Abstract

Based on 2-D protractor property of camera, we proposed a flexible calibration method for zoom camera that used outdoors. It only requires the camera to observe control points once for given zooming settings, when there are several control points at infinity and known the angular distances. Under constraints of image points, the angular distance between their re-projecting vectors and the image of absolute conic (IAC), nonlinear optimization is used to solve parameters of IAC. Then IAC can be uniquely decomposed by the Cholesky factorization, and consequently the intrinsic parameters can be obtained. Towards the factors that affect the accuracy of the calibration, theoretical analysis and computer simulation are carried out respectively consequence in qualitative analysis and quantitative result. On the issues of inaccuracy of principal point, the zooming center is selected to improve the accuracy of calibration. Real data demonstrated the effectiveness of the techniques.

© 2016 Optical Society of America

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References

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  1. S. Ma and Z. Zhang, Computer Vision: Theory and Algorithms (Beijing Sciences, 1998).
  2. G. Zhang, Visual Measurement (Beijing Sciences, 2008).
  3. R. Tsai, “An efficient and accurate camera calibration technique for 3D machine vision,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1986), pp. 364–374.
  4. R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the- shelf TV camera and lenses,” IEEE Trans. Robot. Autom. 3(4), 323–344 (1987).
    [Crossref]
  5. O. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint (MIT Press, 1993).
  6. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. 22(11), 1330–1334 (2000).
    [Crossref]
  7. Z. Wei, L. Cao, and G. Zhang, “A novel 1D target-based calibration method with unknown orientation for structured light vision sensor,” Opt. Laser Technol. 42(4), 570–574 (2010).
    [Crossref]
  8. B. He and Y. Li, “Camera calibration with lens distortion and from vanishing points,” Opt. Eng. 48(1), 013603 (2009).
    [Crossref]
  9. F. Lv, T. Zhao, and R. Nevatia, “Camera calibration from video of a walking human,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1513–1518 (2006).
    [Crossref] [PubMed]
  10. B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogram. Eng. Rem. Sens. 79(6), 555–571 (2013).
    [Crossref]
  11. J. Jin and X. Li, “Efficient camera self-calibration method based on the absolute dual quadric,” J. Opt. Soc. Am. A 30(3), 287–292 (2013).
    [Crossref] [PubMed]
  12. H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
    [Crossref] [PubMed]
  13. J. Davis and X. Chen, “Calibrating pan-tilt cameras in wide-area surveillance networks,” in Proceedings of IEEE International Conference on Computer Vision (IEEE 2003), pp. 144–149.
    [Crossref]
  14. Photo-Sonic, “Mobile multispectral TSPI system,” http://www.photosonics.com/mmts.htm .
  15. J. Kelsey, J. Byrne, M. Cosgrove, S. Seereeram, and R. Mehra, “Vision-based relative pose estimation for autonomous rendezvous and docking,” in Proceedings of IEEE Conference on Aerospace (IEEE 2006), pp. 1–20.
    [Crossref]
  16. V. Lepetit and P. Fua, “Monocular model-based 3D tracking of rigid objects: a survey,” Found. Trends Comput. Graph. Vis. 1(1), 1–89 (2005).
    [Crossref]
  17. S. Sinha and M. Pollefeys, “Pan-tilt-zoom camera calibration and high-resolution mosaic generation,” Comput. Vis. Image Underst. 103(3), 170–183 (2006).
    [Crossref]
  18. R. Willson, Modeling and Calibration of Automated Zoom Lenses, Ph.D. Dissertation (Carnegie Mellon University, 1994).
  19. Z. Wu and R. J. Radke, “Keeping a pan-tilt-zoom camera calibrated,” IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1994–2007 (2013).
    [Crossref] [PubMed]
  20. R. Lenz and R. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3D machine vision metrology,” IEEE Trans. Pattern Anal. 10(5), 713–720 (1988).
    [Crossref]
  21. M. Li and J. Lavest, “Some aspects of zoom lens camera calibration,” IEEE T. Pattern Anal. 18(11), 1105–1110 (1996).
    [Crossref]
  22. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).
  23. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University 2007).
  24. Y. Fei, Error Theory and Data Processing, 6th ed. (China Machine, 2010).
  25. Z. He, Optical Measuring System (National Defense Industry, 2002).
  26. C. Harris and M. Stephens, “A combined corner and edge detector,” in Proceedings of the 4th Alvey Vision Conference (Academic, 1988), pp. 147–151.

2013 (3)

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogram. Eng. Rem. Sens. 79(6), 555–571 (2013).
[Crossref]

J. Jin and X. Li, “Efficient camera self-calibration method based on the absolute dual quadric,” J. Opt. Soc. Am. A 30(3), 287–292 (2013).
[Crossref] [PubMed]

Z. Wu and R. J. Radke, “Keeping a pan-tilt-zoom camera calibrated,” IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1994–2007 (2013).
[Crossref] [PubMed]

2010 (1)

Z. Wei, L. Cao, and G. Zhang, “A novel 1D target-based calibration method with unknown orientation for structured light vision sensor,” Opt. Laser Technol. 42(4), 570–574 (2010).
[Crossref]

2009 (1)

B. He and Y. Li, “Camera calibration with lens distortion and from vanishing points,” Opt. Eng. 48(1), 013603 (2009).
[Crossref]

2007 (1)

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

2006 (2)

F. Lv, T. Zhao, and R. Nevatia, “Camera calibration from video of a walking human,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1513–1518 (2006).
[Crossref] [PubMed]

S. Sinha and M. Pollefeys, “Pan-tilt-zoom camera calibration and high-resolution mosaic generation,” Comput. Vis. Image Underst. 103(3), 170–183 (2006).
[Crossref]

2005 (1)

V. Lepetit and P. Fua, “Monocular model-based 3D tracking of rigid objects: a survey,” Found. Trends Comput. Graph. Vis. 1(1), 1–89 (2005).
[Crossref]

2000 (1)

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

1996 (1)

M. Li and J. Lavest, “Some aspects of zoom lens camera calibration,” IEEE T. Pattern Anal. 18(11), 1105–1110 (1996).
[Crossref]

1988 (1)

R. Lenz and R. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3D machine vision metrology,” IEEE Trans. Pattern Anal. 10(5), 713–720 (1988).
[Crossref]

1987 (1)

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

Cao, L.

Z. Wei, L. Cao, and G. Zhang, “A novel 1D target-based calibration method with unknown orientation for structured light vision sensor,” Opt. Laser Technol. 42(4), 570–574 (2010).
[Crossref]

Fua, P.

V. Lepetit and P. Fua, “Monocular model-based 3D tracking of rigid objects: a survey,” Found. Trends Comput. Graph. Vis. 1(1), 1–89 (2005).
[Crossref]

Harris, C.

C. Harris and M. Stephens, “A combined corner and edge detector,” in Proceedings of the 4th Alvey Vision Conference (Academic, 1988), pp. 147–151.

He, B.

B. He and Y. Li, “Camera calibration with lens distortion and from vanishing points,” Opt. Eng. 48(1), 013603 (2009).
[Crossref]

Hu, H.

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogram. Eng. Rem. Sens. 79(6), 555–571 (2013).
[Crossref]

Jin, J.

Lavest, J.

M. Li and J. Lavest, “Some aspects of zoom lens camera calibration,” IEEE T. Pattern Anal. 18(11), 1105–1110 (1996).
[Crossref]

Lenz, R.

R. Lenz and R. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3D machine vision metrology,” IEEE Trans. Pattern Anal. 10(5), 713–720 (1988).
[Crossref]

Lepetit, V.

V. Lepetit and P. Fua, “Monocular model-based 3D tracking of rigid objects: a survey,” Found. Trends Comput. Graph. Vis. 1(1), 1–89 (2005).
[Crossref]

Li, M.

M. Li and J. Lavest, “Some aspects of zoom lens camera calibration,” IEEE T. Pattern Anal. 18(11), 1105–1110 (1996).
[Crossref]

Li, X.

Li, Y.

B. He and Y. Li, “Camera calibration with lens distortion and from vanishing points,” Opt. Eng. 48(1), 013603 (2009).
[Crossref]

Lv, F.

F. Lv, T. Zhao, and R. Nevatia, “Camera calibration from video of a walking human,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1513–1518 (2006).
[Crossref] [PubMed]

Nevatia, R.

F. Lv, T. Zhao, and R. Nevatia, “Camera calibration from video of a walking human,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1513–1518 (2006).
[Crossref] [PubMed]

Pollefeys, M.

S. Sinha and M. Pollefeys, “Pan-tilt-zoom camera calibration and high-resolution mosaic generation,” Comput. Vis. Image Underst. 103(3), 170–183 (2006).
[Crossref]

Radke, R. J.

Z. Wu and R. J. Radke, “Keeping a pan-tilt-zoom camera calibrated,” IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1994–2007 (2013).
[Crossref] [PubMed]

Sinha, S.

S. Sinha and M. Pollefeys, “Pan-tilt-zoom camera calibration and high-resolution mosaic generation,” Comput. Vis. Image Underst. 103(3), 170–183 (2006).
[Crossref]

Stephens, M.

C. Harris and M. Stephens, “A combined corner and edge detector,” in Proceedings of the 4th Alvey Vision Conference (Academic, 1988), pp. 147–151.

Tsai, R.

R. Lenz and R. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3D machine vision metrology,” IEEE Trans. Pattern Anal. 10(5), 713–720 (1988).
[Crossref]

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

R. Tsai, “An efficient and accurate camera calibration technique for 3D machine vision,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1986), pp. 364–374.

Wei, Z.

Z. Wei, L. Cao, and G. Zhang, “A novel 1D target-based calibration method with unknown orientation for structured light vision sensor,” Opt. Laser Technol. 42(4), 570–574 (2010).
[Crossref]

Wong, K. Y.

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

Wu, B.

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogram. Eng. Rem. Sens. 79(6), 555–571 (2013).
[Crossref]

Wu, Z.

Z. Wu and R. J. Radke, “Keeping a pan-tilt-zoom camera calibrated,” IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1994–2007 (2013).
[Crossref] [PubMed]

Zhang, G.

Z. Wei, L. Cao, and G. Zhang, “A novel 1D target-based calibration method with unknown orientation for structured light vision sensor,” Opt. Laser Technol. 42(4), 570–574 (2010).
[Crossref]

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

Zhang, H.

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

Zhang, Y.

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogram. Eng. Rem. Sens. 79(6), 555–571 (2013).
[Crossref]

Zhang, Z.

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

Zhao, T.

F. Lv, T. Zhao, and R. Nevatia, “Camera calibration from video of a walking human,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1513–1518 (2006).
[Crossref] [PubMed]

Zhu, Q.

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogram. Eng. Rem. Sens. 79(6), 555–571 (2013).
[Crossref]

Comput. Vis. Image Underst. (1)

S. Sinha and M. Pollefeys, “Pan-tilt-zoom camera calibration and high-resolution mosaic generation,” Comput. Vis. Image Underst. 103(3), 170–183 (2006).
[Crossref]

Found. Trends Comput. Graph. Vis. (1)

V. Lepetit and P. Fua, “Monocular model-based 3D tracking of rigid objects: a survey,” Found. Trends Comput. Graph. Vis. 1(1), 1–89 (2005).
[Crossref]

IEEE T. Pattern Anal. (1)

M. Li and J. Lavest, “Some aspects of zoom lens camera calibration,” IEEE T. Pattern Anal. 18(11), 1105–1110 (1996).
[Crossref]

IEEE Trans. Pattern Anal. (2)

R. Lenz and R. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3D machine vision metrology,” IEEE Trans. Pattern Anal. 10(5), 713–720 (1988).
[Crossref]

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

IEEE Trans. Pattern Anal. Mach. Intell. (3)

F. Lv, T. Zhao, and R. Nevatia, “Camera calibration from video of a walking human,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1513–1518 (2006).
[Crossref] [PubMed]

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

Z. Wu and R. J. Radke, “Keeping a pan-tilt-zoom camera calibrated,” IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1994–2007 (2013).
[Crossref] [PubMed]

IEEE Trans. Robot. Autom. (1)

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

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

Opt. Eng. (1)

B. He and Y. Li, “Camera calibration with lens distortion and from vanishing points,” Opt. Eng. 48(1), 013603 (2009).
[Crossref]

Opt. Laser Technol. (1)

Z. Wei, L. Cao, and G. Zhang, “A novel 1D target-based calibration method with unknown orientation for structured light vision sensor,” Opt. Laser Technol. 42(4), 570–574 (2010).
[Crossref]

Photogram. Eng. Rem. Sens. (1)

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogram. Eng. Rem. Sens. 79(6), 555–571 (2013).
[Crossref]

Other (13)

O. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint (MIT Press, 1993).

S. Ma and Z. Zhang, Computer Vision: Theory and Algorithms (Beijing Sciences, 1998).

G. Zhang, Visual Measurement (Beijing Sciences, 2008).

R. Tsai, “An efficient and accurate camera calibration technique for 3D machine vision,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1986), pp. 364–374.

R. Willson, Modeling and Calibration of Automated Zoom Lenses, Ph.D. Dissertation (Carnegie Mellon University, 1994).

J. Davis and X. Chen, “Calibrating pan-tilt cameras in wide-area surveillance networks,” in Proceedings of IEEE International Conference on Computer Vision (IEEE 2003), pp. 144–149.
[Crossref]

Photo-Sonic, “Mobile multispectral TSPI system,” http://www.photosonics.com/mmts.htm .

J. Kelsey, J. Byrne, M. Cosgrove, S. Seereeram, and R. Mehra, “Vision-based relative pose estimation for autonomous rendezvous and docking,” in Proceedings of IEEE Conference on Aerospace (IEEE 2006), pp. 1–20.
[Crossref]

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

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University 2007).

Y. Fei, Error Theory and Data Processing, 6th ed. (China Machine, 2010).

Z. He, Optical Measuring System (National Defense Industry, 2002).

C. Harris and M. Stephens, “A combined corner and edge detector,” in Proceedings of the 4th Alvey Vision Conference (Academic, 1988), pp. 147–151.

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Figures (8)

Fig. 1
Fig. 1 Calibration model.
Fig. 2
Fig. 2 Effect of quantity of control points when focal length is 25mm, (a) shows mean error and RMS error of focal length; (b) shows mean error and RMS error of principal point.
Fig. 3
Fig. 3 Effect of focal length variation, (a) shows the mean error and RMS error of focal length, (b) shows the mean error and RMS error of principal point.
Fig. 4
Fig. 4 Simulated calibration results of two methods. The error in each parameter is plotted as a function of zoom setting. (a) The error in the focal length. (b) The error in the principal point.
Fig. 5
Fig. 5 Determination of zooming center. The upper row illustrates views of camera when it zooms out, and the bottom row illustrates views of camera when it zooms in.
Fig. 6
Fig. 6 The pan, tilt angle and angular distance.
Fig. 7
Fig. 7 Reprojection at f = 24mm, the red circles are images of control points, and green crosses are their re-projection correspondingly.
Fig. 8
Fig. 8 Re-projecting error when f = 24mm, 50mm, 100mm and 200mm.

Tables (1)

Tables Icon

Table 1 Calibration Results

Equations (20)

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

K=[ f x 0 u 0 0 f y v 0 0 0 1 ].
d= K 1 I,
cosθ= d 1 T d 2 d 1 T d 1 d 2 T d 2 = ( K 1 I 1 ) T ( K 1 I 2 ) ( K 1 I 1 ) T ( K 1 I 1 ) ( K 1 I 2 ) T ( K 1 I 2 ) = I 1 T ( K T K 1 ) I 2 I 1 T ( K T K 1 ) I 1 I 2 T ( K T K 1 ) I 2 .
cosθ= I 1 T ω I 2 I 1 T ω I 1 I 2 T ω I 2 .
ω=[ a b/2 d/2 b/2 c e/2 d/2 e/2 h ].
cos θ ij = [ u i u j u i v j + u j v i 2 v i v j u i + u j 2 v i + v j 2 1 ]W [ u i 2 u i v i v i 2 u i v i 1 ]W [ u j 2 u j v j v j 2 u j v j 1 ]W .
W= [ a c d e g ] T = [ 1 f x 2 1 f y 2 2 u 0 f x 2 2 v 0 f y 2 1+ u 0 2 f x 2 + v 0 2 f y 2 ] T , S 1 =[ u i u j v i v j u i + u j 2 v i + v j 2 1 ], S 2 =[ u i 2 v i 2 u i v i 1 ], S 3 =[ u j 2 v j 2 u j v j 1 ],
F= ( S 1 W ) 2 S 2 W S 3 W cos 2 θ=0
F f x =2 S 1 W S 1 W f x ( S 2 W f x S 3 W+ S 2 W S 3 W f x ) cos 2 θ F f y =2 S 1 W S 1 W f y ( S 2 W f y S 3 W+ S 2 W S 3 W f y ) cos 2 θ F u 0 =2 S 1 W S 1 W u 0 ( S 2 W u 0 S 3 W+ S 2 W S 3 W u 0 ) cos 2 θ F v 0 =2 S 1 W S 1 W v 0 ( S 2 W v 0 S 3 W+ S 2 W S 3 W v 0 ) cos 2 θ F u 1 =2 S 1 W S 1 u 1 W( S 2 u 1 W S 3 W+ S 2 W S 3 u 1 W ) cos 2 θ F u 2 =2 S 1 W S 1 u 2 W( S 2 u 2 W S 3 W+ S 2 W S 3 u 2 W ) cos 2 θ F v 1 =2 S 1 W S 1 v 1 W( S 2 v 1 W S 3 W+ S 2 W S 3 v 1 W ) cos 2 θ F v 2 =2 S 1 W S 1 v 2 W( S 2 v 2 W S 3 W+ S 2 W S 3 v 2 W ) cos 2 θ F θ =2 S 2 W S 3 Wcosθsinθ
W f x = f x ' [ 2 / f x 3 0 4 u 0 / f x 3 0 2 u 0 2 / f x 3 ] T W f y = f y ' [ 0 2 / f y 3 0 4 v 0 / f y 3 2 v 0 2 / f y 3 ] T W u 0 = u 0 ' [ 0 0 2 / f x 2 0 2 u 0 / f x 2 ] T W v 0 = v 0 ' [ 0 0 0 2 / f y 2 2 v 0 / f y 2 ] T S 1 u 1 =[ u i2 0 0.5 0 0 ] S 1 u 2 =[ u i1 0 0.5 0 0 ] S 1 v 1 =[ 0 v i2 0 0.5 0 ] S 1 v 2 =[ 0 v i1 0 0.5 0 ] S 1θ =[ 0 0 0 0 0 ] S 2 u 1 =[ 2 u i1 0 1 0 0 ] S 2 u 2 =[ 0 0 0 0 0 ] S 2 v 1 =[ 0 2 v i1 0 1 0 ] S 2 v 2 =[ 0 0 0 0 0 ] S 2θ =[ 0 0 0 0 0 ] S 3 u 1 =[ 0 0 0 0 0 ] S 3 u 2 =[ 2 u i2 0 1 0 0 ] S 3 v 1 =[ 0 0 0 0 0 ] S 3 v 2 =[ 0 2 v i2 0 1 0 ] S 3θ =[ 0 0 0 0 0 ].
Δ u 0 = u 0 u 1 Δ u 1 + u 0 u 2 Δ u 2 + u 0 v 1 Δ v 1 + u 0 v 2 Δ v 2 + u 0 θ Δθ ( F u 1 + F u 2 + F v 1 + F v 2 )Δuv+ F θ Δθ F u 0
Δ u 0 sin 2 θ[ ( u 2 u 0 )+( u 1 u 0 )+( v 2 v 0 )+( v 1 v 0 ) ]Δuv+ f 2 cosθsinθΔθ 2 u 0 u 1 u 2 + cos 2 θ( u 1 u 0 + u 2 u 0 ) = sin 2 θ[ ( u 2 u 0 )+( u 1 u 0 )+( v 2 v 0 )+( v 1 v 0 ) ]Δuv+ f 2 cosθsinθΔθ sin 2 θ( u 1 + u 2 2 u 0 ) =Δuv+ v 1 + v 2 2 v 0 u 1 + u 2 2 u 0 Δuv+ f 2 cosθ sinθ( u 1 + u 2 2 u 0 ) Δθ.
Δ v 0 Δuv+ ( u 1 + u 2 2 u 0 ) ( v 1 + v 2 2 v 0 ) Δuv+ f 2 cosθ sinθ( v 1 + v 2 2 v 0 ) Δθ.
Δ f x f 3 cosθsinθ ( u 1 u 0 )( u 2 u 0 ) cos 2 θ[ ( u 1 u 0 ) 2 + ( u 2 u 0 ) 2 ] Δθ f 3 cosθsinθ -( u 1 u 0 )( u 2 u 0 ) ( u 1 u 2 ) 2 Δθ.
Δ f y f 3 cosθsinθ ( v 1 v 0 )( v 2 v 0 ) cos 2 θ[ ( v 1 v 0 ) 2 + ( v 2 v 0 ) 2 ] Δθ f 3 cosθsinθ -( v 1 v 0 )( v 2 v 0 ) ( v 1 v 2 ) 2 Δθ.
cos θ ij = d i T d j d i T d i d j T d j = d i T d j .
ω= K T K 1 =[ 1 f x 2 0 u 0 f x 2 0 1 f y 2 v 0 f y 2 u 0 f x 2 v 0 f y 2 1+ u 0 2 f x 2 + v 0 2 f y 2 ].
ω 33 =1+ u 0 2 f x 2 + v 0 2 f y 2 =1+ ( u 0 f x 2 ) 2 / ( 1 f x 2 ) + ( v 0 f y 2 ) 2 / ( 1 f y 2 ) ,
g=1+ d 2 4a + e 2 4c .
Δ u 0 = u 0 u 1 Δ u 1 + u 0 u 2 Δ u 2 + u 0 v 1 Δ v 1 + u 0 v 2 Δ v 2 + u 0 θ Δθ ( F u 1 + F u 2 + F v 1 + F v 2 )Δuv+ F θ Δθ F u 0 = { 2[ ( u 1 u 0 )( u 2 u 0 ) f x 2 + ( v 1 v 0 )( v 2 v 0 ) f y 2 +1 ] u 2 u 0 f x 2 2 cos 2 θ{ u 1 u 0 f x 2 [ ( u 2 u 0 ) 2 f x 2 + ( v 2 v 0 ) 2 f y 2 +1 ] } +2[ ( u 1 u 0 )( u 2 u 0 ) f x 2 + ( v 1 v 0 )( v 2 v 0 ) f y 2 +1 ] u 1 u 0 f x 2 2 cos 2 θ{ u 2 u 0 f x 2 [ ( u 1 u 0 ) 2 f x 2 + ( v 1 v 0 ) 2 f y 2 +1 ] } +2[ ( u 1 u 0 )( u 2 u 0 ) f x 2 + ( v 1 v 0 )( v 2 v 0 ) f y 2 +1 ] v 2 v 0 f y 2 2 cos 2 θ{ v 1 v 0 f y 2 [ ( u 2 u 0 ) 2 f x 2 + ( v 2 v 0 ) 2 f y 2 +1 ] } +2[ ( u 1 u 0 )( u 2 u 0 ) f x 2 + ( v 1 v 0 )( v 2 v 0 ) f y 2 +1 ] v 1 v 0 f y 2 2 cos 2 θ{ v 2 v 0 f y 2 [ ( u 1 u 0 ) 2 f x 2 + ( v 1 v 0 ) 2 f y 2 +1 ] } }Δuv +{ 2cosθsinθ[ ( u 1 u 0 ) 2 f x 2 + ( v 1 v 0 ) 2 f y 2 +1 ][ ( u 2 u 0 ) 2 f x 2 + ( v 2 v 0 ) 2 f y 2 +1 ] }Δθ { 2×[ ( u 1 u 0 )( u 2 u 0 ) f x 2 + ( v 1 v 0 )( v 2 v 0 ) f y 2 +1 ]× 2 u 0 u 1 u 2 f x 2 +2 cos 2 θ{ ( u 1 u 0 ) f x 2 [ ( u 2 u 0 ) 2 f x 2 + ( v 2 v 0 ) 2 f y 2 +1 ]+ ( u 2 u 0 ) f x 2 [ ( u 1 u 0 ) 2 f x 2 + ( v 1 v 0 ) 2 f y 2 +1 ] } } sin 2 θ[ ( u 2 u 0 )+( u 1 u 0 )+( v 2 v 0 )+( v 1 v 0 ) ]Δuv+ f x 2 cosθsinθΔθ 2 u 0 u 1 u 2 + cos 2 θ( u 1 u 0 + u 2 u 0 ) = sin 2 θ[ ( u 2 u 0 )+( u 1 u 0 )+( v 2 v 0 )+( v 1 v 0 ) ]Δuv+ f x 2 cosθsinθΔθ sin 2 θ( u 1 + u 2 2 u 0 ) =Δuv+ v 1 + v 2 2 v 0 u 1 + u 2 2 u 0 Δuv+ f x 2 cosθ sinθ( u 1 + u 2 2 u 0 ) Δθ

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