If the noise is treated as statistically independent of the signal, 
the spectra of the image and the noise add. The power spectrum of the 
process is given by 
2 
D. ; 51%) 
®.=| Alu,v) | lo (u,v) + e, (u.vl] ( 
In (5.7) 6, *. and © are the fourier transforms of the autocorrela- 
tion functions of the stochastic processes and A(u,v) the transform 
of the aperture function. 
The noise normally has a greater band width than the image and the 
sampling interval must be adjusted accordingly. There also is the 
possibility of using the aperture of the digitizing system to reduce 
the band width . As mentioned in section 3.2 the aperture in an 
optical system acts as a low-pass filter. A(u,v) in (5.6) or (5.7) 
should be chosen to restrict F or ¢ in the frequency plane in such a 
way that 
f (u,v] : jul» b Iv] > b,! ( 5.8 ) 
where the intervals b , b are related to Ax, Ay by Ax<w/b ,Ay<w/b 
If the digitizing has been done properly it is possible to recover the 
original signal. This is done by interpolation ( Wahl, 1984 ). 
The performance of the system can be estimated from the difference 
between the samples and the original function: 
N N 
2 2 
e = 1/(N-1) [ALF (x.y) - f, v1) ( 5.9 ] 
r 
x=0 y=0 
6. NOISE AND QUANTIZATION 
For the purpose of analyzing the effects of noise on quantization the 
following measures ( Billingsley, 1975 ) are defined 
S/0 
full scale or black to white signal / rms noise ratio 
SNR = actual signal / rms noise 
B = quantizing step/ rms noise 
"Signal" is to be interpreted as the amount of light measured from 
zero = black, or in a film digitizer the film transmittance measured 
from zero = opaque. Of these measures the first two relate signal 
amplitude to rms noise. The last relates the size of the digital step 
to rms noise. For a system with 256 digital levels we will have f 
equal to ( 1 / 256 ) / o since transmittance is measured from zero to 
one. 
The quantizer transforms the signal into a discrete number of steps. 
If no noise is present there is no ambiguity in assigning a given 
signal level its digital number. In the presence of noise the signal 
plus the noise are quantized and from the digital number assigned we 
can not tell the level of the signal. There is a probability of 
assigning a wrong digital number to a given measurement. The probabi- 
lity is measured by the relative areas bounded by the quantizing step 
areas ( fig 6.1 ). 
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