where EC.) denotes the expected-value operator. Assume that the signal 
£(x) and the noise n(x) are uncorrelated. Then the Fourier frequency 
domain specification of the Wiener filter (Andrews and Hunt, 1977) is 
Y(s) = P£(s) (4-13) 
Pe(s) + Pn(s) 
The frequency domain expression for the mean square error is 
MSE = [ Py(s) Y(s) ds (4-14) 
   
Fig 4.2 Wiener smooth filtering. a) original image plus random noise, 
b) Wiener smoothed image 
If the signal to noise ratio is low (Castleman, 1979) eq. (4-14) 
reduces to the approximation of sigma i.e., MSE = [ P(s)ds which can be 
approximated with [£2(x)ax. 1f the Wiener filter is combined with an 
ordinary deconvolution filter (inverse in frequency domain), the 
orginal signal f(x) in the degradation process, ed. (4-2) may be 
restored by applying the filter 
  
Y(s) = H*(s) Pe(s) (4-15a) 
abs(H(s))2 Pg(s) + Pn(S) 
- 1 abs(H(s))2 (4-15b) 
H(s) abs(H(s))^ * Py(s)/PE(S) 
Note that, in the absence of noise (Ph = 0), the above expression 
(4-15b) reduces to the ideal filter 1/8(8). The noise-to-signal power 
ratio term Pp(s)/Pg(s) in (4-15b) may be regarded as a modification 
function which smoothes 1/H(s) in the presence of noise. If the 
statistical properties, i.e., the power spectra of noise and signal are 
unknown, it is common to approximate the power ratio term with a 
constant (Rosenfeld and Kak, 1976). Expression (4-15b) will then 
become 
v(s) = _1 _abs(F(8))2 (4-16) 
H(s) abs(F(s))2 toC 
where C is an arbitrary constant. Wiener filtering provides an optimal 
method to deconvolve an image blurred by noise. However, there are 
problems that limit its effectiveness (Castleman, 1979). The Wiener 
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