Abstract

Fringe projection profilometry is a well-known technique to digitize 3-dimensional (3D) objects and it is widely used in robotic vision and industrial inspection. Probably the single most important problem in single-camera, single-projection profilometry are the shadows and specular reflections generated by the 3D object under analysis. Here a single-camera along with N-fringe-projections is (digital) coherent demodulated in a single-step, solving the shadows and specular reflections problem. Co-phased profilometry coherently phase-demodulates a whole set of N-fringe-pattern perspectives in a single demodulation and unwrapping process. The mathematical theory behind digital co-phasing N-fringe-patterns is mathematically similar to co-phasing a segmented N-mirror telescope.

© 2013 OSA

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References

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  1. K. J. Gasvik, Optical Metrology, 3rd ed. (John Wiley & Sons, 2002).
  2. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Taylor & Francis, 2005).
  3. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010).
    [Crossref]
  4. Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
    [Crossref]
  5. W. H. Su, C. Y. Kuo, C. C. Wang, and C. F. Tu, “Projected fringe profilometry with multiple measurements to form an entire shape,” Opt. Express 16(6), 4069–4077 (2008).
    [Crossref] [PubMed]
  6. X. Liu, X. Peng, H. Chen, D. He, and B. Z. Gao, “Strategy for automatic and complete three-dimensional optical digitization,” Opt. Lett. 37(15), 3126–3128 (2012).
    [Crossref] [PubMed]
  7. S. Lee and L. Q. Bui, “Accurate estimation of the boundary of a structured light pattern,” J. Opt. Soc. Am. A 28(6), 954–961 (2011).
    [Crossref]
  8. R. Gannavarpu and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50(8), iii–x (2012).
    [Crossref]
  9. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
    [Crossref] [PubMed]

2012 (2)

2011 (1)

2010 (2)

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010).
[Crossref]

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

2009 (1)

2008 (1)

Barnes, J. C.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Bui, L. Q.

Chen, H.

Estrada, J. C.

Gannavarpu, R.

R. Gannavarpu and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50(8), iii–x (2012).
[Crossref]

Gao, B. Z.

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010).
[Crossref]

He, D.

Kuo, C. Y.

Lee, S.

Liu, X.

Nguyen, D. A.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Peng, X.

Quiroga, J. A.

Rastogi, P.

R. Gannavarpu and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50(8), iii–x (2012).
[Crossref]

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010).
[Crossref]

Servin, M.

Su, W. H.

Tu, C. F.

Wang, C. C.

Wang, Z.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (2)

Opt. Lasers Eng. (3)

R. Gannavarpu and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50(8), iii–x (2012).
[Crossref]

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010).
[Crossref]

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Opt. Lett. (1)

Other (2)

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

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Taylor & Francis, 2005).

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Figures (7)

Fig. 1
Fig. 1

Standard fringe projection profilometry with a single fringe-projection direction. The projector form a polar-angle θ with respect to the camera-object z axis.

Fig. 2
Fig. 2

In panel (a) we show the Fourier spectrum of the real-valued phase-modulated fringe pattern. In panel (b) we show the frequency-shifted spectrum and the low-pass filtering as a gray disk at the spectral origin.

Fig. 3
Fig. 3

Multiple projector co-phased profilometry experiment. Here a single camera and 3 projectors are shown. All projectors are at the same distance and have the same polar-angle θ.

Fig. 4
Fig. 4

Panel (a) shows the original Fourier spectra of the sum of 4 real-valued fringe patterns. In panel (b) we show the frequency shifted spectra of the sum in Eq. (5). At the spectral center in panel (b), the complex-spectrum of the 4 projected fringe-patterns adds coherently.

Fig. 5
Fig. 5

In panel (a) we show the computer mouse illuminated with white light to see the object, the shadows and specular reflections from the two projector directions. Panel (b) and (c) show the projected fringes.

Fig. 6
Fig. 6

Panel (a) shows the 4-temporal-steps demodulated-phase from the right projection. Panel (b) shows the 4-temporal-steps demodulated-phase from the left projection. Panel (c) shows (already unwrapped) the phase of the coherent-sum of both analytical signals, and we see that the phase-object is recovered without errors due to the shadows and specular bright zones.

Fig. 7
Fig. 7

Single-projector, single-camera, multiple projections co-phased profilometry set-up. Here the camera-object system is rotated as a rigid square-box an azimuth-angle φ, around z. In this way one obtains as many 3D object’s perspective steps as desired. All other relative distances and angles (including the sensitivity polar-angle θ) remain fixed.

Equations (13)

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F(x,y)=1+cos( v 0 x)
I(x,y)=a(x,y)+b(x,y)cos[ ω 0 x+gO(x,y) ]; ω 0 = v 0 cos(θ),g= v 0 sin(θ).
(1/2)b(x,y) e igO(x,y) =LPF[ I(x,y) e i ω 0 x ].
I 0 (x,y)= a 0 (x,y)+ b 0 (x,y)cos[ gO(x,y)+ ω 0 x ], I 1 (x,y)= a 1 (x,y)+ b 1 (x,y)cos[ gO(x,y) ω 0 x ], I 2 (x,y)= a 2 (x,y)+ b 2 (x,y)cos[ gO(x,y)+ ω 0 y ], I 3 (x,y)= a 3 (x,y)+ b 3 (x,y)cos[ gO(x,y) ω 0 y ]; ω 0 = v 0 cos(θ),g= v 0 sin(θ).
A(x,y) e igO(x,y) =LPF[ I 0 (x,y) e i ω 0 x + I 1 (x,y) e i ω 0 x + I 2 (x,y) e i ω 0 y + I 3 (x,y) e i ω 0 y ].
A(x,y) e igO(x,y) =(1/2)[ b 0 (x,y)+ b 1 (x,y)+ b 2 (x,y)+ b 3 (x,y) ] e igO(x,y) .
I n (x,y,mΔ)= a n (x,y)+ b n (x,y)cos{ mΔ+ ω 0 r· d n +gO(x,y) }, withΔ= 2π 3 ,n{0,1,2,3},m{0,1,2}.
A(x,y) e igO(x,y) =I C 1 e i ω 0 r· d 0 +I C 2 e i ω 0 r· d 1 +I C 3 e i ω 0 r· d 2 +I C 4 e i ω 0 r· d 3 I C n (x,y)= I n (0)+ e iΔ I n (Δ)+ e i2Δ I n (2Δ).
I 0 (x,y,mΔ)= a 0 (x,y)+ b 0 (x,y)cos[ mΔ+ c 0 (x,y)+gO(x,y) ],m={0,1,2,3}, I 1 (x,y,mΔ)= a 1 (x,y)+ b 1 (x,y)cos[ mΔ+ c 1 (x,y)+gO(x,y) ],Δ= 2π 4 .
(1/2) b 0 (x,y) e i c 0 (x,y) = I 0 (0)+ I 0 (Δ) e iΔ + I 0 (2Δ) e i2Δ + I 0 (3Δ) e i3Δ , (1/2) b 1 (x,y) e i c 1 (x,y) = I 1 (0)+ I 1 (Δ) e iΔ + I 1 (2Δ) e i2Δ + I 1 (3Δ) e i3Δ .
A(x,y) e igO(x,y) =I C 0 (x,y) e i c 0 (x,y) +I C 1 (x,y) e i c 1 (x,y) , I C 0 (x,y)= I 0 (0)+ I 0 (Δ) e iΔ + I 0 (2Δ) e i2Δ + I 0 (3Δ) e i3Δ , I C 1 (x,y)= I 1 (0)+ I 1 (Δ) e iΔ + I 1 (2Δ) e i2Δ + I 1 (3Δ) e i3Δ .
A(x,y) e igO(x,y) =(1/2)[ b 0 e i n 1 (x,y) + b 1 e i n 2 (x,y) + b 2 e i n 3 (x,y) + b 3 e i n 4 (x,y) ] e igO(x,y) .
A(x,y) e igO(x,y) =(1/2)[ b 0 e i n 1 (x,y) + b 0 e i n 2 (x,y) + b 0 e i n 3 (x,y) + b 0 e i n 4 (x,y) ] e igO(x,y) .

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