Abstract

We describe what we believe is a new phase-shifting algorithm called a double three-step algorithm developed to reduce the measurement error of a three-dimensional shape-measurement system, which is based on digital fringe-projection and phase-shifting techniques. After comparing the performance of different existing phase-shifting algorithms, we present the new double three-step algorithm based on the error analysis of the standard three-step algorithm. In this algorithm, three-step phase shifting is done twice with an initial phase offset of 60° between them, and the two obtained phase maps are averaged to generate the final phase map. Both theoretical and experimental results showed that this new algorithm worked well in significantly reducing the measurement error.

© 2002 Optical Society of America

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  1. D. T. Moore, B. E. Truax, “Phase-locked moire fringe analysis for automated contouring of diffuse surfaces,” Appl. Opt. 18, 91–96 (1979).
    [CrossRef] [PubMed]
  2. H. Takasaki, “Moiré topography,” Appl. Opt. 12, 845–850 (1973).
    [CrossRef] [PubMed]
  3. M. Halioua, R. S. Krishnamurthy, H. Liu, F. P. Chiang, “Projection moiré with moving gratings for automated 3-D topography,” Appl. Opt. 22, 850–855 (1983).
    [CrossRef]
  4. G. Indebetouw, “Profile measurement using projection of running fringes,” Appl. Opt. 17, 2930–2933 (1978).
    [CrossRef] [PubMed]
  5. M. M. Shaw, D. M. Harvey, C. A. Hobson, M. J. Lalor, “Non-contact ranging using dynamic fringe projection,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 22–29 (1989).
    [CrossRef]
  6. Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
    [CrossRef]
  7. T. Maack, R. Kowarschik, “Camera influence on the phase-measuring accuracy of a phase-shifting speckle interferometry,” Appl. Opt. 35, 3514–3524 (1996).
    [CrossRef] [PubMed]
  8. P. L. Wizinowich, “Phase-shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29, 3271–3279 (1990).
    [CrossRef] [PubMed]
  9. B. Trolard, “Speckle noise removal in interference fringes by optoelectronic preprocessing with Epson liquid crystal television,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 126–134 (1996).
    [CrossRef]
  10. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  11. S. Tang, “Self-calibrating five-frame algorithm for phase-shifting interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 91–98 (1996).
    [CrossRef]
  12. M.-H. Daniel, M.-D. Daniel, S. Manuel, “Least square fitting of a sinusoidal signal and its Fourier analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 84–90 (1996).
    [CrossRef]
  13. J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., (Wiley, New York, 1992), pp. 501–598.
  14. S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measurement algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 16, 3101–3105 (1997).
    [CrossRef]
  15. J. Schwider, T. Dresel, B. Manzke, “Some considerations of reduction of reference phase error in phase-stepping interferometry,” Appl. Opt. 38, 655–658 (1999).
    [CrossRef]
  16. C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
    [CrossRef] [PubMed]
  17. P. Hariharan, “Phase-shifting interferometry: minimization of system errors,” Appl. Opt. 39, 967–969 (2000).
  18. Q. Hu, P. S. Huang, Q. Fu, F. P. Chiang, “Calibration of a 3-D surface contouring and ranging system,” in Three-Dimensional Imaging, Optical Metrology, and Inspection V, K. G. Harding, ed., Proc. SPIE3835, 158–166 (1999).
    [CrossRef]

2000

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

P. Hariharan, “Phase-shifting interferometry: minimization of system errors,” Appl. Opt. 39, 967–969 (2000).

1999

1997

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measurement algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 16, 3101–3105 (1997).
[CrossRef]

1996

1994

1990

1987

1983

1979

1978

1973

Chiang, F. P.

M. Halioua, R. S. Krishnamurthy, H. Liu, F. P. Chiang, “Projection moiré with moving gratings for automated 3-D topography,” Appl. Opt. 22, 850–855 (1983).
[CrossRef]

Q. Hu, P. S. Huang, Q. Fu, F. P. Chiang, “Calibration of a 3-D surface contouring and ranging system,” in Three-Dimensional Imaging, Optical Metrology, and Inspection V, K. G. Harding, ed., Proc. SPIE3835, 158–166 (1999).
[CrossRef]

Daniel, M.-D.

M.-H. Daniel, M.-D. Daniel, S. Manuel, “Least square fitting of a sinusoidal signal and its Fourier analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 84–90 (1996).
[CrossRef]

Daniel, M.-H.

M.-H. Daniel, M.-D. Daniel, S. Manuel, “Least square fitting of a sinusoidal signal and its Fourier analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 84–90 (1996).
[CrossRef]

Dresel, T.

Eiju, T.

Fu, Q.

Q. Hu, P. S. Huang, Q. Fu, F. P. Chiang, “Calibration of a 3-D surface contouring and ranging system,” in Three-Dimensional Imaging, Optical Metrology, and Inspection V, K. G. Harding, ed., Proc. SPIE3835, 158–166 (1999).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., (Wiley, New York, 1992), pp. 501–598.

Halioua, M.

Han, G.-S.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measurement algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 16, 3101–3105 (1997).
[CrossRef]

Hariharan, P.

Harvey, D. M.

M. M. Shaw, D. M. Harvey, C. A. Hobson, M. J. Lalor, “Non-contact ranging using dynamic fringe projection,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 22–29 (1989).
[CrossRef]

Hobson, C. A.

M. M. Shaw, D. M. Harvey, C. A. Hobson, M. J. Lalor, “Non-contact ranging using dynamic fringe projection,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 22–29 (1989).
[CrossRef]

Hu, Q.

Q. Hu, P. S. Huang, Q. Fu, F. P. Chiang, “Calibration of a 3-D surface contouring and ranging system,” in Three-Dimensional Imaging, Optical Metrology, and Inspection V, K. G. Harding, ed., Proc. SPIE3835, 158–166 (1999).
[CrossRef]

Huang, P. S.

Q. Hu, P. S. Huang, Q. Fu, F. P. Chiang, “Calibration of a 3-D surface contouring and ranging system,” in Three-Dimensional Imaging, Optical Metrology, and Inspection V, K. G. Harding, ed., Proc. SPIE3835, 158–166 (1999).
[CrossRef]

Hung, Y. Y.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Indebetouw, G.

Joenathan, C.

Kang, M.-G.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measurement algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 16, 3101–3105 (1997).
[CrossRef]

Kim, S.-W.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measurement algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 16, 3101–3105 (1997).
[CrossRef]

Kowarschik, R.

Krishnamurthy, R. S.

Lalor, M. J.

M. M. Shaw, D. M. Harvey, C. A. Hobson, M. J. Lalor, “Non-contact ranging using dynamic fringe projection,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 22–29 (1989).
[CrossRef]

Lin, L.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Liu, H.

Maack, T.

Manuel, S.

M.-H. Daniel, M.-D. Daniel, S. Manuel, “Least square fitting of a sinusoidal signal and its Fourier analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 84–90 (1996).
[CrossRef]

Manzke, B.

Moore, D. T.

Oreb, B. F.

Park, B. G.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Schwider, J.

Shang, H. M.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Shaw, M. M.

M. M. Shaw, D. M. Harvey, C. A. Hobson, M. J. Lalor, “Non-contact ranging using dynamic fringe projection,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 22–29 (1989).
[CrossRef]

Takasaki, H.

Tang, S.

S. Tang, “Self-calibrating five-frame algorithm for phase-shifting interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 91–98 (1996).
[CrossRef]

Trolard, B.

B. Trolard, “Speckle noise removal in interference fringes by optoelectronic preprocessing with Epson liquid crystal television,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 126–134 (1996).
[CrossRef]

Truax, B. E.

Wizinowich, P. L.

Appl. Opt.

P. Hariharan, “Phase-shifting interferometry: minimization of system errors,” Appl. Opt. 39, 967–969 (2000).

H. Takasaki, “Moiré topography,” Appl. Opt. 12, 845–850 (1973).
[CrossRef] [PubMed]

G. Indebetouw, “Profile measurement using projection of running fringes,” Appl. Opt. 17, 2930–2933 (1978).
[CrossRef] [PubMed]

D. T. Moore, B. E. Truax, “Phase-locked moire fringe analysis for automated contouring of diffuse surfaces,” Appl. Opt. 18, 91–96 (1979).
[CrossRef] [PubMed]

M. Halioua, R. S. Krishnamurthy, H. Liu, F. P. Chiang, “Projection moiré with moving gratings for automated 3-D topography,” Appl. Opt. 22, 850–855 (1983).
[CrossRef]

P. L. Wizinowich, “Phase-shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29, 3271–3279 (1990).
[CrossRef] [PubMed]

C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
[CrossRef] [PubMed]

J. Schwider, T. Dresel, B. Manzke, “Some considerations of reduction of reference phase error in phase-stepping interferometry,” Appl. Opt. 38, 655–658 (1999).
[CrossRef]

T. Maack, R. Kowarschik, “Camera influence on the phase-measuring accuracy of a phase-shifting speckle interferometry,” Appl. Opt. 35, 3514–3524 (1996).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
[CrossRef] [PubMed]

Opt. Eng.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measurement algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 16, 3101–3105 (1997).
[CrossRef]

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Other

B. Trolard, “Speckle noise removal in interference fringes by optoelectronic preprocessing with Epson liquid crystal television,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 126–134 (1996).
[CrossRef]

S. Tang, “Self-calibrating five-frame algorithm for phase-shifting interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 91–98 (1996).
[CrossRef]

M.-H. Daniel, M.-D. Daniel, S. Manuel, “Least square fitting of a sinusoidal signal and its Fourier analysis,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2860, 84–90 (1996).
[CrossRef]

J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., (Wiley, New York, 1992), pp. 501–598.

M. M. Shaw, D. M. Harvey, C. A. Hobson, M. J. Lalor, “Non-contact ranging using dynamic fringe projection,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 22–29 (1989).
[CrossRef]

Q. Hu, P. S. Huang, Q. Fu, F. P. Chiang, “Calibration of a 3-D surface contouring and ranging system,” in Three-Dimensional Imaging, Optical Metrology, and Inspection V, K. G. Harding, ed., Proc. SPIE3835, 158–166 (1999).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Configuration of the 3-D shape-measurement system.

Fig. 2
Fig. 2

Picture of the 3-D shape-measurement system.

Fig. 3
Fig. 3

Measured result of a sheet-metal panel with the three-step algorithm.

Fig. 4
Fig. 4

Measured result of the sheet-metal panel with the 2 + 1 algorithm.

Fig. 5
Fig. 5

Measured result of the sheet-metal panel with the 3 + 3 algorithm.

Fig. 6
Fig. 6

Measured result of the sheet-metal panel with the four-step algorithm.

Fig. 7
Fig. 7

Measured result of the sheet-metal panel with the five-step algorithm.

Fig. 8
Fig. 8

Repeatability of the measurement error.

Fig. 9
Fig. 9

Combined response curve of the CCD camera and the projector.

Fig. 10
Fig. 10

Phase error resulting from the nonlinearity error of the camera and the projector.

Fig. 11
Fig. 11

Error waves of the phase maps with four different initial phases.

Fig. 12
Fig. 12

Cross-sectional profile of the final phase map after averaging.

Fig. 13
Fig. 13

Measured result of the sheet-metal panel with the double three-step algorithm.

Equations (20)

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

x=M21+cos2πnp+δ,
fx=a+bx.
fx=a+bM2+M2cos2πnp+δ=a+bM2+bM2cos2πnp+δ.
I1=I01+γcosϕ-120°,
I2=I01+γcosϕ,
I3=I01+γcosϕ+120°.
ϕ=arctan3I1-I32I2-I1+I3.
fx=a+bx+x2.
fx=a+bM2+M2cos2πnp+δ+M2+M2cos2πnp+δ2 =a+bM2+3M28+bM2+M22cos2πnp+δ+M28cos4πnp+2δ.
I=u+v cosϕ+δ+w cos2ϕ+2δ,
I1=u+v cosϕ-120°+w cos2ϕ-240°,
I2=u+v cosϕ+w cos2ϕ,
I3=u+v cosϕ+120°+w cos2ϕ+240°,
tanϕ=3I1-I32I2-I1+I3=v sinϕ-w sin2ϕv cosϕ+w cos2ϕ.
tanΔϕ=tanϕ-ϕ=tanϕ-tanϕ1+tanϕtanϕ=-w sin3ϕw cos3ϕ+v=-sin3ϕcos3ϕ+k,
Δϕ=arctan-sin3ϕcos3ϕ+k,
fx=136.39+3.2485x+0.000862x2.
k=vw=4b+MM=43.2485+0.000862×2550.000862×255=63.115.
Δϕ=arctan-sin3ϕk=-arctansin3ϕk.
Δϕ=arctan-sin3ϕ+180°k=arctansin3ϕk.

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