Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form

"Gaussian curve" redirects here. For the band, see Gaussian Curve (band).

Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value $μ = b$ and variance $σ 2 = c 2$ . In this case, the Gaussian is of the form^[1]

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

$\alpha =-1/2c^{2},$

$\beta =b/c^{2},$

$\gamma =\ln a-(b^{2}/2c^{2}).$

Gaussian functions arise by composing the exponential function with a concave quadratic function:

(Note: $a=1/(\sigma {\sqrt {2\pi }})$ in $\ln a$ , not to be confused with $\alpha =-1/2c^{2}$ )

The Gaussian functions are thus those functions whose logarithm is a concave quadratic function.

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

The function may then be expressed in terms of the FWHM, represented by $w$ :

Alternatively, the parameter $c$ can be interpreted by saying that the two inflection points of the function occur at $x = b \pm c$ .

The full width at tenth of maximum (FWTM) for a Gaussian could be of interest and is

Gaussian functions are analytic, and their limit as $x \to \infty$ is 0 (for the above case of $b = 0$ ).

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

This integral is 1 if and only if ${\textstyle a={\tfrac {1}{c{\sqrt {2\pi }}}}}$ (the normalizing constant), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value $μ = b$ and variance $σ 2 = c 2$ :

These Gaussians are plotted in the accompanying figure.

Gaussian functions centered at zero minimize the Fourier uncertainty principle.

The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: $c^{2}=c_{1}^{2}+c_{2}^{2}$ . The product of two Gaussian probability density functions (PDFs), though, is not in general a Gaussian PDF.

Taking the Fourier transform (unitary, angular-frequency convention) of a Gaussian function with parameters $a = 1$ , $b = 0$ and $c$ yields another Gaussian function, with parameters $c$ , $b = 0$ and $1/c$ .^[2] 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). A physical realization is that of the diffraction pattern: for example, a photographic slide whose transmittance has a Gaussian variation is also a Gaussian function.

The fact that the Gaussian function is an eigenfunction of the continuous Fourier transform allows us to derive the following interesting identity from the Poisson summation formula:

In 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.

statistics

Gaussian functions are the for the (homogeneous and isotropic) diffusion equation (and to the heat equation, which is the same thing), 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/√t and c being linearly related to √t; this time-varying Gaussian is described by the heat kernel. More generally, if the initial mass-density is φ(x), then the mass-density at later times is obtained by taking the convolution of φ with a Gaussian function. The convolution of a function with a Gaussian is also known as a Weierstrass transform.

Green's function

A Gaussian function is the of the ground state of the quantum harmonic oscillator.

wave function

The used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals (see also basis set (chemistry)).

molecular orbitals

Mathematically, the of the Gaussian function can be represented using Hermite functions. For unit variance, the n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial, up to scale.

derivatives

Consequently, Gaussian functions are also associated with the in quantum field theory.

vacuum state

are used in optical systems, microwave systems and lasers.

Gaussian beams

In representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing. Specifically, derivatives of Gaussians (Hermite functions) are used as a basis for defining a large number of types of visual operations.

scale space

Gaussian functions are used to define some types of .

artificial neural networks

In a 2D Gaussian function is used to approximate the Airy disk, describing the intensity distribution produced by a point source.

fluorescence microscopy

In they serve to define Gaussian filters, such as in image processing where 2D Gaussians are used for Gaussian blurs. In digital signal processing, one uses a discrete Gaussian kernel, which may be approximated by the Binomial coefficient^[12] or sampling a Gaussian.

signal processing

In they have been used for understanding the variability between the patterns of a complex training image. They are used with kernel methods to cluster the patterns in the feature space.^[13]

geostatistics

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

Normal distribution

Cauchy distribution

Radial basis function kernel

Mathworld, includes a proof for the relations between c and FWHM

. MathPages.com.