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25 
of negligible width, the intensity distribution is 
characteristic of the lens, and is known as the spread 
function. Mathematically, the spread function can also 
be represented as a summation of sinusoidal intensity 
distributions in the same plane, the relationship being 
defined by the inverse Fourier transform. If the inten 
sity distribution in the image plane is G (u', v') the 
intensity being constant along v*, then 
where D^) is a complex factor giving the phase 0 W) 
and the modulus, T(t^) in relation to a reduced spatial 
frequency facter,«-«?. Thus 
The actual spatial frequency R, in cycles per millimeter, 
in the image plane, is related to^by 
R = 
2 F\ 
CO = 2XRF 
where 1/F = angular cone size of the lens aperture in the 
image plane, and\= wavelength of light in millimeters. A 
is a constant depending on the photometric situation. D(R) 
is known as the optical transfer function, (OTF), T(R) as 
the modulation transfer function (MTF). 
In practical terms, the MTF indicates the relative 
contrast with which a sinusoidal target is imaged, while 
8 (R) gives the lateral shift of the sinusoid in the image 
plane relative to its geometrically derived position. 
(While this phase shift effect is analogous to radial dis 
tortion, in photogrammetric terms, it relates only to the 
specified spatial frequency, whereas the shift of an image 
as normally observed is a consequence of the integrated 
effect for all spatial frequencies.) The linear term of the 
phase shift can be regarded as a radial image distortion, 
higher order terms relating to the quality in terms of image 
blur. The phase effect is normally less important than the 
MTF, since it is zero for a system which has no asymmetric 
aberrations and in general is most marked for higher spatial 
frequencies which are transmitted at low amplitude. 
The MTF can be most directly measured by imaging targets 
which vary sinusoidally in intensity, for a range of spatial 
frequencies, and measuring the image contrast. For this 
purpose contrast is expressed as modulation, M, defined by 
I max - I min 
I max + I min 
(Fig. 5)