Abstract

Structured-light profilometry is a powerful tool to reconstruct the three-dimensional (3D) profile of an object. Accurate profile acquisition is often hindered by not only the nonlinear response (i.e., gamma effect) of electronic devices but also the projection-imaging distortion of lens used in the system. In this paper, a flexible 3D profile reconstruction method based on a nonlinear iterative optimization is proposed to correct the errors caused by the lens distortion. It can be easily extended to measurements for which a more complex projection-imaging distortion model is required. Experimental work shows that the root-mean-square (RMS) error is reduced by eight times and highly accurate results with errors of less than 1‰ can be achieved by the proposed method.

© 2012 Optical Society of America

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2012 (1)

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285, 533–538 (2012).
[CrossRef]

2011 (3)

2010 (11)

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Gamma model and its analysis for phase measurement profilometry,” J. Opt. Soc. Am. A 27, 553–562 (2010).
[CrossRef]

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18, 5229–5244 (2010).
[CrossRef]

H. Cui, W. Liao, X. Cheng, N. Dai, and T. Yuan, “A three-step system calibration procedure with error compensation for 3D shape measurement,” Chin. Opt. Lett. 8, 33–37(2010).
[CrossRef]

L. Huang, P. S. K. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49, 1539–1548 (2010).
[CrossRef]

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35, 1992–1994 (2010).
[CrossRef]

Y. Wen, S. Li, H. Cheng, X. Su, and Q. Zhang, “Universal calculation formula and calibration method in Fourier transform profilometry,” Appl. Opt. 49, 6563–6569 (2010).
[CrossRef]

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe profilometry,” Opt. Lasers Eng. 48, 218–225 (2010).
[CrossRef]

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

S. Zhang, “Recent processes on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

2009 (2)

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

S. Cui and X. Zhu, “A generalized reference-plane-based calibration method in optical triangular profilometry,” Opt. Express 17, 20735–20746 (2009).
[CrossRef]

2008 (2)

F. Da and S. Gai, “Flexible three-dimensional measurement technique based on a digital light processing projector,” Appl. Opt. 47, 377–385 (2008).
[CrossRef]

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[CrossRef]

2007 (3)

2006 (2)

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

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

2004 (2)

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

H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43, 2906–2914 (2004).
[CrossRef]

2000 (2)

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

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

1997 (1)

1983 (1)

Asundi, A.

Bai, P.

Baker, M. J.

Barnes, J.

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe profilometry,” Opt. Lasers Eng. 48, 218–225 (2010).
[CrossRef]

Bothe, T.

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

Brown, G. M.

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

Chen, F.

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

Chen, L.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285, 533–538 (2012).
[CrossRef]

Chen, M.

Chen, W.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

Cheng, H.

Cheng, X.

Chicharo, J. F.

Chihara, K.

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

Chua, P. S. K.

Cui, H.

Cui, S.

Da, F.

Dai, J.

Dai, N.

Datta, A.

A. Datta, J. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” IEEE 12th ICCV Workshops (IEEE, 2009), pp. 1201–1208.

Douxchamps, D.

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

Du, H.

Ekstrand, L.

Flannery, B. P.

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

Gai, S.

Gao, Z.

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

Gorthi, S.

S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Guo, H.

Han, X.

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

Hao, Q.

Hassebrook, L. G.

He, H.

He, X.

Hoang, T.

Huang, L.

Huang, P. S.

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

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

Huntley, J. M.

Juptner, W. P.

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

Kanade, T.

A. Datta, J. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” IEEE 12th ICCV Workshops (IEEE, 2009), pp. 1201–1208.

Kim, J.

A. Datta, J. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” IEEE 12th ICCV Workshops (IEEE, 2009), pp. 1201–1208.

Lau, D. L.

Legarda-Saenz, R.

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

Li, B.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285, 533–538 (2012).
[CrossRef]

Li, J.

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

Li, S.

Li, Z.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[CrossRef]

Liao, W.

Liu, K.

Luo, H.

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

Luu, L.

M. Vo, Z. Wang, L. Luu, and J. Ma, “Advanced geometric camera calibration for machine vision,” Opt. Eng. 50, 110503 (2011).
[CrossRef]

Ma, J.

M. Vo, Z. Wang, L. Luu, and J. Ma, “Advanced geometric camera calibration for machine vision,” Opt. Eng. 50, 110503 (2011).
[CrossRef]

Ma, S.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285, 533–538 (2012).
[CrossRef]

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

Mutoh, K.

Nguyen, D.

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35, 1992–1994 (2010).
[CrossRef]

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe profilometry,” Opt. Lasers Eng. 48, 218–225 (2010).
[CrossRef]

Pan, B.

Press, W. H.

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

Quan, C.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285, 533–538 (2012).
[CrossRef]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

Rastogi, P.

S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Saldner, H. O.

Shan, X.

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

Shi, H.

Shi, Y.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[CrossRef]

Song, M.

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

Su, X.

Takeda, M.

Tay, C. J.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285, 533–538 (2012).
[CrossRef]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

Teukolsky, S. A.

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

Vetterling, W. T.

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

Vo, M.

M. Vo, Z. Wang, L. Luu, and J. Ma, “Advanced geometric camera calibration for machine vision,” Opt. Eng. 50, 110503 (2011).
[CrossRef]

Wang, C.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[CrossRef]

Wang, H.

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

Wang, Y.

Wang, Z.

M. Vo, Z. Wang, L. Luu, and J. Ma, “Advanced geometric camera calibration for machine vision,” Opt. Eng. 50, 110503 (2011).
[CrossRef]

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35, 1992–1994 (2010).
[CrossRef]

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe profilometry,” Opt. Lasers Eng. 48, 218–225 (2010).
[CrossRef]

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

Wen, Y.

Xi, J.

Xu, Y.

Yau, S.

Yuan, T.

Zhang, Q.

Zhang, S.

Y. Xu, L. Ekstrand, J. Dai, and S. Zhang, “Phase error compensation for three-dimensional shape measurement with projector defocusing,” Appl. Opt. 50, 2572–2581 (2011).
[CrossRef]

S. Zhang, “Recent processes on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

S. Zhang and S. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46, 36–43 (2007).
[CrossRef]

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

Zhang, Z.

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

Zhu, F.

Zhu, R.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285, 533–538 (2012).
[CrossRef]

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

Zhu, X.

Appl. Opt. (10)

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
[CrossRef]

H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
[CrossRef]

H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43, 2906–2914 (2004).
[CrossRef]

S. Zhang and S. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46, 36–43 (2007).
[CrossRef]

M. J. Baker, J. Xi, and J. F. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233–1243 (2007).
[CrossRef]

F. Da and S. Gai, “Flexible three-dimensional measurement technique based on a digital light processing projector,” Appl. Opt. 47, 377–385 (2008).
[CrossRef]

L. Huang, P. S. K. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49, 1539–1548 (2010).
[CrossRef]

Y. Wen, S. Li, H. Cheng, X. Su, and Q. Zhang, “Universal calculation formula and calibration method in Fourier transform profilometry,” Appl. Opt. 49, 6563–6569 (2010).
[CrossRef]

F. Zhu, H. Shi, P. Bai, and X. He, “Three-dimensional shape measurement and calibration for fringe projection by considering unequal height of the projector and the camera,” Appl. Opt. 50, 1575–1583 (2011).
[CrossRef]

Y. Xu, L. Ekstrand, J. Dai, and S. Zhang, “Phase error compensation for three-dimensional shape measurement with projector defocusing,” Appl. Opt. 50, 2572–2581 (2011).
[CrossRef]

Chin. J. Lasers (1)

S. Ma, R. Zhu, J. Li, Z. Gao, H. Luo, H. Wang, and X. Shan, “A multi-view measurement system for three-dimensional surface distribution of transient moving target,” Chin. J. Lasers 37, 3091–3097 (2010) (in Chinese).
[CrossRef]

Chin. Opt. Lett. (1)

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

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

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

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

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

Opt. Commun. (1)

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285, 533–538 (2012).
[CrossRef]

Opt. Eng. (5)

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

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

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

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[CrossRef]

M. Vo, Z. Wang, L. Luu, and J. Ma, “Advanced geometric camera calibration for machine vision,” Opt. Eng. 50, 110503 (2011).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (4)

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe profilometry,” Opt. Lasers Eng. 48, 218–225 (2010).
[CrossRef]

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[CrossRef]

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Figures (11)

Fig. 1.
Fig. 1.

Schematic diagram of a CCD-DLP stereo system model.

Fig. 2.
Fig. 2.

Graphical description of a perspective model of the camera with lens distortion.

Fig. 3.
Fig. 3.

Simulated results of a flat object with 10% random intensity noise and 2.0 gamma value. (a) Simulated captured fringe pattern. Error distribution of the object obtained by the (b) LMM and (c) NMM.

Fig. 4.
Fig. 4.

RMS error results of the flat object with various gamma and random intensity noise values using the (a) LMM and (b) NMM.

Fig. 5.
Fig. 5.

Simulated results of a peak-shaped object with 10% random intensity noise and 2.0 gamma value. (a) Simulated captured fringe pattern. Reconstructed profile by the (b) LMM and (c) NMM. Error distribution obtained by the (d) LMM and (e) NMM.

Fig. 6.
Fig. 6.

RMS error results of the peak-shaped object with various gamma and random intensity noise values using the (a) LMM and (b) NMM.

Fig. 7.
Fig. 7.

Calibrated extrinsic parameters results of the CCD-DLP stereo system.

Fig. 8.
Fig. 8.

Reprojection error (in pixel) distribution of the (a) CCD camera and (b) DLP projector.

Fig. 9.
Fig. 9.

Measurement results of a small-size flat plate. (a) Real captured fringe pattern with an ROI. Error distribution of the reconstructed profile by the (b) LMM and (c) NMM.

Fig. 10.
Fig. 10.

Measurement results of a larger size flat plate in the first position. (a) Real captured fringe pattern with an ROI. Error distribution of the reconstructed profile by the (b) LMM and (c) NMM.

Fig. 11.
Fig. 11.

Measurement results of a Buddha model object. (a) Real captured fringe pattern and (b) reconstructed profile by the proposed NMM.

Tables (4)

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Table 1. Intrinsic Parameters of the CCD-DLP Stereo System Used in Simulation

Tables Icon

Table 2. Experimental Calibrated Intrinsic Parameters of the CCD-DLP Stereo System

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Table 3. RMS Error Results (1—linear model method, 2—nonlinear model method) of a Larger Size Flat Plate at Three Different Positions

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Table 4. Distance Measurement Results of the Larger Size Flat Plate

Equations (22)

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

[xCQyCQzCQ]T=[RWCTWC]·[xWQyWQzWQ1]T,
ρ1[uq1vq11]T=AC·[xCQyCQzCQ]T,
AC=[fuαC·fuu00fvv0001],
[xPQyPQzPQ]T=[RWPTWP]·[xWQyWQzWQ1]T,
ρ2[mq2nq21]T=AP·[xPQyPQzPQ]T,
AP=[fmαP·fmm00fnn0001].
{F1(xWQ,yWQ,zWQ,uq1)=0F2(xWQ,yWQ,zWQ,vq1)=0F3(xWQ,yWQ,zWQ,mq2)=0F4(xWQ,yWQ,zWQ,nq2)=0,
xnCq1d=1+kC1(rnCq1)2+kC2(rnCq1)4+kC5(rnCq1)6·xnCq1+dxnCq1,
dxnCq1=2kC3·xnCq1·ynCq1+kC4(rnCq1)2+2(xnCq1)2,
ynCq1d=1+kC1(rnCq1)2+kC2(rnCq1)4+kC5(rnCq1)6·ynCq1+dynCq1,
dynCq1=kC3(rnCq1)2+2(ynCq1)2+2kC4·xnCq1·ynCq1,
xnPq2d=1+kP1(rnPq2)2+kP2(rnPq2)4+kP5(rnPq2)6·xnPq2+dxnPq2,
dxnPq2=2kP3·xnPq2·ynPq2+kP4(rnPq2)2+2(xnPq2)2,
ynPq2d=1+kP1(rnPq2)2+kP2(rnPq2)4+kP5(rnPq2)6·ynPq2+dynPq2,
dynPq2=kP3(rnPq2)2+2(ynPq2)2+2kP4·xnPq2·ynPq2,
{H1(xWQ,yWQ,zWQ,uq1d)=0H2(xWQ,yWQ,zWQ,vq1d)=0H3(xWQ,yWQ,zWQ,mq2d)=0H4(xWQ,yWQ,zWQ,nq2d)=0,
mq2d=Γ(nq2d)
{I1(xWQ,yWQ,zWQ,nq2d)=0I2(xWQ,yWQ,zWQ,nq2d)=0I3(xWQ,yWQ,zWQ,nq2d)=0I4(xWQ,yWQ,zWQ,nq2d)=0,
E(xWQ,yWQ,zWQ,nq2d)=i=14[Ii(xWQ,yWQ,zWQ,nq2d)]2,
(x˜WQ,y˜WQ,z˜WQ)=Min(x˜WQ,y˜WQ,z˜WQ,n˜q2d)|E|,
[RWCTWC]=[0.007200.999960.00556124.0000.999970.007300.00180127.0000.001800.005500.999981113.500],
[RWPTWP]=[0.015200.972800.23140140.1450.999500.021300.02370185.7830.028000.230900.972601109.020].

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