Abstract

An interferometric technique for automated profilometry of diffuse objects has been proposed. It is based on the Fourier-fringe analysis of spatiotemporal specklegrams produced by a wavelength-shift interferometer with a laser diode as a frequency-tunable light source. Unlike conventional moiré techniques the proposed technique permits the objects to have discontinuous height steps and/or surfaces spatially isolated from one another. Experimental results are presented that demonstrate the validity of the principle.

© 1994 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 Second International Workshop on Automatic Processing of Fringe Patterns, Fringe '93, W. Jüptner, W. Osten, eds. (Akademie Verlag, 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).
    [Crossref] [PubMed]
  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. See, for example, G. Häusler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
    [Crossref] [PubMed]
  5. See, for example, G. Seitz, H.-J. Tiziani, “Resolution limits of active triangulation systems by defocusing,” Opt. Eng. 32, 1374–1383 (1993).
    [Crossref]
  6. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [Crossref] [PubMed]
  7. 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]
  8. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [Crossref] [PubMed]
  9. See, for example, M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
    [Crossref]
  10. J. E. Calatroni, P. Sandoz, G. Tribillon, “Surface profiling by means of double spectral modulation,” Appl. Opt. 32, 30–37 (1993).
    [Crossref] [PubMed]
  11. A. A. M. Maas, “Shape measurement using phase shifting speckle interferometry,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 558–568 (1991).
  12. A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with two-wavelength heterodyne speckle interferometers,” Appl. Opt. 24, 2181–2188 (1985).
    [Crossref] [PubMed]
  13. See, for example, Y. Ishii, “Recent developments in laser-diode interferometry,” Opt. Lasers Eng. 14, 293–309 (1991).
    [Crossref]
  14. See, for example, K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985);S. Nakadate, H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985).
    [Crossref] [PubMed]
  15. M. Takeda, M. Kitoh, “Spatiotemporal frequency-multiplex heterodyne interferometry,” J. Opt. Soc. Am. A 7, 1607–1614 (1992).
    [Crossref]
  16. M. Suematsu, M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier-transform technique for fringe analysis,” Appl. Opt. 30, 4046–4055 (1991).
    [Crossref] [PubMed]
  17. S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
    [Crossref]
  18. See, for example, K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
    [Crossref] [PubMed]
  19. K. Hotate, T. Okugawa, “Selective extraction of a two-dimensional optical image by synthesis of the coherence function,” Opt. Lett. 17, 1529–1531 (1992).
    [Crossref] [PubMed]

1993 (2)

See, for example, G. Seitz, H.-J. Tiziani, “Resolution limits of active triangulation systems by defocusing,” Opt. Eng. 32, 1374–1383 (1993).
[Crossref]

J. E. Calatroni, P. Sandoz, G. Tribillon, “Surface profiling by means of double spectral modulation,” Appl. Opt. 32, 30–37 (1993).
[Crossref] [PubMed]

1992 (3)

1991 (2)

1990 (1)

See, for example, M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[Crossref]

1988 (1)

1987 (1)

1985 (3)

1983 (1)

1982 (2)

1970 (1)

Allen, J. B.

Calatroni, J. E.

Creath, K.

Davies, D. E. N.

S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
[Crossref]

Dresel, T.

Fercher, A. F.

Häusler, G.

Heckel, W.

Hotate, K.

Hu, H. Z.

Ina, H.

Ishii, Y.

See, for example, Y. Ishii, “Recent developments in laser-diode interferometry,” Opt. Lasers Eng. 14, 293–309 (1991).
[Crossref]

Itoh, K.

Johnson, W. O.

Kingsley, S. A.

S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
[Crossref]

Kitoh, M.

M. Takeda, M. Kitoh, “Spatiotemporal frequency-multiplex heterodyne interferometry,” J. Opt. Soc. Am. A 7, 1607–1614 (1992).
[Crossref]

Kobayashi, S.

Maas, A. A. M.

A. A. M. Maas, “Shape measurement using phase shifting speckle interferometry,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 558–568 (1991).

Meadows, D. M.

Mutoh, K.

Okugawa, T.

Roddier, C.

Roddier, F.

Sandoz, P.

Seitz, G.

See, for example, G. Seitz, H.-J. Tiziani, “Resolution limits of active triangulation systems by defocusing,” Opt. Eng. 32, 1374–1383 (1993).
[Crossref]

Suematsu, M.

Takeda, M.

Tiziani, H.-J.

See, for example, G. Seitz, H.-J. Tiziani, “Resolution limits of active triangulation systems by defocusing,” Opt. Eng. 32, 1374–1383 (1993).
[Crossref]

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

Tribillon, G.

Venzke, H.

Vry, U.

Appl. Opt. (10)

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).
[Crossref] [PubMed]

M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
[Crossref] [PubMed]

See, for example, G. Häusler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
[Crossref] [PubMed]

T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
[Crossref] [PubMed]

C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
[Crossref] [PubMed]

J. E. Calatroni, P. Sandoz, G. Tribillon, “Surface profiling by means of double spectral modulation,” Appl. Opt. 32, 30–37 (1993).
[Crossref] [PubMed]

A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with two-wavelength heterodyne speckle interferometers,” Appl. Opt. 24, 2181–2188 (1985).
[Crossref] [PubMed]

See, for example, K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985);S. Nakadate, H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985).
[Crossref] [PubMed]

M. Suematsu, M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier-transform technique for fringe analysis,” Appl. Opt. 30, 4046–4055 (1991).
[Crossref] [PubMed]

See, for example, K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
[Crossref] [PubMed]

Electron. Lett. (1)

S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
[Crossref]

Ind. Metrol. (1)

See, for example, M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[Crossref]

J. Opt. Soc. Am. (1)

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

M. Takeda, M. Kitoh, “Spatiotemporal frequency-multiplex heterodyne interferometry,” J. Opt. Soc. Am. A 7, 1607–1614 (1992).
[Crossref]

Opt. Eng. (1)

See, for example, G. Seitz, H.-J. Tiziani, “Resolution limits of active triangulation systems by defocusing,” Opt. Eng. 32, 1374–1383 (1993).
[Crossref]

Opt. Lasers Eng. (1)

See, for example, Y. Ishii, “Recent developments in laser-diode interferometry,” Opt. Lasers Eng. 14, 293–309 (1991).
[Crossref]

Opt. Lett. (1)

Other (2)

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

A. A. M. Maas, “Shape measurement using phase shifting speckle interferometry,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 558–568 (1991).

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Figures (12)

Fig. 1
Fig. 1

Schematic illustration of a wavelength-shift speckle interferometer. f, focal length of the imaging lens.

Fig. 2
Fig. 2

Time variation of a specklegram with temporal frequencies dependent on the height of the object h(x, y). The object shown at the bottom is imaged onto the (xy) plane so that each point in the xy plane corresponds to a point on the object where the height is h(x, y).

Fig. 3
Fig. 3

Temporal-frequency spectra of the specklegram at a point (x, y).

Fig. 4
Fig. 4

Schematic illustration of the experimental system: LD, laser diode; CL, collimator lens; MI, M2, mirrors; D/A converter, digital-to-analog converter.

Fig. 5
Fig. 5

Example of an object with three isolated steel towers with unpolished surfaces and large height discontinuities.

Fig. 6
Fig. 6

Process of data acquisition; 240 frames of time-varying specklegrams are arranged sequentially in a 16 × 15 two-dimensional matrix format. t1, t2, t240: recording times for the first, second, and last frames, respectively.

Fig. 7
Fig. 7

Time-varying specklegrams displayed on a monitor in a 16 × 15 two-dimensional matrix format. Each specklegram has 32 × 32 pixels.

Fig. 8
Fig. 8

Example of the temporal-frequency spectrum of the speckle irradiance variation observed at one fixed location.

Fig. 9
Fig. 9

Height distribution of the object shown in Fig. 5 measured by FTSP. (The height displayed does not scale with the horizontal width.)

Fig. 10
Fig. 10

Example of an object with a hole whose diameter and depth are 2.65 and 21.80 mm, respectively.

Fig. 11
Fig. 11

(a) Height distribution of the object shown in Fig. 10 measured by FTSP. (b) Cross section of the hole. (The height displayed does not scale with the horizontal width.)

Fig. 12
Fig. 12

(a) Correlogram of the coherence-radar technique. (b) Fourier spectrum of the correlogram representing the spectral density of the polychromatic light source.

Equations (31)

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k ( t ) = α t + κ ( t ) ,
g ( x , y ; t ) = a ( x , y ; t ) + b ( x , y ; t ) cos [ 2 k ( t ) l ( x , y ) ] ,
h ( x , y ) = l ( x , y ) l 0 .
f 0 ( x , y ) = α l ( x , y ) / π ,
ϕ ( x , y ; t ) = 2 κ ( t ) l ( x , y )
g ( x , y ; t ) = a ( x , y ; t ) + b ( x , y ; t ) × cos [ 2 π f 0 ( x , y ) t + ϕ ( x , y ; t ) ] .
G ( x , y ; f ) = A ( x , y ; f ) + C [ x , y ; f f 0 ( x , y ) ] + C * { x , y ; [ f + f 0 ( x , y ) ] } ,
c ( x , y ; t ) = ½ b ( x , y ; t ) exp i [ ϕ ( x , y ; t ) ] .
c ( x , y ; t ) exp [ 2 π i f 0 ( x , y ) t ] = ½ b ( x , y ; t ) exp { i [ 2 π f 0 ( x , y ) t + ϕ ( x , y ; t ) ] } .
log { c ( x , y ; t ) exp [ 2 π i f 0 ( x , y ) t ] } = log [ ½ b ( x , y ; t ) ] + i [ 2 π f 0 ( x , y ) t + ϕ ( x , y ; t ) ] .
Φ ( x , y ; t ) + Φ 0 ( x , y ) = 2 k ( t ) l ( x , y )
= 2 π f 0 ( x , y ) t + ϕ ( x , y ; t ) .
Φ ( x , y ; t ) t = 2 l ( x , y ) d k ( t ) d t .
h ( x , y ) = h R × t [ Φ ( x , y ; t ) Φ ( x 0 , y 0 ; t ) ] t [ Φ ( x R , y R ; t ) Φ ( x 0 , y 0 ; t ) ] ,
I ( x , y ; z ) = I ̅ + A [ z h ( x , y ) ] cos { 2 k ̅ [ z h ( x , y ) ] + φ ( z ) }
= k 0 k 1 S ( k ) ( 1 + cos { 2 k [ z h ( x , y ) ] } ) d k ,
I ̅ = k 0 k 1 S ( k ) d k ,
A ( z ) cos [ 2 k ̅ z + φ ( z ) ] = k 0 k 1 S ( k ) cos ( 2 k z ) d k
= sin ( Δ k ) z z × cos ( 2 k ̅ z ) for S ( k ) = 1 ,
Δ h ( x , y ) π / Δ k .
Δ f 0 1 / Δ t = α / Δ k ,
Δ h ( x , y ) = Δ l ( x , y )
= π Δ f 0 ( x , y ) / α
π / Δ k .
Î ( x , y ; f ) = I ( x , y ; z ) exp ( 2 π i f z ) d z
= Î δ ( f ) + 1 2 S ( π f ) exp [ 2 π i h ( x , y ) f ] + 1 2 S ( π f ) exp [ 2 π i h ( x , y ) f ] ,
log { ½ S ( π f ) exp [ 2 π i h ( x , y ) f ] } = log [ ½ S ( π f ) ] i [ 2 π h ( x , y ) f ] ( k 0 / π < f < k 1 / π ) .
Φ ( x , y ; f ) + Φ 0 ( x , y ) = 2 π h ( x , y ) f .
Φ ( x , y ; f ) f = 2 π h ( x , y ) ,
h ( x , y ) = 1 2 π Φ ( x , y ; f ) f
= 1 2 π × k 0 / π k 1 / π [ Φ ( x , y ; f ) f ] S ( π f ) d f k 0 / π k 1 / π S ( π f ) d f ,

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