Abstract

A noncontact, fast, accurate, low-cost, broad-range, full-field, easy-to-implement three-dimensional (3D) shape measurement technique is presented. The technique is based on a generalized fringe projection profilometry setup that allows each system component to be arbitrarily positioned. It employs random phase-shifting, multifrequency projection fringes, ultrafast direct phase unwrapping, and inverse self-calibration schemes to perform 3D shape determination with enhanced accuracy in a fast manner. The relative measurement accuracy can reach 1/10,000 or higher, and the acquisition speed is faster than two 3D views per second. The validity and practicability of the proposed technique have been verified by experiments. Because of its superior capability, the proposed 3D shape measurement technique is suitable for numerous applications in a variety of fields.

© 2009 Optical Society of America

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

2007 (4)

2006 (5)

2005 (4)

L. Chen and C. Quan, “Fringe projection profilometry with nonparallel illumination: a least-squares approach,” Opt. Lett. 30, 2101-2103 (2005).
[CrossRef] [PubMed]

T. Peng, S. Gupta, and K. Lau, “Algorithms for constructing 3-D point clouds using multiple digital fringe patterns,” Comput. Aided Des. Appl. 2, 737-746 (2005).

J. Pan, P. Huang, and F. Chiang, “Color-coded binary fringe projection technique for 3-D shape measurement,” Opt. Eng. 44, 023606 (2005).
[CrossRef]

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).
[CrossRef]

2004 (4)

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

C. Tay, C. Quan, T. Wu, and Y. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43, 1152-1159 (2004).
[CrossRef]

L. Kinell, “Spatiotemporal approach for real-time absolute shape measurements by use of projected fringes,” Appl. Opt. 43, 3018-3027 (2004).
[CrossRef] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671-1673 (2004).
[CrossRef] [PubMed]

2003 (2)

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314(2003).
[CrossRef]

Q. Hu, P. Huang, Q. Fu, and F. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487-493 (2003).
[CrossRef]

2002 (1)

J. Burke, T. Bothe, W. Osten, and C. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312-324(2002).
[CrossRef]

2000 (4)

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

F. Chen, G. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10-22(2000).
[CrossRef]

W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body 3-D measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).
[CrossRef]

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and wavelength scanning,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

1999 (1)

C. Coggrave and J. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573-1581 (1999).
[CrossRef]

1998 (1)

C. Keferstein and M. Marxer, “Testing bench for laser triangulation sensors,” Sens. Rev. 18, 183-187 (1998).
[CrossRef]

1997 (1)

S. McNeill, M. Sutton, Z. Miao, and J. Ma, “Measurement of surface profile using digital image correlation,” Exp. Mech. 37, 13-20 (1997).
[CrossRef]

1996 (2)

W. Osten, W. Nadeborn, and P. Andra, “General hierarchical approach in absolute phase measurement,” Proc. SPIE 2860, 2-13 (1996).
[CrossRef]

W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260(1996).
[CrossRef]

1993 (1)

T. Yoshizawa and T. Tomisawa, “Shadow moiré topography by means of the phase shift method,” Opt. Eng. 32, 1668-1674 (1993).
[CrossRef]

1992 (1)

C. Fraser, “Photogrammetric measurement to one part in a million,” Photogramm. Eng. Remote Sens. 58, 305-310 (1992).

1987 (1)

R. 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, 323-344(1987).
[CrossRef]

Andra, P.

W. Osten, W. Nadeborn, and P. Andra, “General hierarchical approach in absolute phase measurement,” Proc. SPIE 2860, 2-13 (1996).
[CrossRef]

W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260(1996).
[CrossRef]

Bi, H.

Bothe, T.

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

J. Burke, T. Bothe, W. Osten, and C. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312-324(2002).
[CrossRef]

Brown, G.

F. Chen, G. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10-22(2000).
[CrossRef]

Burke, J.

J. Burke, T. Bothe, W. Osten, and C. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312-324(2002).
[CrossRef]

Chen, F.

F. Chen, G. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10-22(2000).
[CrossRef]

Chen, L.

Chen, M.

Chiang, F.

J. Pan, P. Huang, and F. Chiang, “Color-coded binary fringe projection technique for 3-D shape measurement,” Opt. Eng. 44, 023606 (2005).
[CrossRef]

Q. Hu, P. Huang, Q. Fu, and F. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487-493 (2003).
[CrossRef]

Coggrave, C.

C. Coggrave and J. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573-1581 (1999).
[CrossRef]

Du, H.

H. Du and Z. Wang, “Three-dimensional shape measurement with arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32, 2438-2440 (2007).
[CrossRef] [PubMed]

Z. Wang, H. Du, and H. Bi, “Out-of-plane shape determination in fringe projection profilometry,” Opt. Express 14, 12122-12133 (2006).
[CrossRef] [PubMed]

H. Du and Z. Wang, “Real-time 3-D shape measurement with high accuracy and low cost,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Application System Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper JThD49.
[PubMed]

Ferraro, P.

W. Osten and P. Ferraro, “Digital holography and its application in MEMS/MOEMS inspection,” in Optical Inspection of Microsystems, W. Osten, ed. (CRC Press, 2006), p. 351-425.

Fraser, C.

C. Fraser, “Photogrammetric measurement to one part in a million,” Photogramm. Eng. Remote Sens. 58, 305-310 (1992).

Fu, Q.

Q. Hu, P. Huang, Q. Fu, and F. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487-493 (2003).
[CrossRef]

Garcia, V.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314(2003).
[CrossRef]

Guo, H.

Gupta, S.

T. Peng, S. Gupta, and K. Lau, “Algorithms for constructing 3-D point clouds using multiple digital fringe patterns,” Comput. Aided Des. Appl. 2, 737-746 (2005).

Han, B.

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274-280 (2007).
[CrossRef]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671-1673 (2004).
[CrossRef] [PubMed]

He, H.

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).
[CrossRef]

Hess, C.

J. Burke, T. Bothe, W. Osten, and C. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312-324(2002).
[CrossRef]

Hu, Q.

Q. Hu, P. Huang, Q. Fu, and F. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487-493 (2003).
[CrossRef]

Huang, P.

J. Pan, P. Huang, and F. Chiang, “Color-coded binary fringe projection technique for 3-D shape measurement,” Opt. Eng. 44, 023606 (2005).
[CrossRef]

Q. Hu, P. Huang, Q. Fu, and F. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487-493 (2003).
[CrossRef]

Huang, Y.

C. Tay, C. Quan, T. Wu, and Y. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43, 1152-1159 (2004).
[CrossRef]

Huntley, J.

C. Coggrave and J. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573-1581 (1999).
[CrossRef]

Juptner, W.

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

Keferstein, C.

C. Keferstein and M. Marxer, “Testing bench for laser triangulation sensors,” Sens. Rev. 18, 183-187 (1998).
[CrossRef]

Kinell, L.

Kyle, S.

S. Kyle, R. Loser, and D. Warren, “Automated part positioning with the laser tracker,” in Proceedings of the Fifth International Workshop on Accelerator Alignment, ANL/FNAL (1997).

Lau, K.

T. Peng, S. Gupta, and K. Lau, “Algorithms for constructing 3-D point clouds using multiple digital fringe patterns,” Comput. Aided Des. Appl. 2, 737-746 (2005).

Legarda-Sáenz, R.

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

Li, X.

Loser, R.

S. Kyle, R. Loser, and D. Warren, “Automated part positioning with the laser tracker,” in Proceedings of the Fifth International Workshop on Accelerator Alignment, ANL/FNAL (1997).

Luna, E.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314(2003).
[CrossRef]

Ma, J.

S. McNeill, M. Sutton, Z. Miao, and J. Ma, “Measurement of surface profile using digital image correlation,” Exp. Mech. 37, 13-20 (1997).
[CrossRef]

Marxer, M.

C. Keferstein and M. Marxer, “Testing bench for laser triangulation sensors,” Sens. Rev. 18, 183-187 (1998).
[CrossRef]

McNeill, S.

S. McNeill, M. Sutton, Z. Miao, and J. Ma, “Measurement of surface profile using digital image correlation,” Exp. Mech. 37, 13-20 (1997).
[CrossRef]

Miao, Z.

S. McNeill, M. Sutton, Z. Miao, and J. Ma, “Measurement of surface profile using digital image correlation,” Exp. Mech. 37, 13-20 (1997).
[CrossRef]

Nadeborn, W.

W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260(1996).
[CrossRef]

W. Osten, W. Nadeborn, and P. Andra, “General hierarchical approach in absolute phase measurement,” Proc. SPIE 2860, 2-13 (1996).
[CrossRef]

Notni, G.

W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body 3-D measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).
[CrossRef]

Osten, W.

J. Burke, T. Bothe, W. Osten, and C. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312-324(2002).
[CrossRef]

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and wavelength scanning,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245-260(1996).
[CrossRef]

W. Osten, W. Nadeborn, and P. Andra, “General hierarchical approach in absolute phase measurement,” Proc. SPIE 2860, 2-13 (1996).
[CrossRef]

W. Osten and P. Ferraro, “Digital holography and its application in MEMS/MOEMS inspection,” in Optical Inspection of Microsystems, W. Osten, ed. (CRC Press, 2006), p. 351-425.

Pan, J.

J. Pan, P. Huang, and F. Chiang, “Color-coded binary fringe projection technique for 3-D shape measurement,” Opt. Eng. 44, 023606 (2005).
[CrossRef]

Peng, T.

T. Peng, S. Gupta, and K. Lau, “Algorithms for constructing 3-D point clouds using multiple digital fringe patterns,” Comput. Aided Des. Appl. 2, 737-746 (2005).

Peng, X.

Quan, C.

Salas, L.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314(2003).
[CrossRef]

Salinas, J.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314(2003).
[CrossRef]

Schreiber, W.

W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body 3-D measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).
[CrossRef]

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and wavelength scanning,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

Servin, M.

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314(2003).
[CrossRef]

Song, M.

F. Chen, G. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10-22(2000).
[CrossRef]

Sutton, M.

S. McNeill, M. Sutton, Z. Miao, and J. Ma, “Measurement of surface profile using digital image correlation,” Exp. Mech. 37, 13-20 (1997).
[CrossRef]

Tay, C.

C. Tay, C. Quan, T. Wu, and Y. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43, 1152-1159 (2004).
[CrossRef]

Tian, J.

Tomisawa, T.

T. Yoshizawa and T. Tomisawa, “Shadow moiré topography by means of the phase shift method,” Opt. Eng. 32, 1668-1674 (1993).
[CrossRef]

Tsai, R.

R. 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, 323-344(1987).
[CrossRef]

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and wavelength scanning,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

Wang, Z.

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274-280 (2007).
[CrossRef]

H. Du and Z. Wang, “Three-dimensional shape measurement with arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32, 2438-2440 (2007).
[CrossRef] [PubMed]

Z. Wang and H. Bi, “Comments on fringe projection profilometry with nonparallel illumination: a least-squares approach,” Opt. Lett. 31, 1972-1973 (2006).
[CrossRef] [PubMed]

Z. Wang, H. Du, and H. Bi, “Out-of-plane shape determination in fringe projection profilometry,” Opt. Express 14, 12122-12133 (2006).
[CrossRef] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671-1673 (2004).
[CrossRef] [PubMed]

H. Du and Z. Wang, “Real-time 3-D shape measurement with high accuracy and low cost,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Application System Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper JThD49.
[PubMed]

Warren, D.

S. Kyle, R. Loser, and D. Warren, “Automated part positioning with the laser tracker,” in Proceedings of the Fifth International Workshop on Accelerator Alignment, ANL/FNAL (1997).

Wu, T.

C. Tay, C. Quan, T. Wu, and Y. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43, 1152-1159 (2004).
[CrossRef]

Yau, S.

Yoshizawa, T.

T. Yoshizawa and T. Tomisawa, “Shadow moiré topography by means of the phase shift method,” Opt. Eng. 32, 1668-1674 (1993).
[CrossRef]

Yu, Y.

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).
[CrossRef]

Zhang, P.

Zhang, S.

Zhang, Z.

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

Appl. Opt. (4)

Comput. Aided Des. Appl. (1)

T. Peng, S. Gupta, and K. Lau, “Algorithms for constructing 3-D point clouds using multiple digital fringe patterns,” Comput. Aided Des. Appl. 2, 737-746 (2005).

Exp. Mech. (1)

S. McNeill, M. Sutton, Z. Miao, and J. Ma, “Measurement of surface profile using digital image correlation,” Exp. Mech. 37, 13-20 (1997).
[CrossRef]

IEEE J. Robot. Autom. (1)

R. 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, 323-344(1987).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

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

Opt. Eng. (11)

F. Chen, G. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10-22(2000).
[CrossRef]

W. Schreiber and G. Notni, “Theory and arrangements of self-calibrating whole-body 3-D measurement systems using fringe projection technique,” Opt. Eng. 39, 159-169 (2000).
[CrossRef]

L. Salas, E. Luna, J. Salinas, V. Garcia, and M. Servin, “Profilometry by fringe projection,” Opt. Eng. 42, 3307-3314(2003).
[CrossRef]

Q. Hu, P. Huang, Q. Fu, and F. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487-493 (2003).
[CrossRef]

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

C. Tay, C. Quan, T. Wu, and Y. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43, 1152-1159 (2004).
[CrossRef]

T. Yoshizawa and T. Tomisawa, “Shadow moiré topography by means of the phase shift method,” Opt. Eng. 32, 1668-1674 (1993).
[CrossRef]

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and wavelength scanning,” Opt. Eng. 39, 79-85 (2000).
[CrossRef]

J. Pan, P. Huang, and F. Chiang, “Color-coded binary fringe projection technique for 3-D shape measurement,” Opt. Eng. 44, 023606 (2005).
[CrossRef]

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic setup of the proposed technique

Fig. 2
Fig. 2

Projection fringes used in the proposed technique: (a) primary high-frequency fringe pattern and (b) secondary low- frequency fringe pattern.

Fig. 3
Fig. 3

Schematic of the geometric relationship in the proposed system setup

Fig. 4
Fig. 4

Accuracy examination: (a) low-frequency fringe pattern, (b) high-frequency fringe pattern, (c) 2D shape map, (d) 3D shape map, and (e) original and uncropped image.

Fig. 5
Fig. 5

3D shape measurement of a PCB: (a) low-frequency fringe pattern, (b) high-frequency fringe pattern, (c) 2D shape map, and (d) 3D rendered shape map.

Fig. 6
Fig. 6

3D shape measurements of multiple objects: (a) objects, (b) a typical fringe image, (c) 2D shape map, and (d) 3D shape map.

Fig. 7
Fig. 7

3D shape measurement of a rabbit model: (a) three representative fringe patterns from the same view and (b) illustration of the complete 360 ° 3D image.

Fig. 8
Fig. 8

3D shape measurement of a Plexiglas plate: (a) a representative fringe image, (b) 2D shape map, and (c) 3D shape map.

Equations (9)

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{ x P y P } = { z P z A z B z P } { x B y B x A y A } / ( z B z A ) ,
{ x D y D } = { z D z C z P z D } { x P y P x C y C } / ( z P z C ) .
{ x B y B z B } = { x O y O z O } + { x B y B z B } R α , β , γ T ,
{ x D y D z D } = { x O y O z O } + { x D y D z D } R θ , ϕ , ψ T ,
Φ B = Φ B ( x B , y B ) = Φ D ( x D , y D ) = Φ O + 2 π x D / p .
z P = 1 + c 1 Φ B + ( c 2 + c 3 Φ B ) x B + ( c 4 + c 5 Φ B ) y B d 0 + d 1 Φ B + ( d 2 + d 3 Φ B ) x B + ( d 4 + d 5 Φ B ) y B .
z P = 1 + C 1 Φ B + ( C 2 + C 3 Φ B ) I B + ( C 4 + C 5 Φ B ) J B D 0 + D 1 Φ B + ( D 2 + D 3 Φ B ) I B + ( D 4 + D 5 Φ B ) J B .
S = i = 1 m [ 1 + C 1 Φ i + ( C 2 + C 3 Φ i ) I i + ( C 4 + C 5 Φ i ) J i D 0 + D 1 Φ i + ( D 2 + D 3 Φ i ) I i + ( D 4 + D 5 Φ i ) J i z i g ] 2 ,
Φ i u w = Φ i w + INT ( Φ i 1 u w · f i f i - 1 Φ i w 2 π ) · 2 π ( i = 1 , 2 , ... , n ) ,

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