Abstract

The generation of a two-dimensional linear vector space in a coherent optical data processing system and its specific application to the gradient operator is discussed using theoretical and experimental results. The vector space is created by superposing the outputs of two Fourier optical systems having light of mutually orthogonal polarizations. As a result, the total amplitude of the signal in the output plane is a vector sum of the signals from the systems. If each one of the systems performs one of the partial derivative operations of the transverse gradient, and if the inputs to both systems are identical, then the output is the vector sum of the partial derivatives or the transverse gradient operation. The experimental program is heavily oriented toward the realization of the optimum approximation to the jωxî and jωyjˆ filters, rather than using various binary-type filters. Problems with film and lens noise are also discussed.

© 1970 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Inc., New York, 1968).
  2. L. J. Cutrona, E. N. Leith, C. V. Palermo, L. J. Porcello, IRE Trans. Inform. Theory 6, 386 (1960).
    [CrossRef]
  3. L. J. Cutrona, E. N. Leith, L. J. Porcello, Proc. Natl. Electron Conf. 15, 1 (1959).
  4. E. L. O’Neill, IRE Trans. Inform. Theory 2, 56 (1956).
    [CrossRef]
  5. T. M. Holladay, J. D. Gallatin, J. Opt. Soc. Amer. 56, 869 (1966).
    [CrossRef]
  6. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Co., Inc., New York, 1968).
  7. R. G. Eguchi, F. P. Carlson, “Coherent Optical Gradient System,” University of Washington, Tech. Rept., TR 127, November1968.

1966 (1)

T. M. Holladay, J. D. Gallatin, J. Opt. Soc. Amer. 56, 869 (1966).
[CrossRef]

1960 (1)

L. J. Cutrona, E. N. Leith, C. V. Palermo, L. J. Porcello, IRE Trans. Inform. Theory 6, 386 (1960).
[CrossRef]

1959 (1)

L. J. Cutrona, E. N. Leith, L. J. Porcello, Proc. Natl. Electron Conf. 15, 1 (1959).

1956 (1)

E. L. O’Neill, IRE Trans. Inform. Theory 2, 56 (1956).
[CrossRef]

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Co., Inc., New York, 1968).

Carlson, F. P.

R. G. Eguchi, F. P. Carlson, “Coherent Optical Gradient System,” University of Washington, Tech. Rept., TR 127, November1968.

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. V. Palermo, L. J. Porcello, IRE Trans. Inform. Theory 6, 386 (1960).
[CrossRef]

L. J. Cutrona, E. N. Leith, L. J. Porcello, Proc. Natl. Electron Conf. 15, 1 (1959).

Eguchi, R. G.

R. G. Eguchi, F. P. Carlson, “Coherent Optical Gradient System,” University of Washington, Tech. Rept., TR 127, November1968.

Gallatin, J. D.

T. M. Holladay, J. D. Gallatin, J. Opt. Soc. Amer. 56, 869 (1966).
[CrossRef]

Goodman, J. W.

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

Holladay, T. M.

T. M. Holladay, J. D. Gallatin, J. Opt. Soc. Amer. 56, 869 (1966).
[CrossRef]

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. V. Palermo, L. J. Porcello, IRE Trans. Inform. Theory 6, 386 (1960).
[CrossRef]

L. J. Cutrona, E. N. Leith, L. J. Porcello, Proc. Natl. Electron Conf. 15, 1 (1959).

O’Neill, E. L.

E. L. O’Neill, IRE Trans. Inform. Theory 2, 56 (1956).
[CrossRef]

Palermo, C. V.

L. J. Cutrona, E. N. Leith, C. V. Palermo, L. J. Porcello, IRE Trans. Inform. Theory 6, 386 (1960).
[CrossRef]

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. V. Palermo, L. J. Porcello, IRE Trans. Inform. Theory 6, 386 (1960).
[CrossRef]

L. J. Cutrona, E. N. Leith, L. J. Porcello, Proc. Natl. Electron Conf. 15, 1 (1959).

IRE Trans. Inform. Theory (2)

E. L. O’Neill, IRE Trans. Inform. Theory 2, 56 (1956).
[CrossRef]

L. J. Cutrona, E. N. Leith, C. V. Palermo, L. J. Porcello, IRE Trans. Inform. Theory 6, 386 (1960).
[CrossRef]

J. Opt. Soc. Amer. (1)

T. M. Holladay, J. D. Gallatin, J. Opt. Soc. Amer. 56, 869 (1966).
[CrossRef]

Proc. Natl. Electron Conf. (1)

L. J. Cutrona, E. N. Leith, L. J. Porcello, Proc. Natl. Electron Conf. 15, 1 (1959).

Other (3)

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

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Co., Inc., New York, 1968).

R. G. Eguchi, F. P. Carlson, “Coherent Optical Gradient System,” University of Washington, Tech. Rept., TR 127, November1968.

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Figures (16)

Fig. 1
Fig. 1

Optical gradient system.

Fig. 2
Fig. 2

Transfer function of an ideal differentiating filter and a practical realization.

Fig. 3
Fig. 3

Ideal filter decomposed into two practical filters.

Fig. 4
Fig. 4

Initial approximation to ideal differentiating filter.

Fig. 5
Fig. 5

Decomposition of practical differentiating filter.

Fig. 6
Fig. 6

Approximate doublet function.

Fig. 7
Fig. 7

Convolution of π(μ/α) with sin2πWx/πWx.

Fig. 8
Fig. 8

Convolution of (∂/∂x) [π(x/α)] with sinc(Wx).

Fig. 9
Fig. 9

(a) Superimposed signals with orthogonally polarized light. (b) Superimposed signals with parallel light polarizations.

Fig. 10
Fig. 10

Derivative operation using only the sgn(ωx) filter.

Fig. 11
Fig. 11

(a) Hair lines of width 6.1 × 10−3 and 3.3 × 10−3 cm. (b) Derivative of the hair line input image.

Fig. 12
Fig. 12

(a) Binary input image of four wedges, (b) Differentiation of binary scene (a) in one direction, long exposure. (c) Differentiation of binary scene (a) in one direction, short exposure.

Fig. 13
Fig. 13

(a) Arbitrary input object. (b) Derivative of object without a film gate. (c) Derivative of object with a film gate.

Fig. 14
Fig. 14

Amplitude transmission vs exposure curves.

Fig. 15
Fig. 15

Practical filter transfer function.

Fig. 16
Fig. 16

Film curve in collimated light beam to create linear density profile.

Equations (20)

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g T ( x , y ) = g 1 ( x , y ) î + g 2 ( x , y ) j ˆ ,
| g T ( x , y ) | 2 = g 1 ( x , y ) g 1 * ( x , y ) + g 2 ( x , y ) g 2 * ( x , y ) .
g ( x , y ) = ( / x ) g ( x , y ) î + ( / y ) g ( x , y ) j ˆ ,
| g ( x , y ) | 2 = | ( / x ) g ( x , y ) | 2 + | ( / y ) g ( x , y ) | 2 .
g ( x , y ) j ω x G ( ω x , ω y ) î + j ω y G ( ω x , ω y ) j ˆ
F ( ω x , ω y ) = ω x + j ω y ,
g T ( x , y ) = j ( / x ) ( g ( x , y ) + ( / y ) g ( x , y ) .
F 1 { j k ω x G ( ω x , ω y ) r e c t ( ω x / W ) } = k [ g ( x , y ) / x ] * sin c ( W x ) .
F 1 { sgn ( ω x ) r e c t ( ω x / W ) } = j ( sin 2 π W x / π W x ) .
T I = 10 D .
T A = 10 D / 2 .
A n = A min + n ( Δ S / S max ) ( A max A min ) , n = 0,1 , N ,
Δ S = S max / N .
E E max = 1 [ 1 + ( Δ y / Δ x ) 2 ] 1 2 .
( Δ υ / Δ x ) = [ ( E max / E ) 2 1 ] 1 2 .
[ ( Δ y n ) 2 + ( Δ x n ) 2 ] 1 2 = Δ S ,
( Δ y n / Δ x n ) = [ ( E max / E n ) 2 1 ] 1 2 ,
Δ x n = Δ S E n / E max .
y m = n = 0 m Δ y n ,
x m = n = 0 m Δ x n .

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