functional form independent of its position (shift invariant) then the 
equation (2-2) can he expressed as equation (2-1). 
  
f | h | g 
  
  
  
Fig 2.1 A Linear System. 
2.3 The Fourier Transform 
A one-dimensional £orward Fourier transform of a function g(x) is 
defined by 
+ © 
Gta) = Fégla3) = flats) 0727305 dx 
© 
where j 2Y-1 . The operator F defines the Fourier operation. The 
inverse Fourier transform is defined by 
+ 
qa) = FÎ(GCa)) = J'Ets) e* DM dB 
The power spectrum of a function g(x) is defined as 
p(s) = abs(F(g(x))? 
which is a description of the distribution of the energy of the 
function at certain frequencies. The convolution in the spatial domain 
corresponds to multiplication in the frequency domain (and vice versa). 
This means that 
F(g(x)*h(x)) G(s)H(s) 
and 
F(G(s)*H(S)) 
i 
g(x)h(x) 
which in linear system analysis is called the convolution theorem, 
3. MODELLING 
3.1 Image Models 
There are several models for describing a digital image (Rosenfeld, 
1981). Two common models will be described here: statistical and 
deterministic (Andrews and Hunt, 1977). 
3.1.1 Statistical Image Model 
In a statistical model of an image we assume that each component 
(pixel) in the image results from a random variable. Thus, each image 
is the result of à family of random variables, a random field. A 
random field is a two-dimensional case of a random process. Consider 
the random field f(r,w,) where r is a vector in the xy-plane. For à 
given outcome Wj, f(r,wj) isa function over the xy-plane. A sequence 
of such functions is an ensemble. For a given value of r, f(ür,w;) isa 
random variable. The expectation of this random variable mg(r) - 
Et£(r)) is an ensemble average and is called the mean of the random 
- 225 -