Abstract

A fast technique for automatic 3-D shape measurement is proposed and verified by experiments. The technique, based on the principle of phase measurement of the deformed grating pattern which carries the 3-D information of the measured object, can automatically and accurately obtain the phase map or the height information of a measured object at every pixel point without assigning fringe orders and interpreting data in the regions between the fringe orders. Only one image pattern is sufficient for obtaining the phase map. In contrast to the fast Fourier transform based technique, the technique processes a fringe pattern in the real-signal domain instead of the frequency domain by using demodulation and convolution techniques, can process an arbitrary number of pixel points, and is much faster. Theoretical analysis, simulation results, and experimental results are presented.

© 1990 Optical Society of America

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References

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  1. H. Takasaki, “Moire Topography,” Appl. Opt. 9, 1467–1472 (1970).
    [CrossRef] [PubMed]
  2. D. M. Meadows, W. O. Johnson, J. B. Allen, “Generation of Surface Contours by Moire Patterns,” Appl. Opt. 9, 942–947 (1970).
    [CrossRef] [PubMed]
  3. M. Idesawa, T. Yatagui, T. Soma, “Scanning Moire Method and Automatic Measurement of 3-D Shapes,” Appl. Opt. 16, 2152–2162 (1977).
    [CrossRef] [PubMed]
  4. D. T. Moore, B. E. Truax, “Phase-Locked Moire Fringe Analysis for Automated Contouring of Diffuse Surfaces,” Appl. Opt. 18, 91–96 (1979).
    [CrossRef] [PubMed]
  5. H. E. Cline, A. S. Holik, W. E. Lorensen, “Computer-Aided Surface Reconstruction of Interference Contours,” Appl. Opt. 21, 4481–4488 (1982).
    [CrossRef] [PubMed]
  6. H. E. Cline, W. E. Lorensen, A. S. Holik, “Automatic Moire Contouring,” Appl. Opt. 23, 1454–1459 (1984).
    [CrossRef] [PubMed]
  7. G. Indebetouw, “Profile Measurement Using Projection of Running Fringes,” Appl. Opt. 17, 2930–2933 (1978).
    [CrossRef] [PubMed]
  8. M. Takeda, K. Mutoh, “Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  9. V. Srinivasan, H. C. Liu, M. Halioua, “Automated Phase-Measuring Profilometry of 3-D Diffuse Objects,” Appl. Opt. 23, 3105–3108 (1984).
    [CrossRef] [PubMed]
  10. V. Srinivasan, H. C. Liu, M. Halioua, “Automated Phase-Measuring Profilometry: a Phase Mapping Approach,” Appl. Opt. 24, 185–188 (1985).
    [CrossRef] [PubMed]
  11. K. H. Womack, “Interferometric Phase Measurement Using Spatial Synchronous Detection,” Opt. Eng. 23, 391–000 (1984).
    [CrossRef]
  12. Y. Ichioka, M. Inuiya, “Direct Phase Detecting System,” Appl. Opt. 11, 1507–1514 (1972).
    [CrossRef] [PubMed]
  13. W. W. Macy, “Two-Dimensional Fringe-Pattern Analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  14. J. G. Proakis, D. G. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1988), p. 559.

1985 (1)

1984 (3)

1983 (2)

1982 (1)

1979 (1)

1978 (1)

1977 (1)

1972 (1)

1970 (2)

Allen, J. B.

Cline, H. E.

Halioua, M.

Holik, A. S.

Ichioka, Y.

Idesawa, M.

Indebetouw, G.

Inuiya, M.

Johnson, W. O.

Liu, H. C.

Lorensen, W. E.

Macy, W. W.

Manolakis, D. G.

J. G. Proakis, D. G. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1988), p. 559.

Meadows, D. M.

Moore, D. T.

Mutoh, K.

Proakis, J. G.

J. G. Proakis, D. G. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1988), p. 559.

Soma, T.

Srinivasan, V.

Takasaki, H.

Takeda, M.

Truax, B. E.

Womack, K. H.

K. H. Womack, “Interferometric Phase Measurement Using Spatial Synchronous Detection,” Opt. Eng. 23, 391–000 (1984).
[CrossRef]

Yatagui, T.

Appl. Opt. (12)

D. M. Meadows, W. O. Johnson, J. B. Allen, “Generation of Surface Contours by Moire Patterns,” Appl. Opt. 9, 942–947 (1970).
[CrossRef] [PubMed]

H. Takasaki, “Moire Topography,” Appl. Opt. 9, 1467–1472 (1970).
[CrossRef] [PubMed]

Y. Ichioka, M. Inuiya, “Direct Phase Detecting System,” Appl. Opt. 11, 1507–1514 (1972).
[CrossRef] [PubMed]

M. Idesawa, T. Yatagui, T. Soma, “Scanning Moire Method and Automatic Measurement of 3-D Shapes,” Appl. Opt. 16, 2152–2162 (1977).
[CrossRef] [PubMed]

G. Indebetouw, “Profile Measurement Using Projection of Running Fringes,” Appl. Opt. 17, 2930–2933 (1978).
[CrossRef] [PubMed]

D. T. Moore, B. E. Truax, “Phase-Locked Moire Fringe Analysis for Automated Contouring of Diffuse Surfaces,” Appl. Opt. 18, 91–96 (1979).
[CrossRef] [PubMed]

H. E. Cline, A. S. Holik, W. E. Lorensen, “Computer-Aided Surface Reconstruction of Interference Contours,” Appl. Opt. 21, 4481–4488 (1982).
[CrossRef] [PubMed]

W. W. Macy, “Two-Dimensional Fringe-Pattern Analysis,” Appl. Opt. 22, 3898–3901 (1983).
[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]

H. E. Cline, W. E. Lorensen, A. S. Holik, “Automatic Moire Contouring,” Appl. Opt. 23, 1454–1459 (1984).
[CrossRef] [PubMed]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated Phase-Measuring Profilometry of 3-D Diffuse Objects,” Appl. Opt. 23, 3105–3108 (1984).
[CrossRef] [PubMed]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated Phase-Measuring Profilometry: a Phase Mapping Approach,” Appl. Opt. 24, 185–188 (1985).
[CrossRef] [PubMed]

Opt. Eng. (1)

K. H. Womack, “Interferometric Phase Measurement Using Spatial Synchronous Detection,” Opt. Eng. 23, 391–000 (1984).
[CrossRef]

Other (1)

J. G. Proakis, D. G. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1988), p. 559.

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Figures (10)

Fig. 1
Fig. 1

Optical geometry of the projection and recording system.

Fig. 2
Fig. 2

Schematic diagram for data processing procedures.

Fig. 3
Fig. 3

Simulation results—part I.

Fig. 4
Fig. 4

Simulation results—part II.

Fig. 5
Fig. 5

Schematic diagram of the experimental setup.

Fig. 6
Fig. 6

Deformed grating pattern on the object’s surface and reference plane.

Fig. 7
Fig. 7

Unwrapped phase distribution ϕo(x,y) along a row, where y is constant.

Fig. 8
Fig. 8

Distribution of the unwrapped ϕo(x,y) and Δϕ(x,y) where p′ > p.

Fig. 9
Fig. 9

Distribution of unwrapped ϕo(x,y) and Δϕ(x,y) when p′ < p.

Fig. 10
Fig. 10

Reconstruction of the 3-D object’s surface.

Equations (35)

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g p ( x , y ) = n = 0 A n ( x , y ) cos 2 π n x p ,
g r ( x , y ) = R r ( x , y ) n = 0 A n ( x , y ) cos 2 π n p [ x + r ( x ) ] ,
g r ( x , y ) = n = 0 B n ( x , y ) cos [ 2 π n x p + n ϕ r ( x ) ] ,
ϕ r ( x ) = 2 π p r ( x ) .
g o ( x , y ) = R o ( x , y ) n = 0 A n ( x , y ) cos { 2 π n p [ x + r ( x ) + s ( x , y ) ] }
g o ( x , y ) = n = 0 C n ( x , y ) cos [ 2 π n x p + n ϕ o ( x , y ) ] ,
ϕ o ( x , y ) = 2 π p [ r ( x ) + s ( x , y ) ] .
Δ ϕ ( x , y ) = ϕ o ( x , y ) - ϕ r ( x ) = 2 π p [ r ( x ) + s ( x , y ) ] - 2 π p r ( x ) = 2 π p s ( x , y ) ,
g ( x , y ) = n = 0 a n ( x , y ) cos [ 2 π n x p + n ϕ ( x , y ) ] ,
g ( x , y ) cos 2 π x p = n = 0 a n ( x , y ) cos [ 2 π n x p + n ϕ ( x , y ) ] cos 2 π x p = n = 0 1 2 a n ( x , y ) { cos [ 2 π ( n x + x ) p + n ϕ ( x , y ) ] + cos [ 2 π ( n x - x ) p + n ϕ ( x , y ) ] } = a o ( x , y ) cos ( 2 π x p ) + 1 2 a 1 ( x , y ) cos [ 4 π x p + ϕ ( x , y ) ] + 1 2 a 1 ( x , y ) cos ϕ ( x , y ) + 1 2 a 2 ( x , y ) cos [ 6 π x p + 2 ϕ ( x , y ) ] + 1 2 a 2 ( x , y ) cos [ 2 π x p + 2 ϕ ( x , y ) ] + 1 2 a 3 ( x , y ) cos [ 8 π x p + 3 ϕ ( x , y ) ] + 1 2 a 3 ( x , y ) cos [ 4 π x p + 3 ϕ ( x , y ) ] +
x ( n ) = g ( n , y ) cos 2 π n p ,
y ( n ) = k = 0 M - 1 h ( k ) x ( n - k ) = 1 2 a 1 ( n , y ) cos ϕ ( n , y ) .
g 1 ( x , y ) = ½ a 1 ( x , y ) cos ϕ ( x , y ) .
g ( x , y ) sin 2 π x p = n = 0 a n ( x , y ) cos [ 2 π n x p + n ϕ ( x , y ) ] sin 2 π x p = n = 0 1 2 a n ( x , y ) { sin [ 2 π ( n x + x ) p + n ϕ ( x , y ) ] - sin [ 2 π ( n x - x ) p + n ϕ ( x , y ) ] } = a o ( x , y ) sin ( 2 π x p ) + 1 2 a 1 ( x , y ) sin [ 4 π x p + ϕ ( x , y ) ] - 1 2 a 1 ( x , y ) sin ϕ ( x , y ) + 1 2 a 2 ( x , y ) sin [ 6 π x p + 2 ϕ ( x , y ) ] - 1 2 a 2 ( x , y ) sin [ 2 π x p + 2 ϕ ( x , y ) ] + 1 2 a 3 ( x , y ) sin [ 8 π x p + 3 ϕ ( x , y ) ] - 1 2 a 3 ( x , y ) sin [ 4 π x p + 3 ϕ ( x , y ) ] +
g 2 ( x , y ) = - ½ a 1 ( x , y ) sin ϕ ( x , y ) .
g 2 ( x , y ) g 1 ( x , y ) = - 1 2 a 1 ( x , y ) sin ϕ ( x , y ) 1 2 a 1 ( x , y ) cos ϕ ( x , y ) = - tan ϕ ( x , y ) ,
ϕ ( x , y ) = tan - 1 [ - g 2 ( x , y ) g 1 ( x , y ) ] .
Δ ϕ ( x , y ) = ϕ o ( x , y ) - ϕ r ( x ) .
ϕ ( x , y ) = 2 π x ( 1 p - 1 p ) + ϕ ( x , y ) .
ϕ r ( x ) = 2 π x ( 1 p - 1 p ) + ϕ r ( x ) , ϕ o ( x , y ) = 2 π x ( 1 p - 1 p ) + ϕ o ( x , y ) ,
Δ ϕ ( x , y ) = ϕ o ( x , y ) - ϕ r ( x ) = ϕ o ( x , y ) - ϕ r ( x ) .
Z ( x , y ) + L d = Z ( x , y ) b a = Z ( x , y ) s ( x , y ) .
Z ( x , y ) = L p Δ ϕ ( x , y ) 2 π d - p Δ ϕ ( x , y ) .
g ( x ) = A 2 r ( x ) + 2 A π r ( x ) f ( x ) ,
f ( x ) = sin ( 1 ) cos [ 2 π x p + ϕ ( x ) ] - 1 9 sin ( 3 ) cos [ 6 π x p + 3 ϕ ( x ) ] + 1 25 sin ( 5 ) cos [ 10 π x p + 5 ϕ ( x ) ] ,
r ( x ) = 0.4 - 0.25 | sin x 48 | cos x 85 ,
ϕ ( x ) = 0.00000001 x ( x - 125 ) ( x - 250 ) ( x - 500 ) .
0 x 500 , p = 5.3 pixels , a 0 ( x ) = - 0.01 ( x - 500.0 ) x + 80.0 , a 1 ( x ) = 50.0 - 0.000005 x ( x - 250 ) ( x - 450 ) , a 2 ( x ) = 0.00003 x ( x - 350 ) , a 3 ( x ) = 1.5 sin ( x / 20 ) , h ( x ) = 0.00000001 x ( x - 150 ) ( x - 250 ) ( x - 500 ) , g ( x ) = n = 0 3 a n cos [ 2 π n x p + n ϕ ( x ) ] .
h ( 0 ) = h ( 8 ) = 0.16887 , h ( 1 ) = h ( 7 ) = 0.32462 , h ( 2 ) = h ( 6 ) = 0.46602 , h ( 3 ) = h ( 5 ) = 0.56464 , h ( 4 ) = 0.60000.
N ( M - 1 2 + 1 )
ν = 4 N log 2 N N ( M + 1 ) = 2 5 log 2 N ,
| ϕ ( x , y ) x | max < x | 2 π x p + 2 ϕ ( x , y ) | min
| ϕ ( x , y ) x | max < 2 π p - 2 | ϕ ( x , y ) x | max ,
| ϕ ( x , y ) x | max < 2 π 3 p .
| Z ( x , y ) x | max < L 3 d .

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