Abstract

A phase-demodulation method for digital fringe-projection profilometry using the spatial and temporal Nyquist frequencies is presented. It allows to digitize tridimensional surfaces using the highest spatial frequency (π radians per pixel) and consequently with the highest sensitivity for a given digital fringe projector. Working with the highest temporal frequency (π radians per temporal sample), the proposed method rejects the DC component and all even-order distorting harmonics using 2-step phase shifting; this robustness against harmonics is similar to that of the popular 4-step least-squares phase-shifting algorithm. The proposed phase-demodulation method is suitable for the digitization of piecewise continuous surfaces because it does not require spatial low-pass filtering. Gamma calibration is also unnecessary because the projected fringes are binary, and the harmonics produced by the binary profile can be attenuated with a slight defocusing on the digital projector. Viability of the proposed method is supported by experimental results showing complete agreement with the predicted behavior.

© 2017 Optical Society of America

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Gamma correction for digital fringe projection profilometry

Hongwei Guo, Haitao He, and Mingyi Chen
Appl. Opt. 43(14) 2906-2914 (2004)

References

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  1. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
    [Crossref]
  2. R. Kulkarni and P. Rastogi, “Optical measurement techniques - A push for digitization,” Opt. Lasers Eng. 87, 1–17 (2016).
    [Crossref]
  3. S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
    [Crossref]
  4. M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology, Theory, Algorithms, and Applications (Wiley-VCH, 2014).
  5. K. J. Gasvik, Optical Metrology, 3rd ed. (John Wiley & Sons, 2002).
  6. Wikipedia, https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem .
  7. S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34(20), 3080–3082 (2009).
    [Crossref] [PubMed]
  8. T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
    [Crossref] [PubMed]
  9. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
    [Crossref]
  10. D. A. Zweig and R. E. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” Proc. SPIE 1333, 295–302 (1990).
    [Crossref]
  11. M. Johansson, “The Hilbert transform,” M.Sc. Thesis, Växjö University (1999).
  12. M. Servin, J. A. Quiroga, and J. C. Estrada, “Phase-shifting interferometry corrupted by white and non-white additive noise,” Opt. Express 19(10), 9529–9534 (2011).
    [Crossref] [PubMed]
  13. M. Servin, J. M. Padilla, A. Gonzalez, and G. Garnica, “Temporal phase-unwrapping of static surfaces with 2-sensitivity fringe-patterns,” Opt. Express 23(12), 15806–15815 (2015).
    [Crossref] [PubMed]
  14. D. Zheng and F. Da, “Gamma correction for two step phase shifting fringe projection profilometry,” Optik (Stuttg.) 124(13), 1392–1397 (2013).
    [Crossref]
  15. J. Li, X. Y. Su, and L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
    [Crossref]
  16. H. Cui, W. Liao, N. Dai, and X. Cheng, “Linear sinusoidal phase-shifting method resistant to non-sinusoidal phase error,” Chin. Opt. Lett. 10(3), 031201 (2012).
    [Crossref]
  17. P. S. Huang, Q. Hu, F. Jin, and F. P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38(6), 1065–1071 (1999).
    [Crossref]
  18. Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency Selection,” Opt. Express 14(14), 6444–6455 (2006).
    [Crossref] [PubMed]
  19. M. Padilla, M. Servin, and G. Garnica, “Fourier analysis of RGB fringe-projection profilometry and robust phase-demodulation methods against crosstalk distortion,” Opt. Express 24(14), 15417–15428 (2016).
    [Crossref] [PubMed]

2016 (3)

R. Kulkarni and P. Rastogi, “Optical measurement techniques - A push for digitization,” Opt. Lasers Eng. 87, 1–17 (2016).
[Crossref]

S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

M. Padilla, M. Servin, and G. Garnica, “Fourier analysis of RGB fringe-projection profilometry and robust phase-demodulation methods against crosstalk distortion,” Opt. Express 24(14), 15417–15428 (2016).
[Crossref] [PubMed]

2015 (1)

2013 (1)

D. Zheng and F. Da, “Gamma correction for two step phase shifting fringe projection profilometry,” Optik (Stuttg.) 124(13), 1392–1397 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (2)

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[Crossref] [PubMed]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

2009 (1)

2006 (1)

2000 (1)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

1999 (1)

P. S. Huang, Q. Hu, F. Jin, and F. P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38(6), 1065–1071 (1999).
[Crossref]

1990 (2)

J. Li, X. Y. Su, and L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

D. A. Zweig and R. E. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” Proc. SPIE 1333, 295–302 (1990).
[Crossref]

Cheng, X.

Chiang, F. P.

P. S. Huang, Q. Hu, F. Jin, and F. P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38(6), 1065–1071 (1999).
[Crossref]

Cui, H.

Da, F.

D. Zheng and F. Da, “Gamma correction for two step phase shifting fringe projection profilometry,” Optik (Stuttg.) 124(13), 1392–1397 (2013).
[Crossref]

Dai, N.

Dirckx, J. J. J.

S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

Estrada, J. C.

Garnica, G.

Gonzalez, A.

Guo, L. R.

J. Li, X. Y. Su, and L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

Hoang, T.

Hu, Q.

P. S. Huang, Q. Hu, F. Jin, and F. P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38(6), 1065–1071 (1999).
[Crossref]

Huang, P. S.

P. S. Huang, Q. Hu, F. Jin, and F. P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38(6), 1065–1071 (1999).
[Crossref]

Hufnagel, R. E.

D. A. Zweig and R. E. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” Proc. SPIE 1333, 295–302 (1990).
[Crossref]

Jin, F.

P. S. Huang, Q. Hu, F. Jin, and F. P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38(6), 1065–1071 (1999).
[Crossref]

Kulkarni, R.

R. Kulkarni and P. Rastogi, “Optical measurement techniques - A push for digitization,” Opt. Lasers Eng. 87, 1–17 (2016).
[Crossref]

Lei, S.

Li, J.

J. Li, X. Y. Su, and L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

Liao, W.

Nguyen, D.

Padilla, J. M.

Padilla, M.

Pan, B.

Quiroga, J. A.

Rastogi, P.

R. Kulkarni and P. Rastogi, “Optical measurement techniques - A push for digitization,” Opt. Lasers Eng. 87, 1–17 (2016).
[Crossref]

Servin, M.

Su, X. Y.

J. Li, X. Y. Su, and L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

Towers, C. E.

Towers, D. P.

Van der Jeught, S.

S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

Wang, Z.

Zhang, S.

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34(20), 3080–3082 (2009).
[Crossref] [PubMed]

Zhang, Z.

Zheng, D.

D. Zheng and F. Da, “Gamma correction for two step phase shifting fringe projection profilometry,” Optik (Stuttg.) 124(13), 1392–1397 (2013).
[Crossref]

Zweig, D. A.

D. A. Zweig and R. E. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” Proc. SPIE 1333, 295–302 (1990).
[Crossref]

Chin. Opt. Lett. (1)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Opt. Eng. (2)

P. S. Huang, Q. Hu, F. Jin, and F. P. Chiang, “Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring,” Opt. Eng. 38(6), 1065–1071 (1999).
[Crossref]

J. Li, X. Y. Su, and L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (3)

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

R. Kulkarni and P. Rastogi, “Optical measurement techniques - A push for digitization,” Opt. Lasers Eng. 87, 1–17 (2016).
[Crossref]

S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

Opt. Lett. (2)

Optik (Stuttg.) (1)

D. Zheng and F. Da, “Gamma correction for two step phase shifting fringe projection profilometry,” Optik (Stuttg.) 124(13), 1392–1397 (2013).
[Crossref]

Proc. SPIE (1)

D. A. Zweig and R. E. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” Proc. SPIE 1333, 295–302 (1990).
[Crossref]

Other (4)

M. Johansson, “The Hilbert transform,” M.Sc. Thesis, Växjö University (1999).

M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology, Theory, Algorithms, and Applications (Wiley-VCH, 2014).

K. J. Gasvik, Optical Metrology, 3rd ed. (John Wiley & Sons, 2002).

Wikipedia, https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem .

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Figures (12)

Fig. 1
Fig. 1 Basic setup for a single-projector single-camera digital fringe-projector profilometer.
Fig. 2
Fig. 2 Zoomed-in segments of computer-generated open fringes with spatial and temporal carriers. The left side shows fringes below the spatial frequency; the right side shows fringes exactly at Nyquist frequency u0 = π . In both cases ω0 = 2π/8 for illustrative purposes.
Fig. 3
Fig. 3 Temporal quadrature filtering below and exactly at the Nyquist frequency. The vertical arrows represent the discrete-time Fourier transform of I(x,y,t). The blue line is |H3(ω)| for a tunable 3-step PSA having spectral zeroes at ω = {0, ‒ω0}. Below Nyquist we are able to isolate c = (b/2)exp(). On the other hand, when working at the temporal Nyquist frequency, the spectral zero at ω = ‒ω0 would filter-out both analytic signals simultaneously.
Fig. 4
Fig. 4 Zoomed-in segment of computer-generated open fringes with spatial carrier u0 = 2π/Λ and temporal carrier ω0 = 2π/M. Note that only Λ = M produces a non-varying fringes’ profile. In practice, if the fringes’ contrast varies over time it is necessary to rely on many-step PSAs with smooth spectral zeroes on the stop-band for robust quadrature filtering [4].
Fig. 5
Fig. 5 Harmonics-rejection capabilities of the (a) 3-step and (b) 4-step LS-PSAs versus (c) our 2-step algorithm for 1/4<α<1/2 (without loss of generality). The crosses in the third plot represent harmonics rejected by the Hilbert filtering process. The horizontal axes correspond to normalized frequency (ω/ω0) beyond the wrapped range for ease of observation [4].
Fig. 6
Fig. 6 Fringe patterns phase-modulated by a spherical cap. (a) One of the two fringe patters obtained with fringe projection using both Nyquist frequencies, u0 = π and ω0 = π. (b) Zoomed-in section of (a). (c) Low-frequency carrier fringe pattern using u0 = π/20 for visual comparison.
Fig. 7
Fig. 7 Spatial Fourier spectra of the fringe patterns phase-modulated by the spherical cap. (a) corresponds to the raw data, (b) is obtained after temporal filtering, and (c) represents the Hilbert filter that rejects the negative-frequency spectral components. As predicted, the spectral lobes corresponding to the DC term and the odd-order harmonics are gone from (b).
Fig. 8
Fig. 8 Phase maps computed from the analytic signal in Eq. (13). For illustrative purposes, (a) shows the highly wrapped phase before subtracting the spatial carrier. (b) Shows the highly wrapped phase after subtracting the spatial carrier. (c) Shows the phase map proportional to the object’s height profile, computed with spatial unwrapping from (b).
Fig. 9
Fig. 9 Digitization of a combination-square tool. (a) One of two phase-modulated fringe patterns obtained with digital fringe projection using both Nyquist frequencies. (b) Zoomed-in section of (a). (c) One of four low-frequency carrier fringes (using u0 = π/10) to be used as stepping stone on dual-sensitivity temporal phase unwrapping [13].
Fig. 10
Fig. 10 Spatial Fourier spectra of the fringe patterns modulated by the combination square. First panel corresponds to the raw data, (b) is obtained after temporal filtering, and (c) represents the Hilbert filter rejecting the negative-frequency spectral components.
Fig. 11
Fig. 11 Phase maps computed from the searched analytic signal. (a) Highly wrapped phase modulated by the object and the spatial carrier. (b) Wrapped phase after subtracting the spatial carrier. (c) Unwrapped phase map proportional to the searched height profile, computed from (b) and dual-sensitivity temporal phase unwrapping [13].
Fig. 12
Fig. 12 Simultaneous plot of cross-sections taken at the same line from the demodulated phases given by the proposed 2-step demodulation method working at the Nyquist frequencies versus M-step LS-PSAs for M = {3,4,5,6} using Λ = M. Both panels are discussed on the text.

Equations (25)

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f( x p , y p ,t)=0.5+0.5cos( u 0 x p + ω 0 t),
I(x,y,t)=a(x,y)+b(x,y)cos[φ(x,y)+ u c x+ ω 0 t];φ(x,y)= u c tan(θ)s(x,y).
I(x,y,t)=a(x,y)+c(x,y) e i( u c x+ ω 0 t) + c (x,y) e i( u c x+ ω 0 t) , c(x,y)=(1/2)b(x,y)exp[iφ(x,y)].
r 1 (x,y)exp[i φ 1 (x,y)]= e i ω 0 t [I(x,y,t) e i u c x ]h(x,y),
r 2 (x,y)exp[i φ 2 (x,y)]= e i u c x [I(x,y,t)h(t)] t=M = e i u c x m=0 M1 w m I(x,y,m) .
H( ω 0 )=H(0)=0,H( ω 0 )0, ω 0 (0,π).
cos(π x p + ω 0 t)= (1) x p cos( ω 0 t);for x p {1,2,...}.
I(x,y,t)=a(x,y)+b(x,y)cos[φ(x,y)+απx+πt],t=1,2.( u 0 =π, ω 0 =π)
I(x,y,0)=a(x,y)+b(x,y)cos[φ(x,y)+απx], I(x,y,1)=a(x,y)+b(x,y)cos[φ(x,y)+απx+π], I'(x,y)=I(x,y,0)I(x,y,1)=2b(x,y)cos[φ(x,y)+απx].
h 2 (t)=δ(t)δ(t1), H 2 (ω)=1exp(iω).
I'(u,v)=2C(uαπ,v)+2 C (u+απ,v);C(u,v)=F{c(x,y)}.
H(u,v)={ 1ifu>0, 0otherwise.
r 3 (x,y)exp[i φ 3 (x,y)]= e iαπx OSHF[I(x,y,0)I(x,y,1)].( u 0 =π, ω 0 =π)
I n (x,y,t)=I(x,y,t)+n(x,y,t),
r N (x,y)exp[iφ(x,y)+iN(x,y)]= r 3 (x,y)exp[iφ(x,y)]+OSHF[n(x,y,t) h 2 (t)].
H M (ω)= m=0 M2 { 1exp[ i(ω+m ω 0 ) ] } , ω 0 = 2π M .
Φ(x,y)=φ(x,y)+N(x,y).
E[ Φ 2 (x,y) ]=E[ φ 2 (x,y) ]+E[ N 2 (x,y) ] = 4 π 2 Λ 2 tan 2 (θ)E[ s 2 (x,y) ]+ 4 M E[ b 2 (x,y) n 2 (x,y) ].
SN R M = (1/Ω) Ω φ 2 (x,y)dΩ (1/Ω) Ω N 2 (x,y)dΩ = E[ φ 2 (x,y) ] E[ N 2 (x,y) ] =( M Λ 2 ) π 2 tan 2 (θ) Ω s 2 (x,y)dΩ Ω b 2 (x,y) n 2 (x,y)dΩ .
SN R 2 SN R (M3) = (1/2) (1/M) = M 2 >1(forΛ=M).
I(x,y,t)= k=K K b k exp[ik(φ+απx+πt)] ;t=0,1.
I'(x,y)=I(x,y,0)I(x,y,1)= k=K K [1 (1) k ] b k exp[ik(φ+απx)] .
H 2 (ω)I(x,y,ω)=(1 e iω ) k=K K b k e ik(φ+απx) δ(ω ω k ) ; ω k =Arg( e iπk ).
I'(u,v)= k=K K [1 (1) k ] C k (u u k ,v) ; C k (u,v)=F{ b k e ikφ }; u k =Arg( e iπαk ).
r 3 (x,y)exp[i φ 3 (x,y)]= b 1 e iφ + b 3 e ±i3φ + b 5 e ±i5φ + b 7 e ±i7φ +...

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