Abstract

Many optical processes of image formation, image transfer, and image analysis may be represented as one, or a succession of several, linear operations. A linear operation upon a flux distribution function of an n-dimensional argument is defined as one which replaces the value of the function at a point by a linear, weighted average taken over a neighborhood of that point. While such an operation is completely determined by the weighting function used, it is also determined by a “wave-number” spectrum which is a function of an n-dimensional wave-number vector. This wave-number spectrum is the complex conjugate of the n-dimensional Fourier transform of the weighting function. The wave-number spectrum of the flux distribution modified by any number of successive linear operations is the product of the wave-number spectrum of the original distribution, and the wave-number spectra of the several linear operations. An analysis thus performed in wave-number space replaces successive integrations by successive multiplications.

This method of analysis is an extension of the usual method of treating filters in electronic circuits, and may be used to solve problems analogous to those treated in circuit theory. These are: (1) to evaluate the performance of a system; (2) to design a process to search an image for a configuration; (3) to reproduce a picture, with discrimination in favor of a configuration desired, and against others; and (4) to equalize a picture, i.e., to remove image degradation.

© 1952 Optical Society of America

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References

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  1. P. Mertz and F. Gray, Bell Syst. Tech. J. 13, 464 (1934).
    [CrossRef]
  2. G. W. King and A. G. Emslie, J. Opt. Soc. Am. 41, 405 (1951).
    [CrossRef] [PubMed]
  3. N. Wiener, The Extrapolation Interpolation and Smoothing of Stationary Time Series (Technology Press and John Wiley and Sons, Inc., New York, 1949).
  4. F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1937), p. 116.
  5. E. C. Titchmarsh, Theory of Fourier Integrals (Oxford University Press, London, 1948), second edition. N. Wiener, The Fourier Integral (Dover Publications, New York); S. Bochner, Vorlesungen über Fouriersche Integrale (Chelsea Publishing Company, New York, 1948); I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951).

1951 (1)

1934 (1)

P. Mertz and F. Gray, Bell Syst. Tech. J. 13, 464 (1934).
[CrossRef]

Emslie, A. G.

Gray, F.

P. Mertz and F. Gray, Bell Syst. Tech. J. 13, 464 (1934).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1937), p. 116.

King, G. W.

Mertz, P.

P. Mertz and F. Gray, Bell Syst. Tech. J. 13, 464 (1934).
[CrossRef]

Titchmarsh, E. C.

E. C. Titchmarsh, Theory of Fourier Integrals (Oxford University Press, London, 1948), second edition. N. Wiener, The Fourier Integral (Dover Publications, New York); S. Bochner, Vorlesungen über Fouriersche Integrale (Chelsea Publishing Company, New York, 1948); I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951).

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1937), p. 116.

Wiener, N.

N. Wiener, The Extrapolation Interpolation and Smoothing of Stationary Time Series (Technology Press and John Wiley and Sons, Inc., New York, 1949).

Bell Syst. Tech. J. (1)

P. Mertz and F. Gray, Bell Syst. Tech. J. 13, 464 (1934).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (3)

N. Wiener, The Extrapolation Interpolation and Smoothing of Stationary Time Series (Technology Press and John Wiley and Sons, Inc., New York, 1949).

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1937), p. 116.

E. C. Titchmarsh, Theory of Fourier Integrals (Oxford University Press, London, 1948), second edition. N. Wiener, The Fourier Integral (Dover Publications, New York); S. Bochner, Vorlesungen über Fouriersche Integrale (Chelsea Publishing Company, New York, 1948); I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951).

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Figures (4)

Fig. 1
Fig. 1

A multi-stage image formation and reproduction process is illustrated above. An object is first photographed (top) and the photograph is transmitted by facsimile. The facsimile process scans the photograph with an aperture which is the image of a diaphragm which is in front of a phototube. The output of the phototube is amplified and impressed on a cathode ray-tube. If the photographic process is linear (i.e. of unit gamma) and the cathode-ray tube is preceded by linearizing circuits, the process illustrated above can be represented as one single filter function which operates upon the object. This filter function is the ordinary algebraic product of several filter elements, which elements describe the operations of the successive stages of image reproduction.

Fig. 2
Fig. 2

Lens aberrations form a low pass multidimensional filter. The spectral response of this filter is the complex conjugate of the Fourier transform of that function by which the intensity at points in the object plane are weighted to give the intensity at a specified point in the image plane. This weighting function is proportional to the intensity which would be found within the image of a “point source” if the optical system were traversed in reverse sense; i.e., from image plane to object plane.

Fig. 3
Fig. 3

Diffusion of light within the photographic emulsion acts as a low pass filter. The spectral response of this filter is the complex conjugate of the Fourier transform of that function by which the intensity at points in the aerial image are weighted to give the exposure at a specified point of the photograph. In the figure the aerial image is shown as formed on the left-hand side of a diffusing layer. The point O of the photograph, with coordinates (x0, y0), receives an exposure E(x0, y0) which is the integral of the product of the intensity within the aerial image and a weighting function centered at (x0, y0).

Fig. 4
Fig. 4

The combined effect of aberration and diffusion is illustrated here. The exposure at O is derived from the energy within an extended portion of the aerial image; and each point in the aerial image receives energy from an extended region in the object. Hence, the exposure at O is a weighted average of intensity at the object. This weighting function can be computed directly from the weighting functions for aberration and diffusion by the integral: b3(x)=∫b1(x)b2(xy)dy, where b3 is the over-all weighting function, and b1 and b2 are the separate weighting functions. Alternatively, the spectral response B(ω) of the composite filter is the product B1(ω)B2(ω) where B1 and B2 are the spectra of the separate filters. For simplicity of representation, diffusion is shown as occurring only in the vertical direction.

Equations (23)

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M ( ω ) = F ( ω ) + G ( ω ) ,
M ( ω ) = - exp ( i ω x ) m ( x ) d x , F ( ω ) = - exp ( i ω x ) f ( x ) d x , G ( ω ) = - exp ( i ω x ) g ( x ) d x .
m ( x ) = ( 2 π ) - 1 - exp ( - i ω x ) M ( ω ) d ω , etc .
M ( ω ) = - exp ( i ω · x ) m ( x ) d x
m ( x ) = ( 2 π ) - n - exp ( - i ω · x ) M ( ω ) d ω , etc .
ω · x = ω 1 x 1 + ω 2 x 2 + ω n x n
d x = d x 1 d x 2 d x n , d ω = d ω 1 d ω 2 d ω n .
b m ( x ) = - m ( x - y ) b ( y ) d y .
B ( ω ) = - exp ( i ω · y ) b ( y ) d y .
B M ( ω ) = B ( ω ) M ( ω ) .
b 3 b 2 b 1 m ( x ) = - b 3 ( v ) - b 2 ( u ) - b 1 ( y ) m ( x - y - u - v ) d y d u d v = ( 2 π ) - n - exp ( - i ω · x ) × B 3 ( ω ) B 2 ( ω ) B 1 ( ω ) M ( ω ) d ω .
ϕ m ( y ) = - m ( x ) m ( x + y ) d x ,
ϕ m ( y ) = lim α a - n - a / 2 a / 2 - a / 2 a / 2 m ( x ) m ( x + y ) d x .
Φ m ( ω ) = - exp ( i ω · y ) ϕ ( y ) d y .
Φ m ( ω ) = M ( ω ) M ¯ ( ω ) = M ( ω ) 2 ,
( 2 π ) - n - Φ m ( ω ) d ω = - m 2 ( x ) d x ,
Φ b m ( ω ) = B ( ω ) B ¯ ( ω ) Φ m ( ω ) .
A 1 ( ω ) = 1 / B ( ω ) .
( x ) = f ( x ) - a 2 m ( x ) .
A 2 ( ω ) = Φ f ( ω ) / Φ m ( ω ) .
p ( a 3 ) = max [ a 3 m ( x ) ] ,
δ ( x ) = a 3 g ( x ) .
A 3 ( ω ) = F ( ω ) / Φ g ( ω ) .