Abstract

The performance of Fourier-transform profilometry is enhanced by a new technique that is based on spatial frequency multiplexing combined with the Gushov–Solodkin phase unwrapping algorithm. The technique permits the three-dimensional shape measurement of objects that have discontinuous height steps and/or spatially isolated surfaces, which has not been possible by conventional Fourier-transform profilometry. An important feature of the technique is that it requires only a single fringe pattern; the single-shot recording makes possible the instantaneous three-dimensional shape measurement of discontinuous objects in fast motion. Experimental results are presented that demonstrate the validity of the principle.

© 1997 Optical Society of America

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References

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  1. See, for example, H.-J. Tiziani, “Optical techniques for shape measurements,” in Proceedings of the 2nd International Workshop on Automatic Processing of Fringe Patterns, Fringe ’93, W. Jüptner, W. Osten, eds. (Akademie, Berlin, 1993), pp. 165–174.
  2. See, for example, D. M. Meadows, W. O. Johnson, J. B. Allen, “Generation of surface contours by moiré patterns,” Appl. Opt. 9, 942–947 (1970); H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
  3. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  4. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  5. M. Takeda, H. Yamamoto, “Fourier-transform speckle profilometry: three-dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. 33, 7829–7837 (1994).
    [CrossRef] [PubMed]
  6. See, for example, E. Müller, “Fast three-dimensional form measurement,” Opt. Eng. 34, 2754–2755 (1995).
  7. M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, England, 1964), pp. 290–291.
  8. M. Takeda, M. Kitoh, “Spatiotemporal frequency multiplex heterodyne interferometry,” J. Opt. Soc. Am. A 9, 1607–1614 (1992).
    [CrossRef]
  9. J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
    [CrossRef]
  10. V. I. Gushov, Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
    [CrossRef]
  11. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  12. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  13. D. R. Burton, M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. 33, 2939–2948 (1994).
    [CrossRef] [PubMed]
  14. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  15. X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
    [CrossRef]

1995

See, for example, E. Müller, “Fast three-dimensional form measurement,” Opt. Eng. 34, 2754–2755 (1995).

1994

1993

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

1992

1991

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

V. I. Gushov, Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
[CrossRef]

1989

J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

1987

1983

1982

1970

Allen, J. B.

Bone, D. J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, England, 1964), pp. 290–291.

Burton, D. R.

Dresel, T.

Field, J. E.

J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

Gushov, V. I.

V. I. Gushov, Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
[CrossRef]

Häusler, G.

Huntley, J. M.

J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

Ina, H.

Johnson, W. O.

Kitoh, M.

Kobayashi, S.

Lalor, M. J.

Meadows, D. M.

Müller, E.

See, for example, E. Müller, “Fast three-dimensional form measurement,” Opt. Eng. 34, 2754–2755 (1995).

Mutoh, K.

Roddier, C.

Roddier, F.

Solodkin, Y. N.

V. I. Gushov, Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
[CrossRef]

Su, X.-Y.

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Takeda, M.

Tiziani, H.-J.

See, for example, H.-J. Tiziani, “Optical techniques for shape measurements,” in Proceedings of the 2nd International Workshop on Automatic Processing of Fringe Patterns, Fringe ’93, W. Jüptner, W. Osten, eds. (Akademie, Berlin, 1993), pp. 165–174.

Venzke, H.

von Bally, G.

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Vukicevic, D.

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, England, 1964), pp. 290–291.

Yamamoto, H.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

X.-Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Opt. Eng.

J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

See, for example, E. Müller, “Fast three-dimensional form measurement,” Opt. Eng. 34, 2754–2755 (1995).

Opt. Lasers Eng.

V. I. Gushov, Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, England, 1964), pp. 290–291.

See, for example, H.-J. Tiziani, “Optical techniques for shape measurements,” in Proceedings of the 2nd International Workshop on Automatic Processing of Fringe Patterns, Fringe ’93, W. Jüptner, W. Osten, eds. (Akademie, Berlin, 1993), pp. 165–174.

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Figures (13)

Fig. 1
Fig. 1

Spatial frequency multiplexing for the simultaneous acquisition of multiple phase maps with different phase sensitivities; a simple example for twofold multiplexing (K = 2).

Fig. 2
Fig. 2

Height distributions of the object.

Fig. 3
Fig. 3

Simulated fringe pattern for twofold spatial-frequency multiplexing. The periods of the carrier frequencies in the horizontal x direction, 1/fX1 and 1/fX2, are made to be proportional to the relative prime numbers m1 = 3 and m2 = 7.

Fig. 4
Fig. 4

Fourier spectra obtained from the simulated fringe pattern.

Fig. 5
Fig. 5

Wrapped height maps obtained by the Fourier transform method; heights are (A) wrapped moduli Δh1 = 3 mm and (B) Δh2 = 7 mm.

Fig. 6
Fig. 6

Height maps obtained by (a) direct application of the G–S algorithm and (b) the modified G–S algorithm.

Fig. 7
Fig. 7

Spatially frequency-multiplexed fringe pattern projected upon a reference plane (magnified view).

Fig. 8
Fig. 8

Fringe pattern obtained for the combination of continuous and discontinuous objects.

Fig. 9
Fig. 9

Multiplexed spectra separated in the spatial frequency domain.

Fig. 10
Fig. 10

(A) Height distribution wrapped modulo 3 units, (B) the same height distribution wrapped modulo 7 units.

Fig. 11
Fig. 11

Unwrapped height distribution obtained by the G–S method combined with the adaptive height-offset method.

Fig. 12
Fig. 12

Mask generated with a combined criterion of fringe modulations and phase gradients.

Fig. 13
Fig. 13

Final result obtained by the combination of the proposed techniques.

Equations (24)

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Xb1mod m1Xb2mod m2XbKmod mK,
XM1M1b1+M2M2b2++MKMKbK  mod m,
m=m1m2  mK,
mkMk=m,
MkMk1  mod mk,  k=1, 2, , K.
gx, y=rx, yk=1K1+cos2πfX,kx+hx, ytan θ+2πfY,ky
=rx, yk=1K1+cos2πfX,kx+fY,ky+ϕkx, y,
ϕkx, y=2πfX,khx, ytan θ.
hx, yh1x, ymod Δh1hx, yh2x, ymod Δh2hx, yhKx, ymod ΔhK,
hkx, y=ϕkx, y+π2πfX,k tan θ,
Δhk=1fX,k tan θ.
1/fX,1m1=1/fX,2m2==1/fX,KmK=α tan θ  a constant,
Xˆ=hx, y/α  a real number,
bˆk=hkx, y/α  a real number,
mk=Δhk/α  an integer.
Xˆ=X+d,
bˆk=bk+d.
X+db1+dmod m1X+db2+dmod m2   X+dbK+dmod mk,
Xˆ=X+dM1M1b1++MKMKbK+d  mod m.
hx, y=αXˆ.
ΔXˆ=M1M1Δb1++MKMKΔbK.
gkx, y=1/2rx, yexpi2πfX,kx+fY,ky+ϕkx, y.
ĝ0,kx, y=1/2r0x, yexpi2πfX,kx+fY,ky+ϕ0,kx, y,
ĝkx, yĝ0,k*x, y=1/4rx, yr0x, y×expiϕkx, y-ϕ0,kx, y,

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