Abstract

Projection-type moiré contouring can be done without a reference grid by undersampling projected cosine fringes with a charged-coupled-device detector array to eliminate entirely the unwanted sum, shadow, and grid terms of classical moiré methods as well as the spurious moiré fringes that are due to higher harmonics. The technique produces a high-visibility sampled version of the moiré difference contour fringes. Higher-order aliasing can provide increased sensitivity when the same detector array is used. The sampling conditions are formulated as a moiré extension to the Whittaker–Shannon sampling theorem.

© 1984 Optical Society of America

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References

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  1. Lord Rayleigh, Philos. Mag. 47, 81 (1874).Lord Rayleigh, Philos. Mag. 47, 193 (1874).
  2. D. M. Meadows, W. O. Johnson, J. B. Allen, Appl. Opt. 9, 942 (1970).
    [CrossRef] [PubMed]
  3. H. Takasaki, Appl. Opt. 9, 1467 (1970).
    [CrossRef] [PubMed]
  4. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  6. F. Chiang, Exp. Mech. 9, 523 (1969).
    [CrossRef]
  7. J. C. Perrin, A. Thomas, Appl. Opt. 18, 563 (1979).
    [PubMed]
  8. M. Idesawa, T. Yatagai, T. Soma, Appl. Opt. 16, 2152 (1977).
    [CrossRef] [PubMed]

1979 (1)

1977 (1)

1970 (2)

1969 (1)

F. Chiang, Exp. Mech. 9, 523 (1969).
[CrossRef]

1874 (1)

Lord Rayleigh, Philos. Mag. 47, 81 (1874).Lord Rayleigh, Philos. Mag. 47, 193 (1874).

Allen, J. B.

Chiang, F.

F. Chiang, Exp. Mech. 9, 523 (1969).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Idesawa, M.

Johnson, W. O.

Meadows, D. M.

Perrin, J. C.

Rayleigh, Lord

Lord Rayleigh, Philos. Mag. 47, 81 (1874).Lord Rayleigh, Philos. Mag. 47, 193 (1874).

Soma, T.

Takasaki, H.

Thomas, A.

Yatagai, T.

Appl. Opt. (4)

Exp. Mech. (1)

F. Chiang, Exp. Mech. 9, 523 (1969).
[CrossRef]

Philos. Mag. (1)

Lord Rayleigh, Philos. Mag. 47, 81 (1874).Lord Rayleigh, Philos. Mag. 47, 193 (1874).

Other (2)

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (2)

Fig. 1
Fig. 1

Cosinusoidal sampling. Spectrum of A, a cosinusoidal image is convolved with B, a cosinusoidal sampling function spectrum to produce C, the sampled image spectrum. The spectral overlap causes aliasing and produces the moiré difference contour term at low frequency. The other (unwanted) terms are, in order of increasing frequency, the shadow, the grid, and the moiré sum term. The units are relative spectral amplitude and moiré frequency intervals defined as one half of the Nyquist frequency.

Fig. 2
Fig. 2

CCD sampling. A, the spectrum of the same cosiriusoidal image as shown in Fig. 1 is first multiplied by the sine function (dashed line) and then convolved with the comb function B, to yield C, the sampled image spectrum. The spectral overlap causes aliasing and produces an infinite reproduction of the moiré difference spectrum. This can be rewritten as shown in D. Note the absence of the unwanted terms and the increased visibility compared with Fig. 1C. The units are the same as in Fig. 1.

Equations (6)

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f s ( x , y ) = 1 4 { 1 + M cos [ 2 π β ( x γ ) ] + M cos ( 2 π x d ) + M M 2 × cos [ 2 π x ( 1 / β + 1 / d ) γ β ] + M M 2 × cos [ 2 π x ( 1 / β 1 / d ) γ β ] } rect ( array width ) ,
F s ( ξ , η ) = 1 4 { δ ( ξ , η ) + M β 2 δ δ ( β ξ ) δ ( η ) × exp [ i 2 π ( γ β ) ξ ] + M d 2 δ δ ( d ξ ) δ ( η ) + M M β d 4 ( β + d ) δ δ ( ξ 1 / d + 1 / β ) δ ( η ) exp [ i 2 π ( γ β ) ξ ] + M M β d 4 ( β d ) δ δ ( ξ 1 / d 1 / β ) δ ( η ) exp [ i 2 π ( γ β ) ξ ] } * * sinc ( 1 width ) .
f s ( x , y ) = [ f ( x , y ) * * 1 p x p y rect ( x p x , y p y ) ] × comb d x d y ( x d x , y d y ) rect ( x N x d x , y N y d y ) ,
F s ( ξ , η ) = [ F ( ξ , η ) sinc ( p x ξ , p y η ) ] * * comb ( d x ξ , d y η ) * * sinc ( N x d x ξ , N y d y η ) N x d x N y d y .
F s ( ξ , η ) = 1 4 { δ ( ξ , η ) + M 2 ( d x + β ) × δ δ [ ξ / ( 1 / d 1 / β ) ] δ ( η ) × sinc ( p x β ) exp [ i 2 π ( γ β ) ξ ] } * * comb ( d x ξ , d y η ) * * sinc ( N x d x ξ , N y d y η ) N x d x N y d y .
f s ( x , y ) = 1 2 { 1 + M sinc ( p x β ) × cos [ 2 π x ( 1 / d x 1 / β ) γ β ] } × comb d x d y ( x d x , y d y ) rect ( x N x d x , y N y d y ) ,

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