Gaussian function

In mathematics, a Gaussian function (named after Carl Friedrich Gauss) is a function of the form:


 * $$f(x) = a e^{- { (x-b)^2 \over 2 c^2 } }$$

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

The graph of a Gaussian is a characteristic symmetric "bell shape curve" that quickly falls off towards plus/minus infinity. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c controls the width of the "bell".

Gaussian functions are widely used in statistics where they describe the normal distributions, in signal processing where they serve to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used to solve heat equations and diffusion equations and to define the Weierstrass transform.

Properties
Gaussian functions arise by applying the exponential function to a general quadratic function. The Gaussian functions are thus those functions whose logarithm is a quadratic function.

The parameter c is related to the full width at half maximum (FWHM) of the peak according to


 * $$\mathrm{FWHM} = 2 \sqrt{2 \ln(2)}\ c.$$

Alternatively, the parameter c can be interpreted by saying that the two inflection points of the function occur at x=b-c and x=b+c.

Gaussian functions are analytic, and their limit for x&rarr;&plusmn;&infin; is 0. Gaussian functions are among those functions that are elementary but lack elementary antiderivatives; the integral of the Gaussian function is the error function. Nonetheless their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral


 * $$\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}$$

and one obtains


 * $$\int_{-\infty}^\infty a e^{- { (x-b)^2 \over 2 c^2 } }\,dx=ac\cdot\sqrt{2\pi}$$

This integral is 1 if and only if a = 1/(c&radic;(2&pi;)), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value &mu;=b and variance &sigma;2=c2. These Gaussians are graphed in the accompanying figure.

Taking the Fourier transform of a Gaussian function with parameters a, b=0 and c yields another Gaussian function, with parameters ac, b=0 and 1/c. So in particular the Gaussian functions with b=0 and c=1 are kept fixed by the Fourier transform (they are eigenfunctions of the Fourier transform with eigenvalue 1).

The product of two Gaussian functions is again a Gaussian, and the convolution of two Gaussian functions is again a Gaussian.

Two-dimensional Gaussian function


A particular example of a two-dimensional Gaussian function is


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

Here the coefficient A is the amplitude, xo,yo is the center and σx, σy are the x and y spreads of the blob. The figure on the left was created using A = 1, xo = 0, yo = 0, σx = σy = 1.

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


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

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, (xo, yo) = (0, 0), a = c = 1, b = 0.

Meaning of parameters for the general equation
For the general form of the equation the coefficient A is the height of the peak and (xo, yo) is the center of the blob.

If we set


 * $$a = \frac{\cos^2\theta}{2\sigma_x^2} + \frac{\sin^2\theta}{2\sigma_y^2}$$


 * $$b = -\frac{\sin2\theta}{4\sigma_x^2} + \frac{\sin2\theta}{4\sigma_y^2}$$


 * $$c = \frac{\sin^2\theta}{2\sigma_x^2} + \frac{\cos^2\theta}{2\sigma_y^2}$$

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

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)^2/2/sigma_x^2 + sin(theta)^2/2/sigma_y^2; b = -sin(2*theta)/4/sigma_x^2 + sin(2*theta)/4/sigma_y^2 ; c = sin(theta)^2/2/sigma_x^2 + cos(theta)^2/2/sigma_y^2;

[X, Y] = meshgrid(-5:.1:5, -5:.1:5); Z = A*exp( - (a*(X-x0).^2 + 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 computational models of visual system function -- see the articles on scale space and affine shape adaptation.

Also see multivariate normal distribution.

Applications
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.
 * Gaussian functions are closely related to the (homogeneous and isotropic) diffusion equation (and, which is the same thing, to the heat equation), a partial differential equation that describes the time evolution of a mass-density under diffusion. Specifically, if the mass-density at time t=0 is given by a Dirac delta, which essentially means that the mass is initially concentrated in a single point, then the mass-distribution at time t will be given by a Gaussian function, with the parameter a being linearly related to 1/&radic;t and c being linearly related to &radic;t. More generally, if the initial mass-density is &phi;(x), then the mass-density at later times is obtained by taking the convolution of &phi; with a Gaussian function. The convolution of a function with a Gaussian is also known as a Weierstrass transform.
 * A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.
 * The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals (see also basis set (chemistry)).
 * Mathematically, the derivatives of the Gaussian function are used to define 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 smoothing kernels for generating multi-scale representations in computer vision and image processing -- see the article on scale space representation. Specifically, derivatives of Gaussians are used as a basis for defining a large number of types of visual operations.
 * Gaussian functions are used to define some types of artificial neural networks.