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| {{Merge from|Integral of a Gaussian function|date=April 2013}}
| |
| [[Image:Normal Distribution PDF.svg|thumb|360px|right|Normalized Gaussian curves with [[expected value]] μ and [[variance]] σ<sup>2</sup>. The corresponding parameters are ''a'' = 1/(σ√(2π)), ''b'' = μ, ''c'' = σ]]
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| In [[mathematics]], a '''Gaussian function''' (named after [[Carl Friedrich Gauss]]) is a [[function (mathematics)|function]] of the form:
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| :<math>f(x) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }+d</math>
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| for some [[real number|real]] constants ''a'', ''b'', ''c'', ''d'', and ''e'' ≈ 2.71828...([[e (mathematical constant)|Euler's number]]). | |
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| The [[graph of a function|graph]] of a Gaussian is a characteristic symmetric "bell curve" shape that quickly falls off towards zero. The parameter ''a'' is the height of the curve's peak, ''b'' is the position of the center of the peak, and ''c'' (the [[standard deviation]]) controls the width of the "bell".
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| Gaussian functions are widely used in [[statistics]] where they describe the [[normal distribution]]s, in [[signal processing]] where they serve to define [[Gaussian filter]]s, in [[image processing]] where two-dimensional Gaussians are used for [[Gaussian blur]]s, and in [[mathematics]] where they are used to solve [[heat equation]]s and [[diffusion equation]]s and to define the [[Weierstrass transform]].
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| ==Properties==
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| 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.
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| The parameter ''c'' is related to the [[full width at half maximum]] (FWHM) of the peak according to
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| : <math>\mathrm{FWHM} = 2 \sqrt{2 \ln 2}\ c \approx 2.35482 c.</math><ref>Using the [[List of logarithmic identities|logarithmic identity]] <math>\log \left( x \right) = -\log \left( \frac{1}{x} \right)</math>, this expression can be transformed to <math>\mathrm{FWHM} = 2 \sqrt{-2 \ln 0.5}\ c</math>.</ref>
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| Alternatively, the parameter ''c'' can be interpreted by saying that the two [[inflection point]]s of the function occur at ''x'' = ''b'' − ''c'' and ''x'' = ''b'' + ''c''.
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| The '''full width at tenth of maximum''' (FWTM) for a Gaussian could be of interest and is
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| : <math>\mathrm{FWTM} = 2 \sqrt{2 \ln 10}\ c \approx 4.29193 c.</math><ref>Using the [[List of logarithmic identities|logarithmic identity]] <math>\log \left( x \right) = -\log \left( \frac{1}{x} \right)</math>, this expression can be transformed to <math>\mathrm{FWTM} = 2 \sqrt{-2 \ln 0.1}\ c</math>.</ref>
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| Gaussian functions are [[analytic function|analytic]], and their [[limit (mathematics)|limit]] as ''x'' → ∞ is 0.
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| Gaussian functions are among those functions that are [[Elementary function (differential algebra)|elementary]] but lack elementary [[antiderivative]]s; 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]]
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| :<math>\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}</math>
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| and one obtains
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| :<math>\int_{-\infty}^\infty a e^{- { (x-b)^2 \over 2 c^2 } }\,dx=ac\cdot\sqrt{2\pi}.</math>
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| This integral is 1 if and only if ''a'' = 1/(''c''√(2π)), and in this case the Gaussian is the [[probability density function]] of a [[normal distribution|normally distributed]] [[random variable]] with [[expected value]] μ = ''b'' and [[variance]] σ<sup>2</sup> = ''c''<sup>2</sup>:
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| :<math> g(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }. </math>
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| These Gaussians are plotted in the accompanying figure.
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| Gaussian functions centered at zero minimize the Fourier [[Fourier transform#Uncertainty principle|uncertainty principle]].
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| The '''product of two Gaussian functions''' is a Gaussian (warning: the product of two Gaussian ''probability density functions'' is ''not'' in general a Gaussian pdf), and the [[convolution]] of two Gaussian functions is also a Gaussian, with std. deviation being the quadratic mean of the original std. deviations: <math>c = \sqrt{c_{1}^2 + c_{2}^2}</math>.
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| Taking the [[Fourier transform#Other conventions|'''Fourier transform''' (unitary, angular frequency convention)]] of a Gaussian function with parameters ''a'', ''b'' = 0 and ''c'' yields another Gaussian function, with parameters ''<math>\sqrt{2\pi}ac</math>'', ''b'' = 0 and ''<math>\frac{1}{2 \pi c}</math>''.<ref>{{cite web|last=Weisstein|first=Eric W.|title=Fourier Transform--Gaussian|url=http://mathworld.wolfram.com/FourierTransformGaussian.html|publisher=[[MathWorld]]|accessdate=19 December 2013}}</ref> So in particular the Gaussian functions with ''b'' = 0 and ''c'' = <math>\frac{1}{\sqrt{2 \pi}}</math> are kept fixed by the Fourier transform (they are [[eigenfunction]]s of the Fourier transform with eigenvalue 1).
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| <!-- The way the Fourier transform is currently defined in its article (with pi in the exponent, also the way that I prefer), the Gaussian must also have a pi in its exponent. ~~~~ -->
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| <!--
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| Using [[periodic summation]] and [[discretization]] you can construct vectors from the Gaussian function,
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| that behave similarly under the [[Discrete Fourier transform]].
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| Comparing the zeroth coefficient of the Discrete Fourier transform of such a vector
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| with the periodic summation and discretization of the Continuous Fourier transform of the Gaussian yields the interesting identity:
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| -->
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| The fact that the Gaussian function is an eigenfunction of the Continuous Fourier transform
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| allows us to derive the following interesting identity from the [[Poisson summation formula]]:
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| :<math>\sum_{k\in\mathbb{Z}}\exp\left(-\pi\cdot\left(\frac{k}{c}\right)^2\right) = c\cdot\sum_{k\in\mathbb{Z}}\exp(-\pi\cdot(kc)^2).</math>
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| == Two-dimensional Gaussian function ==
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| [[Image:Gaussian 2d.png|thumb|300px|Gaussian curve with a 2-dimensional domain]]
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| 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.
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| A particular example of a two-dimensional Gaussian function is
| |
| <!-- This makes the formula consistent with the 1d formula above -->
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| :<math>f(x,y) = A \exp\left(- \left(\frac{(x-x_o)^2}{2\sigma_x^2} + \frac{(y-y_o)^2}{2\sigma_y^2} \right)\right).</math>
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| Here the coefficient ''A'' is the amplitude, ''x''<sub>o</sub>,y<sub>o</sub> is the center and σ<sub>''x''</sub>, σ<sub>''y''</sub> are the ''x'' and ''y'' spreads of the blob. The figure on the right was created using ''A'' = 1, ''x''<sub>o</sub> = 0, ''y''<sub>o</sub> = 0, σ<sub>''x''</sub> = σ<sub>''y''</sub> = 1.
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| The volume under the Gaussian function is given by
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| :<math>V = \int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y)\,dx dy=2 \pi A \sigma_x \sigma_y.</math>
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| In general, a two-dimensional elliptical Gaussian function is expressed as
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| :<math>f(x,y) = A \exp\left(- \left(a(x - x_o)^2 + 2b(x-x_o)(y-y_o) + c(y-y_o)^2 \right)\right)</math>
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| where the matrix
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| :<math>\left[\begin{matrix} a & b \\ b & c \end{matrix}\right] </math>
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| is [[positive-definite matrix|positive-definite]].
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| Using this formulation, the figure on the right can be created using ''A'' = 1, (''x''<sub>o</sub>, ''y''<sub>o</sub>) = (0, 0), ''a'' = ''c'' = 1/2, ''b'' = 0.
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| === Meaning of parameters for the general equation ===
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| For the general form of the equation the coefficient ''A'' is the height of the peak and (''x''<sub>o</sub>, ''y''<sub>o</sub>) is the center of the blob.
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| If we set
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| :<math>a = \frac{\cos^2\theta}{2\sigma_x^2} + \frac{\sin^2\theta}{2\sigma_y^2}</math>
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| <!-- extra blank line between two lines of displayed TeX, for legibility -->
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| :<math>b = -\frac{\sin2\theta}{4\sigma_x^2} + \frac{\sin2\theta}{4\sigma_y^2}</math>
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| <!-- extra blank line between two lines of displayed TeX, for legibility -->
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| :<math>c = \frac{\sin^2\theta}{2\sigma_x^2} + \frac{\cos^2\theta}{2\sigma_y^2}</math>
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| then we rotate the blob by a clockwise angle <math>\theta</math> (for counterclockwise rotation invert the signs in the b coefficient). This can be seen in the following examples:
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| {|
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| | [[Image:Gaussian 2d 1.svg|thumb|200px|<math>\theta = 0</math>]]
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| | [[Image:Gaussian 2d 2.svg|thumb|200px|<math>\theta = \pi/6</math>]]
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| | [[Image:Gaussian 2d 3.svg|thumb|200px|<math>\theta = \pi/3</math>]]
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| |}
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| Using the following [[GNU Octave|Octave]] code one can easily see the effect of changing the parameters
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| <source lang="matlab">
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| A = 1;
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| x0 = 0; y0 = 0;
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| sigma_x = 1;
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| sigma_y = 2;
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| for theta = 0:pi/100:pi
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| a = cos(theta)^2/2/sigma_x^2 + sin(theta)^2/2/sigma_y^2;
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| b = -sin(2*theta)/4/sigma_x^2 + sin(2*theta)/4/sigma_y^2 ;
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| c = sin(theta)^2/2/sigma_x^2 + cos(theta)^2/2/sigma_y^2;
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| [X, Y] = meshgrid(-5:.1:5, -5:.1:5);
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| Z = A*exp( - (a*(X-x0).^2 + 2*b*(X-x0).*(Y-y0) + c*(Y-y0).^2)) ;
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| surf(X,Y,Z);shading interp;view(-36,36);axis equal;drawnow
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| end
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| </source>
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| 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]].
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| Also see [[multivariate normal distribution]].
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| == Multi-dimensional Gaussian function ==
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| {{main|Multivariate normal distribution}}
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| In an <math>n</math>-dimensional space a Gaussian function can be defined as
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| :<math> | |
| f(x) = \exp(-x^TAx) \;,
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| </math>
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| where <math>x=\{x_1,\dots,x_n\}</math> is a column of <math>n</math> coordinates, <math>A</math> is a [[positive-definite matrix|positive-definite]] <math>n\times n</math> matrix, and <math>{}^T</math> denotes [[transposition (mathematics)|transposition]].
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| The integral of this Gaussian function over the whole <math>n</math>-dimensional space is given as
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| :<math>
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| \int_{\mathbb{R}^n} \exp(-x^TAx) \, dx = \sqrt{\frac{\pi^n}{\det{A}}} \;.
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| </math>
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| It can be easily calculated by diagonalizing the matrix <math>A</math> and changing the integration variables to the eigenvectors of <math>A</math>.
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| More generally a shifted Gaussian function is defined as
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| :<math>
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| f(x) = \exp(-x^TAx+s^Tx) \;,
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| </math>
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| where <math>s=\{s_1,\dots,s_n\}</math> is the shift vector and the matrix <math>A</math> can be assumed to be symmetric, <math>A^T=A</math>, and positive-definite. The following integrals with this function can be calculated with the same technique,
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| :<math>
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| \int_{\mathbb{R}^n} e^{-x^T A x+v^Tx} \, dx = \sqrt{\frac{\pi^n}{\det{A}}} \exp(\frac{1}{4}v^T A^{-1}v)\equiv \mathcal{M}\;.
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| </math>
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| :<math>
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| \int_{\mathbb{R}^n} e^{- x^T A x + v^T x} \left( a^T x \right) \, dx = (a^T u) \cdot
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| \mathcal{M}\;,\; {\rm where}\;
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| u = \frac{1}{2} A^{- 1} v \;.
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| </math>
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| :<math>
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| \int_{\mathbb{R}^n} e^{- x^T A x + v^T x} \left( x^T D x \right) \, dx = \left( u^T D u +
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| \frac{1}{2} {\rm tr} (D A^{- 1}) \right) \cdot \mathcal{M}\;.
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| </math>
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| :<math>
| |
| \begin{align}
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| & \int_{\mathbb{R}^n} e^{- x^T A' x + s'^T x} \left( -
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| \frac{\partial}{\partial x} \Lambda \frac{\partial}{\partial x} \right) e^{-
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| x^T A x + s^T x} \, dx = \\
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| & = \left( 2 {\rm tr} (A' \Lambda A B^{- 1}) + 4 u^T A' \Lambda A u - 2 u^T
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| (A' \Lambda s + A \Lambda s') + s'^T \Lambda s \right) \cdot \mathcal{M}\;,
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| \\ & {\rm where} \;
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| u = \frac{1}{2} B^{- 1} v, v = s + s', B = A + A' \;.
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| \end{align}
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| </math>
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| == Gaussian profile estimation ==
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| A number of fields such as [[Photometry (astronomy)|stellar photometry]], [[Gaussian beam]] characterization, and [[emission spectrum#Emission spectroscopy|emission/absorption line spectroscopy]] work with sampled Gaussian functions and need to accurately estimate the height, position, and width parameters of the function. These are <math>a</math>, <math>b</math>, and <math>c</math> for a 1D Gaussian function, <math>A</math>, <math>(x_0,y_0)</math>, and <math>(\sigma_x,\sigma_y)</math> for a 2D Gaussian function. The most common method for estimating the profile parameters is to take the logarithm of the data and fit a parabola to the resulting data set.<ref name="Guo">[http://dx.doi.org/10.1109/MSP.2011.941846 Hongwei Guo, "A simple algorithm for fitting a Gaussian function," IEEE Sign. Proc. Mag. 28(9): 134-137 (2011).]</ref> While this provides a simple [[least squares]] fitting procedure, the resulting algorithm is biased by excessively weighting small data values, and this can produce large errors in the profile estimate. One can partially compensate for this through [[least squares#Weighted least squares|weighted least squares]] estimation, in which the small data values are given small weights, 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 an iterative procedure in which the weights are updated at each iteration (see [[Iteratively reweighted least squares]]).<ref name="Guo" />
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| Once one has an algorithm for estimating the Gaussian function parameters, it is also important to know how accurate those estimates are. While an 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 use [[Cramér-Rao|Cramér–Rao bound]] theory to obtain an analytical expression for the lower bound on the parameter variances, given some assumptions about the data.<ref name="Hagen1">[http://dx.doi.org/10.1364/AO.46.005374 N. Hagen, M. Kupinski, and E. L. Dereniak, "Gaussian profile estimation in one dimension," Appl. Opt. 46:5374-5383 (2007)]</ref><ref name="Hagen2">[http://dx.doi.org/10.1364/AO.47.006842 N. Hagen and E. L. Dereniak, "Gaussian profile estimation in two dimensions," Appl. Opt. 47:6842-6851 (2008)]</ref>
| |
| # The noise in the measured profile is either [[Independent and identically-distributed random variables|i.i.d.]] Gaussian, or the noise is [[Poisson distribution|Poisson-distributed]].
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| # The spacing between each sampling (i.e. the distance between pixels measuring the data) is uniform.
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| # 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.
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| # 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).
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| When these assumptions are satisfied, the following covariance matrix '''K''' applies for the 1D profile parameters <math>a</math>, <math>b</math>, and <math>c</math> under i.i.d. Gaussian noise and under Poisson noise:<ref name="Hagen1" />
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| :<math> \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} \ ,</math>
| |
| | |
| where <math>\delta_x</math> is the width of the pixels used to sample the function, <math>Q</math> is the quantum efficiency of the detector, and <math>\sigma</math> indicates the standard deviation of the measurement noise. Thus, the individual variances for the parameters are, in the Gaussian noise case,
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| | |
| :<math>\begin{align} \text{var} (a) &= \frac{3 \sigma^2}{2 \sqrt{\pi} \, \delta_x Q^2 c} \\ \text{var} (b) &= \frac{2 \sigma^2 c}{\delta_x \sqrt{\pi} \, Q^2 a^2} \\ \text{var} (c) &= \frac{2 \sigma^2 c}{\delta_x \sqrt{\pi} \, Q^2 a^2} \end{align}</math>
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| | |
| and in the Poisson noise case,
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| | |
| :<math>\begin{align} \text{var} (a) &= \frac{3a}{2 \sqrt{2 \pi} \, c} \\ \text{var} (b) &= \frac{c}{\sqrt{2 \pi} \, a} \\ \text{var} (c) &= \frac{c}{2 \sqrt{2 \pi} \, a}. \end{align} \ </math>
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| | |
| For the 2D profile parameters giving the amplitude <math>A</math>, position <math>(x_0,y_0)</math>, and width <math>(\sigma_x,\sigma_y)</math> of the profile, the following covariance matrices apply:<ref name="Hagen2" />
| |
| | |
| :<math> \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} \ ,</math>
| |
| :<math> \qquad \mathbf{K}_{\text{Poiss}} = \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} \ .</math>
| |
| | |
| where the individual parameter variances are given by the diagonal elements of the covariance matrix.
| |
| | |
| == Discrete Gaussian ==
| |
| {{main|Discrete Gaussian kernel}}
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| | |
| [[File:Discrete Gaussians.svg|thumb|The [[discrete Gaussian kernel]] (black, dashed), compared with the [[sampled Gaussian kernel]] (red, solid) for scales <math>t=.5,1,2,4.</math>]]
| |
| | |
| One may ask for a discrete analog to the Gaussian;
| |
| this is necessary in discrete applications,
| |
| particularly [[digital signal processing]].
| |
| A simple answer is to sample the continuous Gaussian, yielding the [[sampled 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 article [[scale space implementation]].
| |
| | |
| An alternative approach is to use [[discrete Gaussian kernel]]:<ref name="tpl90">[http://www.nada.kth.se/~tony/abstracts/Lin90-PAMI.html Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254.]</ref>
| |
| :<math>T(n, t) = e^{-t} I_n(t)\,</math>
| |
| where <math>I_n(t)</math> denotes the [[modified Bessel function]]s of integer order.
| |
| | |
| This is the discrete analog of the continuous Gaussian in that it is the solution to the discrete [[diffusion equation]] (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation.<ref>Campbell, J, 2007, ''[http://dx.doi.org/10.1016/j.tpb.2007.08.001 The SMM model as a boundary value problem using the discrete diffusion equation]'', Theor Popul Biol. 2007 Dec;72(4):539-46.</ref>
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| | |
| ==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 the [[Green's function]] for 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/√''t'' and ''c'' being linearly related to √''t''. 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]].
| |
| * A Gaussian function is the [[wave function]] of the [[ground state]] of the [[quantum harmonic oscillator]].
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| * The [[molecular orbital]]s used in [[computational chemistry]] can be [[linear combination]]s of Gaussian functions called [[Gaussian orbital]]s (see also [[basis set (chemistry)]]).
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| * Mathematically, the [[derivative]]s of the Gaussian function can be represented using [[Hermite functions]]. The ''n''-th derivative of the Gaussian is the Gaussian function itself multiplied by the ''n''-th [[Hermite polynomial]], up to scale.
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| * Consequently, Gaussian functions are also associated with the [[vacuum state]] in [[quantum field theory]].
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| * [[Gaussian beam]]s are used in optical and microwave systems.
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| * In [[scale space]] 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.
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| * Gaussian functions are used to define some types of [[artificial neural network]]s.
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| * In [[fluorescence microscopy]] a 2D Gaussian function is used to approximate the [[Airy disk]], describing the intensity distribution produced by a [[point source]].
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| * In [[signal processing]] they serve to define [[Gaussian filter]]s, such as in [[image processing]] where 2D Gaussians are used for [[Gaussian blur]]s. In [[digital signal processing]], one uses a [[discrete Gaussian kernel]], which may be defined by sampling a Gaussian, or in a different way.
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| * In [[geostatistics]] 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.<ref>Honarkhah, M and Caers, J, 2010, ''[http://dx.doi.org/10.1007/s11004-010-9276-7 Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling]'', Mathematical Geosciences, 42: 487 - 517</ref>
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| ==See also==
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| *[[Normal distribution]]
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| *[[Lorentzian function]]
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| *[[Integral of a Gaussian function]]
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| {{More footnotes|date=March 2011}}
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| == References ==
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| {{reflist}}
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| ==External links==
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| * [http://mathworld.wolfram.com/GaussianFunction.html Mathworld, includes a proof for the relations between c and FWHM]
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| * [http://www.embege.com/gauss/ JavaScript to create Gaussian convolution kernels]
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| * {{MathPages|id=home/kmath045/kmath045|title=Integrating The Bell Curve}}
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| * [http://github.com/frecker/gaussian-distribution/ Haskell, Erlang and Perl implementation of Gaussian distribution]
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| * [https://upload.wikimedia.org/wikipedia/commons/a/a2/Cumulative_function_n_dimensional_Gaussians_12.2013.pdf Bensimhoun Michael, N-Dimensional Cumulative Function, And Other Useful Facts About Gaussians and Normal Densities (2009)]
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| [[Category:Exponentials]]
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| [[Category:Gaussian function]]
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