Abstract

In this paper, we apply the frequency transfer function formalism to analyze the red, green and blue (RGB) phase-shifting fringe-projection profilometry technique. The phase-shifted patterns in RGB fringe projection are typically corrupted by crosstalk because the sensitivity curves of most projection-recording systems overlap. This crosstalk distortion needs to be compensated in order to obtain high quality measurements. We study phase-demodulation methods for null/mild, moderate, and severe levels of RGB crosstalk. For null/mild crosstalk distortion, we can estimate the searched phase-map using Bruning’s 3-step phase-shifting algorithm (PSA). For moderate crosstalk, the recorded data is usually preprocessed before feeding it into the PSA; alternatively, in this paper we propose a computationally more efficient approach, which combines linear crosstalk compensation with the phase-demodulation algorithm. For severe RGB crosstalk, we expect non-sinusoidal fringes’ profiles (distorting harmonics) and a significant uncertainty on the linear crosstalk calibration (which produces pseudo-detuning error). Analyzing these distorting phenomena, we conclude that squeezing interferometry is the most robust demodulation method for RGB fringe-projection techniques. Finally, we support our conclusions with numerical simulations and experimental results.

© 2016 Optical Society of America

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References

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  1. C. Wust and D. W. Capson, “Surface profile measurement using color fringe projection,” Mach. Vis. Appl. 4(3), 193–203 (1991).
    [Crossref]
  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]
  3. 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]
  4. S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
    [Crossref]
  5. Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
    [Crossref]
  6. J. L. Flores, J. A. Ferrari, G. García Torales, R. Legarda-Saenz, and A. Silva, “Color-fringe pattern profilometry using a generalized phase-shifting algorithm,” Appl. Opt. 54(30), 8827–8834 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  9. T. Katsuta, F. Ando, M. Kida, and M. Kawasaki, “Optical system for tricolor separation,” U.S. Patent: 3 602 637, issued date August 31, 1971.
  10. M. Servin, M. Cywiak, D. Malacara-Hernandez, J. C. Estrada, and J. A. Quiroga, “Spatial carrier interferometry from M temporal phase shifted interferograms: Squeezing Interferometry,” Opt. Express 16(13), 9276–9283 (2008).
    [Crossref] [PubMed]
  11. B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett. 36(6), 996–998 (2011).
    [Crossref] [PubMed]
  12. M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley-VCH, Weinheim, 2014).
  13. J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
    [Crossref]
  14. H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43(14), 2906–2914 (2004).
    [Crossref] [PubMed]
  15. S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34(20), 3080–3082 (2009).
    [Crossref] [PubMed]
  16. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

2015 (1)

2012 (2)

S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
[Crossref]

Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

2011 (1)

2009 (1)

2008 (1)

2006 (1)

2004 (1)

2003 (1)

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[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]

1991 (1)

C. Wust and D. W. Capson, “Surface profile measurement using color fringe projection,” Mach. Vis. Appl. 4(3), 193–203 (1991).
[Crossref]

1982 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Capson, D. W.

C. Wust and D. W. Capson, “Surface profile measurement using color fringe projection,” Mach. Vis. Appl. 4(3), 193–203 (1991).
[Crossref]

Chen, L.

S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
[Crossref]

B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett. 36(6), 996–998 (2011).
[Crossref] [PubMed]

Chen, M.

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]

Cywiak, M.

Estrada, J. C.

Ferrari, J. A.

Flores, J. L.

Gallagher, J. E.

García Torales, G.

Guo, H.

He, H.

Herriott, D. R.

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]

Ina, H.

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]

Kobayashi, S.

Legarda-Saenz, R.

Lei, S.

Li, B.

S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
[Crossref]

B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett. 36(6), 996–998 (2011).
[Crossref] [PubMed]

Ma, S.

S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
[Crossref]

B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett. 36(6), 996–998 (2011).
[Crossref] [PubMed]

Malacara-Hernandez, D.

Quan, C.

S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
[Crossref]

Quiroga, J. A.

Rosenfeld, D. P.

Servin, M.

Silva, A.

Takeda, M.

Tay, C. J.

S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
[Crossref]

Towers, C. E.

Towers, D. P.

Tuya, W.

White, A. D.

Wust, C.

C. Wust and D. W. Capson, “Surface profile measurement using color fringe projection,” Mach. Vis. Appl. 4(3), 193–203 (1991).
[Crossref]

Zhang, S.

Zhang, Z.

Zhang, Z. H.

Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

Zhu, R.

S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
[Crossref]

B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett. 36(6), 996–998 (2011).
[Crossref] [PubMed]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Mach. Vis. Appl. (1)

C. Wust and D. W. Capson, “Surface profile measurement using color fringe projection,” Mach. Vis. Appl. 4(3), 193–203 (1991).
[Crossref]

Opt. Commun. (2)

S. Ma, R. Zhu, C. Quan, B. Li, C. J. Tay, and L. Chen, “Blind phase error suppression for color-encoded digital fringe projection profilometry,” Opt. Commun. 285(7), 1662–1668 (2012).
[Crossref]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

Opt. Eng. (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]

Opt. Express (2)

Opt. Lasers Eng. (1)

Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

Opt. Lett. (2)

Other (3)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

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

T. Katsuta, F. Ando, M. Kida, and M. Kawasaki, “Optical system for tricolor separation,” U.S. Patent: 3 602 637, issued date August 31, 1971.

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Figures (11)

Fig. 1
Fig. 1

Fourier spectra of the temporal-carrier fringe pattern before (on the left side) and after (on the right side) the quadrature linear filtering given by Bruning’s 3-step PSA [7,12].

Fig. 2
Fig. 2

Schematic of a single-camera single-projector setup for fringe projection profilometry.

Fig. 3
Fig. 3

RGB multichannel operation allows us to encode/decode up-to three phase-shifted open-carrier fringes into a single full-color image. Here we show low-frequency fringes for illustrative purposes; a much higher spatial frequency is used on the actual experiments.

Fig. 4
Fig. 4

Left side shows three FTFs given by H(ω)= c ' n exp(inω) using uniformly distributed random values ε mn (0.15,0.15) ; each color represents a different realization. Right side shows the Fourier spectrum of the analytic signal estimated using one of these FTFs.

Fig. 5
Fig. 5

The first row shows the phase map (given by MATLAB’s peaks function) modulating three phase-shifted patterns of open fringes. The second row shows the estimated phase as obtained from the 3-step PSA for different amounts of crosstalk-calibration error. The third row shows the wrapped difference Δφ=( φ φ ^ )mod2π , multiplied by a factor 10 for ease of observation, revealing the predicted double-frequency ripple pattern. The intensity scale for the wrapped phases follows the standard linear mapping from π (black) to π(white).

Fig. 6
Fig. 6

Schematic illustration of Fourier demodulation for the spatial-carrier fringe pattern I S (u,v) in Eq. (24). The black circle represents a binary mask band-passing the searched analytic signal (red pentagon) and rejecting all unwanted distorting harmonic components.

Fig. 7
Fig. 7

Photograph of our test subject being illuminated with RGB phase-shifted open fringes. The decoded RGB images are shown in gray levels.

Fig. 8
Fig. 8

First stage of the squeezing interferometry method as described in Eq. (21).

Fig. 9
Fig. 9

Last steps of the squeezing interferometry method: quadrature filtering of { I S (x',y)} ; wrapped phase estimated as the argument of the analytic signal inside the black circle; and the unwrapped phase proportional to the profile of our test subject.

Fig. 10
Fig. 10

Tridimensional render of the digitized object with RGB fringe projection profilometry. These panels are described in the main text. Units of all axes are in millimeters.

Fig. 11
Fig. 11

Horizontal slices of the unwrapped phase estimated with Burning’s 3-step PSA, the squeezing interferometry method, and the crosstalk-free null test. A global piston was included between each plot for ease of visualization.

Equations (26)

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I(x,y,t)=a(x,y)+b(x,y)cos[φ(x,y)+ ω 0 t],
{I(x,y,t)}=I(x,y,ω)=a(x,y)δ(ω)+ 1 2 b(x,y)[ e iφ(x,y) δ(ω- ω 0 )+ e iφ(x,y) δ(ω+ ω 0 ) ].
H(ω)= n=0 N1 c n exp(inω) ;H( ω 0 )=H(0)=0,H( ω 0 )0.
I(x,y,ω)H(ω)= 1 2 b(x,y)exp[iφ(x,y)]H( ω 0 )δ(ω ω 0 ).
A 0 (x,y)exp[i φ ^ (x,y)]= I(x,y,t)h(t) | t=N1 = n=0 N1 c n I n (x,y) ,
a ^ (x,y)= 1 N n=0 N1 I n (x,y) ; b ^ (x,y)= 2 |H( ω 0 )| | n=0 N1 c n I n (x,y) |; φ ^ (x,y)mod2π= tan 1 { Im[ c n I n (x,y) ] Re[ c n I n (x,y) ] }= tan 1 { b n I n (x,y) a n I n (x,y) }, c n = a n +i b n .
A 0 (x,y)exp[i φ ^ (x,y)]= c n I n (x,y)= I 0 (x,y)+ e i2π/3 I 1 (x,y)+ e i4π/3 I 2 (x,y).
H(ω)= c n e inω =[ 1 e iω ][ 1 e i(ω+2π/3) ].
f 0 (x,y)=255×LPF{ [0.5+0.5cos( u 0 x+0)] 1/γ }, f 1 (x,y)=255×LPF{ [0.5+0.5cos( u 0 x+2π/3)] 1/γ }, f 2 (x,y)=255×LPF{ [0.5+0.5cos( u 0 x+4π/3)] 1/γ }.
I 0 (x,y)=a(x,y)+b(x,y)cos[φ(x,y)+0], I 1 (x,y)=a(x,y)+b(x,y)cos[φ(x,y)+2π/3];φ(x,y)= u 0 tan(θ)[p(x,y)+x], I 2 (x,y)=a(x,y)+b(x,y)cos[φ(x,y)+4π/3].
I ^ 0 (x,y)= I R (x,y), I ^ 1 (x,y)= I G (x,y), I ^ 2 (x,y)= I B (x,y).
[ I R (x,y) I G (x,y) I B (x,y) ]=[ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ][ I 0 (x,y) I 1 (x,y) I 2 (x,y) ].
[ I ^ 0 (x,y) I ^ 1 (x,y) I ^ 2 (x,y) ]=[ B 11 B 12 B 13 B 21 B 22 B 23 B 31 B 32 B 33 ][ I R (x,y) I G (x,y) I B (x,y) ];whereB= A ^ 1 .
A 0 (x,y)exp[i φ ^ (x,y)]=[ 1 e i2π/3 e i4π/3 ][ I ^ 0 (x,y) I ^ 1 (x,y) I ^ 2 (x,y) ]= c n I ^ n (x,y) .
A 0 (x,y)exp[i φ ^ (x,y)]= d 0 I R (x,y)+ d 1 I G (x,y)+ d 2 I B (x,y); [ d 0 d 1 d 2 ]=[ 1 e i2π/3 e i4π/3 ][ B 11 B 12 B 13 B 21 B 22 B 23 B 31 B 32 B 33 ].
A 0 (x,y)exp(i φ ^ )= c ' n I n (x,y) ; [ c ' 0 c ' 1 c ' 2 ] T = [ 1 e i2π/3 e i4π/3 ] T [ 1+ ε 11 ε 12 ε 13 ε 21 1+ ε 22 ε 23 ε 31 ε 32 1+ ε 33 ],
A 0 (x,y) e i φ ^ (x,y) =H(0)a(x,y)+ 1 2 b(x,y){ H(2π/3) e iφ(x,y) +H(2π/3) e iφ(x,y) }.
A 0 (x,y) e i φ ^ (x,y) = 1 2 H(2π/3)b(x,y) e iφ(x,y) { 1+ H(2π/3) H(2π/3) e i2φ(x,y) }.
I(x,y,t)= n=N N b n (x,y)exp{ in[φ(x,y)+(2π/3)t] } , I(x,y,ω)= n=N N b n (x,y)exp[inφ(x,y)]δ(ω ω n ) ; ω n =arg[exp(i2nπ/3)].
I(ω)H(ω)=H(2π/3){ b 1 e iφ + b 2 e i2φ + b 4 e i4φ + b 5 e i5φ +...}δ(ω2π/3) +H(0){ b 0 + b 3 e i3φ + b 3 e i3φ + b 6 e i6φ + b 6 e i6φ +...}δ(ω) +H(2π/3){ b 1 e iφ + b 2 e i2φ + b 4 e i4φ + b 5 e i5φ +...}δ(ω+2π/3).
I S (3x2,y)= I ^ 0 (x,y) I S (3x1,y)= I ^ 1 (x,y) I S (3x0,y)= I ^ 2 (x,y) }(0,0)(x,y)(X,Y),
I S (x',y)= n=N N b n (x',y)exp{ in[φ(x',y)+2πx'/3] } ;(0,0)(x',y)(3X,Y).
I S (u,v)= n=N N { b n (x',y)exp[inφ(x',y)] }δ(u u n ,v); u n =arg[exp(i2nπ/3)].
I S (u,v)={ b 1 e iφ + b 2 e i2φ + b 4 e i4φ + b 5 e i5φ + b 7 e i7φ +... }δ(u2π/3,v) +{ b 0 + b 3 e i3φ + b 3 e i3φ + b 5 e i6φ + b 5 e i6φ +... }δ(u,v) +{ b 1 e iφ + b 2 e i2φ + b 4 e i4φ + b 5 e i5φ + b 7 e i7φ +... }δ(u+2π/3,v).
A 0 (x',y)exp[i φ ^ (x',y)]=exp(i2πx'/3)×QF{ I S (x',y) }.
A ^ =( 0.4334 0.4041 0.0749 0.0791 0.9092 0.3316 0.0007 0.3679 0.9536 ),( d 0 d 1 d 2 )=( 2.2252 - i0.3585 0.4581 + i1.9615 -0.8586 + i0.2542 ).

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