Abstract

Temporal phase unwrapping is an important method for shape measurement in structured light projection. Its measurement errors mainly come from both the camera noise and nonlinearity. Analysis found that least-squares fitting cannot completely eliminate nonlinear errors, though it can significantly reduce the random errors. To further reduce the measurement errors of current temporal phase unwrapping algorithms, in this paper, we proposed a phase averaging method (PAM) in which an additional fringe sequence at the highest fringe density is employed in the process of data processing and the phase offset of each set of the four frames is carefully chosen according to the period of the phase nonlinear errors, based on fast classical temporal phase unwrapping algorithms. This method can decrease both the random errors and the systematic errors with statistical averaging. In addition, the length of the additional fringe sequence can be changed flexibly according to the precision of the measurement. Theoretical analysis and simulation experiment results showed the validity of the proposed method.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (3)

J. D. Tian, X. Peng, and X. B. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46, 336–342 (2008).
[CrossRef]

Z.-H. Xu and X.-Y. Su, “An algorithm of temporal phase unwrapping,” J. Sichuan University (Natural Science Edition) 45, 537–540 (2008).
[CrossRef]

Z. W. Li, C. J. Wang, Y. S. Shi, and Y. Y. Wang, “High precision phase error compensation algorithm for structural light measurement,” Acta Opt. Sin. 28, 1527–1532 (2008).
[CrossRef]

2007 (1)

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 1–9 (2007).
[CrossRef]

2003 (2)

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[CrossRef]

2002 (1)

2001 (1)

2000 (1)

C. R. Coggrave and J. M. Huntley, “Optimization of a shape measurement system based on spatial light modulators,” Opt. Eng. 39, 91–98 (2000).
[CrossRef]

1997 (4)

H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
[CrossRef]

J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[CrossRef]

J. M. Huntley and H. O. Saldner, “Error-reduction methods for shape measurement by temporal phase unwrapping,” J. Opt. Soc. Am. A 14, 3188–3196 (1997).
[CrossRef]

C. Reich, R. Ritter, and J. Thesing, “White light heterodyne principle for 3D-measurement,” Proc. SPIE 3100, 236–224 (1997).
[CrossRef]

1996 (2)

T. Maack and R. Kowarschik, “Camera influence on the phase measuring accuracy of a phase-shifting speckle interferometry,” Appl. Opt. 35, 3514–3524 (1996).
[CrossRef]

B. Trolard, “Speckle noise removal in interference fringes by optoelectronic preprocessing with Epson liquid crystal television,” Proc. SPIE 2860, 126–134 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1990 (1)

Chen, W. Y.

Chiang, F. P.

Coggrave, C. R.

C. R. Coggrave and J. M. Huntley, “Optimization of a shape measurement system based on spatial light modulators,” Opt. Eng. 39, 91–98 (2000).
[CrossRef]

Creath, K.

Hu, Q. Y. J.

Huang, P. S.

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 1–9 (2007).
[CrossRef]

P. S. Huang, Q. Y. J. Hu, and F. P. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41, 4503–4508 (2002).
[CrossRef]

Huntley, J. M.

Kinell, L.

Kowarschik, R.

Li, Z. W.

Z. W. Li, C. J. Wang, Y. S. Shi, and Y. Y. Wang, “High precision phase error compensation algorithm for structural light measurement,” Acta Opt. Sin. 28, 1527–1532 (2008).
[CrossRef]

Maack, T.

Niu, H.

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

Notni, G.

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[CrossRef]

Notni, G. H.

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[CrossRef]

Peng, X.

J. D. Tian, X. Peng, and X. B. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46, 336–342 (2008).
[CrossRef]

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

Reich, C.

C. Reich, R. Ritter, and J. Thesing, “White light heterodyne principle for 3D-measurement,” Proc. SPIE 3100, 236–224 (1997).
[CrossRef]

Ritter, R.

C. Reich, R. Ritter, and J. Thesing, “White light heterodyne principle for 3D-measurement,” Proc. SPIE 3100, 236–224 (1997).
[CrossRef]

Saldner, H. O.

Schmit, J.

Shi, Y. S.

Z. W. Li, C. J. Wang, Y. S. Shi, and Y. Y. Wang, “High precision phase error compensation algorithm for structural light measurement,” Acta Opt. Sin. 28, 1527–1532 (2008).
[CrossRef]

Sjodahl, M.

Su, X.-Y.

Z.-H. Xu and X.-Y. Su, “An algorithm of temporal phase unwrapping,” J. Sichuan University (Natural Science Edition) 45, 537–540 (2008).
[CrossRef]

Tan, Y. S.

Thesing, J.

C. Reich, R. Ritter, and J. Thesing, “White light heterodyne principle for 3D-measurement,” Proc. SPIE 3100, 236–224 (1997).
[CrossRef]

Tian, J. D.

J. D. Tian, X. Peng, and X. B. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46, 336–342 (2008).
[CrossRef]

Trolard, B.

B. Trolard, “Speckle noise removal in interference fringes by optoelectronic preprocessing with Epson liquid crystal television,” Proc. SPIE 2860, 126–134 (1996).
[CrossRef]

Wang, C. J.

Z. W. Li, C. J. Wang, Y. S. Shi, and Y. Y. Wang, “High precision phase error compensation algorithm for structural light measurement,” Acta Opt. Sin. 28, 1527–1532 (2008).
[CrossRef]

Wang, Y. Y.

Z. W. Li, C. J. Wang, Y. S. Shi, and Y. Y. Wang, “High precision phase error compensation algorithm for structural light measurement,” Acta Opt. Sin. 28, 1527–1532 (2008).
[CrossRef]

Wizinowich, P. L.

Xu, Z.-H.

Z.-H. Xu and X.-Y. Su, “An algorithm of temporal phase unwrapping,” J. Sichuan University (Natural Science Edition) 45, 537–540 (2008).
[CrossRef]

Yang, Z.

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

Zhang, S.

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 1–9 (2007).
[CrossRef]

Zhao, H.

Zhao, X. B.

J. D. Tian, X. Peng, and X. B. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46, 336–342 (2008).
[CrossRef]

Acta Opt. Sin. (1)

Z. W. Li, C. J. Wang, Y. S. Shi, and Y. Y. Wang, “High precision phase error compensation algorithm for structural light measurement,” Acta Opt. Sin. 28, 1527–1532 (2008).
[CrossRef]

Appl. Opt. (8)

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

J. Sichuan University (Natural Science Edition) (1)

Z.-H. Xu and X.-Y. Su, “An algorithm of temporal phase unwrapping,” J. Sichuan University (Natural Science Edition) 45, 537–540 (2008).
[CrossRef]

Meas. Sci. Technol. (1)

J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[CrossRef]

Opt. Commun. (1)

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

Opt. Eng. (2)

C. R. Coggrave and J. M. Huntley, “Optimization of a shape measurement system based on spatial light modulators,” Opt. Eng. 39, 91–98 (2000).
[CrossRef]

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 1–9 (2007).
[CrossRef]

Opt. Lasers Eng. (1)

J. D. Tian, X. Peng, and X. B. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46, 336–342 (2008).
[CrossRef]

Proc. SPIE (3)

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[CrossRef]

C. Reich, R. Ritter, and J. Thesing, “White light heterodyne principle for 3D-measurement,” Proc. SPIE 3100, 236–224 (1997).
[CrossRef]

B. Trolard, “Speckle noise removal in interference fringes by optoelectronic preprocessing with Epson liquid crystal television,” Proc. SPIE 2860, 126–134 (1996).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Fourth-order nonlinearity errors of wrapped phase maps in reversed exponential sequence (s=64).

Fig. 2.
Fig. 2.

Distributions of nonlinear error in methods B and D: (a) method D, (b) method B.

Fig. 3.
Fig. 3.

Fourth-order nonlinear errors of averaged phase maps in simple averaging sequence (s=64).

Fig. 4.
Fig. 4.

Fourth-order nonlinear error of averaged phase values in phase averaging sequence.

Fig. 5.
Fig. 5.

Fringe sequence used in SETPU with PAM.

Fig. 6.
Fig. 6.

Fringe sequence used in TFHTPU with PAM.

Fig. 7.
Fig. 7.

Phase errors of methods A–E: (a) method A, (b) method C, (c) method D, (d) method E (n=6), (e) method B, (f) method E (n=22), (g) method E (n=64).

Fig. 8.
Fig. 8.

Fitted curves of phase error in methods A–E.

Fig. 9.
Fig. 9.

Error ratios of methods A–E.

Tables (1)

Tables Icon

Table 1. RMS Errors and Error Ratios of Methods A–E

Equations (29)

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x=M2(1+cos(ϕ+δ)),
f(x)=a+bx+εx2+εx3.
f(x)=a+b[M2(1+cos(ϕ+δ))]+ε[M2(1+cos(ϕ+δ))]2+ε[M2(1+cos(ϕ+δ))]3.
Ii=A+vcos(ϕ+δi)+wcos2(ϕ+δi)+pcos3(ϕ+δi)i=1,2,3,4,
δi=π/2×(i1)i=1,2,3,4.
tan(ϕ)=I4I2I1I3=vsin(ϕ)psin(3ϕ)vcos(ϕ)+pcos(3ϕ),
tan(Δϕ)=tan(ϕ)tan(ϕ)1+tan(ϕ)tan(ϕ)=psin(4ϕ)pcos(4ϕ)+v=sin(4ϕ)cos(4ϕ)+v/p=sin(4ϕ)cos(4ϕ)+k,
Δϕ=ϕϕ=arctan[sin(4ϕ)k](fork1).
δi=π/2×(i1)+π/4i=1,2,3,4,
Δϕ=arctan[sin(4ϕ+π)k]=arctan[sin(4ϕ)k].
ϕw(s)=arctan[I4I2I1I3].
ϕw(s)=arctan[I4I2I1I3]π/4.
{tu=tof original TPU exceptsu=1,2,,n1tv=s,s,s,,sv=n1+1,n1+2,,n1+m,
{tu=1,su=1,2tv=s,s,s,,sv=3,4,5,6,,n,
Φ˜(s)=Φ(t)×s/t=(ωt+εϕ)×s/t=ωs+εϕ×s/t.
{tu=59,64,59,64u=1,2,3,4tv=s,s,s,,sv=5,6,,n,
Φ(t)=ωt+εϕ,
ω^A=Φ(s)/s,
σA=σϕ/s.
ω^B=t=1stΦ(t)/t=1st2.
σB=6σϕ[s(s+1)(2s+1)]1/23σϕ/s3/2.
σC=3σϕ(4s21)1/23σϕ/(2s),
σD=σϕ[s2(log2s2/3)+2s(1/3)]1/2σϕ/(slog2s)fors1.
Φ(t)=ωs+εϕt=3,4,,n,
ω^F=t=3nΦ(t)/(ns).
σF=σϕsm=σϕs(n2),
Φ(t)=ωs+εϕt=1,2,,n.
ω^E=t=1nΦ(t)/(ns).
σE=σϕsn.

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