Abstract

Recently a powerful Fourier transform technique was introduced that was able to extract the phase from only one image. However, because the method is based on the two-dimensional Fourier transform, it inherently suffers from leakage effects. A novel procedure is proposed that does not exhibit this distortion. The procedure uses localized information and estimates both the unknown frequencies and the phases of the fringe pattern (using an interpolated fast Fourier transform method). This allows us to demodulate the fringe pattern without any distortion. The proposed technique is validated on both computer simulations and the profile measurements of a tube.

© 2004 Optical Society of America

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References

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    [CrossRef]
  4. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  5. X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
    [CrossRef]
  6. J. Li, X. Su, L. Guo, “Improved Fourier-transform profilometry for automatic measurement of three-dimension object shapes,” Opt. Eng. 29, 1439–1444 (1990).
    [CrossRef]
  7. J. Yi, S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493–505 (1997).
    [CrossRef]
  8. M. Takeda, Q. Gu, M. Kinoshita, H. Takai, Y. Takahashi, “Frequency-multiplex Fourier-transform profilometry: a single-shot, three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations,” Appl. Opt. 36, 5347–5353 (1997).
    [CrossRef] [PubMed]
  9. J. Lin, X. Su, “Two-dimensional Fourier-transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 11, 3297–3302 (1995).
  10. P. F. Panter, Modulation, Noise and Spectral Analysis (McGraw-Hill, New York, 1965).
  11. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. 13, 2355–2366 (1996).
    [CrossRef]
  12. A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
    [CrossRef]
  13. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), Chap. 6, pp. 194–229.
  14. H. Renders, J. Schoukens, G. Vilain, “High-accuracy spectrum analysis of sampled discrete frequency signals by analytical leakage compensation,” IEEE Trans. Instrum. Meas. IM-33, 287–292 (1984).
    [CrossRef]
  15. V. K. Jain, W. L. Collins, D. C. Davis, “High-accuracy analog measurements via interpolated FFT,” IEEE Trans. Instrum. Meas. IM-28, 113–122 (1979).
    [CrossRef]

2002 (1)

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

2001 (1)

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

1997 (2)

1996 (1)

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. 13, 2355–2366 (1996).
[CrossRef]

1995 (1)

J. Lin, X. Su, “Two-dimensional Fourier-transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 11, 3297–3302 (1995).

1990 (1)

J. Li, X. Su, L. Guo, “Improved Fourier-transform profilometry for automatic measurement of three-dimension object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

1984 (1)

H. Renders, J. Schoukens, G. Vilain, “High-accuracy spectrum analysis of sampled discrete frequency signals by analytical leakage compensation,” IEEE Trans. Instrum. Meas. IM-33, 287–292 (1984).
[CrossRef]

1983 (1)

1982 (1)

1979 (1)

V. K. Jain, W. L. Collins, D. C. Davis, “High-accuracy analog measurements via interpolated FFT,” IEEE Trans. Instrum. Meas. IM-28, 113–122 (1979).
[CrossRef]

1970 (2)

Allen, J. B.

Baldi, A.

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

Bertolino, F.

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

Chen, W.

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

Collins, W. L.

V. K. Jain, W. L. Collins, D. C. Davis, “High-accuracy analog measurements via interpolated FFT,” IEEE Trans. Instrum. Meas. IM-28, 113–122 (1979).
[CrossRef]

Davis, D. C.

V. K. Jain, W. L. Collins, D. C. Davis, “High-accuracy analog measurements via interpolated FFT,” IEEE Trans. Instrum. Meas. IM-28, 113–122 (1979).
[CrossRef]

Fornaro, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. 13, 2355–2366 (1996).
[CrossRef]

Franceschetti, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. 13, 2355–2366 (1996).
[CrossRef]

Ginesu, F.

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

Gu, Q.

Guo, L.

J. Li, X. Su, L. Guo, “Improved Fourier-transform profilometry for automatic measurement of three-dimension object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Huang, S.

J. Yi, S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493–505 (1997).
[CrossRef]

Ina, H.

Jain, V. K.

V. K. Jain, W. L. Collins, D. C. Davis, “High-accuracy analog measurements via interpolated FFT,” IEEE Trans. Instrum. Meas. IM-28, 113–122 (1979).
[CrossRef]

Johnson, W. O.

Kinoshita, M.

Koboyashi, S.

Lanari, R.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. 13, 2355–2366 (1996).
[CrossRef]

Li, J.

J. Li, X. Su, L. Guo, “Improved Fourier-transform profilometry for automatic measurement of three-dimension object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Lin, J.

J. Lin, X. Su, “Two-dimensional Fourier-transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 11, 3297–3302 (1995).

Meadows, D. M.

Mutoh, K.

Panter, P. F.

P. F. Panter, Modulation, Noise and Spectral Analysis (McGraw-Hill, New York, 1965).

Renders, H.

H. Renders, J. Schoukens, G. Vilain, “High-accuracy spectrum analysis of sampled discrete frequency signals by analytical leakage compensation,” IEEE Trans. Instrum. Meas. IM-33, 287–292 (1984).
[CrossRef]

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), Chap. 6, pp. 194–229.

Sansosti, E.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. 13, 2355–2366 (1996).
[CrossRef]

Schoukens, J.

H. Renders, J. Schoukens, G. Vilain, “High-accuracy spectrum analysis of sampled discrete frequency signals by analytical leakage compensation,” IEEE Trans. Instrum. Meas. IM-33, 287–292 (1984).
[CrossRef]

Su, X.

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

J. Lin, X. Su, “Two-dimensional Fourier-transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 11, 3297–3302 (1995).

J. Li, X. Su, L. Guo, “Improved Fourier-transform profilometry for automatic measurement of three-dimension object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Takahashi, Y.

Takai, H.

Takasaki, H.

Takeda, M.

Vilain, G.

H. Renders, J. Schoukens, G. Vilain, “High-accuracy spectrum analysis of sampled discrete frequency signals by analytical leakage compensation,” IEEE Trans. Instrum. Meas. IM-33, 287–292 (1984).
[CrossRef]

Yi, J.

J. Yi, S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493–505 (1997).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Instrum. Meas. (2)

H. Renders, J. Schoukens, G. Vilain, “High-accuracy spectrum analysis of sampled discrete frequency signals by analytical leakage compensation,” IEEE Trans. Instrum. Meas. IM-33, 287–292 (1984).
[CrossRef]

V. K. Jain, W. L. Collins, D. C. Davis, “High-accuracy analog measurements via interpolated FFT,” IEEE Trans. Instrum. Meas. IM-28, 113–122 (1979).
[CrossRef]

J. Opt. Soc. Am. (2)

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. 13, 2355–2366 (1996).
[CrossRef]

M. Takeda, H. Ina, S. Koboyashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Opt. Eng. (2)

J. Li, X. Su, L. Guo, “Improved Fourier-transform profilometry for automatic measurement of three-dimension object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

J. Lin, X. Su, “Two-dimensional Fourier-transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 11, 3297–3302 (1995).

Opt. Lasers Eng. (3)

J. Yi, S. Huang, “Modified Fourier transform profilometry for the measurement of 3-D steep shapes,” Opt. Lasers Eng. 27, 493–505 (1997).
[CrossRef]

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

Other (2)

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), Chap. 6, pp. 194–229.

P. F. Panter, Modulation, Noise and Spectral Analysis (McGraw-Hill, New York, 1965).

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Figures (13)

Fig. 1
Fig. 1

Simulated height distribution h(x, y).

Fig. 2
Fig. 2

(a) Intensity image of the projected fringes with modulation index β = 1.5, (b) frequency spectrum of the intensity image in (a).

Fig. 3
Fig. 3

Height demodulated from the intensity image in Fig. 2: (a) classical 2-D FFT technique, (b) interpolated FFT technique, (c) slice of the image in (a) at x = 0.125, and (d) slice of the image in (b) at x = 0.125. The dotted curves in (c) and (d) represent the true height distribution.

Fig. 4
Fig. 4

(a) Intensity image of the projected fringes with modulation index β = 0.3 and carrier frequency f X = 15.5, (b) frequency spectrum of the intensity image in (a).

Fig. 5
Fig. 5

Height demodulated from the intensity image in Fig. 4: (a) classical 2-D FFT technique, (b) interpolated FFT technique, (c) slice of the image in (a) at x = 0.125, and (d) slice of the image in (b) at x = 0.125. The dotted curves in (c) and (d) represent the true height distribution.

Fig. 6
Fig. 6

(a) Intensity image of the projected fringes with modulation index β = 0.3, carrier frequency f X = 15, and a sinusoidal height distribution with f x = 2.5 and f y = 2.5; (b) frequency spectrum of the intensity image in (a).

Fig. 7
Fig. 7

Height demodulated from the intensity image in Fig. 6: (a) classical FFT technique, (b) interpolated FFT technique, (c) slice of the image in (a) at x = 0.125, and (d) slice of the image in (b) at x = 0.125. The dotted curves in (c) and (d) represent the true height distribution.

Fig. 8
Fig. 8

(a) Height distribution with a step (same as in Fig. 1 except that the height in the lower half part is set to zero), (b) intensity image of the projected fringes in (a), (c) values of the validation criterium R k,l of the proposed interpolated Fourier-transform technique [see Eq. (10)], and (d) locations with a value above 10 shown in black.

Fig. 9
Fig. 9

Height demodulated from the intensity image in Fig. 8(b): (a) classical FFT technique, (b) interpolated FFT technique (the black area around the step denotes the region where no height value is available), (c) slice of the image in (a) at x = 0.125, and (d) slice of the image in (b) at x = 0.125. The dotted curves in (c) and (d) represent the true height distribution.

Fig. 10
Fig. 10

(a) Intensity image with a sinusoidally varying fringe visibility between 0.3 (border) and 1 (image center) and (b) frequency spectrum of the intensity image in (a).

Fig. 11
Fig. 11

Height demodulated from the intensity image in Fig. 10: (a) classical 2-D FFT technique, (b) interpolated FFT technique, (c) slice of the image in (a) at x = 0.125, (d) slice of the image in (b) at x = 0.125. The dotted curves in (c) and (d) represent the true height distribution.

Fig. 12
Fig. 12

(a) Intensity image of the tube with projected fringes, and (b) frequency spectrum of the intensity image in (a).

Fig. 13
Fig. 13

Height demodulated from the intensity image in Fig. 12: (a) classical FFT technique, (b) interpolated FFT technique, (c) slice of (a) at Y = 50, and (d) slice of (b) at Y = 50.

Equations (17)

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

ix, y=rx, y1+cos2πfXx+hx, ytan θ,
hx, y= ϕx, y2πfX tan θ.
Ik, l=m=0M-1n=0N-1 ixm, ynexp-2πιkm/M+ln/N.
i¯xm, yn=1MNk=0M-1l=0N-1I¯k, lexp2πιkm/M+ln/N.
ϕ x, y=arctani¯x, yi¯x, y,
ix= 1Mk=0M-1 Ikexp 2πιkx/M,
ix=A2exp2πιfXx+exp-2πιfXx.
ik,lx=Ak,l sin2πfk,lx+ϕk,l.
Ik,lf=-0.5Ak,lexpιank,l-f+ϕk,l× sinπnk,l-fsinπnk,l-f/M0-exp-ιank,l+f+ϕk,lsinπnk,l+fsinπnk,l+f/M0,
Sx=sinaxsinπxsin πx/M0, Cx=cosaxsinπxsin πx/M0, Ik,lf=Ufk,l+ιVfk,l,  Ak,l expιϕk,l=Xk,l+ιYk,l,
Cnk,l-f-Cnk,l+fXk,l-Snk,l-f-Snk,l+fYk,l=-2Vfk,l, Snk,l-f+Snk,l+fXk,l+Cnk,l-f+Cnk,l+fYk,l=+2Ufk,l.
cospf-cospnk,lcospf+1-cospnk,lsinpf+1sinpf= Uf+1k,lsinpf+1-Vf+1k,lcospf+1Ufk,lsinpf-Vfk,lcospf= Vf+1k,lVfk,l,
nk,l= 1narccosZ2k,lcosnf+1-Z1k,lcosnfZ2k,l-Z1k,l,
Z1k,l=Vfk,lKk,l-cosnfsinnf+Ufk,l, Z2k,l=Vf+1k,lKk,l-cosnf+1sinnf+1+Uf+1k,l, Kk,l= sinnfVf+1k,l-Vfk,l+cosnfUf+1k,l-Ufk,lUf+1k,l-Ufk,l.
ik,lestx=Ak,l sin2πfk,lx+ϕk,l,
Rk,l=100 j=1M0 rk,lxjj=1M0 ik,lxj.
hx, y=cos2πfxxsin2πfyy

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