Abstract

Temporal phase unwrapping is a method of analyzing fringe patterns in which the fringe phase Φ at each pixel is measured and unwrapped as a function of time t. We propose two methods for improving the signal-to-noise ratio of the total phase change by incorporating data from the intermediate phase values. The first involves fitting the expected curve to the measured phase history; the second involves Fourier transformation of the corresponding phasors. The performance of these methods is compared both experimentally, with data from a fringe projector based on a spatial-light modulator, and numerically. It is shown that the optimum performance is given by the Fourier transform method. The best way to use the first method is with Φ decreasing exponentially with time from its maximum value to zero; this provides significant improvements in reliability, accuracy, and computation time compared with the original temporal unwrapping algorithm.

© 1997 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
  5. H. O. Saldner, J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
    [CrossRef] [PubMed]
  6. H. O. Saldner, J. M. Huntley, “Profilometry by temporal phase unwrapping and spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
    [CrossRef]
  7. J. M. Huntley, H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
    [CrossRef]
  8. H. Zhao, W. Chen, Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33, 4497–4500 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. J. Nygårds, Å. Wernersson, “Specular objects in range cameras: reducing ambiguities by motion,” in Proceedings of the 1994 IEEE International Conference on Multisensor Fusion and Integration (IEEE Press, Piscataway, N.J., 1994), pp. 320–329.
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1997 (3)

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

H. O. Saldner, J. M. Huntley, “Profilometry by temporal phase unwrapping and spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

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

1996 (1)

W. Nadeborn, P. Andrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

1995 (1)

1994 (2)

H. Zhao, W. Chen, Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33, 4497–4500 (1994).
[CrossRef] [PubMed]

T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers. Eng. 21, 199–239 (1994).
[CrossRef]

1993 (1)

1990 (1)

1986 (1)

J. M. Huntley, “An image processing system for the analysis of speckle photographs,” J. Phys. E 19, 43–48 (1986).
[CrossRef]

1982 (1)

1974 (1)

Andrä, P.

W. Nadeborn, P. Andrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Brangaccio, D. J.

Brophy, C. P.

Bruning, J. H.

Bryanston-Cross, P. J.

T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers. Eng. 21, 199–239 (1994).
[CrossRef]

Chen, W.

de Groot, P.

Gallagher, J. E.

Herriott, D. R.

Huntley, J. M.

H. O. Saldner, J. M. Huntley, “Profilometry by temporal phase unwrapping and spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

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

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

J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

J. M. Huntley, “An image processing system for the analysis of speckle photographs,” J. Phys. E 19, 43–48 (1986).
[CrossRef]

Ina, H.

Judge, T. R.

T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers. Eng. 21, 199–239 (1994).
[CrossRef]

Kobayashi, S.

Nadeborn, W.

W. Nadeborn, P. Andrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Nygårds, J.

J. Nygårds, Å. Wernersson, “Specular objects in range cameras: reducing ambiguities by motion,” in Proceedings of the 1994 IEEE International Conference on Multisensor Fusion and Integration (IEEE Press, Piscataway, N.J., 1994), pp. 320–329.

Osten, W.

W. Nadeborn, P. Andrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Rosenfeld, D. P.

Saldner, H. O.

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

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

H. O. Saldner, J. M. Huntley, “Profilometry by temporal phase unwrapping and spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

Takeda, M.

Tan, Y.

Wernersson, Å.

J. Nygårds, Å. Wernersson, “Specular objects in range cameras: reducing ambiguities by motion,” in Proceedings of the 1994 IEEE International Conference on Multisensor Fusion and Integration (IEEE Press, Piscataway, N.J., 1994), pp. 320–329.

White, A. D.

Zhao, H.

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

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

J. Phys. E (1)

J. M. Huntley, “An image processing system for the analysis of speckle photographs,” J. Phys. E 19, 43–48 (1986).
[CrossRef]

Meas. Sci. Technol. (1)

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

Opt. Eng. (1)

H. O. Saldner, J. M. Huntley, “Profilometry by temporal phase unwrapping and spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

Opt. Lasers Eng. (1)

W. Nadeborn, P. Andrä, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Opt. Lasers. Eng. (1)

T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers. Eng. 21, 199–239 (1994).
[CrossRef]

Other (1)

J. Nygårds, Å. Wernersson, “Specular objects in range cameras: reducing ambiguities by motion,” in Proceedings of the 1994 IEEE International Conference on Multisensor Fusion and Integration (IEEE Press, Piscataway, N.J., 1994), pp. 320–329.

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Figures (9)

Fig. 1
Fig. 1

Fringe projector for shape measurement with use of synthetic fringes from a spatial-light modulator (SLM). The lines from the projector are cross sections through planes of constant total phase change Ψ, produced by scanning through a total of s fringe patterns.

Fig. 2
Fig. 2

Measured intensity at three pixels [(a), (b), and (c)] as a function of phase-shift index (p, vertical axis) and time (t, horizontal axis).

Fig. 3
Fig. 3

Unwrapped time histories for the three pixels of Fig. 2.

Fig. 4
Fig. 4

Contour maps of the two-dimensional Fourier transforms of the intensity distributions for three pixels shown in Fig. 2. The horizontal and vertical axes are kt and kp, representing spatial frequencies along the t (time) and p (phase-shift index) axes, respectively.

Fig. 5
Fig. 5

Cross sections of the contour maps from Fig. 4 along the line kp=1. The position of the peak provides a direct measure of ω. Vertical axis: 500 arbitrary units corresponds to the contour interval in Fig. 4.

Fig. 6
Fig. 6

Estimated value of ω versus pixel position from measurements on a flat plate. (a)–(e) represent results from methods A–E, respectively.

Fig. 7
Fig. 7

Reliability of unwrapping as a function of ω/π for five methods. Exposure times, (a) 200 ms and (b) 80 ms, correspond to the intermediate- and low-modulation data, respectively.

Fig. 8
Fig. 8

Random errors in estimated value of ω, normalized by the phase error σϕ, as a function of s, for five analysis methods. The discrete symbols are the results of numerical simulations; symbols used are identical to those in Fig. 7. The lines are the theoretical results derived in Sections 3 and 4.

Fig. 9
Fig. 9

Reliability of each of the five methods, as a function of s, for a signal-to-noise ratio of 0.2. The symbols used are identical to those in Fig. 7.

Tables (2)

Tables Icon

Table 1 Theoretical Data Acquisition Times, Computational Effort (Number of Operations Per Pixel), and Random Errors for the Five Analysis Methods A–E

Tables Icon

Table 2 Rms Errors in Calculated ω Values (σA-σE) for Five Analysis Methods A-E, from Data Obtained under Three Different Exposure Levels, Expressed As a Fraction of the Error for Method A, σA

Equations (34)

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Φu(t)=t=1tΔΦw(t, t-1),
ΔΦ(i, j)=Φ(i)-Φ(j)
ΔΦw(i, j)=tan-1ΔI42(i)ΔI13(j)-ΔI13(i)ΔI42(j)ΔI13(i)ΔI13(j)+ΔI42(i)ΔI42(j),
ΔIkl(t)=I(k, t)-I(l, t).
Ψ=Φu(s)
ΔΦu(2t, t)=U[ΔΦw(2t, t), ΔΦu(t, 0)],
ΔΦu(2t, 0)=ΔΦu(2t, t)+ΔΦu(t, 0)
U[Φ1,Φ2]=Φ1-2πNINTΦ1-Φ22π
Φ(t)=ωt+ϕ,
ωˆA=Φu(s)/s,
ωˆB=t=1stΦu(t)t=1st2.
σA=σϕ/s.
σB=6σϕ[s(s+1)(2s+1)]1/23σϕ/s3/2fors1.
ωˆB=t=1sΩB(t)ΔΦw(t, t-1),
ΩB(t)=t=tstt=1st2.
ωˆC=v=0log2 s2vΦu(2v)v=0log2 s22v.
σC=3σϕ(4s2-1)1/23σϕ/2sfors1.
ΔΦu(s-t, s-2t)=U[ΔΦw(s-t, s-2t),ΔΦu(s, s-t)],
ΔΦu(s, s-2t)=ΔΦu(s-t, s-2t)+ΔΦu(s, s-t)
ωˆD=sΦu(s)v=0log2 s-1(s-2v)Φu(s-2v)s2+v=0log2 s-1(s-2v)2.
σD=σϕ[s2(log2 s-2/3)+2s-(1/3)]1/2σϕ/slog2 sfors1.
ωˆD=ΩD(s)ΔΦu(s, s-1)+ν=0log2 s-1ΩD(s-2ν)×ΔΦu(s-2ν, s-2ν+1),
ΩD(t)=s+ν=0ν(s-2ν)
s2+ν=0log2 s-1(s-2ν)2
ΩD(s)=ss2+ν=0log2 s-1(s-2ν)2.
H(kp, kt)=t=1sp=1qI(p, t)exp(-2πi{[kp(p-1)/q]+[kt(t-1)/s]}),
H(1, kt)=t=1sh(t)exp[-2πikt(t-1)/s],
h(t)=ΔI13(t)+iΔI42(t).
ωˆE=2πκ/s.
I(p,t)=cos[ωt+2π(p-1)/4]+I,
t=1, 2, , s,p=1, 2, 3, 4, 
D=M1.2λL,
su=L/1.2λ,
σhσϕL1/4λ3/4.

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