Abstract

A new kind of principle is proposed to measure distorted transparent objects and specular surfaces directly. The method is based on correlation filtering of the object under study with two coded masks in noncoherent Fourier transformation. The information about the phase gradient appears in the form of fringes. The systems are analyzed mathematically. The following properties are shown: The systems can perform in real time and with an extended white light illumination. The various information can be obtained by using different masks. The sensitivity is relatively low and can be altered, marked, continuously varied, and even graded into several sections of different values. Some experiments and possible applications are shown.

© 1982 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 109.
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 119–127.
  3. Liu Liren, Acta Opt. Sinica 1, 213 (1981).
  4. S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J. Stiffler, A. J. Viterbi, Digital Communications with Space Applications (Prentice-Hall, Englewood Cliffs, N.J., 1964).
  5. J. E. Golay, J. Opt. Soc. Am. 61, 272 (1971).
    [CrossRef]
  6. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), pp. 9–75.
    [CrossRef]
  7. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 397–420.
  8. K. Hinsch, Proc. Soc. Photo-Opt. Instrum. Eng. 210, 159 (1979).

1981 (1)

Liu Liren, Acta Opt. Sinica 1, 213 (1981).

1979 (1)

K. Hinsch, Proc. Soc. Photo-Opt. Instrum. Eng. 210, 159 (1979).

1971 (1)

Baumert, L. D.

S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J. Stiffler, A. J. Viterbi, Digital Communications with Space Applications (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 119–127.

Easterling, M. F.

S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J. Stiffler, A. J. Viterbi, Digital Communications with Space Applications (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 397–420.

Golay, J. E.

Golomb, S. W.

S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J. Stiffler, A. J. Viterbi, Digital Communications with Space Applications (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 109.

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), pp. 9–75.
[CrossRef]

Hinsch, K.

K. Hinsch, Proc. Soc. Photo-Opt. Instrum. Eng. 210, 159 (1979).

Liren, Liu

Liu Liren, Acta Opt. Sinica 1, 213 (1981).

Stiffler, J. J.

S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J. Stiffler, A. J. Viterbi, Digital Communications with Space Applications (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Viterbi, A. J.

S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J. Stiffler, A. J. Viterbi, Digital Communications with Space Applications (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 119–127.

Acta Opt. Sinica (1)

Liu Liren, Acta Opt. Sinica 1, 213 (1981).

J. Opt. Soc. Am. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

K. Hinsch, Proc. Soc. Photo-Opt. Instrum. Eng. 210, 159 (1979).

Other (5)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 109.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 119–127.

S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J. Stiffler, A. J. Viterbi, Digital Communications with Space Applications (Prentice-Hall, Englewood Cliffs, N.J., 1964).

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), pp. 9–75.
[CrossRef]

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), pp. 397–420.

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Figures (13)

Fig. 1
Fig. 1

Fourier transform quasi-interferometric system.

Fig. 2
Fig. 2

Equivalent Fourier transform system.

Fig. 3
Fig. 3

Some patterns of correlation filters.

Fig. 4
Fig. 4

Simplified system with (a) coded source and (b) point source.

Fig. 5
Fig. 5

Coded mask with three sensitivities.

Fig. 6
Fig. 6

Patterns of (a) equilateral triangle and (b) its auto-correlation.

Fig. 7
Fig. 7

Mask of rings with equal intervals.

Fig. 8
Fig. 8

System with continuously variable sensitivity.

Fig. 9
Fig. 9

Sensitivity variable systems with (a) single lens, (h) coded source, and (c) point source.

Fig. 10
Fig. 10

Phase gradient contours of a glass bottle with different sensitivities.

Fig. 11
Fig. 11

Contours of phase gradient directions of a glass bottle using (a) simplified system and (b) a system with a correlation filter.

Fig. 12
Fig. 12

Fringes on the water surface.

Fig. 13
Fig. 13

Fringes on a spectacle lens.

Equations (46)

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| F O ( 1 λ f 2 X 2 , 1 λ f 2 Y 2 ) | 2 .
| F O [ 1 λ f 2 ( X 2 - f 2 f 1 X 1 ) , 1 λ f 2 ( Y 2 - f 2 f 1 Y 1 ) ] | 2 .
| h ( X 1 , Y 1 ; X 2 , Y 2 ) | 2 = | F O [ 1 λ f 2 ( X 2 - f 2 f 1 X 1 ) , 1 λ f 2 ( Y 2 - f 2 f 1 Y 1 ) ] | 2 .
I ( X 2 , Y 2 ) = - S 1 ( X 1 , Y 1 ) × | F O [ 1 λ f 2 ( X 2 - f 2 f 1 X 1 ) , 1 λ f 2 ( Y 2 - f 2 f 1 Y 1 ) ] | 2 d X 1 d Y 1 .
I f ( X 2 , Y 2 ) = I ( X 2 , Y 2 ) S 2 ( X 2 , Y 2 ) = S 2 ( X 2 , Y 2 ) - S 1 ( X 1 , Y 1 ) × | F O [ 1 λ f 2 ( X 2 - f 2 f 1 X 1 ) , 1 λ f 2 ( Y 2 - f 2 f 1 Y 2 ) ] | 2 d X 1 d Y 1 .
I f = - S 2 ( X 2 , Y 2 ) S 1 ( f 1 f 2 X 1 , f 1 f 2 Y 1 ) × | F O ( X 2 - X 1 λ f 2 , Y 2 - Y 1 λ f 2 ) | 2 d X 1 d Y 1 = S 2 ( X 2 , Y 2 ) [ S 1 ( f 1 f 2 X 2 , f 1 f 2 Y 2 ) * | F O ( X 2 λ f 2 , Y 2 λ f 2 ) | 2 ]
I f s ( X 2 , Y 2 ) = S 1 ( X 1 , Y 1 ) | F O [ 1 λ f 2 ( X 2 - f 2 f 1 X 1 ) , 1 λ f 2 ( Y 2 - f 2 f 1 Y 2 ) ] | 2 × S 2 ( X 2 , Y 2 ) .
I f d s = S 1 ( f 1 f 2 X 1 , f 1 f 2 Y 1 ) S 2 ( u 2 + X 1 , v 2 + Y 1 ) | F O ( u 2 λ f 2 , v 2 λ f 2 ) | 2 .
I f d = | F O ( u 2 λ f 2 , v 2 λ f 2 ) | 2 - S 2 ( u 2 + X 1 , v 2 + Y 1 ) × S 1 ( f 1 f 2 X 1 , f 1 f 2 Y 1 ) d X 1 d Y 1 = | F O ( u 2 λ f 2 , v 2 λ f 2 ) | 2 S 1 ( f 1 f 2 u 2 , f 1 f 2 v 2 ) * * S 2 ( u 2 , v 2 ) .
I f d ( u 2 , v 2 ) = | F O ( u 2 λ f 2 , v 2 λ f 2 ) | 2 S ( u 2 , v 2 ) ,
S ( u 2 , v 2 ) = S 1 ( f 1 f 2 u 2 , f 1 f 2 v 2 ) * * S 2 ( u 2 , v 2 ) .
I f d ( u 1 , v 1 ) = | F O ( u 1 λ f 1 , v 1 λ f 1 ) | 2 S ( u 1 , v 1 ) ,
S ( u 1 , v 1 ) = S 1 ( u 1 , v 1 ) * * S 2 ( f 2 f 1 u 1 , f 2 f 1 v 1 ) .
O ( X , Y ) = exp [ i ϕ ( X , Y ) ] .
F O ( f x , f y ) = - O ( X , Y ) exp [ - i 2 π ( f x X + f y Y ) ] d X d Y ,
ϕ ( X , Y ) = ϕ ( X 0 , Y 0 ) + ϕ ( X 0 , Y 0 ) X ( X - X 0 ) + ϕ ( X 0 , Y 0 ) Y ( Y - Y 0 ) .
F O ( f x , f y ) = A δ [ f x - ϕ ( X 0 , Y 0 ) 2 π X , f y - ϕ ( X 0 , Y 0 ) 2 π Y ] ,
d ϕ d X = 2 π λ ( n - 1 ) sin α = 2 π λ sin β .
n d r d s = grad ϕ ,             d d s ( n d r d s ) = grad n ,
d d s ( grad ϕ ) = grad n .
grad ϕ = s 2 s 1 grad n d s .
exp [ i ϕ ( X , Y ) ] 1 + i ϕ ( X , Y ) .
C = I max - I min I max + I min = 1 - R 1 + R ,             R = I min I max .
S = tan - 1 ( Δ X 2 f 2 ) or ( Δ X 1 f 1 ) .
F O ( X 0 , Y 0 ) = A δ { [ X 2 λ ¯ f - ϕ ( X 0 , Y 0 , λ ¯ ) 2 π X - X 2 Δ λ λ ¯ 2 f + 2 π X d ϕ ( λ ¯ ) d λ Δ λ ] , [ Y 2 λ ¯ f - ϕ ( X 0 , Y 0 , λ ¯ ) 2 π Y - Y 2 Δ λ λ ¯ 2 f + 2 π Y d ϕ ( λ ¯ ) d λ Δ λ ] } ,
Δ f 1 = - X 2 ( Y 2 ) 2 π f d ϕ ( λ ¯ ) d λ Δ λ .
Δ f 2 = X 2 ( Y 2 ) λ ¯ f Δ λ .
l 3 = f 3 ( l 1 l - f 2 2 - l f 2 ) l 1 ( l - f 3 ) + f 2 ( f 3 - f 2 - l ) ,
M = f 3 f 2 l 1 ( l - f 3 ) + f 2 ( f 3 - f 2 - l ) ,
l 3 = f 3 f 2 f 2 - f 3 ,
M = - f 3 f 2 - f 3 .
f ¯ cut = d 3 2 λ f 2 ,
l 3 = f 3 f 2 2 ( f 2 2 + f 2 f 3 - l 1 f 3 ) ,
M = - f 3 f 2 .
l 3 = f 3 ( 1 + f 3 f 2 ) .
f ¯ cut = d 3 2 λ ( f 2 + f 3 ) .
l 3 = f 2 2 l 1 - f 2 ,
M = - f 2 l 1 - f 2 .
f ¯ cut = d 2 2 λ l 1 .
I ( u 1 , v 1 ) = | F O ( u 1 λ f 1 , v 1 λ f 1 ) | 2 S 1 ( - u 1 , - v 1 ) .
I ( u 2 , v 2 ) = | F O ( u 2 λ f 2 , v 2 λ f 2 ) | 2 S 2 ( u 2 , v 2 ) .
1 f 1 + 1 f 2 = 1 f 1 .
| F O ( X 2 λ D 2 , Y 2 λ D 2 ) | 2 .
I ( u 2 , v 2 ) = | F O ( u 2 λ D 2 , v 2 λ D 2 ) | 2 [ S 1 ( f 1 f 2 u 2 , f 1 f 2 v 2 ) * * S 2 ( u 2 , v 2 ) ] .
I ( u 2 , v 2 ) = | F O ( f 1 u 2 λ f 2 D 1 , f 1 v 2 λ f 2 D 1 ) | 2 [ S 1 ( f 1 f 2 u 2 , f 1 f 2 v 2 ) * * S 2 ( u 2 , v 2 ) ] ,
I ( u 1 , v 1 ) = | F O ( u 1 λ D 1 , v 1 λ D 1 ) | 2 [ S 1 ( u 1 , v 1 ) * * S 2 ( f 2 f 1 u 1 , f 2 f 1 v 1 ) ] .

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