the noise n are modelled as stationary random fields and that noise and 
object are uncorrelated, With a psf in the Fourier frequency domain 
described as linear decreasing (sinc® in the Spatial domain), Frieden 
(1975) shows that the optimum cut-off frequency fe that minimizes the 
mean square error,i.e., minimum abs(F(s)-F(s))2 is 
fc 7 fy (1 - C N(s)/F(s) )1/2) 
where fy is the Nyqvist frequency. A serious problem appears if H has 
zeros at spatial frequencies within the range of interest, Linear 
uniform motion and defocusing are examples of such degradations 
(Sondhi, 1972), Philip (1979) proposes using the derivates of H and G 
in cases where H ig zero. This means that if HO then compute G/H 
else compute G'/H'. T£ Ha) compute the second order derivates and so 
forth. 
2) pol: 
   
Fig 4,1 Inverse filtering, a) original image fig 3.1b 
b) original image fig 3.14 
4.2 Wiener Filtering 
Assume that the original signal f(x), the corresponding degraded signal 
g(x) and the noise n(x) belong to different random fields. The de- 
gradation model can now be formulated in the following way: 
g(x) = f(x) + n(x) (4-9) 
Apply a filter function y(x) to eq. (4-9) 
2(x) = y(x)*g(x) = YOGO*£(x) * y(x)Xn(x) (4-10) 
Such that the output Signal z(x) will be as close as possible to the 
original function f(x), Define the difference between z(x) and f(x) as 
the error signal err(u), i.e.- 
err(x) - f(x») —»2(3) (4-11) 
Given the power Spectrum Pe(x) of the original signal f(x) and the 
power spectrum P,(x) of the noise n(x), determine the impulse response 
y(x) that minimizes the mean Square error 
MSE - E(err2(x)) (4-12) 
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