Abstract

Two methods are proposed to calibrate the revolution axis of a 360 deg, multiview fringe projection system for surface measurement. The first method is based on minimizing the distance between calculated and measured points; the second method is based on minimizing the difference between thus obtained vectors. Both are able to retrieve the revolution axis of a turntable, which is then used to transform surface patches measured at different viewing angles to a common coordinate. In the point-based method, a nonlinear minimization problem has to be solved by the Levenberg–Marquardt algorithm; in the vector-based method, the minimization problem is resolved into several linear equations, and an analytic solution is obtained efficiently. Results of simulation and experiments show that the error of calibration can be less than 0.05 deg for the axis’s orientation and 0.3 mm for the axis’s position (a point on the axis), which is about 0.1% of the measured volume.

© 2013 Optical Society of America

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References

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    [CrossRef]
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2011 (1)

2010 (1)

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

2009 (1)

2008 (1)

P. Zheng, H. Guo, Y. Yu, and M. Chen, “Three-dimensional profile measurement using a flexible new multi-view connection method,” in Proc. SPIE 7155, 715539 (2008).
[CrossRef]

2005 (2)

H. He, M. Chen, H. Guo, and Y. Yu, “Novel multiview connection method based on virtual cylinder for 3-d surface measurement,” Opt. Eng. 44, 083605 (2005).
[CrossRef]

Y. Xu, Q. Yang, and J. Huai, “Calibration of the axis of the turntable in 4-axis laser measuring system and registration of multi-view,” Chinese J. Laser. 32, 659–662 (2005).

2004 (2)

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

H. He, M. Chen, and H. Guo, “Surface measurement of complex objects using connection techniques based on overlapping areas,” Proc. SPIE 5180, 402–412 (2004).
[CrossRef]

2003 (1)

H. Guo and M. Chen, “Multiview connection technique for 360 deg three-dimensional measurement,” Opt. Eng. 42, 900–905 (2003).
[CrossRef]

2002 (1)

R. Sitnik, M. Kujawinska, and J. Woznicki, “Digital fringe projection system for large-volume 360 deg shape measurement,” Opt. Eng. 41, 2443–2449 (2002).

1999 (1)

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

1995 (1)

1994 (1)

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33, 2760–2769 (1994).
[CrossRef]

1992 (1)

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

1991 (1)

1985 (1)

1984 (1)

1983 (1)

Asundi, A.

B. Pan, K. Qian, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34, 416–418 (2009).
[CrossRef]

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

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33, 2760–2769 (1994).
[CrossRef]

Besl, P. J.

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

Bolitho, M.

M. Kazhdan, M. Bolitho, and H. Hoppe, “Poisson surface reconstruction,” in Eurographics Symposium on Geometry Processing (2006), pp. 61–70.

Chan, C. S.

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33, 2760–2769 (1994).
[CrossRef]

Chen, M.

P. Zheng, H. Guo, Y. Yu, and M. Chen, “Three-dimensional profile measurement using a flexible new multi-view connection method,” in Proc. SPIE 7155, 715539 (2008).
[CrossRef]

H. He, M. Chen, H. Guo, and Y. Yu, “Novel multiview connection method based on virtual cylinder for 3-d surface measurement,” Opt. Eng. 44, 083605 (2005).
[CrossRef]

H. He, M. Chen, and H. Guo, “Surface measurement of complex objects using connection techniques based on overlapping areas,” Proc. SPIE 5180, 402–412 (2004).
[CrossRef]

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

H. Guo and M. Chen, “Multiview connection technique for 360 deg three-dimensional measurement,” Opt. Eng. 42, 900–905 (2003).
[CrossRef]

Cheng, X.

Chiang, F. P.

Deguchi, K.

R. Ishiyama, T. Okatani, and K. Deguchi, “Precise 3-d measurement using uncalibrated pattern projection,” in IEEE International Conference on Image Processing, Vol. 1 (2007), pp. 225–228.

DeRose, T.

H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Surface reconstruction from unorganized points,” in Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, Vol. 26 (1992), pp. 71–78.

Duchamp, T.

H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Surface reconstruction from unorganized points,” in Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, Vol. 26 (1992), pp. 71–78.

Farrant, D. I.

Geng, J.

Gorthi, S. S.

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

Guo, H.

P. Zheng, H. Guo, Y. Yu, and M. Chen, “Three-dimensional profile measurement using a flexible new multi-view connection method,” in Proc. SPIE 7155, 715539 (2008).
[CrossRef]

H. He, M. Chen, H. Guo, and Y. Yu, “Novel multiview connection method based on virtual cylinder for 3-d surface measurement,” Opt. Eng. 44, 083605 (2005).
[CrossRef]

H. He, M. Chen, and H. Guo, “Surface measurement of complex objects using connection techniques based on overlapping areas,” Proc. SPIE 5180, 402–412 (2004).
[CrossRef]

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

H. Guo and M. Chen, “Multiview connection technique for 360 deg three-dimensional measurement,” Opt. Eng. 42, 900–905 (2003).
[CrossRef]

Guo, L.

Halioua, M.

He, H.

H. He, M. Chen, H. Guo, and Y. Yu, “Novel multiview connection method based on virtual cylinder for 3-d surface measurement,” Opt. Eng. 44, 083605 (2005).
[CrossRef]

H. He, M. Chen, and H. Guo, “Surface measurement of complex objects using connection techniques based on overlapping areas,” Proc. SPIE 5180, 402–412 (2004).
[CrossRef]

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

Hibino, K.

Hoppe, H.

M. Kazhdan, M. Bolitho, and H. Hoppe, “Poisson surface reconstruction,” in Eurographics Symposium on Geometry Processing (2006), pp. 61–70.

H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Surface reconstruction from unorganized points,” in Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, Vol. 26 (1992), pp. 71–78.

Huai, J.

Y. Xu, Q. Yang, and J. Huai, “Calibration of the axis of the turntable in 4-axis laser measuring system and registration of multi-view,” Chinese J. Laser. 32, 659–662 (2005).

Huang, L.

Ishiyama, R.

R. Ishiyama, T. Okatani, and K. Deguchi, “Precise 3-d measurement using uncalibrated pattern projection,” in IEEE International Conference on Image Processing, Vol. 1 (2007), pp. 225–228.

Kazhdan, M.

M. Kazhdan, M. Bolitho, and H. Hoppe, “Poisson surface reconstruction,” in Eurographics Symposium on Geometry Processing (2006), pp. 61–70.

Krishnamurthy, R. S.

Kujawinska, M.

R. Sitnik, M. Kujawinska, and J. Woznicki, “Digital fringe projection system for large-volume 360 deg shape measurement,” Opt. Eng. 41, 2443–2449 (2002).

Larkin, K. G.

Levoy, M.

S. Rusinkiewiez and M. Levoy, “Efficient variants of the icp algorithm,” in Proceedings of the International Conference on 3D Digital Imaging and Modeling (2001), pp. 145–152.

Liu, H. C.

McDonald, J.

H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Surface reconstruction from unorganized points,” in Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, Vol. 26 (1992), pp. 71–78.

McKay, N. D.

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

Mutoh, K.

Okatani, T.

R. Ishiyama, T. Okatani, and K. Deguchi, “Precise 3-d measurement using uncalibrated pattern projection,” in IEEE International Conference on Image Processing, Vol. 1 (2007), pp. 225–228.

Oreb, B. F.

Pan, B.

Qian, K.

Rastogi, P.

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

Rusinkiewiez, S.

S. Rusinkiewiez and M. Levoy, “Efficient variants of the icp algorithm,” in Proceedings of the International Conference on 3D Digital Imaging and Modeling (2001), pp. 145–152.

Sajan, M. R.

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33, 2760–2769 (1994).
[CrossRef]

Sitnik, R.

R. Sitnik, M. Kujawinska, and J. Woznicki, “Digital fringe projection system for large-volume 360 deg shape measurement,” Opt. Eng. 41, 2443–2449 (2002).

Srinivasan, V.

Stuetzle, W.

H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Surface reconstruction from unorganized points,” in Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, Vol. 26 (1992), pp. 71–78.

Su, X.

Takeda, M.

Woznicki, J.

R. Sitnik, M. Kujawinska, and J. Woznicki, “Digital fringe projection system for large-volume 360 deg shape measurement,” Opt. Eng. 41, 2443–2449 (2002).

Xu, Y.

Y. Xu, Q. Yang, and J. Huai, “Calibration of the axis of the turntable in 4-axis laser measuring system and registration of multi-view,” Chinese J. Laser. 32, 659–662 (2005).

Yang, Q.

Y. Xu, Q. Yang, and J. Huai, “Calibration of the axis of the turntable in 4-axis laser measuring system and registration of multi-view,” Chinese J. Laser. 32, 659–662 (2005).

Yu, Y.

P. Zheng, H. Guo, Y. Yu, and M. Chen, “Three-dimensional profile measurement using a flexible new multi-view connection method,” in Proc. SPIE 7155, 715539 (2008).
[CrossRef]

H. He, M. Chen, H. Guo, and Y. Yu, “Novel multiview connection method based on virtual cylinder for 3-d surface measurement,” Opt. Eng. 44, 083605 (2005).
[CrossRef]

Zheng, P.

P. Zheng, H. Guo, Y. Yu, and M. Chen, “Three-dimensional profile measurement using a flexible new multi-view connection method,” in Proc. SPIE 7155, 715539 (2008).
[CrossRef]

Zhou, W.

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

Adv. Opt. Photon. (1)

Appl. Opt. (5)

Chinese J. Laser. (1)

Y. Xu, Q. Yang, and J. Huai, “Calibration of the axis of the turntable in 4-axis laser measuring system and registration of multi-view,” Chinese J. Laser. 32, 659–662 (2005).

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

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

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

Opt. Eng. (5)

H. He, M. Chen, H. Guo, and Y. Yu, “Novel multiview connection method based on virtual cylinder for 3-d surface measurement,” Opt. Eng. 44, 083605 (2005).
[CrossRef]

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33, 2760–2769 (1994).
[CrossRef]

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

R. Sitnik, M. Kujawinska, and J. Woznicki, “Digital fringe projection system for large-volume 360 deg shape measurement,” Opt. Eng. 41, 2443–2449 (2002).

H. Guo and M. Chen, “Multiview connection technique for 360 deg three-dimensional measurement,” Opt. Eng. 42, 900–905 (2003).
[CrossRef]

Opt. Lasers Eng. (1)

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

Opt. Lett. (1)

Proc. SPIE (2)

H. He, M. Chen, and H. Guo, “Surface measurement of complex objects using connection techniques based on overlapping areas,” Proc. SPIE 5180, 402–412 (2004).
[CrossRef]

P. Zheng, H. Guo, Y. Yu, and M. Chen, “Three-dimensional profile measurement using a flexible new multi-view connection method,” in Proc. SPIE 7155, 715539 (2008).
[CrossRef]

Other (7)

R. Ishiyama, T. Okatani, and K. Deguchi, “Precise 3-d measurement using uncalibrated pattern projection,” in IEEE International Conference on Image Processing, Vol. 1 (2007), pp. 225–228.

R. Hartley and A. Zisserman, eds., Multiple View Geometry in Computer Vision (Cambridge University, 2000).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, eds., Numerical Recipes in C++ (Cambridge University, 2002).

S. Belongie, “Rodrigues’ rotation formula,” From MathWorld–A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/RodriguesRotationFormula.html .

S. Rusinkiewiez and M. Levoy, “Efficient variants of the icp algorithm,” in Proceedings of the International Conference on 3D Digital Imaging and Modeling (2001), pp. 145–152.

H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Surface reconstruction from unorganized points,” in Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, Vol. 26 (1992), pp. 71–78.

M. Kazhdan, M. Bolitho, and H. Hoppe, “Poisson surface reconstruction,” in Eurographics Symposium on Geometry Processing (2006), pp. 61–70.

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Figures (13)

Fig. 1.
Fig. 1.

q0 and q1 are allowed to move in two planes: y=0 and y=100 in the LM minimization process.

Fig. 2.
Fig. 2.

p1, pm,1 and p2, pm,2 are two pairs of measured points before and after rotation. The intersection of their bisecting planes gives an estimate of the revolution axis.

Fig. 3.
Fig. 3.

(a) In simulation 2, the angle α was varied from 0 to 15 deg. (b) In simulation 3, the angle β was varied from 20 to 90 deg.

Fig. 4.
Fig. 4.

Multiview fringe projection system.

Fig. 5.
Fig. 5.

(a),(c) Background image and (b),(d) fringe pattern of the calibration chessboard at two reference positions. A known z-direction shift is introduced by the linear stage.

Fig. 6.
Fig. 6.

Test objects and projected fringe patterns: (a) cylinder, (b) bear model, and (c) angel model.

Fig. 7.
Fig. 7.

Performance with respect to SNR of the measured points. The revolution axis is set parallel to the Y axis, and the rotation of the chessboard is 45 deg.

Fig. 8.
Fig. 8.

Performance with respect to the axis’s orientation. SNR is set to 50 dB, and the rotation of the chessboard is 45 deg.

Fig. 9.
Fig. 9.

Performance with respect to the angle of rotation of the chessboard. The SNR is set to 50 dB. The revolution axis deviates from the Y axis by 3 deg.

Fig. 10.
Fig. 10.

Reconstructed cylinder. (a) Front view. (b) Top view.

Fig. 11.
Fig. 11.

(a) Point cloud of every second surface patch of the bear model. For clarity, not every surface patch is shown. (b and c) Reconstructed bear model at two views.

Fig. 12.
Fig. 12.

Reconstructed angel model at different views.

Fig. 13.
Fig. 13.

Top view of a typical turntable setup with four planes: P1, P2, P3, and P4. The first set of equations involving P1, P2, and P3 gives four solutions: c1, c2, c3, and c4. The second set involving P2, P3, and P4 gives c5, c6, c7, and c8. If the angle of rotation between each pair of planes is the same, there are two identical solutions (c1, c5 and c4, c8), not one, as stated in [13].

Equations (17)

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

(xi,yi,1)T=M·(xw,yw,zw,1)T,
pc,i=f(q0,q1,θ,pi),
D=i=1kpc,ipm,i2,
pc,i=f(q0x,0,q0z,q1x,100,q1z,θ,pi),
vc=R(p2p1),
R=[r11r12r13r21r22r23r31r32r33],
r11=nx2(1cosθ)+cosθ,r12=nxny(1cosθ)nzsinθ,r13=nxnz(1cosθ)+nysinθ,r21=nxny(1cosθ)+nzsinθ,r22=ny2(1cosθ)+cosθ,r23=nynz(1cosθ)nxsinθ,r31=nxnz(1cosθ)nysinθ,r32=nynz(1cosθ)+nxsinθ,r33=nz2(1cosθ)+cosθ,
R=[r11r12r13r21r22r23r31r32r33],
r11=cosθ,r12=nx(1cosθ)nzsinθ,r13=nysinθ,r21=nx(1cosθ)+nzsinθ,r22=1,r23=nz(1cosθ)nxsinθ,r31=nysinθ,r32=nz(1cosθ)+nxsinθ,r33=cosθ.
E=i=1kvc,ivm,i2=i=1kRvivm,i2,
i=1k[a11a12a13a21a22a23a31a32a33]·[nxnynz]=i=1k[b1b2b3],
a11=2vy2(1cosθ)+[vx(1cosθ)vzsinθ]2,a12=vyvz(1cosθ)sinθvxvysin2θ,a13=[vx(1cosθ)vzsinθ][vxsinθ+vz(1cosθ)],a21=a12,a22=(vz2+vx2)sin2θ,a23=vyvzsin2θvxvy(1cosθ)sinθ,a31=a13,a32=a23,a33=2vy2(1cosθ)+[vxsinθ+vz(1cosθ)]2,vy(1cosθ)(vmxvxcosθ)b1=+[vx(1cosθ)vzsinθ](vmyvy)+vysinθ(vmzvzcosθ),b2=vzsinθ(vmxvxcosθ)vxsinθ(vmzvzcosθ),vysinθ(vmxvxcosθ)b3=+[vxsinθ+vz(1cosθ)](vmyvy)+vy(1cosθ)(vmzvzcosθ).
pc,i=R(piq0)+q0,
D=i=1k(IR)q0+Rpipm,i2,
i=1k[c11c12c21c22]·[q0xq0z]=i=1k[d1d2]
c11=(r111)2+r212+r312,c12=(r111)r13+r21r23+r31(r331),c21=c12,c22=r132+r232+(r331)2,d1=ux(r111)+uyr21+uzr31,d2=uxr13+uyr23+uz(r331),
ux=(r11px+r12py+r13pz)pmx,uy=(r21px+r22py+r33pz)pmy,uz=(r31px+r32py+r33pz)pmz.

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