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 wavenumber 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 wavenumber 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.

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  1. P. Mertz and F. Gray, Bell Syst. Tech. J. 13, 464 (1934).
  2. G. W. King and A. G. Emslie, J. Opt. Soc. Am. 41, 405 (1951).
  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, Tleory of Fourier Integrals (Oxford University Press, London, 1948), second edition. N. Wiener, Tc Fourier Integral (Dover Publications, New York); S. Bochner, Vorlesungen fiber Fouriersclie Integrale (Chelsea Publishing Company, New York, 1948); I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951).

Emslie, A. G.

G. W. King and A. G. Emslie, J. Opt. Soc. Am. 41, 405 (1951).

Gray, F.

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

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.

G. W. King and A. G. Emslie, J. Opt. Soc. Am. 41, 405 (1951).

Mertz, P.

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

Titchmarsh, E. C.

E. C. Titchmarsh, Tleory of Fourier Integrals (Oxford University Press, London, 1948), second edition. N. Wiener, Tc Fourier Integral (Dover Publications, New York); S. Bochner, Vorlesungen fiber Fouriersclie 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).

Other

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

G. W. King and A. G. Emslie, J. Opt. Soc. Am. 41, 405 (1951).

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, Tleory of Fourier Integrals (Oxford University Press, London, 1948), second edition. N. Wiener, Tc Fourier Integral (Dover Publications, New York); S. Bochner, Vorlesungen fiber Fouriersclie Integrale (Chelsea Publishing Company, New York, 1948); I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951).

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