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Gaussian function

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Mathematical function

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

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Inmathematics, aGaussian function, often simply referred to as aGaussian, is afunction of the base form $f(x)=\exp(-x^{2})$ and with parametric extension $f(x)=a\exp \left(-{\frac {(x-b)^{2}}{2c^{2}}}\right)$ for arbitraryreal constantsa,b and non-zeroc. It is named after the mathematicianCarl Friedrich Gauss. Thegraph of a Gaussian is a characteristic symmetric "bell curve" shape. The parametera is the height of the curve's peak,b is the position of the center of the peak, andc (thestandard deviation, sometimes called the GaussianRMS width) controls the width of the "bell".

Gaussian functions are often used to represent theprobability density function of anormally distributed random variable withexpected valueμ =b andvarianceσ² =c². In this case, the Gaussian is of the form^[1]

$g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}{\frac {(x-\mu )^{2}}{\sigma ^{2}}}\right).$

Gaussian functions are widely used instatistics to describe thenormal distributions, insignal processing to defineGaussian filters, inimage processing where two-dimensional Gaussians are used forGaussian blurs, and in mathematics to solveheat equations anddiffusion equations and to define theWeierstrass transform. They are also abundantly used inquantum chemistry to formbasis sets.

Properties

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Gaussian functions arise by composing theexponential function with aconcave quadratic function: $f(x)=\exp(\alpha x^{2}+\beta x+\gamma ),$ where

$\alpha =-1/2c^{2},$
$\beta =b/c^{2},$
$\gamma =\ln a-(b^{2}/2c^{2}).$

(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 whoselogarithm is a concave quadratic function.

The parameterc is related to thefull width at half maximum (FWHM) of the peak according to

${\text{FWHM}}=2{\sqrt {2\ln 2}}\,c\approx 2.35482\,c.$

The function may then be expressed in terms of the FWHM, represented byw: $f(x)=ae^{-4(\ln 2)(x-b)^{2}/w^{2}}.$

Alternatively, the parameterc can be interpreted by saying that the twoinflection points of the function occur atx =b ±c.

Thefull width at tenth of maximum (FWTM) for a Gaussian could be of interest and is ${\text{FWTM}}=2{\sqrt {2\ln 10}}\,c\approx 4.29193\,c.$

Gaussian functions areanalytic, and theirlimit asx → ∞ is 0 (for the above case ofb = 0).

Gaussian functions are among those functions that areelementary but lack elementaryantiderivatives; theintegral of the Gaussian function is theerror function:

$\int e^{-x^{2}}\,dx={\frac {\sqrt {\pi }}{2}}\operatorname {erf} x+C.$

Nonetheless, their improper integrals over the whole real line can be evaluated exactly, using theGaussian integral $\int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }},$ and one obtains $\int _{-\infty }^{\infty }ae^{-(x-b)^{2}/(2c^{2})}\,dx=ac\cdot {\sqrt {2\pi }}.$

Normalized Gaussian curves withexpected valueμ andvariance`σ`². The corresponding parameters are ${\textstyle a={\tfrac {1}{\sigma {\sqrt {2\pi }}}}}$ ,`b` =`μ` and`c` =`σ`.

Normalized Gaussian curves withexpected valueμ andvariance`σ`². The corresponding parameters are ${\textstyle a={\tfrac {1}{\sigma {\sqrt {2\pi }}}}}$ ,`b` =`μ` and`c` =`σ`.

This integral is 1 if and only if ${\textstyle a={\tfrac {1}{c{\sqrt {2\pi }}}}}$ (thenormalizing constant), and in this case the Gaussian is theprobability density function of anormally distributed random variable withexpected valueμ =b andvarianceσ² =c²: $g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left({\frac {-(x-\mu )^{2}}{2\sigma ^{2}}}\right).$

These Gaussians are plotted in the accompanying figure.

The product of two Gaussian functions is a Gaussian, and theconvolution 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.

The Fourieruncertainty principle becomes an equality if and only if (modulated) Gaussian functions are considered.^[2]

Taking theFourier transform (unitary, angular-frequency convention) of a Gaussian function with parametersa = 1,b = 0 andc yields another Gaussian function, with parameters $c {\displaystyle c}$ ,b = 0 and $1/c$ .^[3] So in particular the Gaussian functions withb = 0 and $c=1$ are kept fixed by the Fourier transform (they areeigenfunctions of the Fourier transform with eigenvalue 1).A physical realization is that of thediffraction pattern: for example, aphotographic slide whosetransmittance 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^{[clarification needed]} identity from thePoisson summation formula: $\sum _{k\in \mathbb {Z} }\exp \left(-\pi \cdot \left({\frac {k}{c}}\right)^{2}\right)=c\cdot \sum _{k\in \mathbb {Z} }\exp \left(-\pi \cdot (kc)^{2}\right).$

Integral of a Gaussian function

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The integral of an arbitrary Gaussian function is $\int _{-\infty }^{\infty }a\,e^{-(x-b)^{2}/2c^{2}}\,dx=\ a\,|c|\,{\sqrt {2\pi }}.$

An alternative form is $\int _{-\infty }^{\infty }k\,e^{-fx^{2}+gx+h}\,dx=\int _{-\infty }^{\infty }k\,e^{-f{\big (}x-g/(2f){\big )}^{2}+g^{2}/(4f)+h}\,dx=k\,{\sqrt {\frac {\pi }{f}}}\,\exp \left({\frac {g^{2}}{4f}}+h\right),$ wheref must be strictly positive for the integral to converge.

Relation to standard Gaussian integral

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The integral $\int _{-\infty }^{\infty }ae^{-(x-b)^{2}/2c^{2}}\,dx$ for somereal constantsa,b andc > 0 can be calculated by putting it into the form of aGaussian integral. First, the constanta can simply be factored out of the integral. Next, the variable of integration is changed fromx toy =x −b: $a\int _{-\infty }^{\infty }e^{-y^{2}/2c^{2}}\,dy,$ and then to $z=y/{\sqrt {2c^{2}}}$ : $a{\sqrt {2c^{2}}}\int _{-\infty }^{\infty }e^{-z^{2}}\,dz.$

Then, using theGaussian integral identity $\int _{-\infty }^{\infty }e^{-z^{2}}\,dz={\sqrt {\pi }},$

we have $\int _{-\infty }^{\infty }ae^{-(x-b)^{2}/2c^{2}}\,dx=a{\sqrt {2\pi c^{2}}}.$

Two-dimensional Gaussian function

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3d plot of a Gaussian function with a two-dimensional domain

Base form: $f(x,y)=\exp(-x^{2}-y^{2})$

In two dimensions, the power to whiche is raised in the Gaussian function is any negative-definitequadratic form. Consequently, thelevel sets of the Gaussian will always be ellipses.

A particular example of a two-dimensional Gaussian function is $f(x,y)=A\exp \left(-\left({\frac {(x-x_{0})^{2}}{2\sigma _{X}^{2}}}+{\frac {(y-y_{0})^{2}}{2\sigma _{Y}^{2}}}\right)\right).$

Here the coefficientA is the amplitude,x₀, y₀ is the center, andσ_x, σ_y are thex andy spreads of the blob. The figure on the right was created usingA = 1,x₀ = 0,y₀ = 0,σ_x =σ_y = 1.

The volume under the Gaussian function is given by $V=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\,dx\,dy=2\pi A\sigma _{X}\sigma _{Y}.$

In general, a two-dimensional elliptical Gaussian function is expressed as $f(x,y)=A\exp {\Big (}-{\big (}a(x-x_{0})^{2}+2b(x-x_{0})(y-y_{0})+c(y-y_{0})^{2}{\big )}{\Big )},$ where the matrix ${\begin{bmatrix}a&b\\b&c\end{bmatrix}}$ ispositive-definite.

Using this formulation, the figure on the right can be created usingA = 1,(x₀,y₀) = (0, 0),a =c = 1/2,b = 0.

Meaning of parameters for the general equation

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For the general form of the equation the coefficientA is the height of the peak and(x₀,y₀) is the center of the blob.

If we set ${\begin{aligned}a&={\frac {\cos ^{2}\theta }{2\sigma _{X}^{2}}}+{\frac {\sin ^{2}\theta }{2\sigma _{Y}^{2}}},\\b&=-{\frac {\sin \theta \cos \theta }{2\sigma _{X}^{2}}}+{\frac {\sin \theta \cos \theta }{2\sigma _{Y}^{2}}},\\c&={\frac {\sin ^{2}\theta }{2\sigma _{X}^{2}}}+{\frac {\cos ^{2}\theta }{2\sigma _{Y}^{2}}},\end{aligned}}$ then we rotate the blob by a positive, counter-clockwise angle $\theta$ (for negative, clockwise rotation, invert the signs in theb coefficient).^[4]

To get back the coefficients $\theta$ , $\sigma _{X}$ and $\sigma _{Y}$ from $a {\displaystyle a}$ , $b {\displaystyle b}$ and $c {\displaystyle c}$ use

${\begin{aligned}\theta &={\frac {1}{2}}\arctan \left({\frac {2b}{a-c}}\right),\quad \theta \in [-45,45],\\\sigma _{X}^{2}&={\frac {1}{2(a\cdot \cos ^{2}\theta +2b\cdot \cos \theta \sin \theta +c\cdot \sin ^{2}\theta )}},\\\sigma _{Y}^{2}&={\frac {1}{2(a\cdot \sin ^{2}\theta -2b\cdot \cos \theta \sin \theta +c\cdot \cos ^{2}\theta )}}.\end{aligned}}$

Example rotations of Gaussian blobs can be seen in the following examples:

{\displaystyle \theta =-\pi /6} — $\theta =-\pi /6$

{\displaystyle \theta =-\pi /3} — $\theta =-\pi /3$

Using the followingOctave code, one can easily see the effect of changing the parameters:

A=1;x0=0;y0=0;sigma_X=1;sigma_Y=2;[X,Y]=meshgrid(-5:.1:5,-5:.1:5);fortheta=0:pi/100:pia=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);Z=A*exp(-(a*(X-x0).^2+2*b*(X-x0).*(Y-y0)+c*(Y-y0).^2));surf(X,Y,Z);shadinginterp;view(-36,36)waitforbuttonpressend

Such functions are often used inimage processing and in computational models ofvisual system function—see the articles onscale space andaffine shape adaptation.

Also seemultivariate normal distribution.

Higher-order Gaussian or super-Gaussian function or generalized Gaussian function

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A more general formulation of a Gaussian function with a flat-top and Gaussian fall-off can be taken by raising the content of the exponent to a power $P {\displaystyle P}$ : $f(x)=A\exp \left(-\left({\frac {(x-x_{0})^{2}}{2\sigma _{X}^{2}}}\right)^{P}\right).$

This function is known as a super-Gaussian function and is often used for Gaussian beam formulation.^[5] This function may also be expressed in terms of thefull width at half maximum (FWHM), represented byw: $f(x)=A\exp \left(-\ln 2\left(4{\frac {(x-x_{0})^{2}}{w^{2}}}\right)^{P}\right).$

In a two-dimensional formulation, a Gaussian function along $x {\displaystyle x}$ and $y {\displaystyle y}$ can be combined^[6] with potentially different $P_{X}$ and $P_{Y}$ to form a rectangular Gaussian distribution: $f(x,y)=A\exp \left(-\left({\frac {(x-x_{0})^{2}}{2\sigma _{X}^{2}}}\right)^{P_{X}}-\left({\frac {(y-y_{0})^{2}}{2\sigma _{Y}^{2}}}\right)^{P_{Y}}\right).$ or an elliptical Gaussian distribution: $f(x,y)=A\exp \left(-\left({\frac {(x-x_{0})^{2}}{2\sigma _{X}^{2}}}+{\frac {(y-y_{0})^{2}}{2\sigma _{Y}^{2}}}\right)^{P}\right)$

Multi-dimensional Gaussian function

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Main article:Multivariate normal distribution

In an $n {\displaystyle n}$ -dimensional space a Gaussian function can be defined as $f(x)=\exp(-x^{\mathsf {T}}Cx),$ where $x={\begin{bmatrix}x_{1}&\cdots &x_{n}\end{bmatrix}}$ is a column of $n {\displaystyle n}$ coordinates, $C {\displaystyle C}$ is apositive-definite $n\times n$ matrix, and ${}^{\mathsf {T}}$ denotesmatrix transposition.

The integral of this Gaussian function over the whole $n {\displaystyle n}$ -dimensional space is given as $\int _{\mathbb {R} ^{n}}\exp(-x^{\mathsf {T}}Cx)\,dx={\sqrt {\frac {\pi ^{n}}{\det C}}}.$

It can be easily calculated by diagonalizing the matrix $C {\displaystyle C}$ and changing the integration variables to the eigenvectors of $C {\displaystyle C}$ .

More generally a shifted Gaussian function is defined as $f(x)=\exp(-x^{\mathsf {T}}Cx+s^{\mathsf {T}}x),$ where $s={\begin{bmatrix}s_{1}&\cdots &s_{n}\end{bmatrix}}$ is the shift vector and the matrix $C {\displaystyle C}$ can be assumed to be symmetric, $C^{\mathsf {T}}=C$ , and positive-definite. The following integrals with this function can be calculated with the same technique: $\int _{\mathbb {R} ^{n}}e^{-x^{\mathsf {T}}Cx+v^{\mathsf {T}}x}\,dx={\sqrt {\frac {\pi ^{n}}{\det {C}}}}\exp \left({\frac {1}{4}}v^{\mathsf {T}}C^{-1}v\right)\equiv {\mathcal {M}}.$ $\int _{\mathbb {R} ^{n}}e^{-x^{\mathsf {T}}Cx+v^{\mathsf {T}}x}(a^{\mathsf {T}}x)\,dx=(a^{T}u)\cdot {\mathcal {M}},{\text{ where }}u={\frac {1}{2}}C^{-1}v.$ $\int _{\mathbb {R} ^{n}}e^{-x^{\mathsf {T}}Cx+v^{\mathsf {T}}x}(x^{\mathsf {T}}Dx)\,dx=\left(u^{\mathsf {T}}Du+{\frac {1}{2}}\operatorname {tr} (DC^{-1})\right)\cdot {\mathcal {M}}.$ ${\begin{aligned}&\int _{\mathbb {R} ^{n}}e^{-x^{\mathsf {T}}C'x+s'^{\mathsf {T}}x}\left(-{\frac {\partial }{\partial x}}\Lambda {\frac {\partial }{\partial x}}\right)e^{-x^{\mathsf {T}}Cx+s^{\mathsf {T}}x}\,dx\\&\qquad =\left(2\operatorname {tr} (C'\Lambda CB^{-1})+4u^{\mathsf {T}}C'\Lambda Cu-2u^{\mathsf {T}}(C'\Lambda s+C\Lambda s')+s'^{\mathsf {T}}\Lambda s\right)\cdot {\mathcal {M}},\end{aligned}}$ where ${\textstyle u={\frac {1}{2}}B^{-1}v,\ v=s+s',\ B=C+C'.}$

Estimation of parameters

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A number of fields such asstellar photometry,Gaussian beam characterization, andemission/absorption line spectroscopy work with sampled Gaussian functions and need to accurately estimate the height, position, and width parameters of the function. There are three unknown parameters for a 1D Gaussian function (a,b,c) and five for a 2D Gaussian function $(A;x_{0},y_{0};\sigma _{X},\sigma _{Y})$ .

The most common method for estimating the Gaussian parameters is to take the logarithm of the data andfit a parabola to the resulting data set.^[7]^[8] While this provides a simplecurve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem throughweighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use aniteratively reweighted least squares procedure, in which the weights are updated at each iteration.^[8]It is also possible to performnon-linear regression directly on the data, without involving thelogarithmic data transformation; for more options, seeprobability distribution fitting.

Parameter precision

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Once one has an algorithm for estimating the Gaussian function parameters, it is also important to know howprecise those estimates are. Anyleast squares estimation algorithm can provide numerical estimates for the variance of each parameter (i.e., the variance of the estimated height, position, and width of the function). One can also useCramér–Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data.^[9]^[10]

The noise in the measured profile is eitheri.i.d. Gaussian, or the noise isPoisson-distributed.
The spacing between each sampling (i.e. the distance between pixels measuring the data) is uniform.
The peak is "well-sampled", so that less than 10% of the area or volume under the peak (area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region.
The width of the peak is much larger than the distance between sample locations (i.e. the detector pixels must be at least 5 times smaller than the Gaussian FWHM).

When these assumptions are satisfied, the followingcovariance matrixK applies for the 1D profile parameters $a {\displaystyle a}$ , $b {\displaystyle b}$ , and $c {\displaystyle c}$ under i.i.d. Gaussian noise and under Poisson noise:^[9] $\mathbf {K} _{\text{Gauss}}={\frac {\sigma ^{2}}{{\sqrt {\pi }}\delta _{X}Q^{2}}}{\begin{pmatrix}{\frac {3}{2c}}&0&{\frac {-1}{a}}\\0&{\frac {2c}{a^{2}}}&0\\{\frac {-1}{a}}&0&{\frac {2c}{a^{2}}}\end{pmatrix}}\ ,\qquad \mathbf {K} _{\text{Poiss}}={\frac {1}{\sqrt {2\pi }}}{\begin{pmatrix}{\frac {3a}{2c}}&0&-{\frac {1}{2}}\\0&{\frac {c}{a}}&0\\-{\frac {1}{2}}&0&{\frac {c}{2a}}\end{pmatrix}}\ ,$ where $\delta _{X}$ is the width of the pixels used to sample the function, $Q {\displaystyle Q}$ is the quantum efficiency of the detector, and $\sigma$ indicates the standard deviation of the measurement noise. Thus, the individual variances for the parameters are, in the Gaussian noise case, ${\begin{aligned}\operatorname {var} (a)&={\frac {3\sigma ^{2}}{2{\sqrt {\pi }}\,\delta _{X}Q^{2}c}}\\\operatorname {var} (b)&={\frac {2\sigma ^{2}c}{\delta _{X}{\sqrt {\pi }}\,Q^{2}a^{2}}}\\\operatorname {var} (c)&={\frac {2\sigma ^{2}c}{\delta _{X}{\sqrt {\pi }}\,Q^{2}a^{2}}}\end{aligned}}$

and in the Poisson noise case, ${\begin{aligned}\operatorname {var} (a)&={\frac {3a}{2{\sqrt {2\pi }}\,c}}\\\operatorname {var} (b)&={\frac {c}{{\sqrt {2\pi }}\,a}}\\\operatorname {var} (c)&={\frac {c}{2{\sqrt {2\pi }}\,a}}.\end{aligned}}$

For the 2D profile parameters giving the amplitude $A {\displaystyle A}$ , position $(x_{0},y_{0})$ , and width $(\sigma _{X},\sigma _{Y})$ of the profile, the following covariance matrices apply:^[10]

${\begin{aligned}\mathbf {K} _{\text{Gauss}}={\frac {\sigma ^{2}}{\pi \delta _{X}\delta _{Y}Q^{2}}}&{\begin{pmatrix}{\frac {2}{\sigma _{X}\sigma _{Y}}}&0&0&{\frac {-1}{A\sigma _{Y}}}&{\frac {-1}{A\sigma _{X}}}\\0&{\frac {2\sigma _{X}}{A^{2}\sigma _{Y}}}&0&0&0\\0&0&{\frac {2\sigma _{Y}}{A^{2}\sigma _{X}}}&0&0\\{\frac {-1}{A\sigma _{y}}}&0&0&{\frac {2\sigma _{X}}{A^{2}\sigma _{y}}}&0\\{\frac {-1}{A\sigma _{X}}}&0&0&0&{\frac {2\sigma _{Y}}{A^{2}\sigma _{X}}}\end{pmatrix}}\\[6pt]\mathbf {K} _{\operatorname {Poisson} }={\frac {1}{2\pi }}&{\begin{pmatrix}{\frac {3A}{\sigma _{X}\sigma _{Y}}}&0&0&{\frac {-1}{\sigma _{Y}}}&{\frac {-1}{\sigma _{X}}}\\0&{\frac {\sigma _{X}}{A\sigma _{Y}}}&0&0&0\\0&0&{\frac {\sigma _{Y}}{A\sigma _{X}}}&0&0\\{\frac {-1}{\sigma _{Y}}}&0&0&{\frac {2\sigma _{X}}{3A\sigma _{Y}}}&{\frac {1}{3A}}\\{\frac {-1}{\sigma _{X}}}&0&0&{\frac {1}{3A}}&{\frac {2\sigma _{Y}}{3A\sigma _{X}}}\end{pmatrix}}.\end{aligned}}$ where the individual parameter variances are given by the diagonal elements of the covariance matrix.

Discrete Gaussian

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Main article:Discrete Gaussian kernel

Thediscrete Gaussian kernel (solid), compared with thesampled Gaussian kernel (dashed) for scales $t=0.5,1,2,4.$

{\displaystyle t=0.5,1,2,4.} — Thediscrete Gaussian kernel (solid), compared with thesampled Gaussian kernel (dashed) for scales $t=0.5,1,2,4.$

One may ask for a discrete analog to the Gaussian;this is necessary in discrete applications, particularlydigital signal processing. A simple answer is to sample the continuous Gaussian, yielding thesampled Gaussian kernel. However, this discrete function does not have the discrete analogs of the properties of the continuous function, and can lead to undesired effects, as described in the articlescale space implementation.

An alternative approach is to use thediscrete Gaussian kernel:^[11] $T(n,t)=e^{-t}I_{n}(t)$ where $I_{n}(t)$ denotes themodified Bessel functions of integer order.

This is the discrete analog of the continuous Gaussian in that it is the solution to the discretediffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation.^[11]^[12]

Applications

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Gaussian functions appear in many contexts in thenatural sciences, thesocial sciences,mathematics, andengineering. Some examples include:

Instatistics andprobability theory, Gaussian functions appear as the density function of thenormal distribution, which is a limitingprobability distribution of complicated sums, according to thecentral limit theorem.
Gaussian functions are theGreen's function for the (homogeneous and isotropic)diffusion equation (and to theheat equation, which is the same thing), apartial differential equation that describes the time evolution of a mass-density underdiffusion. Specifically, if the mass-density at timet=0 is given by aDirac delta, which essentially means that the mass is initially concentrated in a single point, then the mass-distribution at timet will be given by a Gaussian function, with the parametera being linearly related to 1/√t andc being linearly related to√t; this time-varying Gaussian is described by theheat kernel. More generally, if the initial mass-density is φ(x), then the mass-density at later times is obtained by taking theconvolution of φ with a Gaussian function. The convolution of a function with a Gaussian is also known as aWeierstrass transform.
A Gaussian function is thewave function of theground state of thequantum harmonic oscillator.
Themolecular orbitals used incomputational chemistry can belinear combinations of Gaussian functions calledGaussian orbitals (see alsobasis set (chemistry)).
Mathematically, thederivatives of the Gaussian function can be represented usingHermite functions. For unit variance, then-th derivative of the Gaussian is the Gaussian function itself multiplied by then-thHermite polynomial, up to scale.
Consequently, Gaussian functions are also associated with thevacuum state inquantum field theory.
Gaussian beams are used in optical systems, microwave systems and lasers.
Inscale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations incomputer vision andimage processing. Specifically, derivatives of Gaussians (Hermite functions) are used as a basis for defining a large number of types of visual operations.
Gaussian functions are used to define some types ofartificial neural networks.
Influorescence microscopy a 2D Gaussian function is used to approximate theAiry disk, describing the intensity distribution produced by apoint source.
Insignal processing they serve to defineGaussian filters, such as inimage processing where 2D Gaussians are used forGaussian blurs. Indigital signal processing, one uses adiscrete Gaussian kernel, which may be approximated by theBinomial coefficient^[13] or sampling a Gaussian.
Ingeostatistics they have been used for understanding the variability between the patterns of a complextraining image. They are used with kernel methods to cluster the patterns in the feature space.^[14]