Abstract

We present an adaptive technique for the estimation of nonuniformity parameters of infrared focal-plane arrays that is robust with respect to changes and uncertainties in scene and sensor characteristics. The proposed algorithm is based on using a bank of Kalman filters in parallel. Each filter independently estimates state variables comprising the gain and the bias matrices of the sensor, according to its own dynamic-model parameters. The supervising component of the algorithm then generates the final estimates of the state variables by forming a weighted superposition of all the estimates rendered by each Kalman filter. The weights are computed and updated iteratively, according to the a posteriori-likelihood principle. The performance of the estimator and its ability to compensate for fixed-pattern noise is tested using both simulated and real data obtained from two cameras operating in the mid- and long-wave infrared regime.

© 2006 Optical Society of America

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  1. J. Harris and Y.-M. Chiang, "Nonuniformity correction of infrared image sequences using the constant-statistics constraint," IEEE Trans. Image Process. 8, 1148-1151 (1999).
    [CrossRef]
  2. A. Milton, F. Barone, and M. Kruer, "Influence of nonuniformity on infrared focal plane array performance," Opt. Eng. (Bellingham) 24, 855-862 (1985).
  3. J. Mooney, F. Shepherd, W. Ewing, J. Murguia, and J. Silverman, "Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. (Bellingham) 28, 1151-1161 (1989).
  4. D. Perry and E. Dereniak, "Linear theory of nonuniformity correction in infrared staring sensors," Opt. Eng. (Bellingham) 32, 1854-1859 (1993).
    [CrossRef]
  5. H. Zhou, S. Liu, D. Wang, and Y. Cheng, "Solution for the nonuniformity correction of infrared focal plane arrays," Appl. Opt. 44, 2928-2932 (2005).
    [CrossRef] [PubMed]
  6. E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, and B. Yasuda, "Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery," in Applications of Digital Image Processing XXII, A.G.Tescher, ed., Proc. SPIE 3808, 150-161 (1999).
  7. R. C. Hardie, M. M. Hayat, E. E. Armstrong, and B. Yasuda, "Scene-based nonuniformity correction using video sequences and registration," Appl. Opt. 39, 1241-1250 (2000).
    [CrossRef]
  8. B. Ratliff, M. Hayat, and R. Hardie, "An algebraic algorithm for nonuniformity correction in focal-plane arrays," J. Opt. Soc. Am. A 19, 1737-1747 (2002).
    [CrossRef]
  9. B. Ratliff, M. Hayat, and R. Hardie, "Radiometrically accurate scene-based nonuniformity correction for array sensors," J. Opt. Soc. Am. A 20, 1890-1899 (2002).
    [CrossRef]
  10. B. Ratliff, M. Hayat, and J. Tyo, "Generalized algebraic scene-based nonuniformity correction algorithm," J. Opt. Soc. Am. A 22, 239-249 (2005).
    [CrossRef]
  11. P. Narendra, "Reference-free nonuniformity compensation for IR imaging arrays," in Smart Sensors II, D.F.Barbe, ed., Proc. SPIE 252, 10-17 (1980).
  12. S. Torres and M. Hayat, "Kalman filtering for adaptive nonuniformity correction in infrared focal-plane arrays," J. Opt. Soc. Am. A 20, 470-480 (2003).
    [CrossRef]
  13. S. Torres, J. Pezoa, and M. Hayat, "Scene-based nonuniformity correction for focal plane arrays by the method of the inverse covariance form," Appl. Opt. 42, 5872-5881 (2003).
    [CrossRef] [PubMed]
  14. D. Scribner, M. Kruer, and J. Killiany, "Infrared focal plane array technology," Proc. IEEE 79, 66-85 (1991).
    [CrossRef]
  15. D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
    [CrossRef]
  16. Y.-M. Chiang and J. G. Harris, "An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays," Analog Integr. Circuits Signal Process. 12, 231-238 (1997).
    [CrossRef]
  17. M. Hayat, S. Torres, E. Armstrong, B. Yasuda, and S. Cain, "Statistical algorithm for nonuniformity correction in focal-plane arrays," Appl. Opt. 38, 772-780 (1999).
    [CrossRef]
  18. S. Sims, D. Lianiotis, and D. Magill, "Recursive algorithm for the calculation of the adaptive Kalman filter weighting coefficients," IEEE Trans. Autom. Control 14, 215-218 (1969).
    [CrossRef]
  19. G. C. Holst, CCD Arrays, Cameras and Displays (SPIE, 1996).
  20. R. Hawkes and J. Moore, "Performance bounds for adaptive estimation," Proc. IEEE 64, 1143-1150 (1976).
    [CrossRef]
  21. B. Anderson and J. Moore, Optimal Filtering (Prentice Hall, 1979).
  22. M. F. Tappen and W. T. Freeman, "Comparison of graph cuts with belief propagation for stereo, using identical MRF parameters," in Proceedings of the Ninth IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 900-907.
    [CrossRef]

2005 (2)

2003 (2)

2002 (2)

2000 (1)

1999 (2)

M. Hayat, S. Torres, E. Armstrong, B. Yasuda, and S. Cain, "Statistical algorithm for nonuniformity correction in focal-plane arrays," Appl. Opt. 38, 772-780 (1999).
[CrossRef]

J. Harris and Y.-M. Chiang, "Nonuniformity correction of infrared image sequences using the constant-statistics constraint," IEEE Trans. Image Process. 8, 1148-1151 (1999).
[CrossRef]

1997 (1)

Y.-M. Chiang and J. G. Harris, "An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays," Analog Integr. Circuits Signal Process. 12, 231-238 (1997).
[CrossRef]

1993 (1)

D. Perry and E. Dereniak, "Linear theory of nonuniformity correction in infrared staring sensors," Opt. Eng. (Bellingham) 32, 1854-1859 (1993).
[CrossRef]

1991 (1)

D. Scribner, M. Kruer, and J. Killiany, "Infrared focal plane array technology," Proc. IEEE 79, 66-85 (1991).
[CrossRef]

1989 (1)

J. Mooney, F. Shepherd, W. Ewing, J. Murguia, and J. Silverman, "Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. (Bellingham) 28, 1151-1161 (1989).

1985 (1)

A. Milton, F. Barone, and M. Kruer, "Influence of nonuniformity on infrared focal plane array performance," Opt. Eng. (Bellingham) 24, 855-862 (1985).

1976 (1)

R. Hawkes and J. Moore, "Performance bounds for adaptive estimation," Proc. IEEE 64, 1143-1150 (1976).
[CrossRef]

1969 (1)

S. Sims, D. Lianiotis, and D. Magill, "Recursive algorithm for the calculation of the adaptive Kalman filter weighting coefficients," IEEE Trans. Autom. Control 14, 215-218 (1969).
[CrossRef]

Anderson, B.

B. Anderson and J. Moore, Optimal Filtering (Prentice Hall, 1979).

Armstrong, E.

Armstrong, E. E.

R. C. Hardie, M. M. Hayat, E. E. Armstrong, and B. Yasuda, "Scene-based nonuniformity correction using video sequences and registration," Appl. Opt. 39, 1241-1250 (2000).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, and B. Yasuda, "Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery," in Applications of Digital Image Processing XXII, A.G.Tescher, ed., Proc. SPIE 3808, 150-161 (1999).

Barone, F.

A. Milton, F. Barone, and M. Kruer, "Influence of nonuniformity on infrared focal plane array performance," Opt. Eng. (Bellingham) 24, 855-862 (1985).

Cain, S.

Calufield, J.

D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
[CrossRef]

Cheng, Y.

Chiang, Y.-M.

J. Harris and Y.-M. Chiang, "Nonuniformity correction of infrared image sequences using the constant-statistics constraint," IEEE Trans. Image Process. 8, 1148-1151 (1999).
[CrossRef]

Y.-M. Chiang and J. G. Harris, "An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays," Analog Integr. Circuits Signal Process. 12, 231-238 (1997).
[CrossRef]

Colbert, M.

D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
[CrossRef]

Dereniak, E.

D. Perry and E. Dereniak, "Linear theory of nonuniformity correction in infrared staring sensors," Opt. Eng. (Bellingham) 32, 1854-1859 (1993).
[CrossRef]

Descour, M.

D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
[CrossRef]

Ewing, W.

J. Mooney, F. Shepherd, W. Ewing, J. Murguia, and J. Silverman, "Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. (Bellingham) 28, 1151-1161 (1989).

Freeman, W. T.

M. F. Tappen and W. T. Freeman, "Comparison of graph cuts with belief propagation for stereo, using identical MRF parameters," in Proceedings of the Ninth IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 900-907.
[CrossRef]

Hardie, R.

Hardie, R. C.

R. C. Hardie, M. M. Hayat, E. E. Armstrong, and B. Yasuda, "Scene-based nonuniformity correction using video sequences and registration," Appl. Opt. 39, 1241-1250 (2000).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, and B. Yasuda, "Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery," in Applications of Digital Image Processing XXII, A.G.Tescher, ed., Proc. SPIE 3808, 150-161 (1999).

Harris, J.

J. Harris and Y.-M. Chiang, "Nonuniformity correction of infrared image sequences using the constant-statistics constraint," IEEE Trans. Image Process. 8, 1148-1151 (1999).
[CrossRef]

Harris, J. G.

Y.-M. Chiang and J. G. Harris, "An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays," Analog Integr. Circuits Signal Process. 12, 231-238 (1997).
[CrossRef]

Hawkes, R.

R. Hawkes and J. Moore, "Performance bounds for adaptive estimation," Proc. IEEE 64, 1143-1150 (1976).
[CrossRef]

Hayat, M.

Hayat, M. M.

R. C. Hardie, M. M. Hayat, E. E. Armstrong, and B. Yasuda, "Scene-based nonuniformity correction using video sequences and registration," Appl. Opt. 39, 1241-1250 (2000).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, and B. Yasuda, "Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery," in Applications of Digital Image Processing XXII, A.G.Tescher, ed., Proc. SPIE 3808, 150-161 (1999).

Holst, G. C.

G. C. Holst, CCD Arrays, Cameras and Displays (SPIE, 1996).

Hunt, J.

D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
[CrossRef]

Killiany, J.

D. Scribner, M. Kruer, and J. Killiany, "Infrared focal plane array technology," Proc. IEEE 79, 66-85 (1991).
[CrossRef]

Kruer, M.

D. Scribner, M. Kruer, and J. Killiany, "Infrared focal plane array technology," Proc. IEEE 79, 66-85 (1991).
[CrossRef]

A. Milton, F. Barone, and M. Kruer, "Influence of nonuniformity on infrared focal plane array performance," Opt. Eng. (Bellingham) 24, 855-862 (1985).

D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
[CrossRef]

Lianiotis, D.

S. Sims, D. Lianiotis, and D. Magill, "Recursive algorithm for the calculation of the adaptive Kalman filter weighting coefficients," IEEE Trans. Autom. Control 14, 215-218 (1969).
[CrossRef]

Liu, S.

Magill, D.

S. Sims, D. Lianiotis, and D. Magill, "Recursive algorithm for the calculation of the adaptive Kalman filter weighting coefficients," IEEE Trans. Autom. Control 14, 215-218 (1969).
[CrossRef]

Milton, A.

A. Milton, F. Barone, and M. Kruer, "Influence of nonuniformity on infrared focal plane array performance," Opt. Eng. (Bellingham) 24, 855-862 (1985).

Mooney, J.

J. Mooney, F. Shepherd, W. Ewing, J. Murguia, and J. Silverman, "Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. (Bellingham) 28, 1151-1161 (1989).

Moore, J.

R. Hawkes and J. Moore, "Performance bounds for adaptive estimation," Proc. IEEE 64, 1143-1150 (1976).
[CrossRef]

B. Anderson and J. Moore, Optimal Filtering (Prentice Hall, 1979).

Murguia, J.

J. Mooney, F. Shepherd, W. Ewing, J. Murguia, and J. Silverman, "Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. (Bellingham) 28, 1151-1161 (1989).

Narendra, P.

P. Narendra, "Reference-free nonuniformity compensation for IR imaging arrays," in Smart Sensors II, D.F.Barbe, ed., Proc. SPIE 252, 10-17 (1980).

Perry, D.

D. Perry and E. Dereniak, "Linear theory of nonuniformity correction in infrared staring sensors," Opt. Eng. (Bellingham) 32, 1854-1859 (1993).
[CrossRef]

Pezoa, J.

Ratliff, B.

Sarkady, K.

D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
[CrossRef]

Scribner, D.

D. Scribner, M. Kruer, and J. Killiany, "Infrared focal plane array technology," Proc. IEEE 79, 66-85 (1991).
[CrossRef]

D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
[CrossRef]

Shepherd, F.

J. Mooney, F. Shepherd, W. Ewing, J. Murguia, and J. Silverman, "Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. (Bellingham) 28, 1151-1161 (1989).

Silverman, J.

J. Mooney, F. Shepherd, W. Ewing, J. Murguia, and J. Silverman, "Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. (Bellingham) 28, 1151-1161 (1989).

Sims, S.

S. Sims, D. Lianiotis, and D. Magill, "Recursive algorithm for the calculation of the adaptive Kalman filter weighting coefficients," IEEE Trans. Autom. Control 14, 215-218 (1969).
[CrossRef]

Tappen, M. F.

M. F. Tappen and W. T. Freeman, "Comparison of graph cuts with belief propagation for stereo, using identical MRF parameters," in Proceedings of the Ninth IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 900-907.
[CrossRef]

Torres, S.

Torres, S. N.

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, and B. Yasuda, "Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery," in Applications of Digital Image Processing XXII, A.G.Tescher, ed., Proc. SPIE 3808, 150-161 (1999).

Tyo, J.

Wang, D.

Yasuda, B.

R. C. Hardie, M. M. Hayat, E. E. Armstrong, and B. Yasuda, "Scene-based nonuniformity correction using video sequences and registration," Appl. Opt. 39, 1241-1250 (2000).
[CrossRef]

M. Hayat, S. Torres, E. Armstrong, B. Yasuda, and S. Cain, "Statistical algorithm for nonuniformity correction in focal-plane arrays," Appl. Opt. 38, 772-780 (1999).
[CrossRef]

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, and B. Yasuda, "Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery," in Applications of Digital Image Processing XXII, A.G.Tescher, ed., Proc. SPIE 3808, 150-161 (1999).

Zhou, H.

Analog Integr. Circuits Signal Process. (1)

Y.-M. Chiang and J. G. Harris, "An analog integrated circuit for continuous-time gain and offset calibration of sensor arrays," Analog Integr. Circuits Signal Process. 12, 231-238 (1997).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Autom. Control (1)

S. Sims, D. Lianiotis, and D. Magill, "Recursive algorithm for the calculation of the adaptive Kalman filter weighting coefficients," IEEE Trans. Autom. Control 14, 215-218 (1969).
[CrossRef]

IEEE Trans. Image Process. (1)

J. Harris and Y.-M. Chiang, "Nonuniformity correction of infrared image sequences using the constant-statistics constraint," IEEE Trans. Image Process. 8, 1148-1151 (1999).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Eng. (Bellingham) (3)

A. Milton, F. Barone, and M. Kruer, "Influence of nonuniformity on infrared focal plane array performance," Opt. Eng. (Bellingham) 24, 855-862 (1985).

J. Mooney, F. Shepherd, W. Ewing, J. Murguia, and J. Silverman, "Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. (Bellingham) 28, 1151-1161 (1989).

D. Perry and E. Dereniak, "Linear theory of nonuniformity correction in infrared staring sensors," Opt. Eng. (Bellingham) 32, 1854-1859 (1993).
[CrossRef]

Proc. IEEE (2)

D. Scribner, M. Kruer, and J. Killiany, "Infrared focal plane array technology," Proc. IEEE 79, 66-85 (1991).
[CrossRef]

R. Hawkes and J. Moore, "Performance bounds for adaptive estimation," Proc. IEEE 64, 1143-1150 (1976).
[CrossRef]

Other (6)

B. Anderson and J. Moore, Optimal Filtering (Prentice Hall, 1979).

M. F. Tappen and W. T. Freeman, "Comparison of graph cuts with belief propagation for stereo, using identical MRF parameters," in Proceedings of the Ninth IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 900-907.
[CrossRef]

P. Narendra, "Reference-free nonuniformity compensation for IR imaging arrays," in Smart Sensors II, D.F.Barbe, ed., Proc. SPIE 252, 10-17 (1980).

D. Scribner, K. Sarkady, M. Kruer, J. Calufield, J. Hunt, M. Colbert, and M. Descour, "Adaptive retina-like preprocessing for imaging detector arrays," in Proceedings of the IEEE International Conference on Neural Networks (IEEE, 1993), Vol. 3, pp. 1955-1960.
[CrossRef]

G. C. Holst, CCD Arrays, Cameras and Displays (SPIE, 1996).

E. E. Armstrong, M. M. Hayat, R. C. Hardie, S. N. Torres, and B. Yasuda, "Nonuniformity correction for improved registration and high-resolution image reconstruction in IR imagery," in Applications of Digital Image Processing XXII, A.G.Tescher, ed., Proc. SPIE 3808, 150-161 (1999).

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Figures (4)

Fig. 1
Fig. 1

Image frame 500 from the third block ( k = 3 ) (a) true image, (b) noisy image, (c) corresponding corrected version of the noisy image obtained by the first KF of the bank, (d) corrected version of the noisy image obtained by the fourth KF.

Fig. 2
Fig. 2

Computing time required by the MMAE and its corresponding RMSE obtained versus the subsampling factor used to calculate the a posteriori conditional probabilities.

Fig. 3
Fig. 3

(a) Sample raw image of the fifth block ( k = 5 ) taken from the InSb data set, (b) corrected version of the raw image obtained by the first KF, (c) corrected image obtained by the second KF, (d) corrected frame obtained by the fourth KF.

Fig. 4
Fig. 4

(a) Sample raw image of the first block ( k = 1 ) taken from the HgCdTe data set, (b) corrected version of the raw image obtained by the first KF, (c) corrected frame obtained by the fourth KF, (d) corrected frame obtained by the fifth KF. Note that the image in (d), which has the highest a posteriori probability, offers a slight advantage in performing NUC.

Tables (3)

Tables Icon

Table 1 Spatial Average of the a posteriori Conditional Probabilities p ̂ θ q y k for Each Model a

Tables Icon

Table 2 NUC Performance Parameters Obtained by the MMAE for the Experiment Corresponding to Table 1

Tables Icon

Table 3 Spatial Average of the a posteriori Conditional Probabilities, p ̂ θ q y k , for Each Model When the MMAE is Tracking the Artificial NU Added to a Sequence of Four Blocks of Data a

Equations (18)

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

Y i j ( n ) = A i j T i j ( n ) + B i j + V i j ( n ) ,
X k = Φ X k 1 + W k ,
Y k = H k X k + V k ,
P k = Φ P k 1 Φ T + Q ,
C k = H ¯ P k H ¯ T + R + σ T 2 ( σ A 0 2 + A ¯ 0 ) I l k , l k ,
K k = P k H ¯ T C k 1 ,
P k = ( I 2 , 2 K k H ¯ ) P k ,
X ̂ k = Φ X ̂ k 1 + M ,
X ̂ k = X ̂ k + K k ( Y k H ¯ X ̂ k ) ,
X ̂ 0 = E [ X 0 ] = ( A ¯ 0 B ¯ 0 ) , P 0 = [ σ A 0 2 0 0 σ B 0 2 ] .
X ̂ ̂ k = E [ E [ X k Y 1 , , Y k , Θ ] Y 1 , , Y k ] = E [ X ̂ k ( Θ ) Y 1 , , Y k ] = q = 1 N X ̂ k ( θ q ) P { Θ = θ q Y 1 = y 1 , , Y k = y k } ,
p ̂ θ q y k = f Θ , Y 1 , , Y k ( θ q , y 1 , , y k ) f Y 1 , , Y k ( y 1 , , y k ) = f Y k Y 1 , , Y k 1 , Θ ( y k y 1 , , y k 1 , θ q ) p ̂ θ q y k 1 f Y 1 , , Y k 1 ( y 1 , , y k 1 ) f Y k Y 1 , , Y k 1 ( y k y 1 , , y k 1 ) f Y 1 , , Y k 1 ( y 1 , , y k 1 ) = f Y k Y 1 , , Y k 1 , Θ ( y k y 1 , , y k 1 , θ q ) f Y y Y 1 , , Y k 1 ( Y k y 1 , , y k 1 ) p ̂ θ q y k 1 = f Y k Y 1 , , Y k 1 , Θ ( y k y 1 , , y k 1 , θ q ) d = 1 N f Y k Y 1 , , Y k 1 , Θ ( y k y 1 , , y k 1 , θ d ) p ̂ θ d y k 1 p ̂ θ q y k 1 .
E [ Y k Y 1 , , Y k 1 , θ q ] Y ̂ k ( θ q ) = H ¯ ( θ q ) Φ ( θ q ) X ̂ k 1 ( θ q ) ,
E [ ( Y k ( θ q ) Y ̂ k ( θ q ) ) ( Y k ( θ q ) Y ̂ k ( θ q ) ) T ] = C k ( θ q ) .
f Y k Y 1 , , Y k 1 , Θ ( y k y 1 , , y k 1 , θ q ) = exp { 1 2 [ y k Y ̂ k ( θ q ) ] T C k ( θ q ) 1 [ y k Y ̂ k ( θ q ) ] } 2 π C k ( θ q ) .
E [ r k ( θ q ) r k + n ( θ q ) T ] = H ¯ ( θ q ) ( Φ ( θ q ) P n 1 ( θ q ) Φ T ( θ q ) + Q ( θ q ) + M ( θ q ) M T ( θ q ) ) H ¯ T ( θ q ) ,
Y ¯ = A ¯ 0 T ¯ + B ¯ 0 ,
σ Y 2 = σ A 0 2 ( σ T 0 + T ¯ 2 ) + A ¯ 0 2 σ T 2 + σ B 0 2 ,

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