Gaussian function

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Gaussian curves parametrised by expected value and variance (see normal distribution)

A Gaussian function (named after Carl Friedrich Gauss) is a function of the form:

$f(x) = a e^{-(x-b)^2/c^2}$

for some real constants a > 0, b, and c.

Gaussian functions with c² = 2 are eigenfunctions of the Fourier transform. This means that the Fourier transform of a Gaussian function, f, is not only another Gaussian function but a scalar multiple of f.

Gaussian functions are among those functions that are elementary but lack elementary antiderivatives. Nonetheless their improper integrals over the whole real line can be evaluated exactly (see Gaussian integral):

$\int_{-\infty}^\infty e^{-ax^2}\,dx=\sqrt{\frac{\pi}{a}}.$

1 Two-dimensional Gaussian function
- 1.1 Meaning of parameters for the general equation
2 Applications
3 See also

[edit] Two-dimensional Gaussian function

2-d Gaussian curve

A particular example of a two-dimensional Gaussian function is

$f(x,y) = A e^{-\left( (\frac{x-x_o}{\sigma_x})^2 + (\frac{y-y_o}{\sigma_y})^2\right)}.$

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 left was created using A = 1, x_o = 0, y_o = 0, σ_x = σ_y = 1.

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

$f(x,y) = A \exp \left( - \left(a(x - x_o)^2 + b(x-x_o)(y-y_o) + c(y-y_o)^2 \right) \right)$

where the matrix

$\left[\begin{matrix} a & b \\ b & c \end{matrix}\right]$

is positive-definite.

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

[edit] Meaning of parameters for the general equation

For the general form of the equation the coefficient A is the amplitude and (x_o, y_o) is the center of the blob.

If we set

$a = \left(\frac{\cos\theta}{\sigma_x}\right)^2 + \left(\frac{\sin\theta}{\sigma_y}\right)^2$

$b = -\frac{\sin2\theta}{\sigma_x^2} + \frac{\sin2\theta}{\sigma_y^2}$

$c = \left(\frac{\sin\theta}{\sigma_x}\right)^2 + \left(\frac{\cos\theta}{\sigma_y}\right)^2$

then we rotate the blob by an angle $θ$ . This can be seen in the following examples:

θ = 0

θ = π / 6

θ = π / 3

Using the following MATLAB code one can see the effect of changing the parameters easily

A = 1;
x0 = 0; y0 = 0;
for theta = 0:pi/100:pi
sigma_x = 1;
sigma_y = 2;
a = (cos(theta)/sigma_x)^2 + (sin(theta)/sigma_y)^2;
b = -sin(2*theta)/(sigma_x)^2 + sin(2*theta)/(sigma_y)^2 ;
c = (sin(theta)/sigma_x)^2 + (cos(theta)/sigma_y)^2;

[X, Y] = meshgrid(-5:.1:5, -5:.1:5);
Z = A*exp( - (a*(X-x0).^2 + b*(X-x0).*(Y-y0) + c*(Y-y0).^2)) ;
surf(X,Y,Z);shading interp;view(-36,36);axis equal;drawnow
end

Such functions are often used in image processing and in models of visual system function -- see the articles on scale space and affine shape adaptation.

Also see multivariate normal distribution.

[edit] Applications

The antiderivative of the Gaussian function is the error function.

Gaussian functions appear in many contexts in the natural sciences, the social sciences, mathematics, and engineering. Some examples include:

In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to the central limit theorem.
A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.
The molecular orbitals used in computational chemistry are linear combinations of Gaussian functions called Gaussian orbitals (see also basis set (chemistry)).
Mathematically, the Gaussian function plays an important role in the definition of the Hermite polynomials.
Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory.
Gaussian beams are used in optical and microwave systems,
Gaussian functions are used as pre-smoothing kernels in image processing and computer vision -- see the article on scale space representation.
Gaussian functions are used in some types of artifical neural networks