Abstract

This paper presents a new and effective technique to calibrate a camera without nonlinear iteration optimization. To this end, the centre-of-distortion is accurately estimated firstly. Based on the radial distortion division model, point correspondences between model plane and its image were used to compute the homography and distortion coefficients afterwards. Once the homographies of calibration images are obtained, the camera intrinsic parameters are solved analytically. All the solution techniques applied in this calibration process are non-iterative that do not need any initial guess, with no risk of local minima. Moreover, estimation of the distortion coefficients and intrinsic parameters could be successfully decoupled, yielding the more stable and reliable result. Both simulative and real experiments have been carried out to show that the proposed method is reliable and effective. Without nonlinear iteration optimization, the proposed method is computationally efficient and can be applied to real-time online calibration.

© 2015 Optical Society of America

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References

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  1. C. Ricolfe-Viala, A.-J. Sanchez-Salmeron, and A. Valera, “Calibration of a trinocular system formed with wide angle lens cameras,” Opt. Express 20(25), 27691–27696 (2012).
    [Crossref] [PubMed]
  2. Y. Cui, F. Zhou, Y. Wang, L. Liu, and H. Gao, “Precise calibration of binocular vision system used for vision measurement,” Opt. Express 22(8), 9134–9149 (2014).
    [Crossref] [PubMed]
  3. E. Hall, J. Tio, C. McPherson, and F. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15(12), 42–54 (1982).
    [Crossref]
  4. J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
    [Crossref]
  5. Z. Zhang, D. Zhu, J. Zhang, and Z. Peng, “Improved robust and accurate camera calibration method used for machine vision application,” Opt. Eng. 47(11), 1273–1279 (2008).
    [Crossref]
  6. R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
    [Crossref]
  7. J. Weng, P. R. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
    [Crossref]
  8. J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.
    [Crossref]
  9. J. Heikkilä, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
    [Crossref]
  10. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
    [Crossref]
  11. C. Ricolfe-Viala and A.-J. Sanchez-Salmeron, “Camera calibration under optimal conditions,” Opt. Express 19(11), 10769–10775 (2011).
    [Crossref] [PubMed]
  12. A. Datta, J. S. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” the Ninth IEEE International Workshop on Visual Surveillance (IEEE, 2009), pp. 1201 - 1208.
    [Crossref]
  13. D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
    [Crossref] [PubMed]
  14. T. Rahman and N. Krouglicof, “An efficient camera calibration technique offering robustness and accuracy over a wide range of lens distortion,” IEEE Trans. Image Process. 21(2), 626–637 (2012).
    [Crossref] [PubMed]
  15. F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
    [Crossref]
  16. H. L. R. Hartley, “A non-iterative method for correcting lens distortion from nine-point correspondences,” In Proc. OmniVision’05, ICCV-workshop, (2005).
  17. A. W. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 125–132.
    [Crossref]
  18. J. Salvi, X. Armangué, and J. Batlle, “A comparative review of camera calibrating methods with accuracy evaluation,” Pattern Recognit. 35(7), 1617–1635 (2002).
    [Crossref]
  19. R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with centre of distortion estimation,” in International Conference on Computer Vision (2005), pp. 1834–1841.
    [Crossref]
  20. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).
  21. B. Triggs, “Autocalibration from Planar Scenes,” in European Conference on Computer Vision (1998), pp. 89–105.
  22. Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 892–899 (2004).
    [Crossref] [PubMed]
  23. M. Higuchi, “Precise camera calibration,” http://www.ri.cmu.edu/research_project_detail.html?project_id=617&menu_id=261 .

2014 (1)

2012 (3)

C. Ricolfe-Viala, A.-J. Sanchez-Salmeron, and A. Valera, “Calibration of a trinocular system formed with wide angle lens cameras,” Opt. Express 20(25), 27691–27696 (2012).
[Crossref] [PubMed]

T. Rahman and N. Krouglicof, “An efficient camera calibration technique offering robustness and accuracy over a wide range of lens distortion,” IEEE Trans. Image Process. 21(2), 626–637 (2012).
[Crossref] [PubMed]

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

2011 (1)

2009 (1)

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref] [PubMed]

2008 (2)

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[Crossref]

Z. Zhang, D. Zhu, J. Zhang, and Z. Peng, “Improved robust and accurate camera calibration method used for machine vision application,” Opt. Eng. 47(11), 1273–1279 (2008).
[Crossref]

2004 (1)

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 892–899 (2004).
[Crossref] [PubMed]

2002 (1)

J. Salvi, X. Armangué, and J. Batlle, “A comparative review of camera calibrating methods with accuracy evaluation,” Pattern Recognit. 35(7), 1617–1635 (2002).
[Crossref]

2000 (2)

J. Heikkilä, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
[Crossref]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

1992 (1)

J. Weng, P. R. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

1987 (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

1982 (1)

E. Hall, J. Tio, C. McPherson, and F. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15(12), 42–54 (1982).
[Crossref]

Armangué, X.

J. Salvi, X. Armangué, and J. Batlle, “A comparative review of camera calibrating methods with accuracy evaluation,” Pattern Recognit. 35(7), 1617–1635 (2002).
[Crossref]

Batlle, J.

J. Salvi, X. Armangué, and J. Batlle, “A comparative review of camera calibrating methods with accuracy evaluation,” Pattern Recognit. 35(7), 1617–1635 (2002).
[Crossref]

Chihara, K.

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref] [PubMed]

Cohen, P. R.

J. Weng, P. R. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

Cui, Y.

Y. Cui, F. Zhou, Y. Wang, L. Liu, and H. Gao, “Precise calibration of binocular vision system used for vision measurement,” Opt. Express 22(8), 9134–9149 (2014).
[Crossref] [PubMed]

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Douxchamps, D.

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref] [PubMed]

Fitzgibbon, A. W.

A. W. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 125–132.
[Crossref]

Gao, H.

Hall, E.

E. Hall, J. Tio, C. McPherson, and F. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15(12), 42–54 (1982).
[Crossref]

Hartley, R. I.

R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with centre of distortion estimation,” in International Conference on Computer Vision (2005), pp. 1834–1841.
[Crossref]

Heikkila, J.

J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.
[Crossref]

Heikkilä, J.

J. Heikkilä, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
[Crossref]

Herniou, M.

J. Weng, P. R. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

Kang, S. B.

R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with centre of distortion estimation,” in International Conference on Computer Vision (2005), pp. 1834–1841.
[Crossref]

Krouglicof, N.

T. Rahman and N. Krouglicof, “An efficient camera calibration technique offering robustness and accuracy over a wide range of lens distortion,” IEEE Trans. Image Process. 21(2), 626–637 (2012).
[Crossref] [PubMed]

Liu, L.

Liu, Y.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[Crossref]

McPherson, C.

E. Hall, J. Tio, C. McPherson, and F. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15(12), 42–54 (1982).
[Crossref]

Peng, B.

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Peng, Z.

Z. Zhang, D. Zhu, J. Zhang, and Z. Peng, “Improved robust and accurate camera calibration method used for machine vision application,” Opt. Eng. 47(11), 1273–1279 (2008).
[Crossref]

Rahman, T.

T. Rahman and N. Krouglicof, “An efficient camera calibration technique offering robustness and accuracy over a wide range of lens distortion,” IEEE Trans. Image Process. 21(2), 626–637 (2012).
[Crossref] [PubMed]

Ricolfe-Viala, C.

Sadjadi, F.

E. Hall, J. Tio, C. McPherson, and F. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15(12), 42–54 (1982).
[Crossref]

Salvi, J.

J. Salvi, X. Armangué, and J. Batlle, “A comparative review of camera calibrating methods with accuracy evaluation,” Pattern Recognit. 35(7), 1617–1635 (2002).
[Crossref]

Sanchez-Salmeron, A.-J.

Shi, F.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[Crossref]

Silven, O.

J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.
[Crossref]

Tio, J.

E. Hall, J. Tio, C. McPherson, and F. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15(12), 42–54 (1982).
[Crossref]

Triggs, B.

B. Triggs, “Autocalibration from Planar Scenes,” in European Conference on Computer Vision (1998), pp. 89–105.

Tsai, R. Y.

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

Valera, A.

Wang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[Crossref]

Wang, Y.

Y. Cui, F. Zhou, Y. Wang, L. Liu, and H. Gao, “Precise calibration of binocular vision system used for vision measurement,” Opt. Express 22(8), 9134–9149 (2014).
[Crossref] [PubMed]

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Weng, J.

J. Weng, P. R. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

Zhang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[Crossref]

Z. Zhang, D. Zhu, J. Zhang, and Z. Peng, “Improved robust and accurate camera calibration method used for machine vision application,” Opt. Eng. 47(11), 1273–1279 (2008).
[Crossref]

Zhang, Z.

Z. Zhang, D. Zhu, J. Zhang, and Z. Peng, “Improved robust and accurate camera calibration method used for machine vision application,” Opt. Eng. 47(11), 1273–1279 (2008).
[Crossref]

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 892–899 (2004).
[Crossref] [PubMed]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Zhou, F.

Y. Cui, F. Zhou, Y. Wang, L. Liu, and H. Gao, “Precise calibration of binocular vision system used for vision measurement,” Opt. Express 22(8), 9134–9149 (2014).
[Crossref] [PubMed]

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Zhu, D.

Z. Zhang, D. Zhu, J. Zhang, and Z. Peng, “Improved robust and accurate camera calibration method used for machine vision application,” Opt. Eng. 47(11), 1273–1279 (2008).
[Crossref]

Computer (1)

E. Hall, J. Tio, C. McPherson, and F. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15(12), 42–54 (1982).
[Crossref]

IEEE J. Robot. Autom. (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

IEEE Trans. Image Process. (1)

T. Rahman and N. Krouglicof, “An efficient camera calibration technique offering robustness and accuracy over a wide range of lens distortion,” IEEE Trans. Image Process. 21(2), 626–637 (2012).
[Crossref] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (5)

J. Weng, P. R. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref] [PubMed]

J. Heikkilä, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
[Crossref]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 892–899 (2004).
[Crossref] [PubMed]

Opt. Eng. (1)

Z. Zhang, D. Zhu, J. Zhang, and Z. Peng, “Improved robust and accurate camera calibration method used for machine vision application,” Opt. Eng. 47(11), 1273–1279 (2008).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Pattern Recognit. (2)

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recognit. 41(2), 607–615 (2008).
[Crossref]

J. Salvi, X. Armangué, and J. Batlle, “A comparative review of camera calibrating methods with accuracy evaluation,” Pattern Recognit. 35(7), 1617–1635 (2002).
[Crossref]

Other (8)

R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with centre of distortion estimation,” in International Conference on Computer Vision (2005), pp. 1834–1841.
[Crossref]

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

B. Triggs, “Autocalibration from Planar Scenes,” in European Conference on Computer Vision (1998), pp. 89–105.

M. Higuchi, “Precise camera calibration,” http://www.ri.cmu.edu/research_project_detail.html?project_id=617&menu_id=261 .

A. Datta, J. S. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” the Ninth IEEE International Workshop on Visual Surveillance (IEEE, 2009), pp. 1201 - 1208.
[Crossref]

H. L. R. Hartley, “A non-iterative method for correcting lens distortion from nine-point correspondences,” In Proc. OmniVision’05, ICCV-workshop, (2005).

A. W. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 125–132.
[Crossref]

J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 The mathematical model of camera.
Fig. 2
Fig. 2 The simulated images with the true projection image points and the distorted points: (a) the first simulated image; (b) the second simulated image; (c) the third simulated image.
Fig. 3
Fig. 3 Effects of pixel coordinates noise on intrinsic parameters using the proposed method.
Fig. 4
Fig. 4 Effects of pixel coordinates noise on calibration accuracy.
Fig. 5
Fig. 5 Samples of image used for calibration.
Fig. 6
Fig. 6 Calibration error distribution for testing data achieved by two methods: (a) Zhang's method; (b) Proposed method. Each point marked as ‘ + ’ in the figure denotes the re-projection error of corresponding chessboard corner in the testing images.
Fig. 7
Fig. 7 The computation time of calibration for different number of images achieved by Zhang’s method and the proposed method.

Tables (3)

Tables Icon

Table 1 Comparative result of intrinsic parameters and distortion coefficients

Tables Icon

Table 2 Comparative result of calibration error E r m s 2 evaluated by testing data

Tables Icon

Table 3 The average re-projection error and computation time of calibration for different number of images

Equations (26)

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

ρ m u = A [ R t ] M , with A = [ f u s u 0 0 f v v 0 0 0 1 ]
ρ ( x u y u 1 ) = A ( r 1 r 2 r 3 t ) ( x w y w 0 1 ) = A ( r 1 r 2 t ) ( x w y w 1 )
ρ m u = H M ˜ , with H = A ( r 1 r 2 t )
m u e = ( m d e ) ( 1 + λ 1 r d 2 + λ 2 r d 4 + )
m u e = ( m d e ) 1 + λ 1 r d 2 + λ 2 r d 4 +
[ m d ] T ( [ e ] × H ) M ˜ = 0
[ m d ] T F r M ˜ = 0
e T F r = e T [ e ] × H = 0
m ^ d = ( x ^ d y ^ d 1 ) = ( x d d u 0 y d d v 0 1 ) = ( 1 0 d u 0 0 1 d v 0 0 0 1 ) ( x d y d 1 )
m ^ d = T m d
m ^ u = T m u
[ m ^ d ] T F ^ r M ˜ = 0
[ m d ] T T T F ^ r M ˜ = 0
F r = T T F ^ r
F ^ r = T T F r = ( 1 0 0 0 1 0 d u 0 d v 0 1 ) F r
F ^ r = [ e ^ ] × H ^ = ( 0 1 0 1 0 0 0 0 0 ) H ^
h ^ 1 T = f ^ 2 T , h ^ 2 T = f ^ 1 T
ρ m ^ u = H ^ M ˜
{ x ^ u = x ^ d 1 + λ 1 [ r ^ d ] 2 + λ 2 [ r ^ d ] 4 + y ^ u = y ^ d 1 + λ 1 [ r ^ d ] 2 + λ 2 [ r ^ d ] 4 +
ρ [ x ^ d 1 + λ 1 [ r ^ d ] 2 + λ 2 [ r ^ d ] 4 + y ^ d 1 + λ 1 [ r ^ d ] 2 + λ 2 [ r ^ d ] 4 + 1 ] = [ h ^ 1 T h ^ 2 T h ^ 3 T ] M ˜
[ x ^ d ( h ^ 3 T M ˜ ) ( h ^ 1 T M ˜ ) ( λ 1 [ r ^ d ] 2 + λ 2 [ r ^ d ] 4 + ) y ^ d ( h ^ 3 T M ˜ ) ( h ^ 2 T M ˜ ) ( λ 1 [ r ^ d ] 2 + λ 2 [ r ^ d ] 4 + ) ] = [ h ^ 1 T M ˜ h ^ 2 T M ˜ ]
[ x ^ d [ M ˜ ] T ( f ^ 2 T M ˜ ) [ [ r ^ d ] 2 [ r ^ d ] 4 ] y ^ d [ M ˜ ] T ( f ^ 1 T M ˜ ) [ [ r ^ d ] 2 [ r ^ d ] 4 ] ] [ h ^ 3 λ 1 λ 2 ] = [ f ^ 2 T M ˜ f ^ 1 T M ˜ ]
H = T 1 H ^ = ( 1 0 d u 0 0 1 d v 0 0 0 1 ) H ^
{ h c 1 T A T A 1 h c 2 = 0 h c 1 T A T A 1 h c 1 = h c 2 T A T A 1 h c 2
E r m s 1 = 1 n i = 1 n m r p , i m t r u e , i 2
E r m s 2 = 1 n i = 1 n m r p , i m u d , i 2

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