Abstract

A new type of moiré and its application to topography are discussed. Moiré fringes are generated by observing a grating projected on an object under test with a scanning imaging device. The general equations of the projection-type moiré topography are given. By controlling the phase, the pitch, or the direction of the virtual grating corresponding to the scanning lines of the imaging device, the automatic sign determination of the contour lines is accomplished. By the use of a high precision flying spot scanner with a new type of CRT(DD-tube) and a minicomputer system, an experimental measurement system is developed. Very significant is the fact that a contour line system and a sectional shape of the object are automatically reconstructed and displayed on a color TV monitor.

© 1977 Optical Society of America

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References

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  1. R. Weller, B. M. Shepard, Proc. SESA VI, 35 (1948).
  2. M. Nishida, M. Hondo, Rep. Inst. Phys. Chem. Res. 31, 196 (1955).
  3. P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, New York, 1969).
  4. J. Der Hovanesian, Y. Y. Hung, Appl. Opt. 10, 2734 (1971).
    [CrossRef]
  5. D. M. Meadows, W. O. Johnson, J. B. Allen, Appl. Opt. 9, 942 (1970).
    [CrossRef] [PubMed]
  6. H. Takasaki, Appl. Opt. 9, 1467 (1970).
    [CrossRef] [PubMed]
  7. H. Takasaki, Appl. Opt. 12, 845 (1973).
    [CrossRef] [PubMed]
  8. W. Jaerisch, G. Makosch, Appl. Opt. 12, 1552 (1973).
    [CrossRef] [PubMed]
  9. C. Chiang, Appl. Opt. 14, 177 (1975).
    [PubMed]
  10. P. Benoit, E. Mathieu, J. Hormiére, A. Thomas, Nouv. Rev. Opt. 6, 67 (1975).
    [CrossRef]
  11. Y. Yoshino, Kogaku (Jpn. J. Opt.) 1, 128 (1972).
  12. M. Suzuki, K. Suzuki, Bull. Jpn. Soc. Precis. Eng. 8, 23 (1974).
  13. C. A. Miles, B. S. Speight, J. Phys. E 8, 773 (1975).
    [CrossRef]
  14. B. Dessus, J.-P. Gerardin, P. Mousselet, Opt. Quantum Elec. 7, 15 (1975).
    [CrossRef]
  15. The difficulty of the analysis largely depends on where and how the coordinate system for describing the configuration of the moiré fringes is set. In many papers on the Meadows and Takasaki method, the coordinate system is defined so that the origin is located on the real grating, and two of the three axes are on the grating plane. In the projection-type moiré topography, however, the definition is not unified; in Ref. 10, the origin of the coordinate system is set in the object, and one of the three coordinate axes coincides with the optical axis of the observation system, whereas in Ref. 11, the situation is discussed where the projection and the observation grating are located on a plane, and the two coordinate axes are defined on the plane.
  16. G. Oster, M. Wasserman, C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
    [CrossRef]
  17. M. Suzuki, M. Kanaya, K. Suzuki, Seimitsukikai (J. Jpn. Soc. Precis. Eng.) 42, 386 (1976).
    [CrossRef]
  18. G. Keck, G. Windischbauer, G. Ranninger, Optik 37, 310 (1973).
  19. T. Honda, K. Kamiya, J. Tsujiuchi, Kogaku (Jpn. J. Opt.) 5, 87 (1976).
  20. P. Benoit, E. Mathieu, T. Thomas, Opt. Commun. 15, 392 (1975).
    [CrossRef]
  21. M. Idesawa, T. Yatagai, T. Soma, in Proceedings 3rd International Joint Conference on Pattern Recognition (8–11 November1976, Coronado, Calif.), p. 708.
  22. E. Goto, T. Soma, A. Ono, J. Inst. Telev. Eng. Jpn. 26, 21 (1972).
  23. S. Shibata, T. Soma, E. Goto, A. Ono, Nucl. Inst. Methods 123, 431 (1975).
    [CrossRef]
  24. E. Goto, T. Soma, M. Idesawa, Denshi Tokyo (IEEE Tokyo)79 (1975).

1976

M. Suzuki, M. Kanaya, K. Suzuki, Seimitsukikai (J. Jpn. Soc. Precis. Eng.) 42, 386 (1976).
[CrossRef]

T. Honda, K. Kamiya, J. Tsujiuchi, Kogaku (Jpn. J. Opt.) 5, 87 (1976).

1975

P. Benoit, E. Mathieu, T. Thomas, Opt. Commun. 15, 392 (1975).
[CrossRef]

P. Benoit, E. Mathieu, J. Hormiére, A. Thomas, Nouv. Rev. Opt. 6, 67 (1975).
[CrossRef]

C. A. Miles, B. S. Speight, J. Phys. E 8, 773 (1975).
[CrossRef]

B. Dessus, J.-P. Gerardin, P. Mousselet, Opt. Quantum Elec. 7, 15 (1975).
[CrossRef]

S. Shibata, T. Soma, E. Goto, A. Ono, Nucl. Inst. Methods 123, 431 (1975).
[CrossRef]

E. Goto, T. Soma, M. Idesawa, Denshi Tokyo (IEEE Tokyo)79 (1975).

C. Chiang, Appl. Opt. 14, 177 (1975).
[PubMed]

1974

M. Suzuki, K. Suzuki, Bull. Jpn. Soc. Precis. Eng. 8, 23 (1974).

1973

1972

E. Goto, T. Soma, A. Ono, J. Inst. Telev. Eng. Jpn. 26, 21 (1972).

Y. Yoshino, Kogaku (Jpn. J. Opt.) 1, 128 (1972).

1971

1970

1964

1955

M. Nishida, M. Hondo, Rep. Inst. Phys. Chem. Res. 31, 196 (1955).

1948

R. Weller, B. M. Shepard, Proc. SESA VI, 35 (1948).

Allen, J. B.

Benoit, P.

P. Benoit, E. Mathieu, T. Thomas, Opt. Commun. 15, 392 (1975).
[CrossRef]

P. Benoit, E. Mathieu, J. Hormiére, A. Thomas, Nouv. Rev. Opt. 6, 67 (1975).
[CrossRef]

Chiang, C.

Der Hovanesian, J.

Dessus, B.

B. Dessus, J.-P. Gerardin, P. Mousselet, Opt. Quantum Elec. 7, 15 (1975).
[CrossRef]

Gerardin, J.-P.

B. Dessus, J.-P. Gerardin, P. Mousselet, Opt. Quantum Elec. 7, 15 (1975).
[CrossRef]

Goto, E.

S. Shibata, T. Soma, E. Goto, A. Ono, Nucl. Inst. Methods 123, 431 (1975).
[CrossRef]

E. Goto, T. Soma, M. Idesawa, Denshi Tokyo (IEEE Tokyo)79 (1975).

E. Goto, T. Soma, A. Ono, J. Inst. Telev. Eng. Jpn. 26, 21 (1972).

Honda, T.

T. Honda, K. Kamiya, J. Tsujiuchi, Kogaku (Jpn. J. Opt.) 5, 87 (1976).

Hondo, M.

M. Nishida, M. Hondo, Rep. Inst. Phys. Chem. Res. 31, 196 (1955).

Hormiére, J.

P. Benoit, E. Mathieu, J. Hormiére, A. Thomas, Nouv. Rev. Opt. 6, 67 (1975).
[CrossRef]

Hung, Y. Y.

Idesawa, M.

E. Goto, T. Soma, M. Idesawa, Denshi Tokyo (IEEE Tokyo)79 (1975).

M. Idesawa, T. Yatagai, T. Soma, in Proceedings 3rd International Joint Conference on Pattern Recognition (8–11 November1976, Coronado, Calif.), p. 708.

Jaerisch, W.

Johnson, W. O.

Kamiya, K.

T. Honda, K. Kamiya, J. Tsujiuchi, Kogaku (Jpn. J. Opt.) 5, 87 (1976).

Kanaya, M.

M. Suzuki, M. Kanaya, K. Suzuki, Seimitsukikai (J. Jpn. Soc. Precis. Eng.) 42, 386 (1976).
[CrossRef]

Keck, G.

G. Keck, G. Windischbauer, G. Ranninger, Optik 37, 310 (1973).

Makosch, G.

Mathieu, E.

P. Benoit, E. Mathieu, J. Hormiére, A. Thomas, Nouv. Rev. Opt. 6, 67 (1975).
[CrossRef]

P. Benoit, E. Mathieu, T. Thomas, Opt. Commun. 15, 392 (1975).
[CrossRef]

Meadows, D. M.

Miles, C. A.

C. A. Miles, B. S. Speight, J. Phys. E 8, 773 (1975).
[CrossRef]

Mousselet, P.

B. Dessus, J.-P. Gerardin, P. Mousselet, Opt. Quantum Elec. 7, 15 (1975).
[CrossRef]

Nishida, M.

M. Nishida, M. Hondo, Rep. Inst. Phys. Chem. Res. 31, 196 (1955).

Ono, A.

S. Shibata, T. Soma, E. Goto, A. Ono, Nucl. Inst. Methods 123, 431 (1975).
[CrossRef]

E. Goto, T. Soma, A. Ono, J. Inst. Telev. Eng. Jpn. 26, 21 (1972).

Oster, G.

Ranninger, G.

G. Keck, G. Windischbauer, G. Ranninger, Optik 37, 310 (1973).

Shepard, B. M.

R. Weller, B. M. Shepard, Proc. SESA VI, 35 (1948).

Shibata, S.

S. Shibata, T. Soma, E. Goto, A. Ono, Nucl. Inst. Methods 123, 431 (1975).
[CrossRef]

Soma, T.

S. Shibata, T. Soma, E. Goto, A. Ono, Nucl. Inst. Methods 123, 431 (1975).
[CrossRef]

E. Goto, T. Soma, M. Idesawa, Denshi Tokyo (IEEE Tokyo)79 (1975).

E. Goto, T. Soma, A. Ono, J. Inst. Telev. Eng. Jpn. 26, 21 (1972).

M. Idesawa, T. Yatagai, T. Soma, in Proceedings 3rd International Joint Conference on Pattern Recognition (8–11 November1976, Coronado, Calif.), p. 708.

Speight, B. S.

C. A. Miles, B. S. Speight, J. Phys. E 8, 773 (1975).
[CrossRef]

Suzuki, K.

M. Suzuki, M. Kanaya, K. Suzuki, Seimitsukikai (J. Jpn. Soc. Precis. Eng.) 42, 386 (1976).
[CrossRef]

M. Suzuki, K. Suzuki, Bull. Jpn. Soc. Precis. Eng. 8, 23 (1974).

Suzuki, M.

M. Suzuki, M. Kanaya, K. Suzuki, Seimitsukikai (J. Jpn. Soc. Precis. Eng.) 42, 386 (1976).
[CrossRef]

M. Suzuki, K. Suzuki, Bull. Jpn. Soc. Precis. Eng. 8, 23 (1974).

Takasaki, H.

Theocaris, P. S.

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, New York, 1969).

Thomas, A.

P. Benoit, E. Mathieu, J. Hormiére, A. Thomas, Nouv. Rev. Opt. 6, 67 (1975).
[CrossRef]

Thomas, T.

P. Benoit, E. Mathieu, T. Thomas, Opt. Commun. 15, 392 (1975).
[CrossRef]

Tsujiuchi, J.

T. Honda, K. Kamiya, J. Tsujiuchi, Kogaku (Jpn. J. Opt.) 5, 87 (1976).

Wasserman, M.

Weller, R.

R. Weller, B. M. Shepard, Proc. SESA VI, 35 (1948).

Windischbauer, G.

G. Keck, G. Windischbauer, G. Ranninger, Optik 37, 310 (1973).

Yatagai, T.

M. Idesawa, T. Yatagai, T. Soma, in Proceedings 3rd International Joint Conference on Pattern Recognition (8–11 November1976, Coronado, Calif.), p. 708.

Yoshino, Y.

Y. Yoshino, Kogaku (Jpn. J. Opt.) 1, 128 (1972).

Zwerling, C.

Appl. Opt.

Bull. Jpn. Soc. Precis. Eng.

M. Suzuki, K. Suzuki, Bull. Jpn. Soc. Precis. Eng. 8, 23 (1974).

Denshi Tokyo (IEEE Tokyo)

E. Goto, T. Soma, M. Idesawa, Denshi Tokyo (IEEE Tokyo)79 (1975).

J. Inst. Telev. Eng. Jpn.

E. Goto, T. Soma, A. Ono, J. Inst. Telev. Eng. Jpn. 26, 21 (1972).

J. Opt. Soc. Am.

J. Phys. E

C. A. Miles, B. S. Speight, J. Phys. E 8, 773 (1975).
[CrossRef]

Kogaku (Jpn. J. Opt.)

T. Honda, K. Kamiya, J. Tsujiuchi, Kogaku (Jpn. J. Opt.) 5, 87 (1976).

Y. Yoshino, Kogaku (Jpn. J. Opt.) 1, 128 (1972).

Nouv. Rev. Opt.

P. Benoit, E. Mathieu, J. Hormiére, A. Thomas, Nouv. Rev. Opt. 6, 67 (1975).
[CrossRef]

Nucl. Inst. Methods

S. Shibata, T. Soma, E. Goto, A. Ono, Nucl. Inst. Methods 123, 431 (1975).
[CrossRef]

Opt. Commun.

P. Benoit, E. Mathieu, T. Thomas, Opt. Commun. 15, 392 (1975).
[CrossRef]

Opt. Quantum Elec.

B. Dessus, J.-P. Gerardin, P. Mousselet, Opt. Quantum Elec. 7, 15 (1975).
[CrossRef]

Optik

G. Keck, G. Windischbauer, G. Ranninger, Optik 37, 310 (1973).

Proc. SESA

R. Weller, B. M. Shepard, Proc. SESA VI, 35 (1948).

Rep. Inst. Phys. Chem. Res.

M. Nishida, M. Hondo, Rep. Inst. Phys. Chem. Res. 31, 196 (1955).

Seimitsukikai (J. Jpn. Soc. Precis. Eng.)

M. Suzuki, M. Kanaya, K. Suzuki, Seimitsukikai (J. Jpn. Soc. Precis. Eng.) 42, 386 (1976).
[CrossRef]

Other

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, New York, 1969).

M. Idesawa, T. Yatagai, T. Soma, in Proceedings 3rd International Joint Conference on Pattern Recognition (8–11 November1976, Coronado, Calif.), p. 708.

The difficulty of the analysis largely depends on where and how the coordinate system for describing the configuration of the moiré fringes is set. In many papers on the Meadows and Takasaki method, the coordinate system is defined so that the origin is located on the real grating, and two of the three axes are on the grating plane. In the projection-type moiré topography, however, the definition is not unified; in Ref. 10, the origin of the coordinate system is set in the object, and one of the three coordinate axes coincides with the optical axis of the observation system, whereas in Ref. 11, the situation is discussed where the projection and the observation grating are located on a plane, and the two coordinate axes are defined on the plane.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the scanning moiré topography: O, object; G, projection grating; P, projector; C, camera; S, scanning device; I, image processing unit; M, monitor display.

Fig. 2
Fig. 2

Illustration of different types of moiré fringes: (a) typical deformed grating given by projecting a grating onto a spherical surface; (b) conventional type of the moiré pattern given by superimposing the pattern (a) and a linear reference grating; (c) scanning moiré pattern produced by sampling the pattern (a) along virtual grating lines with a scanning device; (d) modified moiré pattern by smoothing the pattern (c).

Fig. 3
Fig. 3

Geometry for calculating general formulas of the moiré topography.

Fig. 4
Fig. 4

Three-dimensional plot of equiorder surfaces in the case xop = −l, yop = 0, zop = 0; α = 0, β = 0; θ ≠ 0, ϕ ≠ 0; am/Pm = ap/Pg.

Fig. 5
Fig. 5

Three-dimensional plot of equiorder surfaces in the case xop = −l, yop = 0, zop = 0, α = 0, β = 0; θ = 0, ϕ = 0; am/Pm = ap/Pg.

Fig. 6
Fig. 6

Schematic illustration for getting physical interpretation of the transpositions Eqs. (19)(23).

Fig. 7
Fig. 7

Change of contour line levels by changing the phase of the virtual grating. The solid lines and the broken lines correspond to the different contour line systems in different phases of the virtual grating, respectively.

Fig. 8
Fig. 8

Deformed grating given by projecting a grating onto an object. (Ignore the coarse, spurious moiré fringe pattern caused by the halftone screen.)

Fig. 9
Fig. 9

Contour line system given by shifting partially the phase of the virtual grating.

Fig. 10
Fig. 10

Color contour line system and sectional shapes. Color contour lines become dark in the order of blue–green–red. White straight lines indicate base lines, and curved lines corresponding to the base lines are sectional shapes of the object.

Fig. 11
Fig. 11

Color transition rule of contour lines.

Tables (3)

Tables Icon

Table I Error Sensitivities and Estimated Errors for Parameters of the Optical System in the Case when am = ap = a, Pm = Pg = P, α = β = 0, θ = ϕ =0, and yop = zop = 0

Tables Icon

Table II Error Sensitivities and Estimated Errors for Alignment Parametersa

Tables Icon

Table III Error Sensitivities and Estimated Errors due to Miscount of the Fringe Order and Incorrect Measurement of Moiré Fringe

Equations (42)

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n m - n g = n ,
x n = z n x o / a m ,
y n = z n y o / a m ,
z n = a m { P g [ x m - P m ( n + δ ) ] ( x o p k 1 - y o p k 2 + z o p k 3 ) - a p P m ( x o p k 4 + y o p k 5 + z o p k 6 ) } P g [ x m - P m ( n + δ ) ] ( x o k 1 - y o k 2 + a m k 3 ) - a p P m ( x o k 4 + y o k 5 + a m k 6 ) ,
x o = x m cos α - y m sin α ,             y o = x m sin α + y m cos α ,
x n = z n ( x m cos α - y m sin α ) / a m ,
y n = z n ( x m sin α + y m cos α ) / a m ,
z n = h 1 x m - h 2 h 3 x m 2 + h 4 x m y m + h 5 x m + h 6 y m + h 7 ,
z n = g 1 + { g 2 / [ g 3 - g 4 ( n + δ ) ] } ,
d n = z n + 1 - z n             a m a p P m 2 P g [ ( x o p k 1 - y o p k 2 + z o p k 3 ) ( x o k 4 + y o k 5 + a m k 6 ) - ( x o k 1 - y o k 2 + a m k 3 ) ( x o p k 4 + y o p k 5 + z o p k 6 ) ] { P g [ x m - P m ( n + δ ) ] ( x o k 1 - y o k 2 + a m k 3 ) - a p P m ( x o k 4 + y o k 5 + a m k 6 ) } 2 .
z n = - a m l / P m ( n + δ ) ,
x n = z n x m / a m ,
y n = z n y m / a m .
d n - a m l P m ( n + δ ) 2 .
z n = - a m l { cos ϕ - [ x m - P m ( n + δ ) ] sin ϕ / a m } P m ( n + δ ) cos ϕ - { a m + x m [ x m - P m ( n + δ ) ] / a m } sin ϕ .
z n = - a m l P m ( n + δ ) cos θ - ( cos θ - 1 ) x m + { [ x m - P m ( n + δ ) ] y m / a m } sin θ .
z n = - a m l - x m - [ P m ( n + δ ) ] z o p P m ( n + δ ) .
z n = - a m l cos β P m ( n + δ ) - ( 1 - cos β ) x m + y m sin β .
( n + δ ) = ( N + δ a ) + l / P ,
h a = z n - a = - a [ l + P ( N + δ a ) ] P ( N + δ a ) ,
δ a = a / P a
( n + δ ) = ( N + δ b ) + a l / b P ,
h b = z n - b = b ( b - f ) P ( N + δ b ) l f - ( b - f ) P ( N + δ b ) ,
x o = x m cos α - y m sin α ,
y o = x m sin α + y m cos α ,
x = z a m x o = z a m ( x m cos α - y m sin α ) ,
y = z a m y o = z a m ( x m sin α - y m cos α ) .
[ x y z ] = [ cos ϕ 0 - sin ϕ sin θ · sin ϕ cos θ sin θ · cos ϕ cos θ · sin ϕ - sin θ cos θ · cos ϕ ] · [ x - x o p y - y o p z - z o p ] .
x p = x g cos β - y g sin β ;
y p = x g sin β + y g cos β .
x = z a p x p = z a p ( x g cos β - y g sin β ) ;
y = z a p y p = z a p ( x g sin β + y g cos β ) .
x m = P m ( n m + δ m ) ;
x g = P g ( n g + δ g ) .
n = n m - n g ,
δ = δ m - δ g ,
x g = P g P m [ x m - P m ( n + δ ) ] .
( x - x o p ) cos ϕ - ( z - z o p ) sin ϕ = 1 a p · ( x g cos β - y g sin β ) [ ( x - x o p ) cos θ sin ϕ - ( y - y o p ) × sin θ + ( z - z o p ) cos θ cos ϕ ] ;
( x - x o p ) sin θ sin ϕ + ( y - y o p ) cos θ + ( z - z o p ) sin θ cos ϕ = 1 a p · ( x g sin β + y g cos β ) [ ( x - x o p ) cos θ sin ϕ - ( y - y o p ) × sin θ + ( z - z o p ) cos θ cos ϕ ] .
( x - x o p ) ( cos ϕ cos β + sin θ sin ϕ sin β ) + ( y - y o p ) cos θ sin β - ( z - z o p ) ( sin ϕ cos β - sin θ cos ϕ sin β ) = x g a p · [ ( x - x o p ) cos θ sin ϕ - ( y - y o p ) sin θ + ( z - z o p ) cos θ cos ϕ ] .
z = a m { P g [ x m - P m ( n + δ ) ] ( x o p k 1 - y o p k 2 + z o p k 3 ) - a p P m ( x o p k 4 + y o p k 5 + z o p k 6 ) } P g [ x m - P m ( n + δ ) ] ( x o k 1 - y o k 2 + a m k 3 ) - a p P m ( x o k 4 + y o k 5 + a m k 6 ) ,
k 1 = cos θ sin ϕ k 2 = sin θ k 3 = cos θ cos ϕ k 4 = cos ϕ cos β + sin θ sin ϕ sin β k 5 = cos θ sin β k 6 = sin θ cos ϕ sin β - sin ϕ cos β } .

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