Abstract

We investigate the problem of dynamic calibration for our structured light system. First, a method is presented to estimate the rotation matrix and translation vector between the camera and the projector using plane-based homography. Then an approach is introduced to analyze theoretically the error sensitivity in the estimated pose parameters with respect to noise in the projection points. This algorithm is simple and easy to implement. Finally, some numerical simulations and real data experiments are carried out to validate our method.

© 2008 Optical Society of America

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  1. R. Valkenburg and A. McIvor, "Accurate 3D measurement using a structured light system," Image Vis. Comput. 16, 99-110 (1998).
    [CrossRef]
  2. M. Wilczkowiak, P. Sturm, and E. Boyer, "Using geometric constraints through parallelepipeds for calibration and 3D modeling," IEEE Trans. Pattern Anal. Mach. Intell. 27, 194-207 (2005).
    [CrossRef] [PubMed]
  3. Y. F. Li and S. Y. Chen, "Automatic recalibration of a structured light vision system," IEEE Trans. Rob. Autom. 19, 259-268 (2003).
    [CrossRef]
  4. C. Huang, C. Chen, and P. Chung, "An improved algorithm for two-image camera self-calibration and Euclidean structure recovery using absolute quadric," Pattern Recogn. 37, 1713-1722 (2004).
    [CrossRef]
  5. L. Quan and Z. Lan, "Linear N-point camera pose determination," IEEE Trans. Pattern Anal. Mach. Intell. 21, 774-780 (1999).
    [CrossRef]
  6. X-S. Gao, and X.-R. Hou, J. Thang, and H.-F. Cheng, "Complete solution classification for the perspective-three-point problem," IEEE Trans. Pattern Anal. Mach. Intell. 25, 930-943 (2003).
    [CrossRef]
  7. H. C. Longuet-Higgins, "A computer algorithm for reconstructing a scene from two projections," Nature 293, 133-135 (1981).
    [CrossRef]
  8. A. Gruen and T. Huang, Calibration and Orientation of Cameras in Computer Vision (Springer-Verlag, 2001).
  9. O. Faugeras and S. Maybank, "Motion from point matches: multiplicity of solutions," Int. J. Comput. Vis. 4, 225-246 (1990).
    [CrossRef]
  10. R. Hartley, "Estimation of relative camera positions for uncalibrated cameras," in Lecture Notes in Computer Science (Springer-Verlag1992), pp. 579-580.
  11. B. Triggs, "Routines for relative pose of two calibrated cameras from 5 points," Tech. Rep. (INRIA, 2000)http://www.inrialpes.fr/movi/people/Triggs.
  12. D. Nister, "An efficient solution to the five-point relative pose problem," IEEE Trans. Pattern Anal. Mach. Intell. 26, 756-770 (2004).
    [CrossRef]
  13. J. C. Hay, "Optical motion and space perception: an extension of Gibson's analysis," Psychol. Rev. 73, 550-565 (1966).
    [CrossRef] [PubMed]
  14. R. Tsai, T. Huang, and W. Zhu, "Estimating three dimensional motion parameters of a rigid planar patch, II: singular value decomposition," IEEE Trans. Acoust., Speech, Signal Process. 30, 525-534 (1982).
    [CrossRef]
  15. H. C. Longuet-Higgins, "The reconstruction of a plane surface from two perspective projections," Proc. R. Soc. London, Ser. B 227, 399-410 (1986).
    [CrossRef]
  16. Z. Zhang and A. Hanson, "Scaled Euclidean 3D reconstruction based on externally uncalibrated cameras," in Proceedings of the International Symposium on Computer Vision (IEEE, 1995), pp. 37-42.
    [CrossRef]
  17. T. Ueshiba and F. Tomita, "Plane-based calibration algorithm for multi-camera systems via factorization of homography matrices," in Proceedings of the Ninth IEEE, International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 966-973.
    [CrossRef]
  18. A. Raij and M. Pollefeys, "Auto-calibration of multi-projector display walls," in Proceedings of the Ninth IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2004), Vol. 1, pp. 14-17.
  19. T. Okatani and K. Deguchi, "Autocalibration of a projector-camera system," IEEE Trans. Pattern Anal. Mach. Intell. 27, 1845-1855 (2005).
    [CrossRef] [PubMed]
  20. A. Habed and B. Boufama, "Camera self-calibration from bivariate polynomial equations and the coplanarity constraint," Image Vis. Comput. 24, 498-514 (2006).
    [CrossRef]
  21. G. Schweighofer and A. Pinz, "Robust pose estimation from a planar target," IEEE Trans. Pattern Anal. Mach. Intell. 28, 2024-2030 (2006).
    [CrossRef] [PubMed]
  22. R. Mester and M. Muhlich, "Improving motion and orientation estimation using an equilibrated total least squares approach," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), Vol. 2, pp. 929-932.
  23. K. Kanatani and N. Ohta, "Accuracy bounds and optimal computation of homography for image mosaicing applications," in Proceedings of IEEE, International Conference on Computer Vision (IEEE, 1999), Vol. 1, pp. 73-78.
    [CrossRef]
  24. P. Chen and D. Suter, "Homography estimation and heteroscedastic noise--a first order perturbation analysis," Tec. Rep. MECSE-32, (Monash University, 2005).
  25. A. Criminisi, I. Reid, and A. Zisserman, "A plane measuring device," Image Vis. Comput. 17, 625-634 (1999).
    [CrossRef]
  26. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2003), Chap. 5, pp. 132-150.
  27. B. Zhang, Y. F. Li and Y. H. Wu, "Self-recalibration of a structured light system via plane-based homography," Pattern Recogn. 1368-1377 (2007).
    [CrossRef]
  28. R. Hartley, "Chirality," Int. J. Comput. Vis. 26, 41-61 (1998).
    [CrossRef]
  29. J. Weng, T. Huang, and N. Ahuja, "Motion and structure from two perspective views: algorithms, error analysis and error estimation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 451-476 (1989).
    [CrossRef]
  30. C. Lu, D. Hager, and E. Mjolsness, "Fast and globally convergent pose estimation from video images," IEEE Trans. Pattern Anal. Mach. Intell. 22, 610-622 (2000).
    [CrossRef]
  31. P. M. Griffin, L. S. Narasimhan, and S. R. Yee, "Generation of uniquely encoded light patterns for range data acquisition," Pattern Recogn. 25, 609-616 (1992).
    [CrossRef]
  32. Z. Zhang, "A flexible new technique for camera calibration," IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330-1334 (2000).
    [CrossRef]
  33. D. Fofi, J. Salvi, and E. Mouaddib, "Uncalibrated reconstruction: an adaptation to structured light vision," Pattern Recogn. 36, 1631-1644 (2003).
    [CrossRef]

2007 (1)

B. Zhang, Y. F. Li and Y. H. Wu, "Self-recalibration of a structured light system via plane-based homography," Pattern Recogn. 1368-1377 (2007).
[CrossRef]

2006 (2)

A. Habed and B. Boufama, "Camera self-calibration from bivariate polynomial equations and the coplanarity constraint," Image Vis. Comput. 24, 498-514 (2006).
[CrossRef]

G. Schweighofer and A. Pinz, "Robust pose estimation from a planar target," IEEE Trans. Pattern Anal. Mach. Intell. 28, 2024-2030 (2006).
[CrossRef] [PubMed]

2005 (2)

M. Wilczkowiak, P. Sturm, and E. Boyer, "Using geometric constraints through parallelepipeds for calibration and 3D modeling," IEEE Trans. Pattern Anal. Mach. Intell. 27, 194-207 (2005).
[CrossRef] [PubMed]

T. Okatani and K. Deguchi, "Autocalibration of a projector-camera system," IEEE Trans. Pattern Anal. Mach. Intell. 27, 1845-1855 (2005).
[CrossRef] [PubMed]

2004 (2)

C. Huang, C. Chen, and P. Chung, "An improved algorithm for two-image camera self-calibration and Euclidean structure recovery using absolute quadric," Pattern Recogn. 37, 1713-1722 (2004).
[CrossRef]

D. Nister, "An efficient solution to the five-point relative pose problem," IEEE Trans. Pattern Anal. Mach. Intell. 26, 756-770 (2004).
[CrossRef]

2003 (3)

Y. F. Li and S. Y. Chen, "Automatic recalibration of a structured light vision system," IEEE Trans. Rob. Autom. 19, 259-268 (2003).
[CrossRef]

X-S. Gao, and X.-R. Hou, J. Thang, and H.-F. Cheng, "Complete solution classification for the perspective-three-point problem," IEEE Trans. Pattern Anal. Mach. Intell. 25, 930-943 (2003).
[CrossRef]

D. Fofi, J. Salvi, and E. Mouaddib, "Uncalibrated reconstruction: an adaptation to structured light vision," Pattern Recogn. 36, 1631-1644 (2003).
[CrossRef]

2000 (2)

Z. Zhang, "A flexible new technique for camera calibration," IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330-1334 (2000).
[CrossRef]

C. Lu, D. Hager, and E. Mjolsness, "Fast and globally convergent pose estimation from video images," IEEE Trans. Pattern Anal. Mach. Intell. 22, 610-622 (2000).
[CrossRef]

1999 (2)

L. Quan and Z. Lan, "Linear N-point camera pose determination," IEEE Trans. Pattern Anal. Mach. Intell. 21, 774-780 (1999).
[CrossRef]

A. Criminisi, I. Reid, and A. Zisserman, "A plane measuring device," Image Vis. Comput. 17, 625-634 (1999).
[CrossRef]

1998 (2)

R. Hartley, "Chirality," Int. J. Comput. Vis. 26, 41-61 (1998).
[CrossRef]

R. Valkenburg and A. McIvor, "Accurate 3D measurement using a structured light system," Image Vis. Comput. 16, 99-110 (1998).
[CrossRef]

1992 (1)

P. M. Griffin, L. S. Narasimhan, and S. R. Yee, "Generation of uniquely encoded light patterns for range data acquisition," Pattern Recogn. 25, 609-616 (1992).
[CrossRef]

1990 (1)

O. Faugeras and S. Maybank, "Motion from point matches: multiplicity of solutions," Int. J. Comput. Vis. 4, 225-246 (1990).
[CrossRef]

1989 (1)

J. Weng, T. Huang, and N. Ahuja, "Motion and structure from two perspective views: algorithms, error analysis and error estimation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 451-476 (1989).
[CrossRef]

1986 (1)

H. C. Longuet-Higgins, "The reconstruction of a plane surface from two perspective projections," Proc. R. Soc. London, Ser. B 227, 399-410 (1986).
[CrossRef]

1982 (1)

R. Tsai, T. Huang, and W. Zhu, "Estimating three dimensional motion parameters of a rigid planar patch, II: singular value decomposition," IEEE Trans. Acoust., Speech, Signal Process. 30, 525-534 (1982).
[CrossRef]

1981 (1)

H. C. Longuet-Higgins, "A computer algorithm for reconstructing a scene from two projections," Nature 293, 133-135 (1981).
[CrossRef]

1966 (1)

J. C. Hay, "Optical motion and space perception: an extension of Gibson's analysis," Psychol. Rev. 73, 550-565 (1966).
[CrossRef] [PubMed]

IEEE Trans. Acoust., Speech, Signal Process. (1)

R. Tsai, T. Huang, and W. Zhu, "Estimating three dimensional motion parameters of a rigid planar patch, II: singular value decomposition," IEEE Trans. Acoust., Speech, Signal Process. 30, 525-534 (1982).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (9)

D. Nister, "An efficient solution to the five-point relative pose problem," IEEE Trans. Pattern Anal. Mach. Intell. 26, 756-770 (2004).
[CrossRef]

T. Okatani and K. Deguchi, "Autocalibration of a projector-camera system," IEEE Trans. Pattern Anal. Mach. Intell. 27, 1845-1855 (2005).
[CrossRef] [PubMed]

L. Quan and Z. Lan, "Linear N-point camera pose determination," IEEE Trans. Pattern Anal. Mach. Intell. 21, 774-780 (1999).
[CrossRef]

X-S. Gao, and X.-R. Hou, J. Thang, and H.-F. Cheng, "Complete solution classification for the perspective-three-point problem," IEEE Trans. Pattern Anal. Mach. Intell. 25, 930-943 (2003).
[CrossRef]

M. Wilczkowiak, P. Sturm, and E. Boyer, "Using geometric constraints through parallelepipeds for calibration and 3D modeling," IEEE Trans. Pattern Anal. Mach. Intell. 27, 194-207 (2005).
[CrossRef] [PubMed]

G. Schweighofer and A. Pinz, "Robust pose estimation from a planar target," IEEE Trans. Pattern Anal. Mach. Intell. 28, 2024-2030 (2006).
[CrossRef] [PubMed]

J. Weng, T. Huang, and N. Ahuja, "Motion and structure from two perspective views: algorithms, error analysis and error estimation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 451-476 (1989).
[CrossRef]

C. Lu, D. Hager, and E. Mjolsness, "Fast and globally convergent pose estimation from video images," IEEE Trans. Pattern Anal. Mach. Intell. 22, 610-622 (2000).
[CrossRef]

Z. Zhang, "A flexible new technique for camera calibration," IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330-1334 (2000).
[CrossRef]

IEEE Trans. Rob. Autom. (1)

Y. F. Li and S. Y. Chen, "Automatic recalibration of a structured light vision system," IEEE Trans. Rob. Autom. 19, 259-268 (2003).
[CrossRef]

Image Vis. Comput. (3)

A. Habed and B. Boufama, "Camera self-calibration from bivariate polynomial equations and the coplanarity constraint," Image Vis. Comput. 24, 498-514 (2006).
[CrossRef]

A. Criminisi, I. Reid, and A. Zisserman, "A plane measuring device," Image Vis. Comput. 17, 625-634 (1999).
[CrossRef]

R. Valkenburg and A. McIvor, "Accurate 3D measurement using a structured light system," Image Vis. Comput. 16, 99-110 (1998).
[CrossRef]

Int. J. Comput. Vis. (2)

R. Hartley, "Chirality," Int. J. Comput. Vis. 26, 41-61 (1998).
[CrossRef]

O. Faugeras and S. Maybank, "Motion from point matches: multiplicity of solutions," Int. J. Comput. Vis. 4, 225-246 (1990).
[CrossRef]

Nature (1)

H. C. Longuet-Higgins, "A computer algorithm for reconstructing a scene from two projections," Nature 293, 133-135 (1981).
[CrossRef]

Pattern Recogn. (4)

C. Huang, C. Chen, and P. Chung, "An improved algorithm for two-image camera self-calibration and Euclidean structure recovery using absolute quadric," Pattern Recogn. 37, 1713-1722 (2004).
[CrossRef]

D. Fofi, J. Salvi, and E. Mouaddib, "Uncalibrated reconstruction: an adaptation to structured light vision," Pattern Recogn. 36, 1631-1644 (2003).
[CrossRef]

P. M. Griffin, L. S. Narasimhan, and S. R. Yee, "Generation of uniquely encoded light patterns for range data acquisition," Pattern Recogn. 25, 609-616 (1992).
[CrossRef]

B. Zhang, Y. F. Li and Y. H. Wu, "Self-recalibration of a structured light system via plane-based homography," Pattern Recogn. 1368-1377 (2007).
[CrossRef]

Proc. R. Soc. London, Ser. B (1)

H. C. Longuet-Higgins, "The reconstruction of a plane surface from two perspective projections," Proc. R. Soc. London, Ser. B 227, 399-410 (1986).
[CrossRef]

Psychol. Rev. (1)

J. C. Hay, "Optical motion and space perception: an extension of Gibson's analysis," Psychol. Rev. 73, 550-565 (1966).
[CrossRef] [PubMed]

Other (10)

Z. Zhang and A. Hanson, "Scaled Euclidean 3D reconstruction based on externally uncalibrated cameras," in Proceedings of the International Symposium on Computer Vision (IEEE, 1995), pp. 37-42.
[CrossRef]

T. Ueshiba and F. Tomita, "Plane-based calibration algorithm for multi-camera systems via factorization of homography matrices," in Proceedings of the Ninth IEEE, International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 966-973.
[CrossRef]

A. Raij and M. Pollefeys, "Auto-calibration of multi-projector display walls," in Proceedings of the Ninth IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2004), Vol. 1, pp. 14-17.

R. Hartley, "Estimation of relative camera positions for uncalibrated cameras," in Lecture Notes in Computer Science (Springer-Verlag1992), pp. 579-580.

B. Triggs, "Routines for relative pose of two calibrated cameras from 5 points," Tech. Rep. (INRIA, 2000)http://www.inrialpes.fr/movi/people/Triggs.

A. Gruen and T. Huang, Calibration and Orientation of Cameras in Computer Vision (Springer-Verlag, 2001).

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2003), Chap. 5, pp. 132-150.

R. Mester and M. Muhlich, "Improving motion and orientation estimation using an equilibrated total least squares approach," in Proceedings of IEEE International Conference on Image Processing (IEEE, 2001), Vol. 2, pp. 929-932.

K. Kanatani and N. Ohta, "Accuracy bounds and optimal computation of homography for image mosaicing applications," in Proceedings of IEEE, International Conference on Computer Vision (IEEE, 1999), Vol. 1, pp. 73-78.
[CrossRef]

P. Chen and D. Suter, "Homography estimation and heteroscedastic noise--a first order perturbation analysis," Tec. Rep. MECSE-32, (Monash University, 2005).

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Figures (8)

Fig. 1
Fig. 1

Relative pose of the proposed system.

Fig. 2
Fig. 2

Relative errors for the rotation and translation versus number of points for the pose estimation.

Fig. 3
Fig. 3

Relative errors for the rotation and translation as a function of injected noise.

Fig. 4
Fig. 4

Relative errors for the rotation and translation as a function of different random poses.

Fig. 5
Fig. 5

Test on the accuracy of the error prediction method, where data1 denotes the difference between real errors and predicted errors.

Fig. 6
Fig. 6

Test on different pose parameters, where data1 and data2 represent the computed errors and the predicted errors, respectively.

Fig. 7
Fig. 7

Configuration of our structured light system: (a) experimental setup and (b) screen shot of the color-encoded light pattern.

Fig. 8
Fig. 8

Experiment on the man’s head model: (a) man’s head model used for the experiment; (b, c) polygonized results of the points clouds in two different viewpoints; (d, e, f) original feature points and reprojected points, where blue “+” represents original feature points while red “○” represents reprojected points from reconstructed 3D points. They should coincide with each other theoretically.

Tables (3)

Tables Icon

Table 1 Comparison of the Computational Efficiency

Tables Icon

Table 2 Relative Pose of the Vision System

Tables Icon

Table 3 Comparison of the Mean Absolute Errors

Equations (50)

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

m p = σ H m c ,
H = [ h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 ] and its vector be h = ( h 1 , h 2 , h 3 , h 4 , h 5 , h 6 , h 7 , h 8 , h 9 ) T .
A h = 0 ,
A = [ u 1 v 1 1 0 0 0 u 1 u 1 u 1 v 1 u 1 0 0 0 u 1 v 1 1 v 1 u 1 v 1 v 1 v 1 u n u n 1 0 0 0 u n u n u n v n u n 0 0 0 u n v n 1 v n u n v n v n v n ] .
H = λ ( R + t n T ) ,
[ t ] x = [ 0 t 3 t 2 t 3 0 t 1 t 2 t 1 0 ] .
[ t ] x H = λ [ t ] x R .
[ t ] x W [ t ] x = 0 ,
{ a 1 t 1 + b 1 t 2 + c 1 = 0 a 2 t 1 + b 2 t 2 + c 2 = 0 a 3 t 1 + b 3 t 2 + c 3 = 0 } ,
a 1 = w 13 w 23 w 12 w 33 , b 1 = w 11 w 33 w 13 2 ,
c 1 = w 12 w 13 w 11 w 23 ,
a 2 = w 22 w 33 w 23 2 , b 2 = w 13 w 23 w 12 w 33 ,
c 2 = w 12 w 23 w 13 w 22 ,
a 3 = w 22 w 13 w 12 w 23 , b 3 = w 11 w 23 w 12 w 13 ,
c 3 = w 12 2 w 11 w 22 .
H H T = λ 2 I + λ 2 ( R n t T + t n T R T + n T n t t T ) = λ 2 I + λ 2 ( s t T + t s T ) ,
s = R n + n T n 2 t .
det ( W ) = 2 w 12 w 13 w 23 w 23 2 w 11 w 12 2 w 33 w 13 2 w 22 + w 11 w 22 w 33 = 0 .
a 1 b 2 a 2 b 1 = w 33 det ( W ) = 0 ,
c 1 b 2 c 2 b 1 = w 13 det ( W ) = 0 .
{ t 1 = w 13 ± w 13 2 w 11 w 33 w 33 t 2 = ( w 13 w 23 w 12 w 33 ) t 1 + ( w 12 w 13 w 11 w 23 ) w 11 w 33 w 13 2 t 3 = 1 } .
R T C D = 0 ,
B = i = 1 3 B i T B i ,
B i = [ 0 ( C i D i ) T D i C i [ C i + D i ] x ] .
R = [ q 0 2 + q 1 2 q 2 2 q 3 2 2 ( q 1 q 2 q 0 q 3 ) 2 ( q 1 q 3 + q 0 q 2 ) 2 ( q 1 q 2 + q 0 q 3 ) q 0 2 q 1 2 + q 2 2 q 3 2 2 ( q 2 q 3 q 0 q 1 ) 2 ( q 1 q 3 q 0 q 2 ) 2 ( q 2 q 3 + q 0 q 1 ) q 0 2 q 1 2 q 2 2 + q 3 2 ] .
Δ Q = ( A + Δ A ) T ( A + Δ A ) A T A A T Δ A + Δ A T A .
δ Q = ( F Q + G Q ) δ A T ,
F Q = A T I 9 , G Q = [ A 1 1 A 1 2 A 1 2 n A 2 1 A 2 2 A 2 2 n A 9 1 A 9 2 A 9 2 n ] ,
δ h = V Λ Q V T Δ Q v 1 = V Λ Q V T [ v 11 I 9 v 21 I 9 v 91 I 9 ] δ Q = D h δ A T ,
δ S = ( F S + G S ) δ h = D S δ h ,
F S = H T I 3 , G S = [ H 1 1 H 1 2 H 1 3 H 2 1 H 2 2 H 2 3 H 3 1 H 3 2 H 3 3 ] .
δ λ 2 = u 2 T Δ S u 2 = u 2 T [ u 12 I 3 u 22 I 3 u 32 I 3 ] δ S .
δ λ 2 = D λ 2 δ A T ,
δ W = δ S [ e 1 T e 2 T e 3 T ] T δ λ 2 = D W δ A T ,
t 1 ( w 11 , w 13 , w 33 ) t 1 1 2 w 13 2 w 11 w 33 δ w 11 + ( 1 w 33 + w 13 w 33 w 13 2 w 11 w 33 ) δ w 13 + ( w 13 2 w 33 2 w 13 2 w 11 w 33 w 13 w 33 2 ) δ w 33 .
[ δ t 1 δ t 2 ] = [ D t L D t R ] δ W ,
D t L = 1 b 1 2 [ b 1 3 2 0 b 1 2 + w 13 b 1 3 w 33 w 23 b 1 + w 33 b 1 t 2 a 1 b 1 2 ( w 33 t 1 w 13 ) b 1 w 13 a 1 b 1 a 1 b 1 w 33 2 w 13 b 1 t 2 ( w 23 t 1 + w 12 ) b 1 ] ,
D t R = 1 b 1 2 [ 0 0 0 0 0 w 11 b 1 3 2 w 13 b 1 2 2 w 33 2 0 0 ( w 11 w 13 t 1 ) b 1 0 0 w 12 t 1 b 1 + w 11 t 2 b 1 + w 11 a 1 b 1 + 2 w 13 a 1 b 1 2 w 33 2 ] ,
δ t = D t δ A T ,
δ p = D p δ A T ,
δ B = 2 D B δ p ,
δ q = η Λ B η T Δ B q 1 = D q 1 δ p ,
δ R = J q δ q = D R δ A T ,
Γ t = E ( δ t δ t T ) = D t Γ A T D t T ,
Γ R = E ( δ R δ R T ) = D R Γ A T D R T ,
D λ = [ d 31 t 2 d 32 t 1 + 2 c 13 t 2 + d 23 t 1 2 c 12 d 21 2 c 23 t 1 + d 12 d 13 t 2 d 22 t 1 d 31 d 21 t 2 + d 33 t 1 d 12 t 1 + d 11 t 2 + d 33 t 2 d 32 d 13 t 1 d 23 t 2 + d 11 + d 22 d 31 t 2 + d 32 t 1 + 2 c 13 t 2 d 23 t 1 2 c 12 + d 21 2 c 23 t 1 + d 12 d 13 t 2 d 13 t 1 d 23 t 2 d 11 + d 22 d 12 t 1 + d 11 t 2 d 33 t 2 + d 32 d 31 t 2 + d 32 t 1 + 2 c 13 t 2 + d 23 t 1 2 c 12 d 21 2 c 23 t 1 d 12 + d 13 t 2 d 22 t 1 d 31 + d 21 t 2 + d 33 t 1 d 31 t 2 d 32 t 1 + 2 c 13 t 2 d 23 t 1 2 c 12 + d 21 2 c 23 t 1 d 12 + d 13 t 2 ] ,
D t 1 = [ c 13 h 4 + c 23 h 5 d 13 h 4 2 c 23 λ d 23 h 5 d 33 h 6 c 12 h 7 + c 23 h 9 + d 22 h 8 + d 12 h 7 + d 32 h 9 + d 23 λ d 32 λ c 23 h 8 + d 22 λ + c 23 h 6 + d 33 λ c 13 h 6 d 12 λ + c 12 h 9 + c 23 h 7 c 23 h 4 + c 13 h 5 d 13 λ c 12 h 8 c 13 h 4 c 23 h 5 d 13 h 4 2 c 23 λ d 23 h 5 d 33 h 6 c 12 h 7 c 23 h 9 + d 22 h 8 + d 12 h 7 + d 32 h 9 d 23 λ + d 32 λ c 23 h 4 + c 13 h 5 + d 13 λ c 12 h 8 c 13 h 6 d 12 λ c 12 h 9 + c 23 h 7 c 13 h 4 + c 23 h 5 d 13 h 4 2 c 23 λ d 23 h 5 d 33 h 6 + c 12 h 7 c 23 h 9 + d 22 h 8 + d 12 h 7 + d 32 h 9 + d 23 λ + d 32 λ c 23 h 8 d 22 λ + c 23 h 6 + d 33 λ c 13 h 4 c 23 h 5 d 13 h 4 2 c 23 λ d 23 h 5 d 33 h 6 + c 12 h 7 + c 23 h 9 + d 22 h 8 + d 12 h 7 + d 32 h 9 d 23 λ d 32 λ ] ,
D t 2 = [ c 13 h 1 + d 33 h 3 + d 23 h 2 + 2 c 13 λ d 11 h 7 c 13 h 9 d 21 h 8 + d 13 h 1 c 12 h 8 c 23 h 2 d 31 h 9 d 13 λ + d 31 λ c 13 h 8 c 23 h 3 c 12 h 9 d 21 λ d 33 λ + c 13 h 3 + c 11 λ c 13 h 7 c 13 h 2 + c 12 h 7 d 23 λ + c 23 h 1 c 13 h 1 + d 33 h 3 + d 23 h 2 + 2 c 13 λ d 11 h 7 + c 13 h 9 d 21 h 8 + d 13 h 1 + c 12 h 8 + c 23 h 2 d 31 h 9 d 13 λ d 31 λ c 13 h 2 c 12 h 7 d 23 λ c 23 h 1 d 33 λ c 13 h 3 + c 11 λ c 13 h 7 c 13 h 1 + d 33 h 3 + d 23 h 2 + 2 c 13 λ d 11 h 7 + c 13 h 9 d 21 h 8 + d 13 h 1 c 12 h 8 c 23 h 2 d 31 h 9 + d 13 λ d 31 λ c 13 h 8 c 23 h 3 c 12 h 9 + d 21 λ c 13 h 1 + d 33 h 3 + d 23 h 2 + 2 c 13 λ d 11 h 7 c 13 h 9 d 21 h 8 + d 13 h 1 + c 12 h 8 + c 23 h 2 d 31 h 9 + d 13 λ + d 31 λ ] ,
D h L = [ ( d 13 c 13 ) t 2 d 12 + c 12 ( d 23 c 23 ) t 2 d 22 d 33 t 2 d 32 c 23 ( c 13 d 13 ) t 1 + d 11 ( d 23 + c 23 ) t 1 + d 21 + c 12 0 c 23 c 23 t 2 0 c 13 c 23 0 c 13 t 2 c 12 c 13 0 c 23 t 2 c 12 c 13 t 2 0 c 23 t 1 c 12 c 13 t 1 ( d 13 c 13 ) t 2 d 12 + c 12 ( d 23 + c 23 ) t 2 d 22 d 33 t 2 d 32 + c 23 ( c 13 d 13 ) t 1 + d 11 ( d 23 c 23 ) t 1 + d 21 c 12 c 23 t 2 c 12 c 13 t 2 0 c 23 t 1 + c 12 c 13 t 1 c 23 0 c 13 t 2 + c 12 c 13 0 ( d 13 + c 13 ) t 2 d 12 c 12 ( d 23 c 23 ) t 2 d 22 d 33 t 2 d 32 + c 23 ( c 13 d 13 ) t 1 + d 11 ( d 23 + c 23 ) t 1 + d 21 + c 12 0 c 23 c 23 t 2 0 c 13 ( d 13 + c 13 ) t 2 d 12 c 12 ( d 23 + c 23 ) t 2 d 22 d 33 t 2 d 32 c 23 ( c 13 d 13 ) t 1 + d 11 ( d 23 c 23 ) t 1 + d 21 c 12 ] ,
D h R = [ d 33 t 1 + d 31 + c 13 d 12 t 1 d 11 t 2 c 12 t 1 d 22 t 1 d 21 t 2 c 12 t 2 ( d 32 + c 23 ) t 1 ( d 31 + c 13 ) t 2 c 23 t 1 + c 12 0 c 23 t 1 + c 13 t 2 c 12 t 2 c 13 t 1 c 23 t 1 c 13 t 2 0 c 12 t 1 0 c 12 t 2 c 12 t 1 0 d 33 t 1 + d 31 c 13 d 12 t 1 d 11 t 2 c 12 t 1 d 22 t 1 d 21 t 2 + c 12 t 2 ( d 32 c 23 ) t 1 ( d 31 c 13 ) t 2 0 c 12 t 2 c 12 t 1 0 c 13 t 1 c 23 t 1 c 13 t 2 0 c 12 t 1 d 33 t 1 + d 31 c 13 d 12 t 1 d 11 t 2 + c 12 t 1 d 22 t 1 d 21 t 2 c 12 t 2 ( d 32 c 23 ) t 1 ( d 31 c 13 ) t 2 c 23 t 1 + c 12 0 c 23 t 1 c 13 t 2 c 12 t 2 d 33 t 1 + d 31 + c 13 d 12 t 1 d 11 t 2 + c 12 t 1 d 22 t 1 d 21 t 2 + c 12 t 2 ( d 32 + c 23 ) t 1 ( d 31 + c 13 ) t 2 ] .

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