ L x + N x L y + N y ] ,
p L x , L y | I x , I y ( l x , l y | i x , i y ) = m = 0 n = 0 exp [ ( a i x + b ) ] ( a i x + b ) l x l x ! exp [ ( a i y + b ) ] ( a i y + b ) l y l y ! δ ( l x m , l y n )
p N x , N y | I x , I y ( n x , n y | i x , i y ) = p N x , N y ( n x , n y ) = 1 2 π σ exp ( n x 2 2 σ 2 ) 1 2 π σ exp ( n y 2 2 σ 2 ) .
p D x , D y | I x , I y ( d x , d y | i x , i y ) = m = 0 k = 0 P ( l x = m , l y = k , n x = d x m , n y = d y k | i x , i y ) = m = 0 k = 0 ( a i x + b ) m exp [ ( a i x + b ) ] m ! ( a i y + b ) k exp [ ( a i y + b ) ] k ! 1 2 π σ exp ( ( d x m ) 2 2 σ 2 ) 1 2 π σ exp ( ( d y k ) 2 2 σ 2 ) .
p D x , D y ( d x , d y ) = 0 0 p D x , D y | I x , I y ( d x , d y | i x , i y ) p I x , I y ( i x , i y ) d i x d i y = 0 0 m = 0 k = 0 ( a i x + b ) m exp [ ( a i x + b ) ] m ! ( a i y + b ) k exp [ ( a i y + b ) ] k ! 1 2 π σ exp ( ( d x m ) 2 2 σ 2 ) 1 2 π σ exp ( ( d y k ) 2 2 σ 2 ) 1 det ( J ) exp ( J y y i x + J x x i y det ( J ) ) I 0 ( 2 | J x y | det ( J ) i x i y ) d i x d i y .
θ = [ θ 1 θ 2 ] = [ λ 1 λ 2 ] .
F m , k = E [ θ m ln p D ( d ) θ k ln p D ( d ) ] ,
DOP = | λ 1 λ 2 | λ 1 + λ 2 = | θ 1 θ 2 | θ 1 + θ 2 .
var ( DOP ˆ ) ( ψ ( θ ) θ ) T F 1 ψ ( θ ) θ ,
ψ ( θ ) θ = [ ψ ( θ ) θ 1 ψ ( θ ) θ 2 ] .
p D ( d ) = 1 λ 1 λ 2 [ exp ( d λ 1 ) exp ( d λ 2 ) ] .
D = I + N ,
p D ( d ) = d p I ( d n ) p N ( n ) d n = d 1 λ 1 λ 2 [ exp ( d n λ 1 ) exp ( d n λ 2 ) ] 1 2 π σ exp ( n 2 σ 2 ) d n = 1 λ 1 λ 2 [ exp ( 2 λ 1 d σ 2 2 λ 1 2 ) Φ ( d σ σ λ 1 ) exp ( 2 λ 2 d σ 2 2 λ 2 2 ) Φ ( d σ σ λ 2 ) ] .
D = Poisson ( a I + b ) = Poisson [ a ( I x + I y ) + b ] ,
p D | I ( d | i ) = exp [ ( a i + b ) ] ( a i + b ) d d ! .
p D , I ( d , i ) = p D | I ( d | i ) P I ( i ) .
p D ( d ) = 0 p D , I ( d , i ) d i = 0 exp [ ( a i + b ) ] ( a i + b ) d d ! 1 λ 1 λ 2 [ exp ( i λ 1 ) exp ( i λ 2 ) ] d i = exp ( b a λ 1 ) Ei ( d , b , X ) a ( λ 1 λ 2 ) d ! exp ( b a λ 2 ) Ei ( d , b , Y ) a ( λ 1 λ 2 ) d ! ,
Ei ( n , b , z ) = b λ n exp ( z λ ) d λ = { z n 1 Γ ( n + 1 , b z ) b > 0 z n 1 n ! b = 0 } ,
D = Poisson ( a I + b ) + N ,
p L | I ( l , i ) = m = 0 ( a i + b ) l exp [ ( a i + b ) ] l ! δ ( l m ) ,
p N ( n ) = 1 2 π σ exp ( n 2 2 σ 2 ) .
p D , N | I ( d , n | i ) = p L | I ( l , i ) p N ( n ) = m = 0 ( a i + b ) d n exp [ ( a i + b ) ] ( d n ) ! δ ( d n m ) 1 2 π σ exp ( n 2 2 σ 2 ) ,
p D | I ( d | i ) = d x p D , N | I ( d , n | i ) d n = d x m = 0 ( a i + b ) d n exp [ ( a i + b ) ] ( d n ) ! δ ( d n m ) 1 2 π σ exp ( n 2 2 σ 2 ) d n = m = 0 ( a i + b ) m exp [ ( a i + b ) ] m ! 1 2 π σ exp ( ( d m ) 2 2 σ 2 ) .
p D ( d ) = 0 p D | I ( d | i ) p I ( i ) d i = 0 m = 0 ( a i + b ) m exp [ ( a i + b ) ] m ! 1 2 π σ exp ( ( d m ) 2 2 σ 2 ) 1 λ 1 λ 2 [ exp ( i λ 1 ) exp ( i λ 2 ) ] d i = m = 0 1 2 π σ exp ( ( d m ) 2 2 σ 2 ) 0 ( a i + b ) m exp [ ( a i + b ) ] m ! 1 λ 1 λ 2 [ exp ( i λ 1 ) exp ( i λ 2 ) ] d i = m = 0 1 2 π σ exp ( ( d m ) 2 2 σ 2 ) [ exp ( b a λ 1 ) Ei ( m , b , X ) a ( λ 1 λ 2 ) m ! exp ( b a λ 2 ) Ei ( m , b , Y ) a ( λ 1 λ 2 ) m ! ] .
Ei ( n , b , z ) = b λ n exp ( z λ ) d λ = { z n 1 Γ ( n + 1 , b z ) b > 0 z n 1 n ! b = 0 } ,
SNR = μ μ T tr ( Σ ) ,
Σ = E [ ( D μ ) ( D μ ) T ]
SNR = 2 1 + DOP 2 .
SNR G = 1 + 2 J x x J y y σ 2 J x x 2 + J y y 2 + σ 2 ,
SNR P = 1 + 2 J x x J y y I ¯ J x x 2 + J y y 2 + I ¯ .
I ¯ = σ x 2 + σ y 2 = σ 2 ,
I ¯ = σ 0 2 + σ 1 2 + σ 2 2 + σ 3 2 = σ 2 .
S 0 2 = S 1 2 + S 2 2 + S 3 2 .
J S 0 , S 1 , S 2 , S 3 a x , a y , θ x , θ y
p S 0 , S 1 , S 2 , S 3 ( s 0 , s 1 , s 2 , s 3 ) = p S 0 | S 1 , S 2 , S 3 ( s 0 | s 1 , s 2 , s 3 ) p S 1 , S 2 , S 3 ( s 1 , s 2 , s 3 ) .
p S 0 | S 1 , S 2 , S 3 ( s 0 | s 1 , s 2 , s 3 ) = δ ( s 0 s 1 2 + s 2 2 + s 3 2 ) .
p U ( u x , u y ) = 1 π 2 det ( J ) exp ( 1 det ( J ) [ J y y | u x | 2 + J x x | u y | 2 2 Re ( J x y u x * u y ) ] ) .
{ u x r = u x r s 1 = | u x | 2 | u y | 2 s 2 = u x u y * + u x * u y s 3 = j ( u x u y * u x * u y ) } , ,
J ( u x r , s 1 , s 2 , s 3 u x r , u x i , u y r , u y i ) = 8 u x i ( | u x | 2 + | u y | 2 ) ,
p U x r , S 1 , S 2 , S 3 ( u x r , s 1 , s 2 , s 3 ) = ( 8 1 2 ( s 0 + s 1 ) ( u x r ) 2 s 1 2 + s 2 2 + s 3 2 ) 1 1 π 2 det ( J ) exp ( 1 det ( J ) [ J y y s 0 + s 1 2 + J x x s 0 s 1 2 2 Re ( J x y s 2 + j s 3 2 ) ] ) .
p S 1 , S 2 , S 3 ( s 1 , s 2 , s 3 ) = ( s 0 + s 1 ) 2 ( s 0 + s 1 ) 2 p U x r , S 1 , S 2 , S 3 ( u x r , s 1 , s 2 , s 3 ) d u x r = 1 8 π d s 1 2 + s 2 2 + s 3 2 exp [ 1 2 d ( S 0 s 1 2 + s 2 2 + s 3 2 S 1 s 1 S 2 s 2 S 3 s 3 ) ] .
d = det ( J ) = 1 4 ( S 0 2 S 1 2 S 2 2 S 3 2 ) .
p S 0 , S 1 , S 2 , S 3 ( s 0 , s 1 , s 2 , s 3 ) = δ ( s 0 s 1 2 + s 2 2 + s 3 2 ) 8 π d s 0 exp [ 1 2 d ( S 0 s 0 S 1 s 1 S 2 s 2 S 3 s 3 ) ] .
J ( i 0 , i 1 , i 2 , i 3 s 0 , s 1 , s 2 , s 3 ) = 0.012 ,
p I 0 , I 1 , I 2 , I 3 ( i 0 , i 1 , i 2 , i 3 ) = 125 δ ( s 0 s 1 2 + s 2 2 + s 3 2 ) 12 π d s 0 exp [ 1 2 d ( S 0 s 0 S 1 s 1 S 2 s 2 S 3 s 3 ) ] ,
s 0 = i 0 + i 1 + i 2 + i 3 ,
s 1 = 5 3 ( i 0 + i 1 i 2 i 3 ) ,
s 2 = 2.5 ( i 0 i 1 ) ,
s 3 = 2.5 ( i 2 i 3 ) .

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