Abstract

We present an algorithm for estimating the color demixing matrix based on the color fringe patterns captured from the reference plane or the surface of the object. The advantage of this algorithm is that it is a blind approach to calculating the demixing matrix in the sense that no extra images are required for color calibration before performing profile measurement. Simulation and experimental results convince us that the proposed algorithm can significantly reduce the influence of the color cross talk and at the same time improve the measurement accuracy of the color-channel-based phase-shifting profilometry.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. 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]
  3. R. J. Green, J. G. Walker, and D. W. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
    [CrossRef]
  4. X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
    [CrossRef]
  5. 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]
  6. 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]
  7. V. Srinivasan, H. Liu, and M. Halioua, "Automated phase-measuring profilometry of 3-D diffuse objects," Appl. Opt. 23, 3105-3108 (1984).
    [CrossRef] [PubMed]
  8. X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of ronchi grating," Opt. Commun. 94, 561-573 (1992).
    [CrossRef]
  9. H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. (Bellingham) 36, 1799-1805 (1997).
    [CrossRef]
  10. S. Toyooka and Y. Iwaasa, "Automatic profilometry of 3-D diffuse objects by spatial phase detection," Appl. Opt. 25, 1630-1633 (1986).
    [CrossRef] [PubMed]
  11. R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Laser Technol. 26, 393-398 (1994).
    [CrossRef]
  12. A. J. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
    [CrossRef]
  13. J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
    [CrossRef]
  14. P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digitial fringe projection technique for high-speed 3-D surface contouring," Opt. Eng. (Bellingham) 38, 1065-1071 (1999).
    [CrossRef]
  15. L. Kinell, "Multichannel method for absolute shape measurement using projected fringes," Opt. Lasers Eng. 41, 57-71 (2004).
    [CrossRef]
  16. Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "A calibration approach for decoupling colour cross-talk using nonlinear blind signal separation network," in Proceedings of IEEE Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD) (IEEE, 2004), pp. 265-268.
    [PubMed]
  17. F. Berryman, P. Pynsent, and J. Cubillo, "A theoretical comparison of three fringe analysis method for determining the three dimensional shape of an object in the presence of noise," Opt. Lasers Eng. 39, 35-50 (2003).
    [CrossRef]
  18. K. Hibino, B. F. Oreb, and D. I. Farrant, "Phase shifting for nonsinusoidal waveforms with phase-shift errors," J. Opt. Soc. Am. A 12, 761-768 (1995).
    [CrossRef]
  19. J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. (Bellingham) 23, 350-352 (1984).
  20. G. Stoilov and T. Dragostinov, "Phase-stepping interferometry: five-frame algorithm with an aribitrary step," Opt. Lasers Eng. 28, 61-69 (1997).
    [CrossRef]
  21. G.-S. Han and S.-W. Kim, "Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting," Appl. Opt. 33, 7321-7325 (1994).
    [CrossRef] [PubMed]
  22. G. Lai and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822-827 (1991).
    [CrossRef]
  23. X. Chen, M. Gramaglia, and J. A. Yeazell, "Phase-shifting interferometry with uncalibrated phase shifts," Appl. Opt. 39, 585-591 (2000).
    [CrossRef]
  24. H. Kadono, Y. Bitoh, and S. Toyooka, "Statistical interferometry based on a fully developed speckle field: an experimental demonstration with noise analysis," J. Opt. Soc. Am. A 18, 1267-1274 (2001).
    [CrossRef]
  25. C. S. Guo and L. Zhang, "Phase-shifting error and its elimination in phase-shifting digital holography," Opt. Lett. 27, 1687-1689 (2002).
    [CrossRef]
  26. L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003).
    [CrossRef]
  27. L. Z. Cai, Q. Liu, and X. L. Yang, "Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects," Opt. Lett. 29, 183-185 (2004).
    [CrossRef]
  28. Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "Three-dimensional profilometry based on shift estimation of projected fringe patterns," Appl. Opt. 45, 678-687 (2006).
    [CrossRef]

2006 (1)

2004 (2)

2003 (2)

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

L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003).
[CrossRef]

2002 (1)

2001 (3)

H. Kadono, Y. Bitoh, and S. Toyooka, "Statistical interferometry based on a fully developed speckle field: an experimental demonstration with noise analysis," J. Opt. Soc. Am. A 18, 1267-1274 (2001).
[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]

2000 (1)

1999 (2)

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digitial fringe projection technique for high-speed 3-D surface contouring," Opt. Eng. (Bellingham) 38, 1065-1071 (1999).
[CrossRef]

1997 (3)

G. Stoilov and T. Dragostinov, "Phase-stepping interferometry: five-frame algorithm with an aribitrary step," Opt. Lasers Eng. 28, 61-69 (1997).
[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]

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. (Bellingham) 36, 1799-1805 (1997).
[CrossRef]

1995 (2)

A. J. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

K. Hibino, B. F. Oreb, and D. I. Farrant, "Phase shifting for nonsinusoidal waveforms with phase-shift errors," J. Opt. Soc. Am. A 12, 761-768 (1995).
[CrossRef]

1994 (2)

1992 (1)

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

1991 (1)

1988 (1)

R. J. Green, J. G. Walker, and D. W. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

1986 (1)

1984 (2)

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

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. (Bellingham) 23, 350-352 (1984).

1983 (1)

1982 (1)

Berryman, F.

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

Bitoh, Y.

Cai, L. Z.

Castillo, L.

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[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. 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]

Chen, X.

Chiang, F.

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digitial fringe projection technique for high-speed 3-D surface contouring," Opt. Eng. (Bellingham) 38, 1065-1071 (1999).
[CrossRef]

Chicharo, J.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "Three-dimensional profilometry based on shift estimation of projected fringe patterns," Appl. Opt. 45, 678-687 (2006).
[CrossRef]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "A calibration approach for decoupling colour cross-talk using nonlinear blind signal separation network," in Proceedings of IEEE Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD) (IEEE, 2004), pp. 265-268.
[PubMed]

Cubillo, J.

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

Dragostinov, T.

G. Stoilov and T. Dragostinov, "Phase-stepping interferometry: five-frame algorithm with an aribitrary step," Opt. Lasers Eng. 28, 61-69 (1997).
[CrossRef]

Farrant, D. I.

Gramaglia, M.

Green, R. J.

R. J. Green, J. G. Walker, and D. W. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. (Bellingham) 23, 350-352 (1984).

Guo, C. S.

Halioua, M.

Han, G.-S.

Hibino, K.

Ho, Q.

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digitial fringe projection technique for high-speed 3-D surface contouring," Opt. Eng. (Bellingham) 38, 1065-1071 (1999).
[CrossRef]

Hu, Y.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "Three-dimensional profilometry based on shift estimation of projected fringe patterns," Appl. Opt. 45, 678-687 (2006).
[CrossRef]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "A calibration approach for decoupling colour cross-talk using nonlinear blind signal separation network," in Proceedings of IEEE Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD) (IEEE, 2004), pp. 265-268.
[PubMed]

Huang, P.

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digitial fringe projection technique for high-speed 3-D surface contouring," Opt. Eng. (Bellingham) 38, 1065-1071 (1999).
[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.

Iwaasa, Y.

Jin, F.

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digitial fringe projection technique for high-speed 3-D surface contouring," Opt. Eng. (Bellingham) 38, 1065-1071 (1999).
[CrossRef]

Kadono, H.

Kim, S.-W.

Kinell, L.

L. Kinell, "Multichannel method for absolute shape measurement using projected fringes," Opt. Lasers Eng. 41, 57-71 (2004).
[CrossRef]

Kobayashi, S.

Lai, G.

Li, E.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "Three-dimensional profilometry based on shift estimation of projected fringe patterns," Appl. Opt. 45, 678-687 (2006).
[CrossRef]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "A calibration approach for decoupling colour cross-talk using nonlinear blind signal separation network," in Proceedings of IEEE Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD) (IEEE, 2004), pp. 265-268.
[PubMed]

Li, J.

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. (Bellingham) 36, 1799-1805 (1997).
[CrossRef]

Liu, H.

Liu, Q.

Mendoza-Santoyo, F.

A. J. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

Moore, A. J.

A. J. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

Mutoh, K.

Oreb, B. F.

Pynsent, P.

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

Robinson, D. W.

R. J. Green, J. G. Walker, and D. W. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

Rodriguez-Vera, R.

R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Laser Technol. 26, 393-398 (1994).
[CrossRef]

Servin, M.

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Laser Technol. 26, 393-398 (1994).
[CrossRef]

Srinivasan, V.

Stoilov, G.

G. Stoilov and T. Dragostinov, "Phase-stepping interferometry: five-frame algorithm with an aribitrary step," Opt. Lasers Eng. 28, 61-69 (1997).
[CrossRef]

Su, H.

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. (Bellingham) 36, 1799-1805 (1997).
[CrossRef]

Su, X.

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]

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. (Bellingham) 36, 1799-1805 (1997).
[CrossRef]

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Takeda, M.

Toyooka, S.

Villa, J.

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

von Bally, G.

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Vukicevic, D.

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Walker, J. G.

R. J. Green, J. G. Walker, and D. W. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

Xi, J.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "Three-dimensional profilometry based on shift estimation of projected fringe patterns," Appl. Opt. 45, 678-687 (2006).
[CrossRef]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "A calibration approach for decoupling colour cross-talk using nonlinear blind signal separation network," in Proceedings of IEEE Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD) (IEEE, 2004), pp. 265-268.
[PubMed]

Yang, X. L.

Yang, Z.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "Three-dimensional profilometry based on shift estimation of projected fringe patterns," Appl. Opt. 45, 678-687 (2006).
[CrossRef]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "A calibration approach for decoupling colour cross-talk using nonlinear blind signal separation network," in Proceedings of IEEE Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD) (IEEE, 2004), pp. 265-268.
[PubMed]

Yatagai, T.

Yeazell, J. A.

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]

Zhang, L.

Zhang, Q.

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]

Zhou, W.

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Opt. Eng. (Bellingham) (3)

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. (Bellingham) 36, 1799-1805 (1997).
[CrossRef]

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digitial fringe projection technique for high-speed 3-D surface contouring," Opt. Eng. (Bellingham) 38, 1065-1071 (1999).
[CrossRef]

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. (Bellingham) 23, 350-352 (1984).

Opt. Laser Technol. (1)

R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Laser Technol. 26, 393-398 (1994).
[CrossRef]

Opt. Lasers Eng. (8)

A. J. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

R. J. Green, J. G. Walker, and D. W. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

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

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]

G. Stoilov and T. Dragostinov, "Phase-stepping interferometry: five-frame algorithm with an aribitrary step," Opt. Lasers Eng. 28, 61-69 (1997).
[CrossRef]

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

L. Kinell, "Multichannel method for absolute shape measurement using projected fringes," Opt. Lasers Eng. 41, 57-71 (2004).
[CrossRef]

Opt. Lett. (3)

Other (1)

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, "A calibration approach for decoupling colour cross-talk using nonlinear blind signal separation network," in Proceedings of IEEE Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD) (IEEE, 2004), pp. 265-268.
[PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the FPP system.

Fig. 2
Fig. 2

Nonlinear mixing and demixing.

Fig. 3
Fig. 3

Linearly mixed fringe patterns and reconstruction result.

Fig. 4
Fig. 4

Demixed fringe pattern and reconstruction result in the linear cross-talk situation.

Fig. 5
Fig. 5

Simulated fringe patterns and object.

Fig. 6
Fig. 6

Cross section of captured fringe patterns with nonlinear color cross talk and reconstruction result.

Fig. 7
Fig. 7

Object and fringe patterns in the experiment.

Fig. 8
Fig. 8

Surface reconstructed by color channel PSP.

Fig. 9
Fig. 9

Surface reconstructed by BCI.

Fig. 10
Fig. 10

Comparison of the reconstructed surfaces in the experiment in cross-section view.

Tables (1)

Tables Icon

Table 1 Absolute Error and Standard Deviation

Equations (86)

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

s m ( x ) = b 0 + b 1 cos ( 2 π f 0 x + 2 π ( m 1 ) M ) for m = 1 , 2 , , M ,
d m ( x ) = b 0 + b 1 cos ( 2 π f 0 x + 2 π ( m 1 ) M + ϕ ( x ) ) for m = 1 , 2 , , M ,
S = tan ( 2 π f 0 x ) = m = 1 M s m ( x ) sin ( 2 π ( m 1 ) M ) m = 1 M s m ( x ) cos ( 2 π ( m 1 ) M ) ,
D = tan ( 2 π f 0 x + ϕ ( x ) ) = m = 1 M d m ( x ) sin ( 2 π ( m 1 ) M ) m = 1 M d m ( x ) cos ( 2 π ( m 1 ) M ) ,
ϕ ( x ) = u n w r a p ( arctan ( D ) ) u n w r a p ( arctan ( S ) ) ,
h ( x ) = l 0 ϕ ( x ) ϕ ( x ) 2 π f 0 d 0 ,
s = ( s m , n ) M × N ,
b m , 0 = b 0 ( where m = 1 , 2 , , M and b 0 is a constant ) ,
b m , 1 = b 1 ( where m = 1 , 2 , , M and b 1 is a constant ) ,
α m + 1 α m = 2 π M ( where m = 1 , 2 , , M 1 ) ,
s ̂ = As ,
s ̂ m = A m s = a m , 1 ( b 0 + b 1 cos θ ( x ) ) + a m , 2 ( b 0 + b 1 cos ( θ ( x ) + 2 π 3 ) ) + a m , 3 ( b 0 + b 1 cos ( θ ( x ) + 4 π 3 ) ) = b 0 ( a m , 1 + a m , 2 + a m , 3 ) + b 1 [ a m , 1 cos θ ( x ) + a m , 2 ( cos θ ( x ) cos 2 π 3 sin θ ( x ) sin 2 π 3 ) ) [ + a m , 3 ( cos θ ( x ) cos 4 π 3 sin θ ( x ) sin 4 π 3 ) ] = b 0 ( a m , 1 + a m , 2 + a m , 3 ) + b 1 [ ( a m , 1 1 2 a m , 2 1 2 a m , 3 ) cos θ ( x ) 3 2 ( a m , 2 a m , 3 ) sin θ ( x ) ] = b ̂ m , 0 + b ̂ m , 1 cos ( θ ( x ) + α ̂ m ) ,
b ̂ m , 0 = b 0 ( a m , 1 + a m , 2 + a m , 3 ) ,
b ̂ m , 1 = b 1 ( a m , 1 1 2 a m , 2 1 2 a m , 3 ) 2 + 3 4 ( a m , 2 a m , 3 ) 2 = b 1 1 2 [ ( a m , 1 a m , 2 ) 2 + ( a m , 2 a m , 3 ) 2 + ( a m , 1 a m , 3 ) 2 ] ,
tan ( α ̂ m ) = 3 ( a m , 2 a m , 3 ) 2 a m , 1 a m , 2 a m , 3 .
s ̂ m = b ̂ m , 1 cos ( θ ( x ) + α ̂ m ) for m = 1 , 2 , 3 .
s ̃ = W s ̂ = W ( As ) = WAs = Ls ,
s ̃ m = b ̃ m , 1 cos ( θ ( x ) + α ̃ m ) , for m = 1 , 2 , 3 .
b ̃ 1 , 1 = b ̃ 2 , 1 = b ̃ 3 , 1 = b ̃ 1 ,
α ̃ 3 α ̃ 2 = α ̃ 2 α ̃ 1 = 2 π 3 .
s ̂ m ( x ) = k = 0 K b ̂ m , k cos [ k ( θ ( x ) + α ̂ m ) ] , for m = 1 , 2 , 3 ,
s ¯ m ( x ) = b ̂ m , 1 cos ( θ ( x ) + α ̂ m ) , for m = 1 , 2 , 3 ,
q s ̃ ( x ) = 0 for any given x ,
s ̃ ( x ) s ̃ ( x ) = c for any given x ,
J = J 1 + J 2 ,
J 1 = [ q s ̃ ( x ) ] 2 ,
J 2 = [ s ̃ ( x ) s ̃ ( x ) c ] 2 .
W ̂ p + 1 = W ̂ p η W J ( W ̂ p ) ,
W J = W J 1 + W J 2 .
W J 1 = W [ q s ̃ ( x ) ] 2 = W [ q T s ̃ ( x ) ] 2 = 2 [ q T s ̃ ( x ) ] W [ q T s ̃ ( x ) ] = 2 [ q T W s ¯ ( x ) ] W [ q T W s ¯ ( x ) ] = 2 [ q T W s ¯ ( x ) ] [ q s ¯ T ( x ) ] ,
W J 2 = W [ s ̃ ( x ) s ̃ ( x ) c ] 2 = 2 [ s ̃ ( x ) s ̃ ( x ) c ] W [ s ̃ ( x ) s ̃ ( x ) c ] = 2 [ s ̃ T ( x ) s ̃ ( x ) c ] W [ s ̃ T ( x ) s ̃ ( x ) c ] = 2 [ s ̃ T ( x ) s ̃ ( x ) c ] W [ ( W s ¯ ( x ) ) T ( W s ¯ ( x ) ) c ] = 2 [ s ̃ T ( x ) s ̃ ( x ) c ] W [ s ¯ T ( x ) W T W s ¯ ( x ) c ] = 2 [ s ̃ T ( x ) s ̃ ( x ) c ] [ 2 W s ¯ ( x ) s ¯ T ( x ) ] = 4 [ s ¯ T ( x ) W T W s ¯ ( x ) c ] [ W s ¯ ( x ) s ¯ T ( x ) ] .
W J = W J 1 + W J 2 = 2 [ q T W s ¯ ( x ) ] [ q s ¯ T ( x ) ] + 4 [ s ¯ T ( x ) W T W s ¯ ( x ) c ] [ W s ¯ ( x ) s ¯ T ( x ) ] .
W ̂ p + 1 = W ̂ p η { 2 [ q T W p s ¯ ( x ) ] [ q s ¯ T ( x ) ] + 4 [ s ¯ T ( x ) W p T W p s ¯ ( x ) c ] [ W p s ¯ ( x ) s ¯ T ( x ) ] } .
W 1 = [ 1 0 0 0 1 0 0 0 1 ] .
s m ( x ) = 128 + 100 cos ( 2 π f 0 x + 2 π ( m 1 ) 3 ) for m = 1 , 2 , 3 ,
A = [ 0.5 0.1 0.2 0.2 0.6 0.15 0.3 0.2 0.7 ] .
f ( s ) = 128 tanh ( 3 s 128 3 ) tanh ( 3 ) + 128 ,
g ( s ) = s 2 256 ,
s ( x ) = k = 0 P [ p k cos ( k θ ( x ) ) ] ,
g ( s ( x ) ) = k = 0 R [ r k cos ( k θ ( x ) ) ] .
g ( s ) = n = 0 N g n s n = g 0 + g 1 s + g 2 s 2 + + g n s n + + g N s N ,
s m ( x ) = j = 0 Q [ q j cos ( j θ ( x ) ) ] .
s m + 1 ( x ) = s m ( x ) s ( x ) = j = 0 Q [ q j cos ( j θ ( x ) ) ] k = 0 K [ p k cos ( k θ ( x ) ) ] = j = 0 Q k = 0 K q j p k cos ( j θ ( x ) ) cos ( k θ ( x ) ) = 1 2 j = 0 Q k = 0 K q j p k { cos [ ( j + k ) θ ( x ) ] + cos [ ( j k ) θ ( x ) ] } = 1 2 j = 1 Q k = 0 K q j p k cos [ ( j + k ) θ ( x ) ] + 1 2 j = 0 Q k = 0 K q j p k cos [ ( j k ) θ ( x ) ] .
s m + 1 ( x ) = k = 0 R [ i k cos ( k θ ( x ) ) ] .
g n ( g 2 [ g 1 ( s ) ] ) = k = 0 L [ l k cos ( k θ ( x ) ) ] .
s ̃ m = b ̃ m , 1 cos ( θ ( x ) + α ̃ m ) , for m = 1 , 2 , 3 ,
0 α ̃ 1 < α ̃ 2 < α ̃ 3 < 2 π .
b ̃ m , 1 = b ̃ ,
α ̃ 3 α ̃ 2 = α ̃ 2 α ̃ 1 = 2 π 3 ,
q s ̃ ( x ) = 0 for any given x ,
s ̃ ( x ) s ̃ ( x ) = 3 2 b ̃ 2 = c for any given x .
d [ q s ̃ ( x ) ] d x = 0 for any given x ,
d [ s ̃ ( x ) s ̃ ( x ) ] d x = 0 for any given x ;
d θ ( x ) d x m = 1 3 b ̃ m , 1 sin [ θ ( x ) + α ̃ m ] = 0 ,
m = 1 3 d ( b ̃ m , 1 2 sin 2 [ θ ( x ) + α ̃ m ] ) d x = m = 1 3 d ( b ̃ m , 1 2 sin 2 [ θ ( x ) + α ̃ m ] ) d θ d θ ( x ) d x = d θ ( x ) d x m = 1 3 d ( b ̃ m , 1 2 sin 2 [ θ ( x ) + α ̃ m ] ) d θ = d θ ( x ) d x m = 1 3 2 b ̃ m , 1 2 cos [ θ ( x ) + α ̃ m ] sin [ θ ( x ) + α ̃ m ] = d θ ( x ) d x m = 1 3 b ̃ m , 1 2 sin [ 2 θ ( x ) + 2 α ̃ m ] = 0 .
m = 1 3 b ̃ m , 1 sin [ θ ( x ) + α ̃ m ] = 0 ,
m = 1 3 b ̃ m , 1 2 sin [ 2 θ ( x ) + 2 α ̃ m ] = 0 .
m = 1 3 b ̃ m sin ( θ + β m ) = 0 ,
m = 1 3 b ̃ m 2 sin ( 2 θ + 2 β m ) = 0 .
α ̃ 1 α ̃ 1 < α ̃ 2 α ̃ 1 < α ̃ 3 α ̃ 1 < 2 π ,
0 < β 2 < β 3 < 2 π .
b ̃ 2 sin ( β 2 ) + b ̃ 3 sin ( β 3 ) = 0 ,
b ̃ 2 2 sin ( 2 β 2 ) + b ̃ 3 2 sin ( 2 β 3 ) = 0 ,
b ̃ 2 sin ( β 2 ) = b ̃ 3 sin ( β 3 ) ,
b ̃ 2 2 sin ( 2 β 2 ) = b ̃ 3 2 sin ( 2 β 3 ) .
b ̃ 1 2 sin ( 2 θ ) + b ̃ 2 2 sin ( 2 θ + 2 k π ) + b ̃ 3 2 sin ( 2 θ + 2 n π ) = 0 ,
( b ̃ 1 2 + b ̃ 2 2 + b ̃ 3 2 ) sin ( 2 θ ) = 0 .
b ̃ 2 2 sin ( 2 β 2 ) b ̃ 2 sin ( β 2 ) = b ̃ 3 2 sin ( 2 β 3 ) b ̃ 3 sin ( β 3 ) ,
b ̃ 2 cos ( β 2 ) = b ̃ 3 cos ( β 3 ) .
[ b ̃ 2 sin ( β 2 ) ] 2 + [ b ̃ 2 cos ( β 2 ) ] 2 = [ b ̃ 3 sin ( β 3 ) ] 2 + [ b ̃ 3 cos ( β 3 ) ] 2
b ̃ 2 2 = b ̃ 3 2 .
b ̃ 2 = b ̃ 3 .
b ̃ m = 1 3 sin ( θ + β m ) = 0 ,
b ̃ 2 m = 1 3 sin ( 2 θ + 2 β m ) = 0 ,
m = 1 3 sin ( θ + β m ) = 0 ,
m = 1 3 sin ( 2 θ + 2 β m ) = 0 .
sin ( π 2 ) + sin ( π 2 + β 2 ) + sin ( π 2 + β 3 ) = 0 ,
1 + cos ( β 2 ) + cos ( β 3 ) = 0 .
cos ( β 2 ) = cos ( β 3 ) .
1 + 2 cos ( β 2 ) = 0 and 1 + 2 cos ( β 3 ) = 0 ,
cos ( β 2 ) = 1 2 and cos ( β 3 ) = 1 2 .
β 2 = 2 π 3 or 4 π 3 and β 3 = 2 π 3 or 4 π 3 .
α 2 α 1 = β 2 = 2 π 3 ,
α 3 α 2 = ( α 3 α 1 ) ( α 2 α 1 ) = β 3 β 2 = 2 π 3 .
b ̃ 1 = b ̃ 2 = b ̃ 3 = b ̃ ,
α 3 α 2 = α 2 α 1 = 2 π 3 ,

Metrics