Abstract

In order to obtain the correct height reconstruction of a measured object, a projection lens and a camera lens must be placed at equal heights above the reference plane in the traditional Fourier transform profilometry (FTP) method. We propose an improved phase-height mapping formula based on an improved description of the reference fringe and the deformed fringe in FTP when the projection lens and the camera lens are not placed at equal height. With our method, it is easier to obtain the full-field fringe by moving either the projector or the imaging device. In some cases, where the required parallel condition cannot be met, the proposed method offers a flexible way to calculate the height distribution.

© 2007 Optical Society of America

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References

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  1. H. Takasaki, "Moiré topography," Appl. Opt. 9, 1467-1472 (1970).
    [CrossRef] [PubMed]
  2. V. Srinivasan, H. C. Lui, and M. Halioua, "Automatic phase measuring profilometry of 3-D diffuse object," Appl. Opt. 23, 3105-3108 (1984).
    [CrossRef] [PubMed]
  3. 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]
  4. J. Vanherzeele, P. Guillaume, and S. Vanlanduit, "Fourier fringe processing using a regressive Fourier-transform technique," Opt. Lasers Eng. 43, 645-658 (2005).
    [CrossRef]
  5. J. Li, X.-Y. Su, and L.-R. Guo, "Improved Fourier transform profilometry of the automatic measurement of three-dimensional object shapes," Opt. Eng. (Bellingham) 29, 1439-1444 (1990).
    [CrossRef]
  6. W. J. Chen and X. Y. Su, "Error caused by sampling in Fourier transform profilometry," Opt. Eng. (Bellingham) 38, 1029-1034 (1999).
    [CrossRef]
  7. X. Y. Su and W. J. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
    [CrossRef]
  8. V. Zlatko and G. Jadranko, "Phase retrieval errors in standard Fourier fringe analysis of digitally sampled model interferograms," Appl. Opt. 44, 6940-6947 (2005).
    [CrossRef]
  9. F. J. Cuevas, M. Servin, and R. Rodriguez-Vera, "Depth object recovery using radial basis functions," Opt. Commun. 163, 270-277 (1999).
    [CrossRef]
  10. F. J. Cuevas, M. Servin, and O. N. Stavroudis, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
    [CrossRef]
  11. G. Dinesh, J. Joby, and S. Kehar, "Object reconstruction in multilayer neural network based profilometry using grating structure comprising two regions with different spatial periods," Opt. Lasers Eng. 42, 179-192 (2004).
    [CrossRef]
  12. H. W. Guo, H. T. He, Y. J. Yu, and M. Y. Chen, "Least-squares calibration method for fringe projection profilometry," Opt. Eng. (Bellingham) 44, 033603 (2005).
    [CrossRef]
  13. X. L. Zhang, Y. C. Lin, M. R. Zhao, X. B. Niu, and Y. G. Huang, "Calibration of a fringe projection profilometry system using virtual phase calibrating model planes," J. Opt. A, Pure Appl. Opt. 7, 192-197 (2005).
    [CrossRef]
  14. H. Y. Liu, W. H. Su, K. Reichard, and S. Z. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-801 (2003).
    [CrossRef]
  15. W. S. Zhou and X. Y. Su, "A direct mapping algorithm for phase-measuring profilometry," J. Mod. Opt. 41, 89-94 (1994).
    [CrossRef]
  16. G. S. Spagnolo, G. Guattari, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, "Contouring of artwork surface by fringe projection and FFT analysis," Opt. Lasers Eng. 33, 141-156 (2000).
    [CrossRef]
  17. C. G. Quan, C. J. Tay, and L. J. Chen, "Fringe-density estimation by continuous wavelet transform," Appl. Opt. 44, 2359-2365 (2005).
    [CrossRef] [PubMed]
  18. K. J. Gåsvik, "Optical techniques," in Interferogram Analysis, D.W.Robinson and G.T.Reid, eds. (IOP, 1993), pp. 49-93.
  19. W. J. Chen, P. Bu, S. Z. Zheng, and X. Y. Su, "Study on Fourier transforms profilometry based on bi-color projecting," Opt. Laser Technol. 39, 821-827 (2007).
    [CrossRef]

2007 (1)

W. J. Chen, P. Bu, S. Z. Zheng, and X. Y. Su, "Study on Fourier transforms profilometry based on bi-color projecting," Opt. Laser Technol. 39, 821-827 (2007).
[CrossRef]

2005 (5)

C. G. Quan, C. J. Tay, and L. J. Chen, "Fringe-density estimation by continuous wavelet transform," Appl. Opt. 44, 2359-2365 (2005).
[CrossRef] [PubMed]

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

V. Zlatko and G. Jadranko, "Phase retrieval errors in standard Fourier fringe analysis of digitally sampled model interferograms," Appl. Opt. 44, 6940-6947 (2005).
[CrossRef]

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

X. L. Zhang, Y. C. Lin, M. R. Zhao, X. B. Niu, and Y. G. Huang, "Calibration of a fringe projection profilometry system using virtual phase calibrating model planes," J. Opt. A, Pure Appl. Opt. 7, 192-197 (2005).
[CrossRef]

2004 (1)

G. Dinesh, J. Joby, and S. Kehar, "Object reconstruction in multilayer neural network based profilometry using grating structure comprising two regions with different spatial periods," Opt. Lasers Eng. 42, 179-192 (2004).
[CrossRef]

2003 (1)

H. Y. Liu, W. H. Su, K. Reichard, and S. Z. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-801 (2003).
[CrossRef]

2001 (1)

X. Y. Su and W. J. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

2000 (2)

F. J. Cuevas, M. Servin, and O. N. Stavroudis, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
[CrossRef]

G. S. Spagnolo, G. Guattari, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, "Contouring of artwork surface by fringe projection and FFT analysis," Opt. Lasers Eng. 33, 141-156 (2000).
[CrossRef]

1999 (2)

F. J. Cuevas, M. Servin, and R. Rodriguez-Vera, "Depth object recovery using radial basis functions," Opt. Commun. 163, 270-277 (1999).
[CrossRef]

W. J. Chen and X. Y. Su, "Error caused by sampling in Fourier transform profilometry," Opt. Eng. (Bellingham) 38, 1029-1034 (1999).
[CrossRef]

1994 (1)

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

1990 (1)

J. Li, X.-Y. Su, and L.-R. Guo, "Improved Fourier transform profilometry of the automatic measurement of three-dimensional object shapes," Opt. Eng. (Bellingham) 29, 1439-1444 (1990).
[CrossRef]

1984 (1)

1983 (1)

1970 (1)

Appl. Opt. (5)

J. Mod. Opt. (1)

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

J. Opt. A, Pure Appl. Opt. (1)

X. L. Zhang, Y. C. Lin, M. R. Zhao, X. B. Niu, and Y. G. Huang, "Calibration of a fringe projection profilometry system using virtual phase calibrating model planes," J. Opt. A, Pure Appl. Opt. 7, 192-197 (2005).
[CrossRef]

Opt. Commun. (3)

H. Y. Liu, W. H. Su, K. Reichard, and S. Z. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-801 (2003).
[CrossRef]

F. J. Cuevas, M. Servin, and R. Rodriguez-Vera, "Depth object recovery using radial basis functions," Opt. Commun. 163, 270-277 (1999).
[CrossRef]

F. J. Cuevas, M. Servin, and O. N. Stavroudis, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
[CrossRef]

Opt. Eng. (Bellingham) (3)

J. Li, X.-Y. Su, and L.-R. Guo, "Improved Fourier transform profilometry of the automatic measurement of three-dimensional object shapes," Opt. Eng. (Bellingham) 29, 1439-1444 (1990).
[CrossRef]

W. J. Chen and X. Y. Su, "Error caused by sampling in Fourier transform profilometry," Opt. Eng. (Bellingham) 38, 1029-1034 (1999).
[CrossRef]

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

Opt. Laser Technol. (1)

W. J. Chen, P. Bu, S. Z. Zheng, and X. Y. Su, "Study on Fourier transforms profilometry based on bi-color projecting," Opt. Laser Technol. 39, 821-827 (2007).
[CrossRef]

Opt. Lasers Eng. (4)

X. Y. Su and W. J. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

G. S. Spagnolo, G. Guattari, C. Sapia, D. Ambrosini, D. Paoletti, and G. Accardo, "Contouring of artwork surface by fringe projection and FFT analysis," Opt. Lasers Eng. 33, 141-156 (2000).
[CrossRef]

G. Dinesh, J. Joby, and S. Kehar, "Object reconstruction in multilayer neural network based profilometry using grating structure comprising two regions with different spatial periods," Opt. Lasers Eng. 42, 179-192 (2004).
[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]

Other (1)

K. J. Gåsvik, "Optical techniques," in Interferogram Analysis, D.W.Robinson and G.T.Reid, eds. (IOP, 1993), pp. 49-93.

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Figures (14)

Fig. 1
Fig. 1

Optical geometry. The projection lens (Grating) and the camera lens (CCD) are not placed at equal heights above the reference plane; on the contrary they form an angle α. The optical axis of the projector lens still crosses the optical axis of the camera lens at point O on the reference plane (R). D is an arbitrary point on the measured object, its coordinates are ( x , y ) , d is the distance between the exit pupil of the projector and the entrance pupil of the imaging system, and L is the distance between the entrance pupil of the imaging system and the reference plane. The angle between the optical axes of the projector and the CCD is θ. In addition, A I 1 O = β , I 2 C O = δ , D A C = γ .

Fig. 2
Fig. 2

Simulated object.

Fig. 3
Fig. 3

Deformed fringe distribution when α = 0 ° .

Fig. 4
Fig. 4

Calculated height error when α = 0 ° .

Fig. 5
Fig. 5

Calculated height error when α = 10 °

Fig. 6
Fig. 6

Calculated height error when α = 20 ° .

Fig. 7
Fig. 7

Calculated height error when α = 10 ° .

Fig. 8
Fig. 8

Calculated height error when α = 20 ° .

Fig. 9
Fig. 9

Height error distribution using the traditional method when α = 20 ° .

Fig. 10
Fig. 10

Error distributions of the traditonal method and the improved method.

Fig. 11
Fig. 11

Reference fringe captured by a CCD camera.

Fig. 12
Fig. 12

Deformed fringe modulated by a 3D object.

Fig. 13
Fig. 13

Retrieved height using our method.

Fig. 14
Fig. 14

Retrieved height using the traditional method.

Equations (19)

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I 1 ( x , y ) = I 0 ( x , y ) [ 1 + γ cos 2 π x f ( x ) ] ,
I 1 O ¯ = L + d sin α cos θ .
f ( x ) = f cos θ [ 1 x sin ( 2 θ ) L + d sin α ] ,
I 1 ( x , y ) = I 0 ( x , y ) ( 1 + γ cos { 2 π f cos θ [ x x 2 sin ( 2 θ ) L + d sin α ] } ) .
I 2 ( x , y ) = I 0 ( x , y ) ( 1 + γ cos { 2 π f cos θ [ x x 2 sin ( 2 θ ) L + d sin α ] ψ ( x , y ) } ) ,
C A ¯ B D ¯ = I 1 P ¯ D F ¯ ,
P G ¯ = x d sin α L B D ¯ .
So I 1 P ¯ = I 1 G ¯ P G ¯ = d cos α x d sin α L B D ¯ .
C A ¯ = B D ¯ L + d sin α B D ¯ ( d cos α x d sin α L B D ¯ ) .
ψ ( x , y ) = 2 π f cos θ B D ¯ L + d sin α B D ¯ ( d cos α x d sin α L B D ¯ ) .
I 1 ( x , y ) = I 0 ( x , y ) { 1 + γ cos [ 2 π f 0 x + ϕ 0 ( x , y ) ] } ,
I 2 ( x , y ) = I 0 ( x , y ) { 1 + γ cos [ 2 π f 0 x + ϕ ( x , y ) ] } ,
Δ ϕ ( x , y ) = ϕ ( x , y ) ϕ 0 ( x , y ) .
B D ¯ = C A ¯ cot γ cot δ ,
cot δ = ϕ C 2 π f 0 L .
cot γ = d cos α + ϕ D 2 π f 0 L + d sin α .
C A ¯ = ϕ D C 2 π f 0 .
B D ¯ = ϕ D C L ( L + d sin α ) 2 π f 0 d L cos α + L ϕ D C d ϕ C sin α .
B D ¯ = L ϕ D C 2 π f 0 d + ϕ D C ,

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