Abstract

A method automatically processing a projected grating to profile a 3-D diffuse object is proposed. A deformed grating pattern projected on the object is regarded as a phase modulated pattern with a constant spatial carrier frequency. To retrieve phase modulation, acquired data in a microcomputer are sinusoidally fitted using a phase detection algorithm similar to that used in communication techniques. High sensitivity measurements of height distribution can be done using simple optical geometry. Random and systematic errors inherent in the method are discussed in detail.

© 1986 Optical Society of America

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References

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  1. H. Takasaki, “Moire Topogrpahy,” Appl. Opt. 9, 1467 (1970).
    [CrossRef] [PubMed]
  2. M. Idesawa, T. Yatagai, T. Soma, “Scanning Moire Method and Automatic Measurement of 3-D Shapes,” Appl. Opt. 16, 2152 (1977).
    [CrossRef] [PubMed]
  3. J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 409–437.
  4. 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 (1982).
    [CrossRef]
  5. M. Takeda, K. Mutoh, “Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes,” Appl. Opt. 22, 3977 (1983).
    [CrossRef] [PubMed]
  6. V. Srinivasan, H. C. Liu, M. Halioua, “Automated Phase-Measuring Profilometry of 3-D Diffuse Objects,” Appl. Opt. 23, 3105 (1984).
    [CrossRef] [PubMed]
  7. S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
    [CrossRef]
  8. M. Takeda, “Subfringe Interferometry Fundamentals,” Jpn. J. Opt. 13, 55 (1984).

1984 (3)

S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
[CrossRef]

M. Takeda, “Subfringe Interferometry Fundamentals,” Jpn. J. Opt. 13, 55 (1984).

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

1983 (1)

1982 (1)

1977 (1)

1970 (1)

Bruning, J. H.

J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 409–437.

Halioua, M.

Idesawa, M.

Ina, H.

Kobayashi, S.

Liu, H. C.

Mutoh, K.

Soma, T.

Srinivasan, V.

Takasaki, H.

Takeda, M.

Tominaga, M.

S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
[CrossRef]

Toyooka, S.

S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
[CrossRef]

Yatagai, T.

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Figures (6)

Fig. 1
Fig. 1

Geometry of projecting and imaging a grating pattern on the object: P, exit pupil of the projection optics; I, entrance pupil of the imaging optics; R, plane perpendicular to the optical axis of the imaging optics; Q, plane perpendicular to the projection optics; O, intersecting point of the two optical axes.

Fig. 2
Fig. 2

Phase modulation in reference plane R along the x axis, expressed in Eq. (11).

Fig. 3
Fig. 3

Schematic diagram of the experimental setup.

Fig. 4
Fig. 4

Resultant height distribution of a palster bust.

Fig. 5
Fig. 5

Comparison of profiles of a cross section obtained by the SPD (circles) and the contact profilometer (solid line).

Fig. 6
Fig. 6

Plot of relative phase error caused by neglecting the phase gradient, expressed in Eq. (21).

Equations (24)

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g ( x , y ) = a ( x , y ) + b ( x , y ) cos [ 2 π f 0 x + φ ( x , y ) ] ,
φ i ( x ) = α i + β i x .
g i ( x ) = a i + b i cos [ α i + ( 2 π f 0 + β i ) x ] .
S [ g i ] = i / f 0 ( i + 1 ) / f 0 g i ( x ) sin ( 2 π f 0 x ) d x = b i ( 1 4 π f 0 + β i 1 β i ) sin φ i ¯ sin ( β i / 2 f 0 ) ,
C [ g i ] = i / f 0 ( i + 1 ) / f 0 g i ( x ) cos ( 2 π f 0 x ) d x = b i ( 1 4 π f 0 + β i + 1 β i ) cos φ i ¯ sin ( β i / 2 f 0 ) ,
φ i ¯ = α i + 2 i + 1 2 f 0 β i
S [ g i ] C [ g i ] = 2 π f 0 2 π f 0 + β i tan φ i ¯ .
φ i ¯ = tan 1 S [ g i ] C [ g i ] .
g 0 ( x ) = a + b cos [ 2 π f 0 x + φ 0 ( x ) ] .
B C ¯ = O B ¯ O C ¯ = x 2 sin ϑ cos ϑ l 0 + x sin ϑ cos ϑ .
φ 0 ( x ) = 2 π f 0 B C ¯ = 2 π f 0 x 2 sin ϑ cos ϑ l 0 + x sin ϑ cos ϑ .
C D ¯ h ( x , y ) = d 0 l 0 h ( x , y ) .
φ ( x , y ) = 2 π f 0 C D ¯ .
h ( x , y ) = l 0 C D ¯ d 0 + C D ¯ = l 0 φ ( x , y ) 2 π d 0 f 0 + φ ( x , y ) .
φ ( x , y ) = 2 π f 0 B D ¯ .
φ ( x , y ) = φ ( x , y ) φ 0 ( x ) .
σ r = 1 S n ,
β i 2 π f 0 = , S i C i = K .
φ i ¯ = tan 1 [ ( 1 + ) K ] .
φ i ¯ = tan 1 K K 1 + K 2 .
Δ φ = K 1 + K 2 .
p ( K ) = 1 π ( 1 + K 2 ) .
p ( Δ φ / ) = 1 2 π 1 4 ( Δ φ / ) 2 .
σ s = | Δ φ | 2 = i / 8 ,

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