Abstract

We propose a universal calculation formula of Fourier transform profilometry and give a strict theoretical analysis about the phase–height mapping relation. As the request on the experimental setup of the universal calculation formula is unconfined, the projector and the camera can be located arbitrarily to get better fringe information, which makes the operation flexible. The phase–height calibration method under the universal condition is proposed, which can avoid measuring the system parameters directly. It makes the system easy to manipulate and improves the measurement velocity. A computer simulation and experiment are conducted to verify its validity. The calculation formula and calibration method have been applied to measure an object of 22.00mm maximal height. The relative error of the measurement result is only 0.59%. The experimental results prove that the three-dimensional shape of tested objects can be reconstructed exactly by using the calculation formula and calibration method, and the system has better universality.

© 2010 Optical Society of America

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References

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  1. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  2. B. Zhao and A. Asundi, “Discussion on spatial resolution and sensitivity of Fourier transform fringe detection,” Opt. Eng. 39, 2715–2719 (2000).
    [CrossRef]
  3. W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
    [CrossRef]
  4. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
    [CrossRef]
  5. J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645–658 (2005).
    [CrossRef]
  6. X. Mao, W. Chen, and X. Su, “Improved Fourier transform profilometry,” Appl. Opt. 46, 664–668 (2007).
    [CrossRef] [PubMed]
  7. W. Zhou and X. Su, “A direct mapping algorithm for phase-measuring profilometry,” J. Mod. Opt. 41, 89–94(1994).
    [CrossRef]
  8. W. Li, X. Su, and L. Su, “A practical coordinate mapping method for phase-measuring profilometry,” Proc. SPIE 3558, 125–130 (1998).
    [CrossRef]

2007 (1)

2005 (2)

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645–658 (2005).
[CrossRef]

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

2001 (1)

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

2000 (1)

B. Zhao and A. Asundi, “Discussion on spatial resolution and sensitivity of Fourier transform fringe detection,” Opt. Eng. 39, 2715–2719 (2000).
[CrossRef]

1998 (1)

W. Li, X. Su, and L. Su, “A practical coordinate mapping method for phase-measuring profilometry,” Proc. SPIE 3558, 125–130 (1998).
[CrossRef]

1994 (1)

W. Zhou and X. Su, “A direct mapping algorithm for phase-measuring profilometry,” J. Mod. Opt. 41, 89–94(1994).
[CrossRef]

1983 (1)

Asundi, A.

B. Zhao and A. Asundi, “Discussion on spatial resolution and sensitivity of Fourier transform fringe detection,” Opt. Eng. 39, 2715–2719 (2000).
[CrossRef]

Cao, Y.

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

Chen, W.

X. Mao, W. Chen, and X. Su, “Improved Fourier transform profilometry,” Appl. Opt. 46, 664–668 (2007).
[CrossRef] [PubMed]

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

Guillaume, P.

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645–658 (2005).
[CrossRef]

Li, W.

W. Li, X. Su, and L. Su, “A practical coordinate mapping method for phase-measuring profilometry,” Proc. SPIE 3558, 125–130 (1998).
[CrossRef]

Mao, X.

Mutoh, K.

Su, L.

W. Li, X. Su, and L. Su, “A practical coordinate mapping method for phase-measuring profilometry,” Proc. SPIE 3558, 125–130 (1998).
[CrossRef]

Su, X.

X. Mao, W. Chen, and X. Su, “Improved Fourier transform profilometry,” Appl. Opt. 46, 664–668 (2007).
[CrossRef] [PubMed]

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

W. Li, X. Su, and L. Su, “A practical coordinate mapping method for phase-measuring profilometry,” Proc. SPIE 3558, 125–130 (1998).
[CrossRef]

W. Zhou and X. Su, “A direct mapping algorithm for phase-measuring profilometry,” J. Mod. Opt. 41, 89–94(1994).
[CrossRef]

Takeda, M.

Vanherzeele, J.

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645–658 (2005).
[CrossRef]

Vanlanduit, S.

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645–658 (2005).
[CrossRef]

Xiang, L.

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

Zhang, Q.

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

Zhao, B.

B. Zhao and A. Asundi, “Discussion on spatial resolution and sensitivity of Fourier transform fringe detection,” Opt. Eng. 39, 2715–2719 (2000).
[CrossRef]

Zhou, W.

W. Zhou and X. Su, “A direct mapping algorithm for phase-measuring profilometry,” J. Mod. Opt. 41, 89–94(1994).
[CrossRef]

Appl. Opt. (2)

J. Mod. Opt. (1)

W. Zhou and X. Su, “A direct mapping algorithm for phase-measuring profilometry,” J. Mod. Opt. 41, 89–94(1994).
[CrossRef]

Opt. Eng. (1)

B. Zhao and A. Asundi, “Discussion on spatial resolution and sensitivity of Fourier transform fringe detection,” Opt. Eng. 39, 2715–2719 (2000).
[CrossRef]

Opt. Lasers Eng. (3)

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

J. Vanherzeele, P. Guillaume, and S. Vanlanduit, “Fourier fringe processing using a regressive Fourier-transform technique,” Opt. Lasers Eng. 43, 645–658 (2005).
[CrossRef]

Proc. SPIE (1)

W. Li, X. Su, and L. Su, “A practical coordinate mapping method for phase-measuring profilometry,” Proc. SPIE 3558, 125–130 (1998).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Universal optical geometry of the FTP.

Fig. 2
Fig. 2

Geometric relation of the system sketch map.

Fig. 3
Fig. 3

Sketch map of plane PEJB.

Fig. 4
Fig. 4

Simulations: (a) reference fringe; (b) deformed fringe; (c) continuous phase distribution of the object; (d) middle row of the phase of different standard reference planes; (e) 3D height map of the measurement result; (f) middle section of the result on the x axis.

Fig. 5
Fig. 5

Simulations: (a) calculated height error using our method; (b) calculated height error using traditional calculation formula of FTP; (c) middle rows of the two reconstructed height error distributions.

Fig. 6
Fig. 6

Calibration: (a) middle row of the continuous phase of the six standard reference planes (after phase unwrapping); (b) distribution of the system constant a ( x , y ) ; (c) distribution of the system constant b ( x , y ) ; (d) distribution of the system constant c ( x , y ) .

Fig. 7
Fig. 7

Experiment: (a) reference fringe; (b) deformed fringe; (c) continuous phase distribution of the object; (d) 3D height map of the measurement result; (e) middle section of the result on the y axis.

Equations (17)

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I 0 ( x , y ) = I 0 { 1 + cos [ 2 π x f 0 + φ 0 ( x , y ) ] } ,
I ( x , y ) = I 0 { 1 + cos [ 2 π x f 0 + φ ( x , y ) ] } .
Δ φ ( x , y ) = φ ( x , y ) φ 0 ( x , y ) .
x G = O C ¯ 2 G C ¯ 2 = l · 1 / cos 2 β 1 / cos 2 δ .
s = P O ¯ 2 + C O ¯ 2 2 · P O ¯ · C O ¯ · cos ω = ( r / cos ξ ) 2 + ( l / cos β ) 2 2 r l cos ω / cos ξ cos β .
B I ¯ = h cos δ l | l · 1 / cos 2 β 1 / cos 2 δ x B | ,
D E ¯ = ( r h · cos δ ) l | l · 1 / cos 2 β 1 / cos 2 δ x B | .
P D ¯ = P F ¯ ( B G ¯ B I ¯ ) = s 2 ( r - l ) 2 cos 2 δ ( 1 h cos δ l ) | l · 1 / cos 2 β 1 / cos 2 δ x B | .
P E ¯ = P D ¯ + D E ¯ = s 2 ( r L ) 2 cos 2 δ + ( r l ) l | l · 1 / cos 2 β 1 / cos 2 δ x B | .
A B ¯ P E ¯ = N I ¯ N D ¯ = h r / cos δ h .
A B ¯ = h · [ s 2 ( r l ) 2 cos 2 δ + ( r l ) l | l · 1 / cos 2 β 1 / cos 2 δ x B | ] r / cos δ h ' .
Δ φ = 2 π f 0 · A B ¯ .
h = h cos δ .
h = r · Δ φ 2 π f 0 [ s 2 ( r l ) 2 cos 2 δ + ( r l ) l | l · 1 / cos 2 β 1 / cos 2 δ x B | ] + Δ φ ,
1 h ( x , y ) = a ( x , y ) + b ( x , y ) · 1 Δ φ ( x , y ) .
b ( x , y ) = 2 π f 0 [ s 2 ( r l ) 2 cos 2 δ + ( r l ) l | l · 1 / cos 2 β 1 / cos 2 δ x B | ] r .
1 h ( x , y ) = a ( x , y ) + b ( x , y ) · 1 Δ φ ( x , y ) + c ( x , y ) · 1 Δ φ 2 ( x , y ) .

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