Abstract

This letter presents an improved phase referencing technique, called Method of Multiple References, for optical profilometry. Based on a lookup table, the method eliminates several major drawbacks of single-reference Fourier Transform Interferometry by enabling surface error correction for steep slopes and step discontinuities, and by allowing mapping of multiple discrete objects using a single image set. The algorithm is tested using a fiber optic coupler-based FTI system and shown to have RMS surface error less than 0.03mm.

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References

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  1. F. Chen, G. Brown, and M. M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
    [CrossRef]
  2. T. L. Pennington, H. Xiao, R. May, and A. Wang, “Miniaturized 3-D surface profilometer using a fiber optic coupler,” Opt. Laser Technol. 33(5), 313–320 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. T. L. Pennington, “Miniaturized 3-D Mapping System Using a Fiber Optic Coupler as a Young's Double Pinhole Interferometer,” PhD, Electrical Engineering, Virginia Tech, Blacksburg (2000).
  6. D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4(1), 267–280 (1987).
    [CrossRef]
  7. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28(16), 3268–3270 (1989).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  9. H. O. Saldner and J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36(2), 610–615 (1997).
    [CrossRef]
  10. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21(14), 2470 (1982).
    [CrossRef] [PubMed]
  11. W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983).
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2001 (1)

T. L. Pennington, H. Xiao, R. May, and A. Wang, “Miniaturized 3-D surface profilometer using a fiber optic coupler,” Opt. Laser Technol. 33(5), 313–320 (2001).
[CrossRef]

2000 (1)

F. Chen, G. Brown, and M. M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

1997 (1)

H. O. Saldner and J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36(2), 610–615 (1997).
[CrossRef]

1994 (1)

T. Judge, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21(4), 199–239 (1994).
[CrossRef]

1989 (1)

1987 (1)

1984 (1)

1983 (2)

1982 (1)

Brown, G.

F. Chen, G. Brown, and M. M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

Chen, F.

F. Chen, G. Brown, and M. M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

Cline, H. E.

Ghiglia, D. C.

Holik, A. S.

Huntley, J. M.

H. O. Saldner and J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36(2), 610–615 (1997).
[CrossRef]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28(16), 3268–3270 (1989).
[CrossRef] [PubMed]

Itoh, K.

Judge, T.

T. Judge, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21(4), 199–239 (1994).
[CrossRef]

Lorensen, W. E.

Macy, W. W.

Mastin, G. A.

May, R.

T. L. Pennington, H. Xiao, R. May, and A. Wang, “Miniaturized 3-D surface profilometer using a fiber optic coupler,” Opt. Laser Technol. 33(5), 313–320 (2001).
[CrossRef]

Mutoh, K.

Pennington, T. L.

T. L. Pennington, H. Xiao, R. May, and A. Wang, “Miniaturized 3-D surface profilometer using a fiber optic coupler,” Opt. Laser Technol. 33(5), 313–320 (2001).
[CrossRef]

Romero, L. A.

Saldner, H. O.

H. O. Saldner and J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36(2), 610–615 (1997).
[CrossRef]

Song, M. M.

F. Chen, G. Brown, and M. M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

Takeda, M.

Wang, A.

T. L. Pennington, H. Xiao, R. May, and A. Wang, “Miniaturized 3-D surface profilometer using a fiber optic coupler,” Opt. Laser Technol. 33(5), 313–320 (2001).
[CrossRef]

Xiao, H.

T. L. Pennington, H. Xiao, R. May, and A. Wang, “Miniaturized 3-D surface profilometer using a fiber optic coupler,” Opt. Laser Technol. 33(5), 313–320 (2001).
[CrossRef]

Appl. Opt. (5)

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

Opt. Eng. (2)

H. O. Saldner and J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36(2), 610–615 (1997).
[CrossRef]

F. Chen, G. Brown, and M. M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

Opt. Laser Technol. (1)

T. L. Pennington, H. Xiao, R. May, and A. Wang, “Miniaturized 3-D surface profilometer using a fiber optic coupler,” Opt. Laser Technol. 33(5), 313–320 (2001).
[CrossRef]

Opt. Lasers Eng. (1)

T. Judge, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21(4), 199–239 (1994).
[CrossRef]

Other (1)

T. L. Pennington, “Miniaturized 3-D Mapping System Using a Fiber Optic Coupler as a Young's Double Pinhole Interferometer,” PhD, Electrical Engineering, Virginia Tech, Blacksburg (2000).

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic of fiber the coupler-based Fourier Transform Interferometry setup used in this demonstration. (b) Image of fringes projected onto a flat calibration target.

Fig. 2
Fig. 2

Graphical representation of lookup table for MoMR, based on data taken from calibration of the fiber optic coupler-based FTI system in Fig. 1. The value of x0 (λ1 ) varies as the surface height is changed, but the peak difference Δx0 remains relatively constant. Nominal fringe orders, assigned during calibration, are listed above each data line.

Fig. 3
Fig. 3

(a) Interferogram projection on a multi-surface image. A single fixed reference fringe and its resulting limited mapping area are shown, as are the multiple alternate references allowed by MoMR. (b) Contour map demonstrating multi-surface reconstruction. Limestone particles (same as (a)) are bounded and profiled using 3 reference fringes.

Fig. 6
Fig. 6

(a) Uncorrected surface profile after initial processing with reference fringe located in center region (equivalent to fixed fringe referencing). (b) Surface profile after MoMR correction. The areas to the left of the steep slope (upper) and right of the step discontinuity (lower) are now correctly represented.

Fig. 4
Fig. 4

Simulated two-sided FFT of a single data row containing a localized 2:1 slope. A Hann window, symmetric about the center frequency of the fringe pattern, is used as the FTI filter. The presence of the steep slope adds wide sidebands to the center carrier, and portions of the signal are attenuated by the edges of the filter.

Fig. 5
Fig. 5

Determination of data ranges for MoMR surface error correction. Each reference fringe is attributed to a region of data on one side of an error feature, eliminating the propagation of errors. Reference regions are determined by (a) comparison with an adjacent error-free row, or (b) identification of discontinuities from the slope of the reconstructed surface map. The majority of real surfaces fall into case (a), but (b) applies to the test surface in Fig. 6.

Tables (1)

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Table 1 Comparison to physical measurement: MoMR-FTI accuracy and resolution

Equations (6)

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φ ( x , z ) = 4 π a λ × z ( x ) sin β + x cos β d P + x sin β z ( x ) cos β
φ ( x ) = tan 1 [ Im { c ( x ) } / Re { c ( x ) } ]
φ S ( N ) ( y ) = φ U ( x 0 ( y ) , y ) N 2 π φ r e f : N ( x , y ) = φ U ( x , y ) φ S ( N ) ( y )
f ( x ) = 1 2 π d φ d x 2 a λ d P [ cos β + d z d x sin β ]
Δ z max = λ d P 2 a sin β 1 S m m cot β     = 2.1 m m
φ e r r ( y ) = φ S ( N ) ( y ) φ S ( M ) ( y )

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