Abstract

The predictions of success rate and depth uncertainty for the negative exponential sequence used for temporal phase unwrapping of shape data are generalized to include the effect of a reduced sequence and speckle noise in single-channel and multichannel systems, respectively. To cope with the reduction of the sequence, a scaling factor is introduced. A thorough investigation is made of the performance of this algorithm, called the reduced temporal phase-unwrapping algorithm. Two different approaches are considered: a single-channel approach in which all the necessary images are acquired sequentially in time and a multichannel approach in which the three channels of a color CCD camera are used to carry the phase-stepped images for each fringe density in parallel. The performance of these two approaches are investigated by numerical simulations. The simulations are based on a physical model in which the speckle contrast, the fringe modulation, and random noise are considered the sources of phase errors. Expressions are found that relate the physical quantities to phase errors for the single-channel and the multichannel approaches. In these simulations the single-channel approach was found to be the most robust. Expressions that relate the measurement accuracy and the unwrapping reliability, respectively, with the reduction of the fringe sequence were also found. As expected, the measurement accuracy is not affected by a shorter fringe sequence, whereas a significant reduction in the unwrapping reliability is found as compared with the complete negative exponential sequence.

© 2001 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  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]
  4. H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
    [CrossRef]
  5. J. M. Huntley, H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
    [CrossRef]
  6. C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1999 (2)

C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
[CrossRef]

J. M. Huntley, “Simple model for image-plane polychromatic speckle contrast,” Appl. Opt. 38, 2212–2215 (1999).
[CrossRef]

1998 (1)

1997 (4)

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

H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a 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]

1994 (3)

1993 (1)

1990 (1)

Brophy, C. P.

Burke, J.

Chen, W.

Chiang, F. P.

P. S. Huang, Q. Ho, F. Jin, F. P. Chiang, “Color-encoded fringe projection and phase shifting for 3D surface contouring,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyimesi, SPIE3407, 477–482 (1998).

Coggrave, C. R.

C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
[CrossRef]

Dorsch, R. G.

Häusler, G.

Helmers, H.

Herrmann, J. M.

Ho, Q.

P. S. Huang, Q. Ho, F. Jin, F. P. Chiang, “Color-encoded fringe projection and phase shifting for 3D surface contouring,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyimesi, SPIE3407, 477–482 (1998).

Hu, Y.-Q.

Huang, P. S.

P. S. Huang, Q. Ho, F. Jin, F. P. Chiang, “Color-encoded fringe projection and phase shifting for 3D surface contouring,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyimesi, SPIE3407, 477–482 (1998).

Huntley, J. M.

C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
[CrossRef]

J. M. Huntley, “Simple model for image-plane polychromatic speckle contrast,” Appl. Opt. 38, 2212–2215 (1999).
[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]

H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a 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]

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

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

Jin, F.

P. S. Huang, Q. Ho, F. Jin, F. P. Chiang, “Color-encoded fringe projection and phase shifting for 3D surface contouring,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyimesi, SPIE3407, 477–482 (1998).

Saldner, H.

Saldner, H. O.

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

H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a 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]

Tan, Y.

Zhao, H.

Appl. Opt. (7)

J. Opt. Soc. Am. A (2)

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. (2)

C. R. Coggrave, J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).
[CrossRef]

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

Other (1)

P. S. Huang, Q. Ho, F. Jin, F. P. Chiang, “Color-encoded fringe projection and phase shifting for 3D surface contouring,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyimesi, SPIE3407, 477–482 (1998).

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Figures (6)

Fig. 1
Fig. 1

Basic idea of temporal phase unwrapping. The constant increase in phase in a given pixel as a function of the number of projected fringes is shown as a straight line. Marked with circles are the evenly sampled version in which no phase jumps larger than 2π are introduced; the solid circles correspond to the negative exponential sequence.

Fig. 2
Fig. 2

Basic triangulation setup for the measurement of shape. A fringe pattern is modulated by a video projector and projected onto the diffusely reflecting object. The irradiance of the projected pattern is acquired by a video camera positioned at an angle from the projector.

Fig. 3
Fig. 3

Random phase errors in the single-channel approach for the four-frame (circle) and the three-frame (cross) approaches. The lines indicate Eq. (10) for the two different approaches. The results for the four-frame approach have been translated 0.05 rad for display purposes. Only results considered successful are shown.

Fig. 4
Fig. 4

Random phase errors in the three-channel approach. The line indicates Eq. (11). Only results considered successful are shown.

Fig. 5
Fig. 5

Random errors for the fitted, s f , and unfitted, s ω, versions of the reduced temporal phase-unwrapping algorithm as a function of phase noise. Both full and reduced sequencies are included. Solid and dashed–dotted lines are the relations given by the assymptotical form of Eqs. (13) and (12), respectively. Circles, numerical results from the fringe sequence s = 16; x’s, results from the fringe sequence s = 64.

Fig. 6
Fig. 6

Success rate, S.R., as a function of the phase noise for different values of the scaling factor T. Discrete symbols are the results from the simulations, and the solid curves come from the expression in Eq. (14) for different values of T.

Tables (2)

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Table 1 Fringe Sequences Used in the Simulations

Tables Icon

Table 2 Parameter Settings Used in the Simulations

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

t=2k-1, s,
ΔΦus-ti, s-ti+1=UΔΦws-ti, s-ti+1, ΔΦus, s-ti,
ΔΦus, s-ti+1=ΔΦus, s-ti+ΔΦus-ti, s-ti+1,
UΦ1, Φ2=Φ1-2π NINTΦ1-TΦ22π
T=ti+1-titi
ωˆ=sΦus+i=1m-1s-tiΦus-tis2+i=1m-1s-ti2,
Iregj=ImodjIsj+2Rjj,
Imodj=1+M cosωtx+αj
Isj=1Ni=1ceilN k|a|i2,
k=N-floorNif i=ceilN1otherwise,
sΦ=Ra1C+a2M,
sΦ=C0.15+0.76M+R6.91R+0.24M-3.31C.
sw=sΦs
sf=sΦs2+i=1m-1s-ti21/2
S.R.=exp-sΦ3T2.14,
SΦgS.R., T=3-4 ln S.RT2.1,

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