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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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