Abstract

The measurement of an object’s shape using projected fringe patterns needs a relation between the measured phase and the object’s height. Among various methods, the Fourier transform profilometry proposed by Takeda and Mutoh [Appl. Opt. 22, 3977–3982 (1983)] is widely used in the literature. Rajoub et al. have shown that the reference relation given by Takeda is erroneous [J. Opt. A. Pure Appl. Opt. 9, 66–75 (2007)]. This paper follows from Rajoub’s study. Our results for the phase agree with Rajoub’s results for both parallel- and crossed-optical-axes geometries and for either collimated or noncollimated projection. Our two main results are: (i) we show experimental evidence of the error in Takeda’s formula and (ii) we explain the error in Takeda’s derivation and we show that Rajoub’s argument concerning Takeda’s error is not correct.

© 2009 Optical Society of America

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References

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  1. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  2. 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]
  3. J. Yi and S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493-505 (1997).
    [CrossRef]
  4. B. Zhao and A. Asundi, “Discussion on spatial resolution and sensitivity of Fourier transform fringe detection,” Opt. Eng. 39, 2715-2719 (2000).
    [CrossRef]
  5. X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36, 49-64 (2001).
    [CrossRef]
  6. Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at the free surface,” Opt. Laser Technol. 34, 107-113 (2002).
    [CrossRef]
  7. F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Lasers Eng. 39, 35-50 (2003).
    [CrossRef]
  8. F. Berryman, P. Pynsent, and J. Cubillo, “The effect of windowing in Fourier transform profilometry applied to noisy images,” Opt. Lasers Eng. 41, 815-825 (2004).
    [CrossRef]
  9. J. Zhong and J. WengSpatial carrier-fringe pattern analysis by means of wavelet transform,” Appl. Opt. 43, 4993-4998 (2004).
    [CrossRef] [PubMed]
  10. 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]
  11. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482-489 (2006).
    [CrossRef]
  12. X. Mao, W. Chen, and X. Su, “Improved Fourier transform profilometry,” Appl. Opt. 46, 664-668 (2007).
    [CrossRef] [PubMed]
  13. X. Mao, W. Chen, X. Su, G. Xu, and X. Bian, “Fourier transform profilometry based on a projecting-imaging model,” J. Opt. Soc. Am. A 24, 3735-3740 (2007).
    [CrossRef]
  14. E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier transform profilometry,” Opt. Lasers Eng. 46, 106-116 (2008).
    [CrossRef]
  15. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263-284 (2001).
    [CrossRef]
  16. B. A. Rajoub and M. J. Lalor, “A new phase-to-height model for measuring object shape using collimated projections of structured light,” J. Opt. A Pure Appl. Opt. 7, S368-S375 (2005).
    [CrossRef]
  17. B. A. Rajoub, M. J. Lalor, D. R. Burton, and S. A. Karout, “A new model for measuring object shape using non-collimated fringe-pattern projections,” J. Opt. A Pure Appl. Opt. 9, S66-S75 (2007).
    [CrossRef]
  18. Eq. (36) in is rather intricate. For the sake of clarity, it is worth mentioning that our expression for φ(Y) gives the same result as their Eq. (36) owing to the correspondences between Rajoub's notations and our notations: yfp=−D, zfp=Lp, yop=−D−fpsinθ, zop=Lp+fpcosθ, zfc=Lc, yoc=0, zoc=Lc+fc, Yiyoc=Y, zi=h, and ω′=ωp/cosθ.
  19. The ray bC′ has the direction of the vector (x′,y′,h−Lc) and the ray aC′ has the direction of the vector (x,y,h−Lc). The cross product of both vectors has to vanish and the first component of the cross product is xh+(x′−x)Lc. We get δxx′−x=−xh/L
  20. 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]
  21. P. Cobelli, A. Maurel, V. Pagneux, and P. Petitjeans, “Fast global measurement of water waves by Fourier transform profilometry,” submitted to Exp. in Fluids .
  22. We have used the unwrapping algorithm unwrap from MATLAB (The MathWorks, 2007).
  23. The useful relations are tan⁡(θ−β)=Ob′¯cosθ/L and tanβ=(D+y′)/(L−h), with y′/(L−h)=y/L. β is defined in Fig. 13.
  24. C. Quan, C. J. Tay, H. M. Shang, and P. J. Bryanston-Cross “Contour measurement by fibre optic fringe projection and Fourier transform analysis,” Opt. Commun. 118, 47983(1995).
    [CrossRef]
  25. F. Lilley, “An optical 3-D body surface measurement system to improve radiotherapy treatment of cancer,” Ph.D. thesis, Faculty of General Engineering (Liverpool John Moores University, 1999).
  26. L. C. Fang and L. Yang “A new approach to high precision 3-D measuring system,” Image Vis. Comput. 17, 80514(1999).

2008 (1)

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier transform profilometry,” Opt. Lasers Eng. 46, 106-116 (2008).
[CrossRef]

2007 (3)

2006 (1)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482-489 (2006).
[CrossRef]

2005 (2)

B. A. Rajoub and M. J. Lalor, “A new phase-to-height model for measuring object shape using collimated projections of structured light,” J. Opt. A Pure Appl. Opt. 7, S368-S375 (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]

2004 (2)

F. Berryman, P. Pynsent, and J. Cubillo, “The effect of windowing in Fourier transform profilometry applied to noisy images,” Opt. Lasers Eng. 41, 815-825 (2004).
[CrossRef]

J. Zhong and J. WengSpatial carrier-fringe pattern analysis by means of wavelet transform,” Appl. Opt. 43, 4993-4998 (2004).
[CrossRef] [PubMed]

2003 (1)

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

2002 (1)

Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at the free surface,” Opt. Laser Technol. 34, 107-113 (2002).
[CrossRef]

2001 (2)

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

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

2000 (2)

B. Zhao and A. Asundi, “Discussion on spatial resolution and sensitivity of Fourier transform fringe detection,” Opt. Eng. 39, 2715-2719 (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 (1)

L. C. Fang and L. Yang “A new approach to high precision 3-D measuring system,” Image Vis. Comput. 17, 80514(1999).

1997 (1)

J. Yi and S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493-505 (1997).
[CrossRef]

1995 (1)

C. Quan, C. J. Tay, H. M. Shang, and P. J. Bryanston-Cross “Contour measurement by fibre optic fringe projection and Fourier transform analysis,” Opt. Commun. 118, 47983(1995).
[CrossRef]

1983 (1)

1982 (1)

Accardo, G.

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]

Ambrosini, D.

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]

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]

Berryman, F.

F. Berryman, P. Pynsent, and J. Cubillo, “The effect of windowing in Fourier transform profilometry applied to noisy images,” Opt. Lasers Eng. 41, 815-825 (2004).
[CrossRef]

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

Bian, X.

Bryanston-Cross, P. J.

C. Quan, C. J. Tay, H. M. Shang, and P. J. Bryanston-Cross “Contour measurement by fibre optic fringe projection and Fourier transform analysis,” Opt. Commun. 118, 47983(1995).
[CrossRef]

Burton, D. R.

B. A. Rajoub, M. J. Lalor, D. R. Burton, and S. A. Karout, “A new model for measuring object shape using non-collimated fringe-pattern projections,” J. Opt. A Pure Appl. Opt. 9, S66-S75 (2007).
[CrossRef]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482-489 (2006).
[CrossRef]

Busca, G.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier transform profilometry,” Opt. Lasers Eng. 46, 106-116 (2008).
[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]

Chao, Y.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Chen, W.

X. Mao, W. Chen, X. Su, G. Xu, and X. Bian, “Fourier transform profilometry based on a projecting-imaging model,” J. Opt. Soc. Am. A 24, 3735-3740 (2007).
[CrossRef]

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]

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Cobelli, P.

P. Cobelli, A. Maurel, V. Pagneux, and P. Petitjeans, “Fast global measurement of water waves by Fourier transform profilometry,” submitted to Exp. in Fluids .

Cubillo, J.

F. Berryman, P. Pynsent, and J. Cubillo, “The effect of windowing in Fourier transform profilometry applied to noisy images,” Opt. Lasers Eng. 41, 815-825 (2004).
[CrossRef]

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

Fang, L. C.

L. C. Fang and L. Yang “A new approach to high precision 3-D measuring system,” Image Vis. Comput. 17, 80514(1999).

Gdeisat, M. A.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482-489 (2006).
[CrossRef]

Guattari, G.

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]

Huang, S.

J. Yi and S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493-505 (1997).
[CrossRef]

Ina, H.

Karout, S. A.

B. A. Rajoub, M. J. Lalor, D. R. Burton, and S. A. Karout, “A new model for measuring object shape using non-collimated fringe-pattern projections,” J. Opt. A Pure Appl. Opt. 9, S66-S75 (2007).
[CrossRef]

Kobayashi, S.

Lalor, M. J.

B. A. Rajoub, M. J. Lalor, D. R. Burton, and S. A. Karout, “A new model for measuring object shape using non-collimated fringe-pattern projections,” J. Opt. A Pure Appl. Opt. 9, S66-S75 (2007).
[CrossRef]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482-489 (2006).
[CrossRef]

B. A. Rajoub and M. J. Lalor, “A new phase-to-height model for measuring object shape using collimated projections of structured light,” J. Opt. A Pure Appl. Opt. 7, S368-S375 (2005).
[CrossRef]

Lilley, F.

F. Lilley, “An optical 3-D body surface measurement system to improve radiotherapy treatment of cancer,” Ph.D. thesis, Faculty of General Engineering (Liverpool John Moores University, 1999).

Mao, X.

Maurel, A.

P. Cobelli, A. Maurel, V. Pagneux, and P. Petitjeans, “Fast global measurement of water waves by Fourier transform profilometry,” submitted to Exp. in Fluids .

Mutoh, K.

Pagneux, V.

P. Cobelli, A. Maurel, V. Pagneux, and P. Petitjeans, “Fast global measurement of water waves by Fourier transform profilometry,” submitted to Exp. in Fluids .

Paoletti, D.

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]

Petitjeans, P.

P. Cobelli, A. Maurel, V. Pagneux, and P. Petitjeans, “Fast global measurement of water waves by Fourier transform profilometry,” submitted to Exp. in Fluids .

Pynsent, P.

F. Berryman, P. Pynsent, and J. Cubillo, “The effect of windowing in Fourier transform profilometry applied to noisy images,” Opt. Lasers Eng. 41, 815-825 (2004).
[CrossRef]

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

Quan, C.

C. Quan, C. J. Tay, H. M. Shang, and P. J. Bryanston-Cross “Contour measurement by fibre optic fringe projection and Fourier transform analysis,” Opt. Commun. 118, 47983(1995).
[CrossRef]

Rajoub, B. A.

B. A. Rajoub, M. J. Lalor, D. R. Burton, and S. A. Karout, “A new model for measuring object shape using non-collimated fringe-pattern projections,” J. Opt. A Pure Appl. Opt. 9, S66-S75 (2007).
[CrossRef]

B. A. Rajoub and M. J. Lalor, “A new phase-to-height model for measuring object shape using collimated projections of structured light,” J. Opt. A Pure Appl. Opt. 7, S368-S375 (2005).
[CrossRef]

Sapia, C.

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]

Shang, H. M.

C. Quan, C. J. Tay, H. M. Shang, and P. J. Bryanston-Cross “Contour measurement by fibre optic fringe projection and Fourier transform analysis,” Opt. Commun. 118, 47983(1995).
[CrossRef]

Spagnolo, G. S.

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]

Su, X.

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

X. Mao, W. Chen, X. Su, G. Xu, and X. Bian, “Fourier transform profilometry based on a projecting-imaging model,” J. Opt. Soc. Am. A 24, 3735-3740 (2007).
[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]

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

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Su, X.-Y.

Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at the free surface,” Opt. Laser Technol. 34, 107-113 (2002).
[CrossRef]

Takeda, M.

Tay, C. J.

C. Quan, C. J. Tay, H. M. Shang, and P. J. Bryanston-Cross “Contour measurement by fibre optic fringe projection and Fourier transform analysis,” Opt. Commun. 118, 47983(1995).
[CrossRef]

Weng, J.

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]

Xu, G.

Yang, L.

L. C. Fang and L. Yang “A new approach to high precision 3-D measuring system,” Image Vis. Comput. 17, 80514(1999).

Yi, J.

J. Yi and S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493-505 (1997).
[CrossRef]

Zappa, E.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier transform profilometry,” Opt. Lasers Eng. 46, 106-116 (2008).
[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]

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Zhang, Q.-C.

Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at the free surface,” Opt. Laser Technol. 34, 107-113 (2002).
[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]

Zhong, J.

Appl. Opt. (3)

Exp. in Fluids (1)

P. Cobelli, A. Maurel, V. Pagneux, and P. Petitjeans, “Fast global measurement of water waves by Fourier transform profilometry,” submitted to Exp. in Fluids .

Image Vis. Comput. (1)

L. C. Fang and L. Yang “A new approach to high precision 3-D measuring system,” Image Vis. Comput. 17, 80514(1999).

J. Opt. A Pure Appl. Opt. (2)

B. A. Rajoub and M. J. Lalor, “A new phase-to-height model for measuring object shape using collimated projections of structured light,” J. Opt. A Pure Appl. Opt. 7, S368-S375 (2005).
[CrossRef]

B. A. Rajoub, M. J. Lalor, D. R. Burton, and S. A. Karout, “A new model for measuring object shape using non-collimated fringe-pattern projections,” J. Opt. A Pure Appl. Opt. 9, S66-S75 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482-489 (2006).
[CrossRef]

C. Quan, C. J. Tay, H. M. Shang, and P. J. Bryanston-Cross “Contour measurement by fibre optic fringe projection and Fourier transform analysis,” Opt. Commun. 118, 47983(1995).
[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. Laser Technol. (1)

Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at the free surface,” Opt. Laser Technol. 34, 107-113 (2002).
[CrossRef]

Opt. Lasers Eng. (8)

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

F. Berryman, P. Pynsent, and J. Cubillo, “The effect of windowing in Fourier transform profilometry applied to noisy images,” Opt. Lasers Eng. 41, 815-825 (2004).
[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]

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

J. Yi and S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493-505 (1997).
[CrossRef]

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier transform profilometry,” Opt. Lasers Eng. 46, 106-116 (2008).
[CrossRef]

X. Su and W. 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]

Other (5)

F. Lilley, “An optical 3-D body surface measurement system to improve radiotherapy treatment of cancer,” Ph.D. thesis, Faculty of General Engineering (Liverpool John Moores University, 1999).

We have used the unwrapping algorithm unwrap from MATLAB (The MathWorks, 2007).

The useful relations are tan⁡(θ−β)=Ob′¯cosθ/L and tanβ=(D+y′)/(L−h), with y′/(L−h)=y/L. β is defined in Fig. 13.

Eq. (36) in is rather intricate. For the sake of clarity, it is worth mentioning that our expression for φ(Y) gives the same result as their Eq. (36) owing to the correspondences between Rajoub's notations and our notations: yfp=−D, zfp=Lp, yop=−D−fpsinθ, zop=Lp+fpcosθ, zfc=Lc, yoc=0, zoc=Lc+fc, Yiyoc=Y, zi=h, and ω′=ωp/cosθ.

The ray bC′ has the direction of the vector (x′,y′,h−Lc) and the ray aC′ has the direction of the vector (x,y,h−Lc). The cross product of both vectors has to vanish and the first component of the cross product is xh+(x′−x)Lc. We get δxx′−x=−xh/L

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Figures (16)

Fig. 1
Fig. 1

Reproduction of Takeda’s representation in crossed- optical-axes geometry. A fringe pattern is projected onto a reference surface (as point a) and on a deformed surface (point b). The corresponding phase variation in the intensity recorded by the camera is Δ φ ( y ) (2D object analysis is presented in Takeda’s paper [2]).

Fig. 2
Fig. 2

Optical setup.

Fig. 3
Fig. 3

In general, the rays C b and b C are not in a vertical plane. We define b 0 ( 0 , y , h ) as the projection of b ( x , y , h ) in the plane x = 0 (which contains O, C, and C ). In this vertical plane x = 0 , the angles α and β, respectively, measure ( O C b 0 ^ ) and ( O C b 0 ^ ) . These angles measure also the positions of B and A on the projector and on the camera grating. The angle θ = ( c C O ^ ) measures the inclination of the projector’s axis with respect to the camera’s axis.

Fig. 4
Fig. 4

Optical setup for the projector and the camera at the same distance L from the reference plane R, (a) in the parallel- and (b) in the crossed-optical-axes geometries.

Fig. 5
Fig. 5

Experimental configuration: Σ is at distance h ( y ) from the reference plane R with h ( 0 < y y 0 3 cm ) = y , h ( 0 < y y 0 3 cm ) = y zero, otherwise. I is the image plane of the projector. In the experiments, L = 105.2 cm and θ can vary, and for θ = 0 ° (parallel-optical-axes geometry), D = 18 cm .

Fig. 6
Fig. 6

Experimental intensity variations I ( x , y ) captured by the camera (a) in the parallel-optical-axes geometry with D = 18 cm and L = 105.2 cm and (b) in the crossed-optical-axes geometry with D = 70.7 cm ( θ = 33.9 ° ), L = 105.2 cm , and y 0 = 0 .

Fig. 7
Fig. 7

Intensity variations I ( y ) in (a) the parallel- and (b) the crossed-optical-axes geometries. The curves correspond to the averages over the x direction of the 3D plot in Fig. 6.

Fig. 8
Fig. 8

(a) Intensity variations I 0 ( y ) for fringe projection onto the reference plane in the absence of the triangular prism and (b) the corresponding unwrapped phase φ 0 ( y ) (see [23]). Experiments correspond to θ = 33.9 ° with L = 105.2 cm . The points are the experimental data (only one point of each 150 points is indicated for visibility), the solid curve corresponds to our Eq. (2.12) and the dashed curve corresponds to Eq. 2.13 from [12, 13, 20].

Fig. 9
Fig. 9

(a) Signals I 0 ( y ) and I ( y ) for fringe projections on the reference plane R and on the Σ plane. (b) Unwrapped phase difference Δ φ ( y ) and (c) reconstructed height h ( y ) using Eqs. (2.9, 2.11). The experiment is conducted in the parallel-optical-axes geometry with L = 105.2 cm , D = 18 cm , and y 0 = 24 cm .

Fig. 10
Fig. 10

Height reconstruction h ( y ) for various y 0 values. Solid curves correspond to our phase-to-height relation and dashed curves correspond to Takeda’s phase-to-height relation. The experimental configuration is the same as in Fig. 9.

Fig. 11
Fig. 11

Reconstructed height h ( y ) in the crossed-optical-axes geometry (a) for θ = 33.9 ° and varying the y 0 position of the triangle and (b) for y 0 = 16 cm and varying θ = 0 , 18.1 and 41 ° . Solid curves correspond to our height reconstruction from Eq. (3.3) with Eq. (1.2). Dashed curves are the height reconstructed using Takeda’s relation Eq. (1.1).

Fig. 12
Fig. 12

(a) Reproduction of Takeda’s representation and (b) the same representation including useful additional points: b is the point of the ray B b on the plane I, b a is the point of ray b B on the plane R ( O y ) , and b 0 is the point on plane R with b 0 b parallel to the projector’s optical axis C O . Similar construction is used to define a a , a 0 , and a .

Fig. 13
Fig. 13

Same representation as in Fig. 12b considering the ray A a B instead of the ray B a A . Now the ray A B is seen by the camera as coming from point a = b a . Otherwise, the same definitions for the points a , b and a 0 , b 0 as in Fig. 12b are used.

Fig. 14
Fig. 14

System geometry used to calculate the fringe spacing p n in the crossed-optical-axes geometry.

Fig. 15
Fig. 15

Open circles, experimental fringe spacing deduced from Fig. 7 in the parallel- and the crossed-optical-axes geometries (the crossed-axes geometry gives higher fringe spacing). The solid curves correspond to our expression in Eq. (A5) and in the body of the text.

Fig. 16
Fig. 16

Collimated projection in the (a) crossed- and (b) parallel-optical-axes geometries.

Equations (49)

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h ( y ) = L Δ φ ( y ) Δ φ ( y ) ω 0 D ,
h ( x , y ) = L Δ φ Δ φ ω 0 D ,
x = x h L x , y = y h L y ;
h ( x , y ) = L Δ φ ( 1 + sin 2 θ y / D ) 2 Δ φ ( 1 + sin 2 θ y / D ) [ 1 sin 2 θ ( 1 y / D ) ] ω 0 D ;
h ( x , y ) = L Δ φ ω 0 y ;
h ( x , y ) = L Δ φ ω sin θ ( L cot θ y ) .
I ( X , Y ) = 1 + cos φ ( X , Y ) .
φ 0 ( X , Y ) = ω p Y A .
φ ( X , Y ) = ω p Y B .
tan α = Y B f p = ( D + y ) tan θ ( L p h ) tan θ ( D + y ) + L p h ,
tan β = Y f c = y L c h .
Y B = f p D ( L c h ) Y / f c tan θ ( L p h ) L p h + tan θ [ D ( L c h ) Y / f c ] ,
φ ( X , Y ) = ω c cos θ ( 1 h / L c ) Y + G c [ ( L p h ) tan θ D ] ( 1 h / L p ) tan θ L p 1 [ ( 1 h / L c ) Y / G c D ] , φ 0 ( X , Y ) = ω c cos θ Y + G c [ L p tan θ D ] 1 tan θ L p 1 [ Y / G c D ] .
φ ( x , y ) = ω cos θ y L p tan θ + D + h / L c ( L c tan θ y ) 1 + tan θ ( D + y ) / L p h / L p ( 1 + tan θ y / L c ) , φ 0 ( x , y ) = ω cos θ y L p tan θ + D 1 + tan θ ( D + y ) / L p .
x = x h L c x , y = y h L c y ,
φ ( x , y ) = ω y ω D L L h ( x , y ) , φ 0 ( x , y ) = ω y ω D .
Δ φ ( x , y ) = ω D h ( x , y ) L h ( x , y ) .
φ ( x , y ) = ω cos θ y + h ( x , y ) / L ( D y ) 1 + sin 2 θ y / D h ( x , y ) / L [ 1 sin 2 θ ( 1 y / D ) ] , φ 0 ( x , y ) = ω cos θ y 1 + sin 2 θ y / D .
φ 0 S ( x , y ) = ω cos θ [ 1 2 sin θ cos θ y / L ] .
Δ φ ( x , y ) = ω 0 D h ( x , y ) ( 1 + sin 2 θ y / D ) [ 1 + sin 2 θ y / D h ( x , y ) / L ( 1 sin 2 θ ( 1 y / D ) ) ] ,
Δ φ T ( x , y ) = ω 0 D h ( x , y ) L h ( x , y ) .
Δ φ = Δ φ T C ( θ , h / L , y / D ) ,
C ( θ , h / L , y / D ) = [ 1 + sin 2 θ ( y D h L h ) ] 1 [ 1 + sin 2 θ y D ] 1 .
Δ φ = Δ φ T [ 1 + O ( θ 2 h / L , θ 2 y / D ) ] .
h ( y ) = L Δ φ ( y ) Δ φ ( y ) ω 0 D ,
h ( y ) = L Δ φ ( y ) ( 1 + sin 2 θ y ) 2 Δ φ ( y ) ( 1 + sin 2 θ y / D ) [ 1 sin 2 θ ( 1 y / D ) ] ω 0 D .
h T ( y ) = h ( y ) A 2 + ( A B + h ( y ) ) h ( y ) / L ,
φ = ω p Y B .
φ = ω O b ¯ = ω 0 O b a ¯ + ω 0 ( O b ¯ cos θ O b a ¯ ) .
φ ( y ) = ω 0 y + ω 0 b a b 0 ¯ ,
φ 0 ( y ) = ω 0 y + ω 0 a b 0 ¯ ,
Δ φ T ( y ) = ω 0 b a a ¯ = ω 0 h D L h .
Δ φ = ω 0 ( O b a ¯ O a ¯ ) + ω 0 ( b a b 0 ¯ a b 0 ¯ ) = 0 ,
φ ( y ) = ω 0 y + ω 0 a b 0 ¯ ,
φ 0 ( y ) = ω 0 y + ω 0 a a 0 ¯ ,
Δ φ ( y ) = ω 0 a 0 b 0 ¯ .
tan α n = O w n ¯ L / cos θ , O w n ¯ = n p I ,
tan ( θ + α n ) = C y n ¯ L .
p n + 1 = L [ tan ( θ + α n + 1 ) tan ( θ + α n ) ] ,
p n + 1 = L cos 2 θ tan α n + 1 tan α n ( 1 tan θ tan α n ) ( 1 tan θ tan α n + 1 ) .
p n + 1 = p I cos θ 1 ( 1 n sin θ p I / L ) [ 1 ( n + 1 ) sin θ p I / L ] .
p n p 0 cos θ 1 + sin θ cos θ y n + 1 / L 1 tan θ y w n / L ,
p n p 0 cos θ 1 + sin θ cos θ [ tan ( θ + α n + 1 ) tan θ ] 1 sin θ cos θ n p 0 / L ,
φ ( x , y ) = ω p Y B ,
φ ( x , y ) = ω p cos θ y + ω p sin θ h ( 1 cot θ y / L ) .
φ 0 ( x , y ) = ω p cos θ y
Δ φ ( x , y ) = ω sin θ h ( 1 cot θ y / L ) ,
φ 0 ( x , y ) = ω p ( D + y ) , φ ( x , y ) = ω p ( D + y ) ,
Δ φ ( x , y ) = ω h ( x , y ) L y .

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