Abstract

A simple and robust technique for absolute phase measurement based on number theory is presented. The new, to our knowledge, technique, which is compared with the Gushov–Solodkin algorithm, surmounts the shortcomings in the Gushov–Solodkin algorithm. The technique permits the three-dimensional shape measurement of objects that have discontinuous height steps and has resulted in a new and more powerful method of measuring surface absolute profile. Experimental results are presented that demonstrate the validity of the principle.

© 2001 Optical Society of America

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References

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  1. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 2977–3982 (1983).
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfield, A. D. White, D. J. Brangaccio, “Digital wavefront measurement interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  3. D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
    [CrossRef]
  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. W. Nadeborn, P. Andraï, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
    [CrossRef]
  6. V. I. Gushov, Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
    [CrossRef]
  7. 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, 5347–5354 (1997).
    [CrossRef] [PubMed]
  8. J. Zhong, M. Wang, “Phase unwrapping by a lookup table method: application to phase maps with singular points,” Opt. Eng. 38, 2075–2081 (1999).
    [CrossRef]

1999 (1)

J. Zhong, M. Wang, “Phase unwrapping by a lookup table method: application to phase maps with singular points,” Opt. Eng. 38, 2075–2081 (1999).
[CrossRef]

1997 (2)

1996 (1)

W. Nadeborn, P. Andraï, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

1995 (1)

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

1991 (1)

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

1983 (1)

1974 (1)

Andraï, P.

W. Nadeborn, P. Andraï, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Atkinson, J. T.

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Burton, D. R.

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

Gallagher, J. E.

Goodall, A. J.

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

Gu, Q.

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]

Herriott, D. R.

Huntley, J. M.

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]

Kinoshita, M.

Lalor, M. J.

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

Mutoh, K.

Nadeborn, W.

W. Nadeborn, P. Andraï, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Osten, W.

W. Nadeborn, P. Andraï, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Rosenfield, D. P.

Saldner, H. O.

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]

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]

Takahashi, Y.

Takai, H.

Takeda, M.

Wang, M.

J. Zhong, M. Wang, “Phase unwrapping by a lookup table method: application to phase maps with singular points,” Opt. Eng. 38, 2075–2081 (1999).
[CrossRef]

White, A. D.

Zhong, J.

J. Zhong, M. Wang, “Phase unwrapping by a lookup table method: application to phase maps with singular points,” Opt. Eng. 38, 2075–2081 (1999).
[CrossRef]

Appl. Opt. (3)

Opt. Eng. (2)

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. Zhong, M. Wang, “Phase unwrapping by a lookup table method: application to phase maps with singular points,” Opt. Eng. 38, 2075–2081 (1999).
[CrossRef]

Opt. Lasers Eng. (3)

D. R. Burton, A. J. Goodall, J. T. Atkinson, M. J. Lalor, “The use of carrier frequency shifting for the elimination of phase discontinuities in Fourier transform profilometry,” Opt. Lasers Eng. 23, 245–257 (1995).
[CrossRef]

W. Nadeborn, P. Andraï, W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

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

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Figures (14)

Fig. 1
Fig. 1

Optical geometry for grating projection profilometry.

Fig. 2
Fig. 2

Fringe pattern with f x,2 obtained by means of rotating an angle θ to the fringe pattern with f x,1.

Fig. 3
Fig. 3

Absolute phase values for Δϕ1(x):Δφ2(x):Δφ3(x) = 3:5 :32/2.

Fig. 4
Fig. 4

Wrapped phrase values (a) Δϕ1(x), (b) Δϕ2(x), and (c) Δϕ3(x) with maximal random error δϕ = ±1/20.

Fig. 5
Fig. 5

Absolute phase values obtained with the G–S algorithm (a) Δφ1(x) and (b) Δφ2(x).

Fig. 6
Fig. 6

Absolute phase values obtained with the new technique (a) Δφ1(x) and (b) Δφ3(x).

Fig. 7
Fig. 7

Values of Δϕ1ϕ3 where Δϕ1ϕ3 = (Δϕ1 - 2Δϕ3)/2π.

Fig. 8
Fig. 8

Schematic diagram of optical setup. IMP-PC, personal computer.

Fig. 9
Fig. 9

Carrier frequency fringe patterns projected upon a reference plane (a) with f x,1 and (b) with f x,2.

Fig. 10
Fig. 10

Fringe patterns projected upon an object surface (a) with f x,1 and (b) with f x,2.

Fig. 11
Fig. 11

Wrapped phase distribution (a) Δϕ1(x, y) and (b) Δϕ2(x, y) for a section (row, 300).

Fig. 12
Fig. 12

Values of Δϕ1ϕ2= [Δϕ1(x, y) - μΔϕ2(x, y)]/2π for a section (row, 300).

Fig. 13
Fig. 13

Values of absolute phase (a) Δφ1(x, y) and (b) Δφ2(c, y) for a section (row, 300).

Fig. 14
Fig. 14

Absolute phase distributions obtained by the proposed technique (a) Δφ1(x, y) and (b) Δφ2(x, y).

Tables (1)

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Table 1 Values of T(n 1, n 3) where T(n 1, n 3) = 2 n 3 - n 1

Equations (43)

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I0kx, y=a0x, y+b0x, ycos2πfx,kx+2πfy,ky+φ0kx, y.
Ikx, y=ax, y+bx, ycos2πfx,kx+2πfy,ky+φkx, y,
Δϕkx, y=ϕkx, y-ϕ0kx, y,
Δφkx, y=φkx, y-φ0kx, y=2nkπ+Δφkx, y,
hx, y=lΔφkx, yΔφkx, y+2πfx,kd.
Xb1mod m1, Xb2mod m2,  Xbkmod mk.
XM1M1b1+M2M2b2++MkMkbk  mod m,
m=m1m2  mk,
mkMk=m,
MkMk1 mod mk.
Δφkx, ymk/2π=mknk+1/2πmkΔϕkx, y.
hx, y=lΔφ1x, yΔφ1x, y+2πfx,1d=lΔφ2x, yΔφ2x, y+2πfx,2d.
Δφ1x, yΔφ2(x, y)=fx,1fx,2.
Δφkx, y/fx,k=β,
1/fx,1m1=1/fx,2m2==1/fx,kmk=α.
β2πα=mknk-1+mkΔϕkx, y+2π2π,
Xˆ=β2πα, bˆk=mkΔϕkx, y+2π2π,  ifΔϕkx, y0,
bˆk=mkΔϕkx, y2π,  ifΔϕkx, y>0.
Xˆ=X+,
bˆk=bk+,
Xˆ=X+M1M1b1+M2M2b2++MkMkbk+  mod m.
Δφkx, y=2παfx,kXˆ.
hx, y=FΔφ=F2nπ+Δϕ.
hx, y=F12n1π+Δϕ1=F22n2π+Δϕ2.
F12n1π+Δϕ1-F22n2π+Δϕ2=0,
Δφ1Δφ2=2n1π+Δϕ12n2π+Δϕ2=μ,
μn2-n1=Δϕ1-μΔϕ22π.
Tn1, n2=μn2-n1,
Δϕ1ϕ2=Δϕ1-μΔϕ22π.
Δϕ1ϕ2=Δϕ1+δϕ1-μΔϕ2+δϕ22π,
Δϕ1ϕ2-Tn1, n2<τ,
Δϕ1ϕ2=Tn1, n2.
ΔT=|Ti, i - Tj, j|.
0τΔTmin/2,
Δϕ1x, y+δϕ1x, y-μΔϕ2x, y+δϕ2x, y/2π-Tn1, n2<τ,
|δϕ1x, y-μδϕ2x, y|<πΔTmin.
fx,1/fx,2=μ.
fx,1/fx,2=1/cos θ.
12πΔϕ1-μ+δμΔϕ2-μ+δμn2-n1<τ.
n2+Δϕ22πδμ<ΔTmin2,
fx,1fx,2=a0+1a1+1+1an+,
fx,1fx,2=a0+1a1+1+1an=m2m1,
fx,1fx,2=1cos20° 15 20.

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