Abstract

Optical differentiation and integration are not only interesting from the standpoint of optical computation, but are also important in picture processing. Based on optical correlation, or convolution, and the applications of difference equations, a technique for synthesizing the differentiation and integration filters is presented. This technique compares favorably with other methods which now exist.

© 1971 Optical Society of America

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References

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  1. J. B. DeVelis and G. O. Reynolds, Theory and Applications of Holography (Addison–Wesley, Reading, Mass., 1967), p. 154.
  2. A. Rosenfeld, Picture Processing by Computer (Academic, New York, 1969), Sec. 6.5, p. 94.
  3. T. S. Huang and O. J. Tretiak, in Optical and Electro-optical Information Processing, edited by J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C. J. Koester, and A. Vanderburgh (M.I.T. Press, Cambridge, Mass., 1965), Ch. 3, p. 45.
  4. A. W. Lohmann and D. P. Paris, Appl. Opt. 7, 651 (1968).
    [CrossRef] [PubMed]
  5. A. Vander Lugt, Opt. Acta 15, 1 (1968).
    [CrossRef]
  6. R. G. Eguchi and F. P. Carlson, Appl. Opt. 9, 687 (1970).
    [CrossRef] [PubMed]
  7. R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1968).
  8. In a somewhat different approach Trabka and Roetling [E. A. Trabka and P. G. Roetling, J. Opt. Soc. Am. 54, 1242 (1964)] have also arrived at similar expressions for the differential operators discussed in this section.
    [CrossRef]
  9. A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).
  10. These techniques for obtaining the negative sign have also been successfully applied to synthesizing a spatial filter for the combined operation of subtraction and correlation [S. K. Yao and S. H. Lee, J. Opt. Soc. Am. 60, 1547A (1970)].

1970 (2)

These techniques for obtaining the negative sign have also been successfully applied to synthesizing a spatial filter for the combined operation of subtraction and correlation [S. K. Yao and S. H. Lee, J. Opt. Soc. Am. 60, 1547A (1970)].

R. G. Eguchi and F. P. Carlson, Appl. Opt. 9, 687 (1970).
[CrossRef] [PubMed]

1968 (2)

1964 (2)

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1968).

Carlson, F. P.

DeVelis, J. B.

J. B. DeVelis and G. O. Reynolds, Theory and Applications of Holography (Addison–Wesley, Reading, Mass., 1967), p. 154.

Eguchi, R. G.

Huang, T. S.

T. S. Huang and O. J. Tretiak, in Optical and Electro-optical Information Processing, edited by J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C. J. Koester, and A. Vanderburgh (M.I.T. Press, Cambridge, Mass., 1965), Ch. 3, p. 45.

Lee, S. H.

These techniques for obtaining the negative sign have also been successfully applied to synthesizing a spatial filter for the combined operation of subtraction and correlation [S. K. Yao and S. H. Lee, J. Opt. Soc. Am. 60, 1547A (1970)].

Lohmann, A. W.

Paris, D. P.

Reynolds, G. O.

J. B. DeVelis and G. O. Reynolds, Theory and Applications of Holography (Addison–Wesley, Reading, Mass., 1967), p. 154.

Roetling, P. G.

Rosenfeld, A.

A. Rosenfeld, Picture Processing by Computer (Academic, New York, 1969), Sec. 6.5, p. 94.

Trabka, E. A.

Tretiak, O. J.

T. S. Huang and O. J. Tretiak, in Optical and Electro-optical Information Processing, edited by J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C. J. Koester, and A. Vanderburgh (M.I.T. Press, Cambridge, Mass., 1965), Ch. 3, p. 45.

Vander Lugt, A.

A. Vander Lugt, Opt. Acta 15, 1 (1968).
[CrossRef]

A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

Yao, S. K.

These techniques for obtaining the negative sign have also been successfully applied to synthesizing a spatial filter for the combined operation of subtraction and correlation [S. K. Yao and S. H. Lee, J. Opt. Soc. Am. 60, 1547A (1970)].

Appl. Opt. (2)

IEEE Trans. (1)

A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

J. Opt. Soc. Am. (2)

These techniques for obtaining the negative sign have also been successfully applied to synthesizing a spatial filter for the combined operation of subtraction and correlation [S. K. Yao and S. H. Lee, J. Opt. Soc. Am. 60, 1547A (1970)].

In a somewhat different approach Trabka and Roetling [E. A. Trabka and P. G. Roetling, J. Opt. Soc. Am. 54, 1242 (1964)] have also arrived at similar expressions for the differential operators discussed in this section.
[CrossRef]

Opt. Acta (1)

A. Vander Lugt, Opt. Acta 15, 1 (1968).
[CrossRef]

Other (4)

J. B. DeVelis and G. O. Reynolds, Theory and Applications of Holography (Addison–Wesley, Reading, Mass., 1967), p. 154.

A. Rosenfeld, Picture Processing by Computer (Academic, New York, 1969), Sec. 6.5, p. 94.

T. S. Huang and O. J. Tretiak, in Optical and Electro-optical Information Processing, edited by J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C. J. Koester, and A. Vanderburgh (M.I.T. Press, Cambridge, Mass., 1965), Ch. 3, p. 45.

R. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill, New York, 1968).

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Figures (7)

F. 1
F. 1

Experimental result for ∂f/∂x.

F. 2
F. 2

Object pattern for differentiation.

F. 3
F. 3

Experimental result for ∂f/∂y.

F. 4
F. 4

Experimental result for ∂f/∂x+∂f/∂y.

F. 5
F. 5

Experimental result for 2f/∂x2+2f/∂y2.

F. 6
F. 6

Experimental result for 2f/∂x∂y.

F. 7
F. 7

Integration, (a) Object pattern f(x,y). (b) Experimental result for −∞xf(ζ,η). Note that only the left half of the response represents integration, (c) A microdensitometer trace across (b).

Equations (13)

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Δ x f = [ f ( x + h , y ) f ( x , y ) ] / h ,
f ( ζ , η ) δ [ ζ ( x + a ) , η ( y + b ) ] d ζ d η = f ( x + a , y + b ) .
Δ x f = 1 h f ( ζ , η ) [ δ ( ζ x h , η y ) δ ( ζ x , η y ) ] d ζ d η ( 1 / h ) f ( x , y ) * g 1 ( x , y ) ,
f / x = lim h 0 Δ x f .
f / y = lim h 0 [ ( 1 / h ) f ( x , y ) * g 2 ( x , y ) ] ,
g 2 ( x , y ) = δ ( x , y + h ) δ ( x , y ) ; f x + f y = lim h 0 [ 1 h f ( x , y ) * g 3 ( x , y ) ] ,
g 3 ( x , y ) = δ ( x + h , y ) + δ ( x , y + h ) 2 δ ( x , y ) ; 2 f x 2 + 2 f y 2 = lim h 0 [ 1 h 2 f ( x , y ) * g 4 ( x , y ) ] ,
g 4 ( x , y ) = δ ( x + h , y ) + δ ( x h , y ) + δ ( x , y + h ) + δ ( x , y h ) 4 δ ( x , y ) ;
2 f / x y = lim h 0 [ ( 1 / h 2 ) f ( x , y ) * g 5 ( x , y ) ] ,
g 5 ( x , y ) = δ ( x + h , y + h ) + δ ( x , y ) δ ( x , y + h ) δ ( x + h , y ) .
f ( x , y ) * u ( x , 0 ) = f ( ζ , η ) u ( x ζ , η ) d ζ d η = x f ( ζ , η ) d ζ .
f ( x , y ) * u ( x , y ) = f ( ζ , η ) u ( x ζ , y η ) d ζ d η = x y f ( ζ , η ) d ζ d η .
g 1 ( x , y ) = δ ( x + h , y ) δ ( x , y ) .