Abstract

We present a new method for improving the measurement of three-dimensional (3-D) shapes by using color information of the measured scene as an additional parameter. The widest used algorithms for 3-D surface measurement by use of structured fringe patterns are phase stepping and Fourier fringe analysis. There are a number of problems and limitations inherent in these algorithms that include: that the phase maps produced are wrapped modulo 2π, that in some cases the acquired fringe pattern does not fill the field of view, that there may be spatially isolated areas, and that there is often invalid and/or noisy data. The new method presented to our knowledge for the first time here uses multiple colored fringe patterns, which are projected at different angles onto the measured scene. These patterns are analyzed with a specially adapted multicolor version of the standard Fourier fringe analysis method. In this way a number of the standard difficulties outlined above are addressed.

© 2002 Optical Society of America

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References

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  1. W. Liu, Z. Wang, G. Mu, Z. Fang, “Color-coded projection grating method for shape measurement with a single exposure,” Appl. Opt. 39, 3504–3508 (2000).
    [CrossRef]
  2. G. Hausler, D. Ritter, “Parallel three-dimensional sensing by color-coded triangulation,” Appl. Opt. 32, 7164–7169 (1993).
    [CrossRef] [PubMed]
  3. P. Carre, “Installation et utilisation du compateur photoelectrique et inteferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  4. J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  5. J. C. Wyant, “Phase Measurement System for Adaptive Optics,” AGARD Conf. Proc. 300, 65–71 (1981).
  6. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  7. Y. Surrel, “Design of algorithm for phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  8. H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
    [CrossRef]
  9. K. Creath, J. Schmit, “Errors in spatial phase-stepping techniques,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kusawinska, K. Patorski, eds., Proc. SPIE2340, 170–176 (1994).
    [CrossRef]
  10. P. S. Huang, Q. Hu, F. Jin, F.-P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38, 1065–1071 (1999).
    [CrossRef]
  11. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  12. D. J. Bone, H. A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [CrossRef] [PubMed]
  13. D. R. Burton, M. J. Lalor, “Managing some of the problems of Fourier fringe analysis,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 149–160 (1989).
    [CrossRef]
  14. W. Macy, “Two-dimensional fringe pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef]
  15. J. D. Folley, A. van Dam, S. T. Feiner, J. F. Hughes, “Computer graphics: principles and practice,” Second Edition in C (Addison-Wesley, Reading, Mass., 1996).
  16. R. W. Schafer, A. V. Oppenheim, Digital Signal Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  17. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 25, 1653–1660 (1986).
  18. M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
    [CrossRef]
  19. D. Bone, “Fourier fringe analysis: the dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  20. H. Zhao, W. Chen, Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33, 4497–4500 (1994).
    [CrossRef] [PubMed]
  21. J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  22. M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust unwrapper for two-dimensional images,” in Vision Systems: Sensors, Sensor Systems, and Components, O. Loffeld, ed., Proc. SPIE2784, 106–110 (1996).
    [CrossRef]
  23. J.-L. Li, X.-Y. Su, J.-T. Li, “Phase unwrapping algorithm based on reliability and edge detection,” Opt. Eng. 36, 1685–1690 (1997).
    [CrossRef]
  24. X.-Y. Su, L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier transform profilometry,” Opt. Eng. 40, 637–643 (2001).
    [CrossRef]
  25. X. Su, G. Von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141 (1993).
    [CrossRef]
  26. D. R. Burton, M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. 33, 2939–2948 (1994).
    [CrossRef] [PubMed]

2001 (1)

X.-Y. Su, L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

2000 (1)

1999 (2)

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

P. S. Huang, Q. Hu, F. Jin, F.-P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38, 1065–1071 (1999).
[CrossRef]

1997 (1)

J.-L. Li, X.-Y. Su, J.-T. Li, “Phase unwrapping algorithm based on reliability and edge detection,” Opt. Eng. 36, 1685–1690 (1997).
[CrossRef]

1996 (2)

1994 (2)

1993 (3)

1991 (1)

1987 (1)

1986 (2)

1983 (1)

1982 (1)

1981 (1)

J. C. Wyant, “Phase Measurement System for Adaptive Optics,” AGARD Conf. Proc. 300, 65–71 (1981).

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du compateur photoelectrique et inteferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Arevalillo Herráez, M. A.

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[CrossRef]

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust unwrapper for two-dimensional images,” in Vision Systems: Sensors, Sensor Systems, and Components, O. Loffeld, ed., Proc. SPIE2784, 106–110 (1996).
[CrossRef]

Bachor, H. A.

Bone, D.

Bone, D. J.

Brangaccio, D. J.

Bruning, J. H.

Burton, D. R.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[CrossRef]

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

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust unwrapper for two-dimensional images,” in Vision Systems: Sensors, Sensor Systems, and Components, O. Loffeld, ed., Proc. SPIE2784, 106–110 (1996).
[CrossRef]

D. R. Burton, M. J. Lalor, “Managing some of the problems of Fourier fringe analysis,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 149–160 (1989).
[CrossRef]

Carre, P.

P. Carre, “Installation et utilisation du compateur photoelectrique et inteferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Chen, W.

Chiang, F.-P.

P. S. Huang, Q. Hu, F. Jin, F.-P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38, 1065–1071 (1999).
[CrossRef]

Clegg, D. B.

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[CrossRef]

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust unwrapper for two-dimensional images,” in Vision Systems: Sensors, Sensor Systems, and Components, O. Loffeld, ed., Proc. SPIE2784, 106–110 (1996).
[CrossRef]

Creath, K.

K. Creath, J. Schmit, “Errors in spatial phase-stepping techniques,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kusawinska, K. Patorski, eds., Proc. SPIE2340, 170–176 (1994).
[CrossRef]

Eiju, T.

Fang, Z.

Feiner, S. T.

J. D. Folley, A. van Dam, S. T. Feiner, J. F. Hughes, “Computer graphics: principles and practice,” Second Edition in C (Addison-Wesley, Reading, Mass., 1996).

Folley, J. D.

J. D. Folley, A. van Dam, S. T. Feiner, J. F. Hughes, “Computer graphics: principles and practice,” Second Edition in C (Addison-Wesley, Reading, Mass., 1996).

Gallagher, J. E.

Hariharan, P.

Hausler, G.

Herriot, D. R.

Hu, Q.

P. S. Huang, Q. Hu, F. Jin, F.-P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38, 1065–1071 (1999).
[CrossRef]

Huang, P. S.

P. S. Huang, Q. Hu, F. Jin, F.-P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38, 1065–1071 (1999).
[CrossRef]

Hughes, J. F.

J. D. Folley, A. van Dam, S. T. Feiner, J. F. Hughes, “Computer graphics: principles and practice,” Second Edition in C (Addison-Wesley, Reading, Mass., 1996).

Huntley, J. M.

Ina, H.

Jin, F.

P. S. Huang, Q. Hu, F. Jin, F.-P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38, 1065–1071 (1999).
[CrossRef]

Kobayashi, S.

Lalor, M. J.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[CrossRef]

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

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust unwrapper for two-dimensional images,” in Vision Systems: Sensors, Sensor Systems, and Components, O. Loffeld, ed., Proc. SPIE2784, 106–110 (1996).
[CrossRef]

D. R. Burton, M. J. Lalor, “Managing some of the problems of Fourier fringe analysis,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 149–160 (1989).
[CrossRef]

Li, J.-L.

J.-L. Li, X.-Y. Su, J.-T. Li, “Phase unwrapping algorithm based on reliability and edge detection,” Opt. Eng. 36, 1685–1690 (1997).
[CrossRef]

Li, J.-T.

J.-L. Li, X.-Y. Su, J.-T. Li, “Phase unwrapping algorithm based on reliability and edge detection,” Opt. Eng. 36, 1685–1690 (1997).
[CrossRef]

Liu, W.

Macy, W.

Mu, G.

Oppenheim, A. V.

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Oreb, B. F.

Ritter, D.

Rosenfeld, D. P.

Saldner, H.

Sandeman, R. J.

Schafer, R. W.

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Schmit, J.

K. Creath, J. Schmit, “Errors in spatial phase-stepping techniques,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kusawinska, K. Patorski, eds., Proc. SPIE2340, 170–176 (1994).
[CrossRef]

Su, X.

X. Su, G. Von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141 (1993).
[CrossRef]

Su, X.-Y.

X.-Y. Su, L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

J.-L. Li, X.-Y. Su, J.-T. Li, “Phase unwrapping algorithm based on reliability and edge detection,” Opt. Eng. 36, 1685–1690 (1997).
[CrossRef]

Surrel, Y.

Takeda, M.

Tan, Y.

van Dam, A.

J. D. Folley, A. van Dam, S. T. Feiner, J. F. Hughes, “Computer graphics: principles and practice,” Second Edition in C (Addison-Wesley, Reading, Mass., 1996).

Von Bally, G.

X. Su, G. Von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141 (1993).
[CrossRef]

Vukicevic, D.

X. Su, G. Von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141 (1993).
[CrossRef]

Wang, Z.

White, A. D.

Wyant, J. C.

J. C. Wyant, “Phase Measurement System for Adaptive Optics,” AGARD Conf. Proc. 300, 65–71 (1981).

Xue, L.

X.-Y. Su, L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

Zhang, H.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Zhao, H.

AGARD Conf. Proc. (1)

J. C. Wyant, “Phase Measurement System for Adaptive Optics,” AGARD Conf. Proc. 300, 65–71 (1981).

Appl. Opt. (13)

J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

W. Macy, “Two-dimensional fringe pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
[CrossRef]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 25, 1653–1660 (1986).

D. J. Bone, H. A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
[CrossRef] [PubMed]

D. Bone, “Fourier fringe analysis: the dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

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

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

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[CrossRef]

Y. Surrel, “Design of algorithm for phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

G. Hausler, D. Ritter, “Parallel three-dimensional sensing by color-coded triangulation,” Appl. Opt. 32, 7164–7169 (1993).
[CrossRef] [PubMed]

W. Liu, Z. Wang, G. Mu, Z. Fang, “Color-coded projection grating method for shape measurement with a single exposure,” Appl. Opt. 39, 3504–3508 (2000).
[CrossRef]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

Metrologia (1)

P. Carre, “Installation et utilisation du compateur photoelectrique et inteferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Commun. (1)

X. Su, G. Von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141 (1993).
[CrossRef]

Opt. Eng. (4)

P. S. Huang, Q. Hu, F. Jin, F.-P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38, 1065–1071 (1999).
[CrossRef]

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

J.-L. Li, X.-Y. Su, J.-T. Li, “Phase unwrapping algorithm based on reliability and edge detection,” Opt. Eng. 36, 1685–1690 (1997).
[CrossRef]

X.-Y. Su, L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

Other (5)

K. Creath, J. Schmit, “Errors in spatial phase-stepping techniques,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kusawinska, K. Patorski, eds., Proc. SPIE2340, 170–176 (1994).
[CrossRef]

J. D. Folley, A. van Dam, S. T. Feiner, J. F. Hughes, “Computer graphics: principles and practice,” Second Edition in C (Addison-Wesley, Reading, Mass., 1996).

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1975).

D. R. Burton, M. J. Lalor, “Managing some of the problems of Fourier fringe analysis,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. SPIE1163, 149–160 (1989).
[CrossRef]

M. A. Arevalillo Herráez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust unwrapper for two-dimensional images,” in Vision Systems: Sensors, Sensor Systems, and Components, O. Loffeld, ed., Proc. SPIE2784, 106–110 (1996).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Flow diagram describing stages of colored structured light method based on FFA.

Fig. 2
Fig. 2

(a) Object illuminated by two colored fringe patterns, (d) and (e) show the wrapped phase information from the red channel (b) and the blue channel (c).

Fig. 3
Fig. 3

Unwrapping source images with shadows by using a standard Schafer unwrapper, where (a) is the red channel and (b) is the blue channel.

Fig. 4
Fig. 4

Unwrapping source images with shadow by using an improved unwrapper, where (a) is the red channel and (b) is the blue channel.

Fig. 5
Fig. 5

The unwrapped red (a) and blue (b) channels after tilt removal.

Fig. 6
Fig. 6

Flow diagram describing stages of a masking calculation algorithm based on sinusoidal shift.

Fig. 7
Fig. 7

View of original fringe pattern (a), 180° shifted fringe pattern (b), result of combined original and shifted fringe patterns (c), and calculated mask with phase information (d).

Fig. 8
Fig. 8

Reconstructed surface height from red and blue channels.

Fig. 9
Fig. 9

Flow diagram of a surface reconstruction algorithm.

Fig. 10
Fig. 10

Typical example of the crossed areas from the red and blue channels.

Fig. 11
Fig. 11

Different results of the surface reconstruction: (a) is the ideally reconstructed surface, (b) is the reconstructed surface profile with constant error, (c) is the reconstructed surface profile with nonconstant error.

Fig. 12
Fig. 12

Different optical subsystem configurations using two video projectors: (a) shows configuration with α12 = 90° and (b) shows an 180° angle between the two projectors.

Fig. 13
Fig. 13

Fringes-contoured female mannequin thorax (a), separated red (b) and blue (c) fringe patterns projected onto the female mannequin thorax.

Fig. 14
Fig. 14

Grayscale maps and binary masks of the measured surface of the female mannequin thorax from red (a) and blue (b) color channels.

Fig. 15
Fig. 15

3-D view of the reconstructed surface of the female mannequin thorax from two color channels.

Fig. 16
Fig. 16

Fringes-contoured object with spatial isolated areas in color channels (a), separated red (b) and blue (c) fringe patterns projected onto the scene.

Fig. 17
Fig. 17

Masked areas in red (a) and blue (b) channels.

Fig. 18
Fig. 18

Rendered 3-D view of the measured surface: (a) is the measured surface from the red channel and (b) is the measured surface from the blue channel.

Fig. 19
Fig. 19

3-D view of the reconstructed surface with spatial isolated areas.

Equations (15)

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

R=arx, y+brx, ycos2πfrxx+fryy, G=agx, y+bgx, ycos2πfgxx+fgyy, B=abx, y+bbx, ycos2πfbxx+fbyy,
R=arx, y+n=1 br,nx, ycos2πnfrxx+fryy+nϕrx, y, G=agx, y+n=1 bg,nx, ycos2πnfgxx+fgyy+nϕgx, y, B=abx, y+n=1 bb,nx, ycos2πnfbxx+fbyy+nϕbx, y,
imx, y=amx, y+n=112 bm,nx, y×(expi[2πnfmxx+nϕmx, y+exp-i[2πnfmxx+nϕmx, y)
imx, y=amx, y+n=1qm,nx, y×expi2πnfmxx+qm,n*x, yexp-i2πnfmxx,
qm,nx, y=12 bm,nx, yexpinϕmx, y, n=1, 2,
qm,n*x, y=12 bm,nx, yexp-inϕmx, y, n=1, 2,
Imf=Amf+n=1Qm,nfx-nfmx+Qm,n*fx+nfmx,
ϕmx, y=tan-1Imqm,lx, yexpi2πfmxReqm,lx, yexpi2πfmx.
hrx, y=ϕrx, ykr,
hgx, y=ϕgx, ykg,
hbx, y=ϕbx, ykb,
Mmx, y=0, where the phase is invalid1, where the phase is valid.
νx, y=i0x, y2+i180x, y21/2.
Hx, y=Shrx, yhgx, yhbx, y,
Kt=1mi=1m Ri-1mj=1m Bj,

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