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The combination of the aperture function and the pulses concentrated 
at the sampling points models the real situation. Due to physical 
restrictions sampling at a point can not be achieved. Thus the sample 
always will be taken over a finite area rather than at a point. 
The image has a limited extension in space and the expression ( 5. 2 ) 
has to be multiplied by a window function defined by: 
1 for |x| < A and [y|< B 
(x,y)= ( 3.3.) 
rect,, 
0 otherwise 
and we get : fax. v) f(x,y) rect, (x,y) i: 5.4) 
x/2'1y/2 
A combination of all these steps gives an expression for sampling a 
signal ( image ) limited in space with a finite aperture: 
{x.y} (5.5) 
qoo yo Ceocyrtaooyn[ I r TI rect 1 
x/2'‘ y/2 
This can also be formulated in the frequency plane: 
UI Ey (Flu vA Gu 138] M MEIETNET * 
af cs 1 1 
(sinc(_xu) sinc(_yv)] ( 5.6 ] 
the functions f(x,y) and rect(x,y) describe properties of the image, 
i.e. the distribution of grey levels and the extension of the image. 
a(x,y) and the 6-function model properties of the digitizing system. 
A complete description of the image should include the effect of the 
grain noise. A noise term should be added so that we, rather than 
f(x,y), write f(x,y) + n(x,y) or f(x,y)n,. The fact that our original 
image is degraded by noise is one of our main concerns since grain 
noise affects both the choice of sampling interval and the accuracy of 
the grey level quantization. 
The sampled values are usally represented as pixels. This means that 
each sample is represented by a square of the size of the sampling 
interval with the intensity equal to the amplitude in the sample 
point. This can be interpreted as a convolution of tbe,discrete signal 
with a constant impulse response over the interval in < Ax/2 lyl < 
ày/2 ). As a consequence of this convolution the frequency spectrum of 
the digital image will be distorted. 
5.2 h óc Mo 
In the preceding section the image is treated as a deterministic 
function. In the analysis of a digitizing system it is sometimes 
advantageous to use a stochastic model. The reason is the stochastic 
nature of the noise and the possibility of using statistical 
performance and quality measures. In any system designed for a certain 
Class of images, e.g. aerial photographs, each image f(x,y) can be 
treated as a realization of a two-dimensional stochastic process 
Fix.y). 
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