Abstract

When Fourier methods are used to predict the response of an imaging system to very small objects, the concept most directly related to the size of the object or its image is not spatial frequency, but bandwidth. The importance of this distinction is easily demonstrated, using the convolution theorem and the properties of Dirac delta-functions. For example, it is shown that the contrast of the standard three-bar target is not reduced to zero when its number of “lines/mm” equals the high-frequency cutoff of the appropriate transfer function. Misleading conclusions can be avoided by using the correct Fourier integral representation of the object, instead of the series approximation.

© 1965 Optical Society of America

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Figures (6)

Fig. 1
Fig. 1

Derivation of Fourier transform pair representing the results of Young’s double-slit experiment (in em amplitude units).

Fig. 2
Fig. 2

Transform pairs as in Fig. 1, but with all right-hand members squared, representing Young’s interference pattern (in radiant units); and the corresponding self-convolution of the aperture function.

Fig. 3
Fig. 3

Standard Air Force resolution chart, showing typical three-bar targets.

Fig. 4
Fig. 4

One-dimensional frequency spectrum of standard three-bar target derived by same technique used in Figs. 1 and 2 for generating transform pairs. Here the left-hand members may be regarded as spatial patterns and the right-hand members as frequency spectra (since Fourier transform pairs of real, even functions are interchangeable).

Fig. 5
Fig. 5

Frequency spectrum of three-bar target object with bar spacing 1/N (dotted), and image (solid) formed by rectangular, aperture-limited optical system with cutoff frequency of N cycles/mm.

Fig. 6
Fig. 6

Input and output patterns corresponding to Fig. 5—note residual peaks identifiable with original bars, even though the output spectrum contains zero energy at frequencyN I max - I min I max + I min = 7 %.

Equations (1)

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V = I max - I min I max + I min .

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