Abstract

For a three-dimensional shape measurement system with a single projector and multiple cameras, registering patches from different cameras is crucial. Registration usually involves a complicated and time-consuming procedure. We propose a new method that can robustly match different patches via absolute phase without significantly increasing its cost. For y and z coordinates, the transformations from one camera to the other are approximated as third-order polynomial functions of the absolute phase. The x coordinates involve only translations and scalings. These functions are calibrated and only need to be determined once. Experiments demonstrated that the alignment error is within RMS 0.7mm.

© 2008 Optical Society of America

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  1. J. Salvi, J. Pagès , and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849(2004).
  2. S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45, 083601 (2006)
  3. R. Legarda-Sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464-471 (2004).
  4. C. Reich, “Photogrammetrical matching of point clouds for 3D measurement of complex objects,” Proc. SPIE 3520, 100-110(1998).
  5. R. Massen, J. Gässler, C. Konz, and H. Richter, “From a million of dull point coordinates to an intelligent CAD description,” Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W. Osten, eds. (Wiley, 1993), pp. 194-203.
  6. H. Schöfeld, G. Häusler, and S. Karbacher, “Reverse engineering using optical 3D sensors,” Proc. SPIE 3313, 115-125(1998).
  7. P. J. Neugebauer, “Reconstruction of real-world objects via simultaneous registration and robust combination of multiple range images,” Int. J. Shape Model. 3, 71-90 (1997).
  8. P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
    [CrossRef]
  9. Y. Chen and G. Medioni, “Object modelling by registration of multiple range images,” Image Vision Comput. 10, 145-155(1992).
  10. H. A. Beyer, V. Uffenkamp, and G. van der Vlugt, “Quality control in industry with digital photogrammetry,” in Optical 3D-Measurement Techniques III (Wichmann, 1995), pp. 29-38.
  11. V. Kirschner, W. Schreiber, R. Kowarschik, and G. Notni, “Self-calibrating shape measuring system based on fringe projection,” Proc. SPIE 3102, 5-13 (1997).
  12. H. Kühmstedt, G. Notni, W. Schreiber, and J. Gerber, “Full-hemisphere automatic optical 3D measurement system,” Proc. SPIE 3100, 261-265 (1997).
  13. A. K. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38, 339-344 (1999).
  14. S. Zhang and S.-T. Yau, “High-resolution, real-time 3D absolute coordinate measurement based on a phase-shifting method,” Opt. Express 14, 2644-2649 (2006).
    [CrossRef]
  15. W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).
  16. V. Yalla and L. G. Hassebrook, “A novel geometric calibration technique for scalable multi-projector displays,” Technical Report CSP-06-010 (University of Kentucky, 2006).
  17. Y. Wang, K. Liu, Q. Hao, D. Lau, and L. G. Hassebrook, “Multicamera phase measuring profilometry for accurate depth measurement,” Proc. SPIE 6555, 655509 (2007).
  18. S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601(2006).
  19. P. S. Huang and S. Zhang, “Fast three-step phase shifting algorithm,” Appl. Opt. 45, 5086-5091 (2006).
    [CrossRef]
  20. S. Zhang and S.-T. Yau, “Three-dimensional shape measurement using a structured light system with dual cameras,” Opt. Eng. 47, (1), 013604 (2008).
  21. D. Malacara, ed., ,i>Optical Shop Testing (Wiley, 1992).
  22. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  23. S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50-57 (2007).
    [CrossRef]
  24. P. S. Huang and X. Han, “On improving the accuracy of structured light systems,” Proc. SPIE 6382, 63820H (2006).

2008

S. Zhang and S.-T. Yau, “Three-dimensional shape measurement using a structured light system with dual cameras,” Opt. Eng. 47, (1), 013604 (2008).

2007

Y. Wang, K. Liu, Q. Hao, D. Lau, and L. G. Hassebrook, “Multicamera phase measuring profilometry for accurate depth measurement,” Proc. SPIE 6555, 655509 (2007).

S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50-57 (2007).
[CrossRef]

2006

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601(2006).

P. S. Huang and X. Han, “On improving the accuracy of structured light systems,” Proc. SPIE 6382, 63820H (2006).

S. Zhang and S.-T. Yau, “High-resolution, real-time 3D absolute coordinate measurement based on a phase-shifting method,” Opt. Express 14, 2644-2649 (2006).
[CrossRef]

P. S. Huang and S. Zhang, “Fast three-step phase shifting algorithm,” Appl. Opt. 45, 5086-5091 (2006).
[CrossRef]

S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45, 083601 (2006)

2004

R. Legarda-Sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464-471 (2004).

J. Salvi, J. Pagès , and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849(2004).

2000

W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).

1999

A. K. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38, 339-344 (1999).

1998

C. Reich, “Photogrammetrical matching of point clouds for 3D measurement of complex objects,” Proc. SPIE 3520, 100-110(1998).

H. Schöfeld, G. Häusler, and S. Karbacher, “Reverse engineering using optical 3D sensors,” Proc. SPIE 3313, 115-125(1998).

1997

P. J. Neugebauer, “Reconstruction of real-world objects via simultaneous registration and robust combination of multiple range images,” Int. J. Shape Model. 3, 71-90 (1997).

V. Kirschner, W. Schreiber, R. Kowarschik, and G. Notni, “Self-calibrating shape measuring system based on fringe projection,” Proc. SPIE 3102, 5-13 (1997).

H. Kühmstedt, G. Notni, W. Schreiber, and J. Gerber, “Full-hemisphere automatic optical 3D measurement system,” Proc. SPIE 3100, 261-265 (1997).

1992

P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
[CrossRef]

Y. Chen and G. Medioni, “Object modelling by registration of multiple range images,” Image Vision Comput. 10, 145-155(1992).

Asundi, A. K.

A. K. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38, 339-344 (1999).

Batlle, J.

J. Salvi, J. Pagès , and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849(2004).

Besl, P. J.

P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
[CrossRef]

Beyer, H. A.

H. A. Beyer, V. Uffenkamp, and G. van der Vlugt, “Quality control in industry with digital photogrammetry,” in Optical 3D-Measurement Techniques III (Wichmann, 1995), pp. 29-38.

Bothe, T.

R. Legarda-Sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464-471 (2004).

Chen, Y.

Y. Chen and G. Medioni, “Object modelling by registration of multiple range images,” Image Vision Comput. 10, 145-155(1992).

Gässler, J.

R. Massen, J. Gässler, C. Konz, and H. Richter, “From a million of dull point coordinates to an intelligent CAD description,” Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W. Osten, eds. (Wiley, 1993), pp. 194-203.

Gerber, J.

H. Kühmstedt, G. Notni, W. Schreiber, and J. Gerber, “Full-hemisphere automatic optical 3D measurement system,” Proc. SPIE 3100, 261-265 (1997).

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Han, X.

P. S. Huang and X. Han, “On improving the accuracy of structured light systems,” Proc. SPIE 6382, 63820H (2006).

Hao, Q.

Y. Wang, K. Liu, Q. Hao, D. Lau, and L. G. Hassebrook, “Multicamera phase measuring profilometry for accurate depth measurement,” Proc. SPIE 6555, 655509 (2007).

Hassebrook, L. G.

Y. Wang, K. Liu, Q. Hao, D. Lau, and L. G. Hassebrook, “Multicamera phase measuring profilometry for accurate depth measurement,” Proc. SPIE 6555, 655509 (2007).

V. Yalla and L. G. Hassebrook, “A novel geometric calibration technique for scalable multi-projector displays,” Technical Report CSP-06-010 (University of Kentucky, 2006).

Häusler, G.

H. Schöfeld, G. Häusler, and S. Karbacher, “Reverse engineering using optical 3D sensors,” Proc. SPIE 3313, 115-125(1998).

Huang, P. S.

P. S. Huang and S. Zhang, “Fast three-step phase shifting algorithm,” Appl. Opt. 45, 5086-5091 (2006).
[CrossRef]

S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45, 083601 (2006)

P. S. Huang and X. Han, “On improving the accuracy of structured light systems,” Proc. SPIE 6382, 63820H (2006).

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601(2006).

Jüptner, W. P.

R. Legarda-Sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464-471 (2004).

Karbacher, S.

H. Schöfeld, G. Häusler, and S. Karbacher, “Reverse engineering using optical 3D sensors,” Proc. SPIE 3313, 115-125(1998).

Kirschner, V.

V. Kirschner, W. Schreiber, R. Kowarschik, and G. Notni, “Self-calibrating shape measuring system based on fringe projection,” Proc. SPIE 3102, 5-13 (1997).

Konz, C.

R. Massen, J. Gässler, C. Konz, and H. Richter, “From a million of dull point coordinates to an intelligent CAD description,” Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W. Osten, eds. (Wiley, 1993), pp. 194-203.

Kowarschik, R.

V. Kirschner, W. Schreiber, R. Kowarschik, and G. Notni, “Self-calibrating shape measuring system based on fringe projection,” Proc. SPIE 3102, 5-13 (1997).

Kühmstedt, H.

H. Kühmstedt, G. Notni, W. Schreiber, and J. Gerber, “Full-hemisphere automatic optical 3D measurement system,” Proc. SPIE 3100, 261-265 (1997).

Lau, D.

Y. Wang, K. Liu, Q. Hao, D. Lau, and L. G. Hassebrook, “Multicamera phase measuring profilometry for accurate depth measurement,” Proc. SPIE 6555, 655509 (2007).

Legarda-Sáenz, R.

R. Legarda-Sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464-471 (2004).

Li, X.

Liu, K.

Y. Wang, K. Liu, Q. Hao, D. Lau, and L. G. Hassebrook, “Multicamera phase measuring profilometry for accurate depth measurement,” Proc. SPIE 6555, 655509 (2007).

Massen, R.

R. Massen, J. Gässler, C. Konz, and H. Richter, “From a million of dull point coordinates to an intelligent CAD description,” Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W. Osten, eds. (Wiley, 1993), pp. 194-203.

McKay, H. D.

P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
[CrossRef]

Medioni, G.

Y. Chen and G. Medioni, “Object modelling by registration of multiple range images,” Image Vision Comput. 10, 145-155(1992).

Neugebauer, P. J.

P. J. Neugebauer, “Reconstruction of real-world objects via simultaneous registration and robust combination of multiple range images,” Int. J. Shape Model. 3, 71-90 (1997).

Notni, G.

W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).

V. Kirschner, W. Schreiber, R. Kowarschik, and G. Notni, “Self-calibrating shape measuring system based on fringe projection,” Proc. SPIE 3102, 5-13 (1997).

H. Kühmstedt, G. Notni, W. Schreiber, and J. Gerber, “Full-hemisphere automatic optical 3D measurement system,” Proc. SPIE 3100, 261-265 (1997).

Pagès , J.

J. Salvi, J. Pagès , and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849(2004).

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Reich, C.

C. Reich, “Photogrammetrical matching of point clouds for 3D measurement of complex objects,” Proc. SPIE 3520, 100-110(1998).

Richter, H.

R. Massen, J. Gässler, C. Konz, and H. Richter, “From a million of dull point coordinates to an intelligent CAD description,” Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W. Osten, eds. (Wiley, 1993), pp. 194-203.

Salvi, J.

J. Salvi, J. Pagès , and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849(2004).

Schöfeld, H.

H. Schöfeld, G. Häusler, and S. Karbacher, “Reverse engineering using optical 3D sensors,” Proc. SPIE 3313, 115-125(1998).

Schreiber, W.

W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).

V. Kirschner, W. Schreiber, R. Kowarschik, and G. Notni, “Self-calibrating shape measuring system based on fringe projection,” Proc. SPIE 3102, 5-13 (1997).

H. Kühmstedt, G. Notni, W. Schreiber, and J. Gerber, “Full-hemisphere automatic optical 3D measurement system,” Proc. SPIE 3100, 261-265 (1997).

Uffenkamp, V.

H. A. Beyer, V. Uffenkamp, and G. van der Vlugt, “Quality control in industry with digital photogrammetry,” in Optical 3D-Measurement Techniques III (Wichmann, 1995), pp. 29-38.

van der Vlugt, G.

H. A. Beyer, V. Uffenkamp, and G. van der Vlugt, “Quality control in industry with digital photogrammetry,” in Optical 3D-Measurement Techniques III (Wichmann, 1995), pp. 29-38.

Wang, Y.

Y. Wang, K. Liu, Q. Hao, D. Lau, and L. G. Hassebrook, “Multicamera phase measuring profilometry for accurate depth measurement,” Proc. SPIE 6555, 655509 (2007).

Yalla, V.

V. Yalla and L. G. Hassebrook, “A novel geometric calibration technique for scalable multi-projector displays,” Technical Report CSP-06-010 (University of Kentucky, 2006).

Yau, S.-T.

Zhang, S.

S. Zhang and S.-T. Yau, “Three-dimensional shape measurement using a structured light system with dual cameras,” Opt. Eng. 47, (1), 013604 (2008).

S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50-57 (2007).
[CrossRef]

P. S. Huang and S. Zhang, “Fast three-step phase shifting algorithm,” Appl. Opt. 45, 5086-5091 (2006).
[CrossRef]

S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45, 083601 (2006)

S. Zhang and S.-T. Yau, “High-resolution, real-time 3D absolute coordinate measurement based on a phase-shifting method,” Opt. Express 14, 2644-2649 (2006).
[CrossRef]

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601(2006).

Zhou, W.

A. K. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38, 339-344 (1999).

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
[CrossRef]

Image Vision Comput.

Y. Chen and G. Medioni, “Object modelling by registration of multiple range images,” Image Vision Comput. 10, 145-155(1992).

Int. J. Shape Model.

P. J. Neugebauer, “Reconstruction of real-world objects via simultaneous registration and robust combination of multiple range images,” Int. J. Shape Model. 3, 71-90 (1997).

Opt. Eng.

S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45, 083601 (2006)

R. Legarda-Sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464-471 (2004).

A. K. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38, 339-344 (1999).

W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601(2006).

S. Zhang and S.-T. Yau, “Three-dimensional shape measurement using a structured light system with dual cameras,” Opt. Eng. 47, (1), 013604 (2008).

Opt. Express

Pattern Recogn.

J. Salvi, J. Pagès , and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849(2004).

Proc. SPIE

H. Schöfeld, G. Häusler, and S. Karbacher, “Reverse engineering using optical 3D sensors,” Proc. SPIE 3313, 115-125(1998).

C. Reich, “Photogrammetrical matching of point clouds for 3D measurement of complex objects,” Proc. SPIE 3520, 100-110(1998).

V. Kirschner, W. Schreiber, R. Kowarschik, and G. Notni, “Self-calibrating shape measuring system based on fringe projection,” Proc. SPIE 3102, 5-13 (1997).

H. Kühmstedt, G. Notni, W. Schreiber, and J. Gerber, “Full-hemisphere automatic optical 3D measurement system,” Proc. SPIE 3100, 261-265 (1997).

P. S. Huang and X. Han, “On improving the accuracy of structured light systems,” Proc. SPIE 6382, 63820H (2006).

Y. Wang, K. Liu, Q. Hao, D. Lau, and L. G. Hassebrook, “Multicamera phase measuring profilometry for accurate depth measurement,” Proc. SPIE 6555, 655509 (2007).

Other

V. Yalla and L. G. Hassebrook, “A novel geometric calibration technique for scalable multi-projector displays,” Technical Report CSP-06-010 (University of Kentucky, 2006).

D. Malacara, ed., ,i>Optical Shop Testing (Wiley, 1992).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

R. Massen, J. Gässler, C. Konz, and H. Richter, “From a million of dull point coordinates to an intelligent CAD description,” Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W. Osten, eds. (Wiley, 1993), pp. 194-203.

H. A. Beyer, V. Uffenkamp, and G. van der Vlugt, “Quality control in industry with digital photogrammetry,” in Optical 3D-Measurement Techniques III (Wichmann, 1995), pp. 29-38.

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Figures (13)

Fig. 1
Fig. 1

One projection line will be imaged to yield different lines for different cameras viewed from different viewing angles.

Fig. 2
Fig. 2

System setup for structured light with one projector and dual cameras.

Fig. 3
Fig. 3

Photograph of the real measurement system.

Fig. 4
Fig. 4

Matching process.

Fig. 5
Fig. 5

Calibration images of two cameras. The white line is an absolute phase line with absolute phase value of 30 π rad . (a) Left camera image. (b) Right camera image.

Fig. 6
Fig. 6

Polynomial fittings of δ y and δ z with respect to ϕ a : (a)  δ y ( ϕ a ) , (b)  δ z ( ϕ a ) .

Fig. 7
Fig. 7

Polynomial fittings of δ x with respect to ϕ a .

Fig. 8
Fig. 8

Absolute phase lines, from bottom to top, with an absolute phase value of ϕ 2 = 0 π , 24 π , ... , 44 π . (a) Before alignment. (b) After alignment.

Fig. 9
Fig. 9

Corner coordinates of the checkerboard. (a) Before matching. (b) After matching.

Fig. 10
Fig. 10

Corners of the checkerboard.

Fig. 11
Fig. 11

Corner coordinates of the checkerboard in a 2D plane. (a) Before matching. (b) After matching (RMS 0.7 mm ).

Fig. 12
Fig. 12

Measurement result for a more complicated object. (a) 3D geometry from the left-hand camera. (b) 3-D geometry obtained from the right-hand camera. (c) Before matching, two geometries rendered in the same scene. (d) After matching, two geometries rendered together. (e) After matching, two geometries rendered from another viewing angle. (f) After matching, two geometries rendered in wireframe mode.

Fig. 13
Fig. 13

2D photos of head sculpture from two cameras. (a) 2D photo from left-hand camera. (b) 2D photo from right-hand camera.

Equations (18)

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

I 1 = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) 2 π / 3 ] ,
I 2 = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) ] ,
I 3 = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) + 2 π / 3 ] ,
ϕ ( x , y ) = tan 1 [ 3 ( I 1 I 3 ) 2 I 2 I 1 I 3 ] .
s c l [ u c l , v c l , 1 ] T = M l [ x w , y w , z w , 1 ] T ,
s c r [ u c r , v c r , 1 ] T = M r [ x w , y w , z w , 1 ] T ,
s c p [ u c p , v c p , 1 ] T = M p [ x w , y w , z w , 1 ] T ,
δ x ( ϕ a ) = x l x r = f x ( ϕ a ) ,
δ y ( ϕ a ) = y l y r = f y ( ϕ a ) ,
δ z ( ϕ a ) = y l y r = f z ( ϕ a ) ,
δ x ( ϕ k ) = i = 1 N x i r / N j = 1 M x j l / M ,
δ y ( ϕ k ) = i = 1 N y i r / N j = 1 M y j l / M ,
δ z ( ϕ k ) = i = 1 N z i r / N j = 1 M z j l / M .
x = x r + f x ( ϕ a ) ,
y = y r + f y ( ϕ a ) ,
z = z r + f z ( ϕ a ) .
δ y ϕ a = f y ( ϕ a ) = c y 0 + c y 1 ϕ a + c y 2 ϕ a 2 + c y 3 ϕ a 3 ,
δ z ϕ a = f z ( ϕ a ) = c z 0 + c z 1 ϕ a + c z 2 ϕ a 2 + c z 3 ϕ a 3 .

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