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The 1D Gaussian filter is:

(1) 
The impulse response of the 1D Gaussian Filter is given by:

(2) 
Properties of the Gaussian Filter
 An important property of the Gaussian function is that the fourier of the Gaussian is itself a Gaussian

(3) 
 The width of the Gaussian increases as increases
Figure 3:
Effect of parameter sigma on the Gaussian function

 and are inversely related i.e, the bandwidth of the filter is inversely related to .
The impulse response of the 2D Gaussian filter is:

(4) 
The frequency response of the 2D Gaussian Filter is given by:

(5) 
=
where,
Figure 4:
Plot of frequency response of the 2D Gaussian

As can be seen from the figure, the frequency response of the 2D Gaussian, it is a low pass filter.
Next: Gaussian and Laplacian Pyramids
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