The basis function for the generation of kernels (convolution matrices) for image low-pass filtering or smoothing is the Gaussian distribution. The function has the form:

with parameters x, mean and standard deviation.

Different parameters can lead to different “bell shaped” curves like the following ones:

Considering that we often will work with Gaussian functions centered at zero (i.e. mean=0), we can simplify the previous formula with the following ones.

One dimension:

Two dimensions:

N dimensions: