Abstract

An improved method is proposed to perform calibration-based fringe projected profilometry using a two-frequency fringe pattern for the 3D shape measurements of objects with large discontinuous height steps. A fabrication scheme for the two-frequency pattern is described as well. The proposed method offers following major advantages: (1) only one phase measurement needed for operation, (2) easiness for calibration, (3) robust performance, especially for automatic phase unwrapping, and (4) more flexible data acquisition for complex objects. This makes it possible for a single-shot measurement of dynamic objects with discontinuities. Both theoretical descriptions and experimental demonstrations are provided.

© 2006 Optical Society of America

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  1. G. Indebetouw, "Profile measurement using projection of running fringes," Appl. Opt. 17, 2930-2933 (1978).
    [CrossRef] [PubMed]
  2. V. Srinivasan, H. C. Liu, and M. Halioua, "Automated phase-measuring profilometry of 3-D diffuse objects," Appl. Opt. 23, 3105-3108 (1984).
    [CrossRef] [PubMed]
  3. M. Takeda and K. Motoh, "Fourier transform profilometry for the automatic measurement of 3-D object shaped," Appl. Opt. 22, 3977-3982 (1983).
    [CrossRef] [PubMed]
  4. X. Su, and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
    [CrossRef]
  5. K. G. Larkin and B. F. Oreb, "Design and assessment of symmetrical phase-shifting algorithms," J. Opt. Soc. Am. A 9, 1740-1748 (1992).
    [CrossRef]
  6. V. Y. Su, G von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
    [CrossRef]
  7. Y. Surrel, "Design of algorithms for phase measurements by the use of phase stepping," Appl. Opt. 35, 51-60 (1996).
    [CrossRef] [PubMed]
  8. F. Chen, G. M. Brown, and M. Song, "Overview of three-dimensional shape measurement using optical methods," Opt. Eng. 39, 10-22 (2000).
    [CrossRef]
  9. W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003).
    [CrossRef]
  10. D. C. Ghiglia and L. A. Romero, "Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods," J. Opt. Soc. Am. A 11, 107 (1994).
    [CrossRef]
  11. K. A. Stetson, J. Wahid, and P. Gauthier, "Noise-immune phase unwrapping by use of calculated wrap regions," Appl. Opt. 36, 4830-4838 (1997).
    [CrossRef] [PubMed]
  12. A. Collaro, G. Franceschetti, F. Palmieri, and M. S. Ferreiro, "Phase unwrapping by means of genetic algorithms," J. Opt. Soc. Am. A 15, 407-418 (1998).
    [CrossRef]
  13. J.-J. Chyou, S.-J. Chen, and Y.-K. Chen, "Two-dimensional phase unwrapping with a multichannel least-mean-square algorithm," Appl. Opt. 43, 5655-5661 (2004).
    [CrossRef] [PubMed]
  14. J. M. Huntley and H. O. Saldner, "Temporal phase-unwrapping algorithm for automated inteferogram analysis," Appl. Opt. 32, 3047-3052 (1993).
    [CrossRef] [PubMed]
  15. H. O. Saldner and J. M. Huntley, "Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector," Opt. Eng. 36, 610-615 (1997).
    [CrossRef]
  16. K. Creath, "Step height measurement using two-wavelength phase-shifting interferometry," Appl. Opt. 26, 2810-2816 (1987).
    [CrossRef] [PubMed]
  17. H. Zhao, W. Chen, and Y. Tan, "Phase-unwrapping algorithm for the measurement of three-dimensional object shapes," Appl. Opt. 33, 4497-4500 (1994).
    [CrossRef] [PubMed]
  18. D. R. Burton and M. J. Lalor, "Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping," Appl. Opt. 33, 2939-2948 (1994)
    [CrossRef] [PubMed]
  19. Y. Hao, Y. Zhao, and D. Li, "Multifrequency grating projection profilometry based on the nonlinear excess fraction method," Appl. Opt. 38, 4106-4110 (1999).
    [CrossRef]
  20. E. B. Li, X. Peng, J. Xi, J. F. Chicharo, J. Q. Yao, and D.W. Zhang, "Multi-frequency and multiple phase-shift sinusoidal fringe projection for 3D profilometry," Opt. Express 13, 1561-1569 (2005).
    [CrossRef] [PubMed]
  21. M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilomery: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," Appl. Opt. 36, 5347-5354 (1997).
    [CrossRef] [PubMed]
  22. J. L. Li, H. J. Su, and X. Y. Su, "Two-frequency grating used in phase-measuring profilometry," Appl. Opt. 36, 277-280 (1997).
    [CrossRef] [PubMed]
  23. H. Liu, W. H. Su, K. R., and S. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-80 (2003).
    [CrossRef]
  24. L. B. Jackson, Digital Filters and Signal Processing (Toppan, 1996), Chap. 6.

2005 (1)

2004 (1)

2003 (2)

W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003).
[CrossRef]

H. Liu, W. H. Su, K. R., and S. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-80 (2003).
[CrossRef]

2001 (1)

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

2000 (1)

F. Chen, G. M. Brown, and M. Song, "Overview of three-dimensional shape measurement using optical methods," Opt. Eng. 39, 10-22 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (4)

1996 (1)

1994 (3)

1993 (2)

J. M. Huntley and H. O. Saldner, "Temporal phase-unwrapping algorithm for automated inteferogram analysis," Appl. Opt. 32, 3047-3052 (1993).
[CrossRef] [PubMed]

V. Y. Su, G von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
[CrossRef]

1992 (1)

1987 (1)

1984 (1)

1983 (1)

1978 (1)

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, "Overview of three-dimensional shape measurement using optical methods," Opt. Eng. 39, 10-22 (2000).
[CrossRef]

Burton, D. R.

Chen, F.

F. Chen, G. M. Brown, and M. Song, "Overview of three-dimensional shape measurement using optical methods," Opt. Eng. 39, 10-22 (2000).
[CrossRef]

Chen, S.-J.

Chen, W.

Chen, Y.-K.

Chicharo, J. F.

Chyou, J.-J.

Collaro, A.

Creath, K.

Ferreiro, M. S.

Franceschetti, G.

Gauthier, P.

Ghiglia, D. C.

Gu, Q.

Halioua, M.

Hao, Y.

Huntley, J. M.

H. O. Saldner and J. M. Huntley, "Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector," Opt. Eng. 36, 610-615 (1997).
[CrossRef]

J. M. Huntley and H. O. Saldner, "Temporal phase-unwrapping algorithm for automated inteferogram analysis," Appl. Opt. 32, 3047-3052 (1993).
[CrossRef] [PubMed]

Indebetouw, G.

Kinoshita, M.

Lalor, M. J.

Larkin, K. G.

Li, D.

Li, E. B.

Li, J. L.

Liu, H.

H. Liu, W. H. Su, K. R., and S. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-80 (2003).
[CrossRef]

W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003).
[CrossRef]

Liu, H. C.

Motoh, K.

Oreb, B. F.

Palmieri, F.

Peng, X.

Reichard, K.

W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003).
[CrossRef]

Romero, L. A.

Saldner, H. O.

H. O. Saldner and J. M. Huntley, "Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector," Opt. Eng. 36, 610-615 (1997).
[CrossRef]

J. M. Huntley and H. O. Saldner, "Temporal phase-unwrapping algorithm for automated inteferogram analysis," Appl. Opt. 32, 3047-3052 (1993).
[CrossRef] [PubMed]

Song, M.

F. Chen, G. M. Brown, and M. Song, "Overview of three-dimensional shape measurement using optical methods," Opt. Eng. 39, 10-22 (2000).
[CrossRef]

Srinivasan, V.

Stetson, K. A.

Su, H. J.

Su, V. Y.

V. Y. Su, G von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
[CrossRef]

Su, W. H.

H. Liu, W. H. Su, K. R., and S. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-80 (2003).
[CrossRef]

W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003).
[CrossRef]

Su, X.

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

Su, X. Y.

Surrel, Y.

Takahashi, Y.

Takai, H.

Takeda, M.

Tan, Y.

Wahid, J.

Xi, J.

Yao, J. Q.

Yin, S.

W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003).
[CrossRef]

Yu, F. T. S.

W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003).
[CrossRef]

Zhang, D.W.

Zhao, H.

Zhao, Y.

Appl. Opt. (13)

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

V. Srinivasan, H. C. Liu, and M. Halioua, "Automated phase-measuring profilometry of 3-D diffuse objects," Appl. Opt. 23, 3105-3108 (1984).
[CrossRef] [PubMed]

M. Takeda and K. Motoh, "Fourier transform profilometry for the automatic measurement of 3-D object shaped," Appl. Opt. 22, 3977-3982 (1983).
[CrossRef] [PubMed]

Y. Surrel, "Design of algorithms for phase measurements by the use of phase stepping," Appl. Opt. 35, 51-60 (1996).
[CrossRef] [PubMed]

K. A. Stetson, J. Wahid, and P. Gauthier, "Noise-immune phase unwrapping by use of calculated wrap regions," Appl. Opt. 36, 4830-4838 (1997).
[CrossRef] [PubMed]

J.-J. Chyou, S.-J. Chen, and Y.-K. Chen, "Two-dimensional phase unwrapping with a multichannel least-mean-square algorithm," Appl. Opt. 43, 5655-5661 (2004).
[CrossRef] [PubMed]

J. M. Huntley and H. O. Saldner, "Temporal phase-unwrapping algorithm for automated inteferogram analysis," Appl. Opt. 32, 3047-3052 (1993).
[CrossRef] [PubMed]

K. Creath, "Step height measurement using two-wavelength phase-shifting interferometry," Appl. Opt. 26, 2810-2816 (1987).
[CrossRef] [PubMed]

H. Zhao, W. Chen, and Y. Tan, "Phase-unwrapping algorithm for the measurement of three-dimensional object shapes," Appl. Opt. 33, 4497-4500 (1994).
[CrossRef] [PubMed]

D. R. Burton and M. J. Lalor, "Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping," Appl. Opt. 33, 2939-2948 (1994)
[CrossRef] [PubMed]

Y. Hao, Y. Zhao, and D. Li, "Multifrequency grating projection profilometry based on the nonlinear excess fraction method," Appl. Opt. 38, 4106-4110 (1999).
[CrossRef]

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilomery: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," Appl. Opt. 36, 5347-5354 (1997).
[CrossRef] [PubMed]

J. L. Li, H. J. Su, and X. Y. Su, "Two-frequency grating used in phase-measuring profilometry," Appl. Opt. 36, 277-280 (1997).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

V. Y. Su, G von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
[CrossRef]

H. Liu, W. H. Su, K. R., and S. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-80 (2003).
[CrossRef]

Opt. Eng. (3)

F. Chen, G. M. Brown, and M. Song, "Overview of three-dimensional shape measurement using optical methods," Opt. Eng. 39, 10-22 (2000).
[CrossRef]

W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003).
[CrossRef]

H. O. Saldner and J. M. Huntley, "Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector," Opt. Eng. 36, 610-615 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

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

Other (1)

L. B. Jackson, Digital Filters and Signal Processing (Toppan, 1996), Chap. 6.

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Figures (10)

Fig. 1.
Fig. 1.

Quantization scheme used for generating a digital sinusoidal grating from its continuous counterpart.

Fig. 2.
Fig. 2.

Patterns resulted from an area-encoding scheme for representing discrete transmittance levels.

Fig. 3.
Fig. 3.

Appearance of a designed digital two-frequency pattern.

Fig. 4.
Fig. 4.

Fourier spectrum of the image of a digital two-frequency pattern.

Fig. 5.
Fig. 5.

Coordinate systems in a non-telecentric projected fringe measurement system.

Fig. 6.
Fig. 6.

Appearance of a two-frequency fringe pattern projected to the testing object. A 12-bit camera with 1024×1024 pixels was used to record the fringes.

Fig. 7.
Fig. 7.

Phase map of (a) the higher frequency, and (b) the lower frequency. Note that the phases were within the interval between -π and π. Phase values close to π, were displayed in the bright regions; while areas shown in dark indicated that the phases were close to -π, as shown in the color bar.

Fig. 8.
Fig. 8.

Unwrapped phase map from Fig. 7. (a) The higher frequency, and (b) the lower frequency. Phases were mapped to the corresponding colors: the brighter one represented the larger phase value.

Fig. 9.
Fig. 9.

Unwrapped phase map of the higher frequency. Errors while phase unwrapping have been corrected.

Fig. 10.
Fig. 10.

3D profile of the testing sample measured from the proposed scheme.

Equations (20)

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

t ( x ) = 1 2 + 1 4 cos ( 2 π d x ) + 1 4 cos ( 2 π Pd x )
t ̅ ( n ) = 1 w nw ( n + 1 ) w [ t ( x ) ] dx
= 1 2 + 1 4 sinc ( π N ) cos ( 2 n + 1 N π ) + 1 4 sinc ( π PN ) cos ( 2 n + 1 PN π ) ,
t n = [ M · t ̅ ( n ) ] M ,
I ( x , y ) = a ( x , y ) + h ( x , y ) cos φ h ( x , y ) + l ( x , y ) cos φ l ( x , y ) ,
{ φ h ( x , y ) = 2 π d x + Δ φ ( x , y ) , φ l ( x , y ) = 2 π pd x + Δ φ ( x , y ) P
I ( x , y ) = a ( x , y ) + h ˜ ( x , y ) exp ( j 2 π d x ) + h ˜ * ( x , y ) exp ( j 2 π d x )
+ l ˜ ( x , y ) exp ( j 2 π d x ) + l ˜ * ( x , y ) exp ( j 2 π d x ) ,
h ˜ ( x , y ) = 1 2 h ( x , y ) exp [ j Δ φ ( x , y ) ] ,
l ˜ ( x , y ) = 1 2 l ( x , y ) exp ( j Δ φ ( x , y ) P ) .
F ( f x , y ) = A ( f x , y ) + H ˜ ( f x 1 d , y ) + H ˜ * ( f x + 1 d , y )
+ L ˜ ( f x 1 Pd , y ) + L ˜ * ( f x + 1 Pd , y ) ,
s h ( x , y ) = h ( x , y ) exp ( j 2 π d x ) exp [ j Δ φ ( x , y ) ] ,
s l ( x , y ) = l ( x , y ) exp ( j 2 π Pd x ) exp [ j Δ φ ( x , y ) P ] .
Φ h ( x , y ) = tan 1 { Im [ s h ( x , y ) ] Re [ s h ( x , y ) ] } ,
Φ l ( x , y ) = tan 1 { Im [ s l ( x , y ) ] Re [ s l ( x , y ) ] } .
Q = [ Unwrap { Φ h } P · Unwrap { Φ l } 2 π ] ,
{ x = a 1 z + a o y = b 1 z + b o ,
z = m 1 φ + m 0 n 1 φ + 1 + 0 i 6 [ g i h i φ n ' 1 φ + 1 ( m 1 φ + m 0 n 1 φ + 1 ) i ] ,
z ( φ ) = i = 0 n c i φ i + i = 0 m c i φ i ,

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