Abstract

Digital fringe projection profilometry utilizes a digital video projector as a structured light source and thus gains great flexibility. However, the gamma nonlinearity of the video projector inevitably decreases the accuracy and resolution of the measurement. We propose a gamma-correction technique based on statistical analysis of the fringe images. The technique allows one to estimate the value of gamma from the normalized cumulative histogram of the fringe images. By iterating the two steps, gamma estimation and phase evaluation, the actual gamma value can be calculated. At the same time the phase distribution of the fringe pattern can be solved with higher accuracy. In so doing, neither photometric calibration nor knowledge of the device is required. Both computer simulation and experiment are carried out to demonstrate the validity of this technique.

© 2004 Optical Society of America

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  1. V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry of a 3-D diffuse object,” Appl. Opt. 23, 3105–3108 (1984).
    [CrossRef]
  2. V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. 24, 185–188 (1985).
    [CrossRef] [PubMed]
  3. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  4. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  5. S. Tang, Y. Y. Hung, “Fast profilometer for the automatic measurement of 3-D object shapes,” Appl. Opt. 29, 3012–3018 (1990).
    [CrossRef] [PubMed]
  6. M. Pirga, M. Kujawińska, “Modified procedure for automatic surface topography,” Measurement 13, 191–197 (1994).
    [CrossRef]
  7. J.-F. Lin, X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
    [CrossRef]
  8. M. S. Mermelstein, D. L. Feldhun, L. G. Shirley, “Video-rate surface profiling with acousto-optic accordion fringe interferometry,” Opt. Eng. 39, 106–113 (2000).
    [CrossRef]
  9. G. S. Spagnolo, D. Ambrosini, “Diffractive optical element-based profilometer for surface inspection,” Opt. Eng. 40, 44–52 (2001).
    [CrossRef]
  10. F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
    [CrossRef]
  11. P. S. Huang, F. Jin, F.-P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
    [CrossRef]
  12. P. S. Huang, Q. Hu, F.-P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38, 1065–1071 (1999).
    [CrossRef]
  13. C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
    [CrossRef]
  14. G. Sansoni, M. Carocci, R. Rodella, “Three-dimensional vision based on a combination of gray-code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt. 38, 6565–6573 (1999).
    [CrossRef]
  15. S. Kakunai, T. Sakamoto, K. Iwata, “Profile measurement taken with liquid-crystal grating,” Appl. Opt. 38, 2824–2828 (1999).
    [CrossRef]
  16. Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
    [CrossRef]
  17. C. A. Poynton, “Gamma and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video,” SMPTE J. 102, 1099–1108 (1993).
    [CrossRef]
  18. H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10, 1428–1433 (2001).
    [CrossRef]
  19. K. A. Stetson, W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [CrossRef]
  20. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  21. W.-S. Li, X.-Y. Su, “Phase unwrapping algorithm based on phase fitting reliability in structured light projection,” Opt. Eng. 41, 1365–1372 (2002).
    [CrossRef]
  22. The fringe patterns are typically not the natural images described in Ref. 18, and the processing operation is different and can be described as follows: First, we measure and remove the additional background brightness from the images. Then we sample the full range of possible gamma values and calculate the high-order correlations by using the method described in Ref. 18. Last, we search the gamma value corresponding to the maximum of the high-order correlations.
  23. M. Rivera, J. L. Marroquin, S. Botello, M. Servin, “Robust spatiotemporal quadrature filter for multiphase stepping,” Appl. Opt. 39, 284–292 (2000).
    [CrossRef]

2002 (1)

W.-S. Li, X.-Y. Su, “Phase unwrapping algorithm based on phase fitting reliability in structured light projection,” Opt. Eng. 41, 1365–1372 (2002).
[CrossRef]

2001 (3)

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10, 1428–1433 (2001).
[CrossRef]

G. S. Spagnolo, D. Ambrosini, “Diffractive optical element-based profilometer for surface inspection,” Opt. Eng. 40, 44–52 (2001).
[CrossRef]

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

2000 (3)

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

M. S. Mermelstein, D. L. Feldhun, L. G. Shirley, “Video-rate surface profiling with acousto-optic accordion fringe interferometry,” Opt. Eng. 39, 106–113 (2000).
[CrossRef]

M. Rivera, J. L. Marroquin, S. Botello, M. Servin, “Robust spatiotemporal quadrature filter for multiphase stepping,” Appl. Opt. 39, 284–292 (2000).
[CrossRef]

1999 (5)

G. Sansoni, M. Carocci, R. Rodella, “Three-dimensional vision based on a combination of gray-code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt. 38, 6565–6573 (1999).
[CrossRef]

S. Kakunai, T. Sakamoto, K. Iwata, “Profile measurement taken with liquid-crystal grating,” Appl. Opt. 38, 2824–2828 (1999).
[CrossRef]

P. S. Huang, F. Jin, F.-P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[CrossRef]

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

C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
[CrossRef]

1996 (1)

1995 (1)

J.-F. Lin, X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
[CrossRef]

1994 (1)

M. Pirga, M. Kujawińska, “Modified procedure for automatic surface topography,” Measurement 13, 191–197 (1994).
[CrossRef]

1993 (1)

C. A. Poynton, “Gamma and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video,” SMPTE J. 102, 1099–1108 (1993).
[CrossRef]

1990 (1)

1985 (2)

1984 (2)

V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry of a 3-D diffuse object,” Appl. Opt. 23, 3105–3108 (1984).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1983 (1)

Ambrosini, D.

G. S. Spagnolo, D. Ambrosini, “Diffractive optical element-based profilometer for surface inspection,” Opt. Eng. 40, 44–52 (2001).
[CrossRef]

Botello, S.

Brohinsky, W. R.

Burton, D. R.

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Carocci, M.

Chiang, F.-P.

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

P. S. Huang, F. Jin, F.-P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[CrossRef]

Coggrave, C. R.

C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
[CrossRef]

Farid, H.

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10, 1428–1433 (2001).
[CrossRef]

Feldhun, D. L.

M. S. Mermelstein, D. L. Feldhun, L. G. Shirley, “Video-rate surface profiling with acousto-optic accordion fringe interferometry,” Opt. Eng. 39, 106–113 (2000).
[CrossRef]

Halioua, M.

Hu, Q.

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

P. S. Huang, F. Jin, F.-P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[CrossRef]

Hung, Y. Y.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

S. Tang, Y. Y. Hung, “Fast profilometer for the automatic measurement of 3-D object shapes,” Appl. Opt. 29, 3012–3018 (1990).
[CrossRef] [PubMed]

Huntley, J. M.

C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
[CrossRef]

Iwata, K.

Jin, F.

P. S. Huang, F. Jin, F.-P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[CrossRef]

Kakunai, S.

Kujawinska, M.

M. Pirga, M. Kujawińska, “Modified procedure for automatic surface topography,” Measurement 13, 191–197 (1994).
[CrossRef]

Lalor, M. J.

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Li, W.-S.

W.-S. Li, X.-Y. Su, “Phase unwrapping algorithm based on phase fitting reliability in structured light projection,” Opt. Eng. 41, 1365–1372 (2002).
[CrossRef]

Lin, J.-F.

J.-F. Lin, X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
[CrossRef]

Lin, L.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Liu, H. C.

Marroquin, J. L.

Mermelstein, M. S.

M. S. Mermelstein, D. L. Feldhun, L. G. Shirley, “Video-rate surface profiling with acousto-optic accordion fringe interferometry,” Opt. Eng. 39, 106–113 (2000).
[CrossRef]

Mutoh, K.

Park, B. G.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Pirga, M.

M. Pirga, M. Kujawińska, “Modified procedure for automatic surface topography,” Measurement 13, 191–197 (1994).
[CrossRef]

Poynton, C. A.

C. A. Poynton, “Gamma and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video,” SMPTE J. 102, 1099–1108 (1993).
[CrossRef]

Rivera, M.

Rodella, R.

Sakamoto, T.

Sansoni, G.

Servin, M.

Shang, H. M.

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Shirley, L. G.

M. S. Mermelstein, D. L. Feldhun, L. G. Shirley, “Video-rate surface profiling with acousto-optic accordion fringe interferometry,” Opt. Eng. 39, 106–113 (2000).
[CrossRef]

Spagnolo, G. S.

G. S. Spagnolo, D. Ambrosini, “Diffractive optical element-based profilometer for surface inspection,” Opt. Eng. 40, 44–52 (2001).
[CrossRef]

Srinivasan, V.

Stetson, K. A.

Su, X.-Y.

W.-S. Li, X.-Y. Su, “Phase unwrapping algorithm based on phase fitting reliability in structured light projection,” Opt. Eng. 41, 1365–1372 (2002).
[CrossRef]

J.-F. Lin, X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
[CrossRef]

Surrel, Y.

Takeda, M.

Tang, S.

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Wu, F.

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Zhang, H.

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Appl. Opt. (9)

IEEE Trans. Image Process. (1)

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10, 1428–1433 (2001).
[CrossRef]

Measurement (1)

M. Pirga, M. Kujawińska, “Modified procedure for automatic surface topography,” Measurement 13, 191–197 (1994).
[CrossRef]

Opt. Commun. (1)

F. Wu, H. Zhang, M. J. Lalor, D. R. Burton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Opt. Eng. (8)

W.-S. Li, X.-Y. Su, “Phase unwrapping algorithm based on phase fitting reliability in structured light projection,” Opt. Eng. 41, 1365–1372 (2002).
[CrossRef]

J.-F. Lin, X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34, 3297–3302 (1995).
[CrossRef]

M. S. Mermelstein, D. L. Feldhun, L. G. Shirley, “Video-rate surface profiling with acousto-optic accordion fringe interferometry,” Opt. Eng. 39, 106–113 (2000).
[CrossRef]

G. S. Spagnolo, D. Ambrosini, “Diffractive optical element-based profilometer for surface inspection,” Opt. Eng. 40, 44–52 (2001).
[CrossRef]

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

C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Y. Y. Hung, L. Lin, H. M. Shang, B. G. Park, “Practical three-dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).
[CrossRef]

Opt. Lasers Eng. (1)

P. S. Huang, F. Jin, F.-P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[CrossRef]

SMPTE J. (1)

C. A. Poynton, “Gamma and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video,” SMPTE J. 102, 1099–1108 (1993).
[CrossRef]

Other (1)

The fringe patterns are typically not the natural images described in Ref. 18, and the processing operation is different and can be described as follows: First, we measure and remove the additional background brightness from the images. Then we sample the full range of possible gamma values and calculate the high-order correlations by using the method described in Ref. 18. Last, we search the gamma value corresponding to the maximum of the high-order correlations.

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Figures (12)

Fig. 1
Fig. 1

Signal waveform and CDF curve as changed by gamma nonlinearity: (a) response curves of system with different gamma values, (b) input sinusoidal signal, (c) output waveforms, (d) curves of CDFs.

Fig. 2
Fig. 2

Values of LUTs with, solid curve, parameters a = b = 0.5 and, dashed curve, parameters a = 0.529 and b = 0.314.

Fig. 3
Fig. 3

Measurement system.

Fig. 4
Fig. 4

Predefined phase distribution (a) with the carrier frequency and (b) without the carrier frequency.

Fig. 5
Fig. 5

Synthetic fringe patterns with a phase shift of (a) 0°, (b) 120°, and (c) 240°.

Fig. 6
Fig. 6

Phase errors (rad) of, solid line, the suggested approach and, dotted lines, the conventional phase-shifting algorithm, when the actual gamma is 2.2.

Fig. 7
Fig. 7

Relationship between the estimated gamma and number of iterations. (The actual gamma is 2.2.)

Fig. 8
Fig. 8

Sections of the fringe images corrupted by additive Gaussian noises. The actual gamma is 2.2, and the noise SDs are (a) 0.005, (b) 0.01, (c) 0.02, and (d) 0.05. The estimated gamma values are 2.1865, 2.1857, 2.1905, and 2.1728, respectively.

Fig. 9
Fig. 9

Fringe image processing: (a) a deformed fringe pattern, (b) a wrapped phase map, (c) an unwrapped phase map, (d) the phase difference.

Fig. 10
Fig. 10

Three-dimensional reconstruction result of the measured surface.

Fig. 11
Fig. 11

Relationship between the estimated gamma and the number of iterations.

Fig. 12
Fig. 12

Luminance nonlinearity of the LCD projector: solid curve, suggested algorithm (gamma = 1.4802); circles, photometric calibration result.

Tables (1)

Tables Icon

Table 1 Results of Simulations in the Absence of Noise

Equations (24)

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w=uγ,
ux=a+b cos2πfx+ϕ, x-, +,
wx=a+b cos2πfx+ϕγ, x-, +,
Fw=Pwxw=1-1πarccosw1/γ-a/b,a-bγwa+bγ.
Hs=1Nn=0N-1 ρn, where ρn=0if sn>s1if sns.
w=s-smina+bγ-a-bγsmax-smin+a-bγ,
Hs=Hw-a-bγsmax-smina+bγ-a-bγ+smin=Hww.
Hww=1-1πarccosw1/γˆ-a/b.
Hs=1-1πarccos1b×s-smina+bγˆ-a-bγˆsmax-smin+a-bγˆ1/γˆ-ab.
Hsmax+smin2=1-1πarccos1b×a+bγˆ+a-bγˆ21/γˆ-ab.
gnx, y=a+b cos2πx/p+2πn/N,
Ini, j=Ri, ja+b cosϕi, j+2πn/Nγ+Bi, j,
ϕˆ0i, j=-arctann=0N-1 Ini, jsin2πn/Nn=0N-1 Ini, jcos2πn/N,
Rˆ0i, j=2Nbn=0N-1 Ini, jsin2πn/N2+n=0N-1 Ini, jcos2πn/N21/2,
Bˆ0i, j=1Nn=0N-1 Ini, j-aRˆi, j.
Jn0i, j=Ini, j-Bˆ0i, j/Rˆ0i, j,
HJmaxk+Jmink/2=1Mi,j1Nn=0N-1 ρni, j,
ρni, j=0if Jnki, j>Jmaxk+Jmink/21if Jnki, jJmaxk+Jmink/2
Ini, j=Îni, j+Îni, jRi, j ΔRi, j+Îni, jBi, j ΔBi, j+Îni, jϕi, j Δϕi, j+,,
ΔRk+1i, ja+b cosϕˆki, j+2πn/Nγˆk +ΔBk+1i, j+Δϕk+1i, jRˆki, jγˆkba +b cosϕˆki, j+2πn/Nγˆk-1-sinϕˆki, j +2πn/N =Ini, j-Rˆki, ja+b cosϕˆki, j+2πn/Nγˆk-Bˆki, j.
ϕˆk+1i, j=ϕˆki, j+Δϕk+1i, j,
Rˆk+1i, j=Rˆki, j+ΔRk+1i, j,
Bˆk+1i, j=Bˆki, j+ΔBk+1i, j.
Jnk+1i, j=Ini, j-Bˆk+1i, j/Rˆk+1i, j.

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