The Gaussian filter is a linear filter that is usually used as a smoother . The output of the gaussian filter at the moment is the weighted mean of the input values, and the weights are defined by formula
is the "distance" in time from the current moment;
is the parameter of the Gaussian filter;
is the normalization constant chosen to make the sum of all weights equal to the unit value.
If you plot the values of against , then the plot coincides with the famous bell-like curve describing the density of the gaussian distribution . This explains the word "gaussian" in the name of the filter.
The gaussian filter is completely defined by a single parameter . The greater the value of , the wider the window function , and, hence, the greater the degree of smoothing.
Besides the one-dimensional gaussian filter described above, there are extensions to the case of two dimensions, say, . Such two-dimensional gaussian filters are widely used in image processing .