One form of smoothing the image is to convolve the image intensity
with a Gaussian. Let us temporarily pretend that the image is infinite
and that the image intensity is 
. The Gaussian is a function
of the form
and convolution with it is the image processing equivalent of low pass filtering: why do you think this is? The higher the sigma the more low pass the filter. Now the filtered image is
One effect of filtering with a Gaussian is to reduce the amount of
noise, because of the low pass characteristic of the Gaussian.
The following is the image of Figure 1 with noise added
on to it (Figure (2 )
followed by filtering with a Gaussian with 
Figure
(3).
Figure 2: The original image corrupted with noise
Figure 3: The noisy image filtered by a Gaussian of variance 3
Additional examples are available at here.
