Abstract

Grating interferometric imaging has some unique advantages. The theory developed has been for 1-D objects only. The 2-D imaging methods reported are either complex or have low SNR. Two new methods for 2-D interferometric imaging are presented in this paper. The first is that an object composed of discrete points is imaged with a grating interferometer composed of three or more 1-D grating interferometers of which the grating line directions are different. The second is that a 2-D continuous distribution object is imaged with a 1-D grating interferometer by sampling line byline. These two methods are simple and may be practical for real time processing.

© 1990 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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1986 (2)

A. M. Tai, “Passive Synthetic Aperture Imaging Using an Achromatic Grating Interferometer,” Appl. Opt. 25, 3179–3190 (1986).
[CrossRef] [PubMed]

L. Zaichun, Z. Yimo, “Further Studies of Holographic Techniques for Imaging through Fog,” Acta Optica Sinica, 6, 606–611 (1986).

1984 (4)

1981 (2)

1980 (1)

1979 (1)

1977 (1)

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217–219 (1977).
[CrossRef]

1975 (1)

Aleksoff, C. C.

Alferness, R.

Chang, B. J.

Chang, J. S.

Chen, H.

Cheng, Y. S.

Collins, G.

Khoo, I.

Leith, E. N.

Swanson, G. R.

Tai, A. M.

Wynn, T.

Yimo, Z.

L. Zaichun, Z. Yimo, “Further Studies of Holographic Techniques for Imaging through Fog,” Acta Optica Sinica, 6, 606–611 (1986).

Zaichun, L.

L. Zaichun, Z. Yimo, “Further Studies of Holographic Techniques for Imaging through Fog,” Acta Optica Sinica, 6, 606–611 (1986).

Acta Optica Sinica (1)

L. Zaichun, Z. Yimo, “Further Studies of Holographic Techniques for Imaging through Fog,” Acta Optica Sinica, 6, 606–611 (1986).

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217–219 (1977).
[CrossRef]

Opt. Lett. (3)

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Figures (8)

Fig. 1
Fig. 1

Grating interferometer.

Fig. 2
Fig. 2

Fringe box of a grating interferometer.

Fig. 3
Fig. 3

Formation of an image from the interferogram.

Fig. 4
Fig. 4

Grating interferometer composed of coaxial 1-D grating interferometer.

Fig. 5
Fig. 5

Optical arrangement of the experiments.

Fig. 6
Fig. 6

Imaging a circular array of eight points with a grating interferometer by rotating the object twice.

Fig. 7
Fig. 7

Imaging a 12-point circularly symmetric object with a grating interferometer by rotating the object three times.

Fig. 8
Fig. 8

Imaging a continuous distribution object with a 1-D grating interferometer by sampling line by line.

Equations (21)

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I = ½ + ½ cos 4 π f 1 ( x - z sin θ ) .
I = ½ S ( θ ) d θ + ½ S ( θ ) cos [ 4 π f 1 ( x - z sin θ ) ] d θ ,
I ( z ) = I o + ½ S ( θ ) cos [ 4 π f 1 ( sin α - θ cos α ) z ] d θ ,
F - 1 { I ( z ) } = I ( z ) exp ( j 2 π x f z λ F ) d z = 1 2 I o ( x f λ F ) + 1 2 S ( x f 2 f 1 λ cos α F - tan α ) + 1 2 S ( x f 2 f 1 λ cos α F + tan α ) ,
u ( ξ ) = C S ( ξ a ) ,
u = S ( θ x , θ y ) exp ( j 2 π θ x λ x ) exp ( j 2 π θ y λ y ) ,
I 1 = I o + ½ S ( θ x , θ y ) cos [ 4 π f 1 ( x - θ x z ) ] d θ x d θ y ,
I 2 = I o + ½ S ( θ x , θ y ) cos [ 4 π f 1 ( y - θ y z ) ] d θ x d θ y ,
I 3 = I o + ½ S ( θ x , θ y ) cos { 4 π f 1 [ sin γ x + cos γ y - z ( sin γ θ x + cos γ θ y ) ] } d θ x d θ y ,
S ( θ x , θ y ) = k S k ( θ x - θ x k , θ y - θ y k ) .
θ x 1 θ x 2 θ x N , θ y 1 θ y 2 θ y N ,
I 1 = I o + ½ Σ S k cos [ 4 π f 1 ( x - θ x k z ) ] ,
I 2 = I o + ½ Σ S k cos [ 4 π f 1 ( y - θ y k z ) ] ,
I 3 = I o + ½ Σ S k cos { 4 π f 1 [ sin γ x + cos γ y - ( sin γ θ x k + cos γ θ y k ) z ] } .
u 1 ( ξ ) = C 1 k S k δ ( ξ - a θ x k ) ,
u 2 ( η ) = C 2 k S k δ ( η - a θ y k ) ,
u 3 ( ξ , η ) = C 3 k S k δ ( η - tan γ ξ - a θ y k + tan γ a θ x k ) .
u ( ξ , η ) = C [ k S k 3 δ ( ξ - a θ x k ) δ ( η - a θ y k ) δ ( η - tan γ ξ - a θ y k + tan γ a θ x k ) + k 1 S k 2 S 1 δ ( ξ - a θ x k ) δ ( η - a θ y k ) δ ( η - tan γ ξ - a θ y 1 + tan γ a θ x 1 ) + i j k S i S j S k δ ( ξ - a θ x i ) × δ ( η - a θ y j ) δ ( η - tan γ ξ - a θ y k + tan γ a θ x k ) ] .
tan γ θ y k - θ y j θ x k - θ x i , i , j , k = 1 , 2 , , N and i j k , or i = j k ,
u i ( ξ , η ) = C k S k 3 δ ( ξ - a θ x k , η - a θ y k ) .
u ( ξ , η ) = C [ k S k 2 δ ( ξ - a θ x k ) δ ( η - a θ y k ) + i j S i S j δ ( ξ - a θ x i ) δ ( η - a θ y j ) ] .

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