Gaussian Function - Two-dimensional Gaussian Function

Two-dimensional Gaussian Function

In two dimensions, the power to which e is raised in the Gaussian function is any negative-definite quadratic form. Consequently, the level sets of the Gaussian will always be ellipses.

A particular example of a two-dimensional Gaussian function is

Here the coefficient A is the amplitude, x_o,y_o is the center and σ_x, σ_y are the x and y spreads of the blob. The figure on the right was created using A = 1, x_o = 0, y_o = 0, σ_x = σ_y = 1.

In general, a two-dimensional elliptical Gaussian function is expressed as

where the matrix

is positive-definite.

Using this formulation, the figure on the right can be created using A = 1, (x_o, y_o) = (0, 0), a = c = 1/2, b = 0.