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Gaussian function

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</style><div role="note" class="hatnote navigation-not-searchable">"Gaussian curve" redirects here. For the band, see <a href="Gaussian_Curve_(band)" title="Gaussian Curve (band)">Gaussian Curve (band)</a>.</div>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a Gaussian function, often simply referred to as a Gaussian, is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of the base form

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\exp(-x^{2})}</annotation>

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and with parametric extension

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<annotation encoding="application/x-tex">{\displaystyle f(x)=a\exp \left(-{\frac {(x-b)^{2}}{2c^{2}}}\right)}</annotation>

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for arbitrary <a href="Real_number" title="Real number">real</a> constants a, b and non-zero c. It is named after the mathematician <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>. The <a href="Graph_of_a_function" title="Graph of a function">graph</a> of a Gaussian is a characteristic symmetric "<a href="Normal_distribution" title="Normal distribution">bell curve</a>" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the <a href="Standard_deviation" title="Standard deviation">standard deviation</a>, sometimes called the Gaussian <a href="Root_mean_square" title="Root mean square">RMS</a> width) controls the width of the "bell".

Gaussian functions are often used to represent the <a href="Probability_density_function" title="Probability density function">probability density function</a> of a <a href="Normal_distribution" title="Normal distribution">normally distributed</a> <a href="Random_variable" title="Random variable">random variable</a> with <a href="Expected_value" title="Expected value">expected value</a> <var>μ</var> = <var>b</var> and <a href="Variance" title="Variance">variance</a> <var>σ</var>2 = <var>c</var>2. In this case, the Gaussian is of the form<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}{\frac {(x-\mu )^{2}}{\sigma ^{2}}}\right).}</annotation>

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Gaussian functions are widely used in <a href="Statistics" title="Statistics">statistics</a> to describe the <a href="Normal_distribution" title="Normal distribution">normal distributions</a>, in <a href="Signal_processing" title="Signal processing">signal processing</a> to define <a href="Gaussian_filter" title="Gaussian filter">Gaussian filters</a>, in <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a> where two-dimensional Gaussians are used for <a href="Gaussian_blur" title="Gaussian blur">Gaussian blurs</a>, and in mathematics to solve <a href="Heat_equation" title="Heat equation">heat equations</a> and <a href="Diffusion_equation" title="Diffusion equation">diffusion equations</a> and to define the <a href="Weierstrass_transform" title="Weierstrass transform">Weierstrass transform</a>. They are also abundantly used in <a href="Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a> to form <a href="Basis_set_(chemistry)" title="Basis set (chemistry)">basis sets</a>.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Gaussian functions arise by composing the <a href="Exponential_function" title="Exponential function">exponential function</a> with a <a href="Concave_function" title="Concave function">concave</a> <a href="Quadratic_function" title="Quadratic function">quadratic function</a>:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\exp(\alpha x^{2}+\beta x+\gamma ),}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\exp(\alpha x^{2}+\beta x+\gamma ),}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \alpha =-1/2c^{2},}</annotation>

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</math><img src="./d220cfff4d181e24ce8459c8e22e3a2cfedae703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.59ex; height:3.176ex;" alt="{\displaystyle \alpha =-1/2c^{2},}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \beta =b/c^{2},}</annotation>

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</math><img src="./7a8585899c3a6176c62bbe9c490295164d1dfd1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.298ex; height:3.176ex;" alt="{\displaystyle \beta =b/c^{2},}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \gamma =\ln a-(b^{2}/2c^{2}).}</annotation>

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</math><img src="./a1da47b68ef58200382d133f70c18a31263fe43e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.651ex; height:3.176ex;" alt="{\displaystyle \gamma =\ln a-(b^{2}/2c^{2}).}" loading="lazy"></li></ul>

(Note: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1/(\sigma {\sqrt {2\pi }})}">

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<annotation encoding="application/x-tex">{\displaystyle a=1/(\sigma {\sqrt {2\pi }})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ln a}</annotation>

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</math><img src="./f2c526a8f272c427b1556d5817a9372c38103f88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.556ex; height:2.176ex;" alt="{\displaystyle \ln a}" loading="lazy">, not to be confused with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =-1/2c^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =-1/2c^{2}}</annotation>

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The Gaussian functions are thus those functions whose <a href="Logarithm" title="Logarithm">logarithm</a> is a concave quadratic function.

The parameter c is related to the <a href="Full_width_at_half_maximum" title="Full width at half maximum">full width at half maximum</a> (FWHM) of the peak according to

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<annotation encoding="application/x-tex">{\displaystyle {\text{FWHM}}=2{\sqrt {2\ln 2}}\,c\approx 2.35482\,c.}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle f(x)=ae^{-4(\ln 2)(x-b)^{2}/w^{2}}.}</annotation>

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Alternatively, the parameter c can be interpreted by saying that the two <a href="Inflection_point" title="Inflection point">inflection points</a> of the function occur at <var>x</var> = <var>b</var> ± <var>c</var>.

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

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<annotation encoding="application/x-tex">{\displaystyle {\text{FWTM}}=2{\sqrt {2\ln 10}}\,c\approx 4.29193\,c.}</annotation>

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Gaussian functions are <a href="Analytic_function" title="Analytic function">analytic</a>, and their <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a> as <var>x</var> → ∞ is 0 (for the above case of <var>b</var> = 0).

Gaussian functions are among those functions that are <a href="Elementary_function_(differential_algebra)" class="mw-redirect" title="Elementary function (differential algebra)">elementary</a> but lack elementary <a href="Antiderivative" title="Antiderivative">antiderivatives</a>; the <a href="Integral" title="Integral">integral</a> of the Gaussian function is the <a href="Error_function" title="Error function">error function</a>:

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<annotation encoding="application/x-tex">{\displaystyle \int e^{-x^{2}}\,dx={\frac {\sqrt {\pi }}{2}}\operatorname {erf} x+C.}</annotation>

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Nonetheless, their improper integrals over the whole real line can be evaluated exactly, using the <a href="Gaussian_integral" title="Gaussian integral">Gaussian integral</a>

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }},}</annotation>

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and one obtains

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }ae^{-(x-b)^{2}/(2c^{2})}\,dx=ac\cdot {\sqrt {2\pi }}.}</annotation>

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This integral is 1 if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a={\tfrac {1}{c{\sqrt {2\pi }}}}}">

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<annotation encoding="application/x-tex">{\textstyle a={\tfrac {1}{c{\sqrt {2\pi }}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left({\frac {-(x-\mu )^{2}}{2\sigma ^{2}}}\right).}</annotation>

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These Gaussians are plotted in the accompanying figure.

The product of two Gaussian functions is a Gaussian, and the <a href="Convolution" title="Convolution">convolution</a> of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=c_{1}^{2}+c_{2}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle c^{2}=c_{1}^{2}+c_{2}^{2}}</annotation>

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</math><img src="./056b9701268a3889fa592382eec0f784f7c3eca3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.122ex; height:3.343ex;" alt="{\displaystyle c^{2}=c_{1}^{2}+c_{2}^{2}}" loading="lazy">. The product of two Gaussian probability density functions (PDFs), though, is not in general a Gaussian PDF.

The Fourier <a href="Fourier_transform#Uncertainty_principle" title="Fourier transform">uncertainty principle</a> becomes an equality if and only if (modulated) Gaussian functions are considered.<a href="#cite_note-2">[2]</a>

Taking the <a href="Fourier_transform#Angular_frequency_(ω)" title="Fourier transform">Fourier transform (unitary, angular-frequency convention)</a> of a Gaussian function with parameters <var>a</var> = 1, <var>b</var> = 0 and <var>c</var> yields another Gaussian function, with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle 1/c}</annotation>

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</math><img src="./ea4ccea7e8b651b718456ef2203efc251a58ca84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.332ex; height:2.843ex;" alt="{\displaystyle 1/c}" loading="lazy">.<a href="#cite_note-3">[3]</a> So in particular the Gaussian functions with <var>b</var> = 0 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=1}">

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<annotation encoding="application/x-tex">{\displaystyle c=1}</annotation>

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</math><img src="./3e3467f9e219a5ea38a30da5c3a02c2c23f61a79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=1}" loading="lazy"> are kept fixed by the Fourier transform (they are <a href="Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the Fourier transform with eigenvalue 1).

A physical realization is that of the <a href="Fraunhofer_diffraction#Diffraction_by_an_aperture_with_a_Gaussian_profile" title="Fraunhofer diffraction">diffraction pattern</a>: for example, a <a href="Photographic_slide" class="mw-redirect" title="Photographic slide">photographic slide</a> whose <a href="Transmittance" title="Transmittance">transmittance</a> 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 <a href="Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a>:

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<annotation encoding="application/x-tex">{\displaystyle \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).}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Integral_of_a_Gaussian_function">Integral of a Gaussian function</h2></div>

The integral of an arbitrary Gaussian function is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }a\,e^{-(x-b)^{2}/2c^{2}}\,dx=\ a\,|c|\,{\sqrt {2\pi }}.}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }a\,e^{-(x-b)^{2}/2c^{2}}\,dx=\ a\,|c|\,{\sqrt {2\pi }}.}</annotation>

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An alternative form is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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),}">

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<annotation encoding="application/x-tex">{\displaystyle \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),}</annotation>

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where f must be strictly positive for the integral to converge.

<div class="mw-heading mw-heading3"><h3 id="Relation_to_standard_Gaussian_integral">Relation to standard Gaussian integral</h3></div>

The integral

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }ae^{-(x-b)^{2}/2c^{2}}\,dx}</annotation>

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for some <a href="Real_number" title="Real number">real</a> constants a, b and c > 0 can be calculated by putting it into the form of a <a href="Gaussian_integral" title="Gaussian integral">Gaussian integral</a>. First, the constant a can simply be factored out of the integral. Next, the variable of integration is changed from x to <var>y</var> = <var>x</var> − b:

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<annotation encoding="application/x-tex">{\displaystyle a\int _{-\infty }^{\infty }e^{-y^{2}/2c^{2}}\,dy,}</annotation>

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and then to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=y/{\sqrt {2c^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle z=y/{\sqrt {2c^{2}}}}</annotation>

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</math><img src="./ea45eea3e85058df7aef3594445c10d68796ee4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.052ex; height:3.509ex;" alt="{\displaystyle z=y/{\sqrt {2c^{2}}}}" loading="lazy">:

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a{\sqrt {2c^{2}}}\int _{-\infty }^{\infty }e^{-z^{2}}\,dz.}</annotation>

</semantics>

</math>

Then, using the <a href="Gaussian_integral" title="Gaussian integral">Gaussian integral identity</a>

</mrow>

</mrow>

</msubsup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }e^{-z^{2}}\,dz={\sqrt {\pi }},}</annotation>

</semantics>

</math>

we have

</mrow>

</mrow>

</msubsup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }ae^{-(x-b)^{2}/2c^{2}}\,dx=a{\sqrt {2\pi c^{2}}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Two-dimensional_Gaussian_function">Two-dimensional Gaussian function</h2></div>

Base form:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x,y)=\exp(-x^{2}-y^{2})}</annotation>

</semantics>

</math>

In two dimensions, the power to which e is raised in the Gaussian function is any negative-definite quadratic form. Consequently, the <a href="Level_set" title="Level set">level sets</a> of the Gaussian will always be ellipses.

A particular example of a two-dimensional Gaussian function is

<mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 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).}</annotation>

</semantics>

</math>

Here the coefficient A is the amplitude, x0, y0 is the center, and σx, σy are the x and y spreads of the blob. The figure on the right was created using A = 1, x0 = 0, y0 = 0, σx = σy = 1.

The volume under the Gaussian function is given by

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\,dx\,dy=2\pi A\sigma _{X}\sigma _{Y}.}</annotation>

</semantics>

</math>

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

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 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 )},}</annotation>

</semantics>

</math>

where the matrix

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a&b\\b&c\end{bmatrix}}}</annotation>

</semantics>

</math>

is <a href="Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite</a>.

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

<div class="mw-heading mw-heading3"><h3 id="Meaning_of_parameters_for_the_general_equation">Meaning of parameters for the general equation</h3></div>

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

If we set

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

</semantics>

</math>then we rotate the blob by a positive, counter-clockwise angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> (for negative, clockwise rotation, invert the signs in the b coefficient).<a href="#cite_note-4">[4]</a>

To get back the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{X}}</annotation>

</semantics>

</math><img src="./380c53c60c8301a5c80924b66363d831dfa80b9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.96ex; height:2.009ex;" alt="{\displaystyle \sigma _{X}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{Y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{Y}}</annotation>

</semantics>

</math><img src="./6a9fb889441f514e155f65e77dc5b7c7a5a84f35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.814ex; height:2.009ex;" alt="{\displaystyle \sigma _{Y}}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

</semantics>

</math>

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

<table>

<td>

</td>

<td>

</td>

<td>

</td></tr></tbody></table>

Using the following <a href="GNU_Octave" title="GNU Octave">Octave</a> code, one can easily see the effect of changing the parameters:

<div class="mw-highlight mw-highlight-lang-octave mw-content-ltr" dir="ltr"><pre>A = 1;

x0 = 0; y0 = 0;

sigma_X = 1;

sigma_Y = 2;

[X, Y] = meshgrid(-5:.1:5, -5:.1:5);

for theta = 0:pi/100:pi

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);

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)

waitforbuttonpress

end

</pre></div>

Such functions are often used in <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a> and in computational models of <a href="Visual_system" title="Visual system">visual system</a> function—see the articles on <a href="Scale_space" title="Scale space">scale space</a> and <a href="Affine_shape_adaptation" title="Affine shape adaptation">affine shape adaptation</a>.

Also see <a href="Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a>.

<div class="mw-heading mw-heading3"><h3 id="Higher-order_Gaussian_or_super-Gaussian_function_or_generalized_Gaussian_function">Higher-order Gaussian or super-Gaussian function or generalized Gaussian function</h3></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">:

<mrow>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=A\exp \left(-\left({\frac {(x-x_{0})^{2}}{2\sigma _{X}^{2}}}\right)^{P}\right).}</annotation>

</semantics>

</math>

This function is known as a super-Gaussian function and is often used for Gaussian beam formulation.<a href="#cite_note-5">[5]</a> This function may also be expressed in terms of the <a href="Full_width_at_half_maximum" title="Full width at half maximum">full width at half maximum</a> (FWHM), represented by w:

<mrow>

<mrow>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=A\exp \left(-\ln 2\left(4{\frac {(x-x_{0})^{2}}{w^{2}}}\right)^{P}\right).}</annotation>

</semantics>

</math>

In a two-dimensional formulation, a Gaussian function along <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> can be combined<a href="#cite_note-6">[6]</a> with potentially different <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{X}}</annotation>

</semantics>

</math><img src="./8348dd8ce7e6f7f4778ee01fa5bdc7b828afd98c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.125ex; height:2.509ex;" alt="{\displaystyle P_{X}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{Y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{Y}}</annotation>

</semantics>

</math><img src="./e52a028cc165bfc94caedb9464a32c36bf49130a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.978ex; height:2.509ex;" alt="{\displaystyle P_{Y}}" loading="lazy"> to form a rectangular Gaussian distribution:

<mrow>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 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).}</annotation>

</semantics>

</math>

or an elliptical Gaussian distribution:

<mrow>

<mrow>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 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)}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Multi-dimensional_Gaussian_function">Multi-dimensional Gaussian function</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal distribution</a></div>

In an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\exp(-x^{\mathsf {T}}Cx),}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\begin{bmatrix}x_{1}&\cdots &x_{n}\end{bmatrix}}}">

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x={\begin{bmatrix}x_{1}&\cdots &x_{n}\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./4f4b39485615a7021d575449c1db8e4db6598346.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.774ex; height:2.843ex;" alt="{\displaystyle x={\begin{bmatrix}x_{1}&\cdots &x_{n}\end{bmatrix}}}" loading="lazy"> is a column of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> coordinates, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

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</math><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"> matrix, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{\mathsf {T}}}">

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<annotation encoding="application/x-tex">{\displaystyle {}^{\mathsf {T}}}</annotation>

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</math><img src="./de0c33e315a8e2cbf2daec6394ccc13017d41fd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.351ex; height:2.509ex;" alt="{\displaystyle {}^{\mathsf {T}}}" loading="lazy"> denotes <a href="Transpose" title="Transpose">matrix transposition</a>.

The integral of this Gaussian function over the whole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}\exp(-x^{\mathsf {T}}Cx)\,dx={\sqrt {\frac {\pi ^{n}}{\det C}}}.}</annotation>

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It can be easily calculated by diagonalizing the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> and changing the integration variables to the eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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More generally a shifted Gaussian function is defined as

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\exp(-x^{\mathsf {T}}Cx+s^{\mathsf {T}}x),}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\begin{bmatrix}s_{1}&\cdots &s_{n}\end{bmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle s={\begin{bmatrix}s_{1}&\cdots &s_{n}\end{bmatrix}}}</annotation>

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</math><img src="./dd3233a130f2fe87345da79299c17ea20d9486d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.056ex; height:2.843ex;" alt="{\displaystyle s={\begin{bmatrix}s_{1}&\cdots &s_{n}\end{bmatrix}}}" loading="lazy"> is the shift vector and the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> can be assumed to be symmetric, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\mathsf {T}}=C}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\mathsf {T}}=C}</annotation>

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</math><img src="./bae4b4c07a3e33573dd84bdc5c911fa0009e9682.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.014ex; height:2.676ex;" alt="{\displaystyle C^{\mathsf {T}}=C}" loading="lazy">, and positive-definite. The following integrals with this function can be calculated with the same technique:

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<annotation encoding="application/x-tex">{\displaystyle \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}}.}</annotation>

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<mtext> where </mtext>

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<annotation encoding="application/x-tex">{\displaystyle \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.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \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}}.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u={\frac {1}{2}}B^{-1}v,\ v=s+s',\ B=C+C'.}">

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<annotation encoding="application/x-tex">{\textstyle u={\frac {1}{2}}B^{-1}v,\ v=s+s',\ B=C+C'.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Estimation_of_parameters">Estimation of parameters</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Normal_distribution#Estimation_of_parameters" title="Normal distribution">Normal distribution § Estimation of parameters</a></div>

A number of fields such as <a href="Photometry_(astronomy)" title="Photometry (astronomy)">stellar photometry</a>, <a href="Gaussian_beam" title="Gaussian beam">Gaussian beam</a> characterization, and <a href="Emission_spectrum#Emission_spectroscopy" title="Emission spectrum">emission/absorption line spectroscopy</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A;x_{0},y_{0};\sigma _{X},\sigma _{Y})}">

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<annotation encoding="application/x-tex">{\displaystyle (A;x_{0},y_{0};\sigma _{X},\sigma _{Y})}</annotation>

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The most common method for estimating the Gaussian parameters is to take the logarithm of the data and <a href="Polynomial_fitting" class="mw-redirect" title="Polynomial fitting">fit a parabola</a> to the resulting data set.<a href="#cite_note-Caruana_Searle_Heller_Shupack_1986_pp._1162–1167-7">[7]</a><a href="#cite_note-Guo-8">[8]</a> While this provides a simple <a href="Curve_fitting" title="Curve fitting">curve fitting</a> 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 through <a href="Weighted_least_squares" title="Weighted least squares">weighted least squares</a> 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 an <a href="Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">iteratively reweighted least squares</a> procedure, in which the weights are updated at each iteration.<a href="#cite_note-Guo-8">[8]</a>

It is also possible to perform <a href="Non-linear_regression" class="mw-redirect" title="Non-linear regression">non-linear regression</a> directly on the data, without involving the <a href="Logarithmic_data_transformation" class="mw-redirect" title="Logarithmic data transformation">logarithmic data transformation</a>; for more options, see <a href="Probability_distribution_fitting" title="Probability distribution fitting">probability distribution fitting</a>.

<div class="mw-heading mw-heading3"><h3 id="Parameter_precision">Parameter precision</h3></div>

Once one has an algorithm for estimating the Gaussian function parameters, it is also important to know how <a href="Accuracy_and_precision" title="Accuracy and precision">precise</a> those estimates are. Any <a href="Least_squares" title="Least squares">least squares</a> 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 use <a href="Cram%C3%A9r%E2%80%93Rao_bound" title="Cramér–Rao bound">Cramér–Rao bound</a> theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data.<a href="#cite_note-Hagen1-9">[9]</a><a href="#cite_note-Hagen2-10">[10]</a>

<ol><li>The noise in the measured profile is either <a href="Independent_and_identically-distributed_random_variables" class="mw-redirect" title="Independent and identically-distributed random variables">i.i.d.</a> Gaussian, or the noise is <a href="Poisson_distribution" title="Poisson distribution">Poisson-distributed</a>.</li>

<li>The spacing between each sampling (i.e. the distance between pixels measuring the data) is uniform.</li>

<li>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.</li>

<li>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).</li></ol>

When these assumptions are satisfied, the following <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a> K applies for the 1D profile parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> under i.i.d. Gaussian noise and under Poisson noise:<a href="#cite_note-Hagen1-9">[9]</a>

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<annotation encoding="application/x-tex">{\displaystyle \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}}\ ,}</annotation>

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</math><img src="./dc60167425d241916b486aa52fd5303529661a4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.665ex; height:2.676ex;" alt="{\displaystyle \delta _{X}}" loading="lazy"> is the width of the pixels used to sample the function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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</math><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> is the quantum efficiency of the detector, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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and in the Poisson noise case,

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}">

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</math><img src="./29c296094af9a1c665425debeac5eaab99a37a04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}" loading="lazy">, and width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sigma _{X},\sigma _{Y})}">

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<annotation encoding="application/x-tex">{\displaystyle (\sigma _{X},\sigma _{Y})}</annotation>

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</math><img src="./8bae74535026cc5d469194d6485c8f2fbc68f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.616ex; height:2.843ex;" alt="{\displaystyle (\sigma _{X},\sigma _{Y})}" loading="lazy"> of the profile, the following covariance matrices apply:<a href="#cite_note-Hagen2-10">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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where the individual parameter variances are given by the diagonal elements of the covariance matrix.

<div class="mw-heading mw-heading2"><h2 id="Discrete_Gaussian">Discrete Gaussian</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Discrete_Gaussian_kernel" class="mw-redirect" title="Discrete Gaussian kernel">Discrete Gaussian kernel</a></div>

One may ask for a discrete analog to the Gaussian;

this is necessary in discrete applications, particularly <a href="Digital_signal_processing" title="Digital signal processing">digital signal processing</a>. A simple answer is to sample the continuous Gaussian, yielding the <a href="Sampled_Gaussian_kernel" class="mw-redirect" title="Sampled Gaussian kernel">sampled Gaussian kernel</a>. 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 <a href="Scale_space_implementation" title="Scale space implementation">scale space implementation</a>.

An alternative approach is to use the <a href="Discrete_Gaussian_kernel" class="mw-redirect" title="Discrete Gaussian kernel">discrete Gaussian kernel</a>:<a href="#cite_note-lin90-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle T(n,t)=e^{-t}I_{n}(t)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle I_{n}(t)}</annotation>

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</math><img src="./f5330f8efe654b11ae57849d376898006b123144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.89ex; height:2.843ex;" alt="{\displaystyle I_{n}(t)}" loading="lazy"> denotes the <a href="Modified_Bessel_function" class="mw-redirect" title="Modified Bessel function">modified Bessel functions</a> of integer order.

This is the discrete analog of the continuous Gaussian in that it is the solution to the discrete <a href="Diffusion_equation" title="Diffusion equation">diffusion equation</a> (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation.<a href="#cite_note-lin90-11">[11]</a><a href="#cite_note-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Gaussian functions appear in many contexts in the <a href="Natural_sciences" class="mw-redirect" title="Natural sciences">natural sciences</a>, the <a href="Social_sciences" class="mw-redirect" title="Social sciences">social sciences</a>, <a href="Mathematics" title="Mathematics">mathematics</a>, and <a href="Engineering" title="Engineering">engineering</a>. Some examples include:

<ul><li>In <a href="Statistics" title="Statistics">statistics</a> and <a href="Probability_theory" title="Probability theory">probability theory</a>, Gaussian functions appear as the density function of the <a href="Normal_distribution" title="Normal distribution">normal distribution</a>, which is a limiting <a href="Probability_distribution" title="Probability distribution">probability distribution</a> of complicated sums, according to the <a href="Central_limit_theorem" title="Central limit theorem">central limit theorem</a>.</li>

<li>Gaussian functions are the <a href="Green's_function" title="Green's function">Green's function</a> for the (homogeneous and isotropic) <a href="Diffusion_equation" title="Diffusion equation">diffusion equation</a> (and to the <a href="Heat_equation" title="Heat equation">heat equation</a>, which is the same thing), a <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a> that describes the time evolution of a mass-density under <a href="Diffusion" title="Diffusion">diffusion</a>. Specifically, if the mass-density at time t=0 is given by a <a href="Dirac_delta" class="mw-redirect" title="Dirac delta">Dirac delta</a>, 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 <a href="Heat_kernel" title="Heat kernel">heat kernel</a>. More generally, if the initial mass-density is φ(x), then the mass-density at later times is obtained by taking the <a href="Convolution" title="Convolution">convolution</a> of φ with a Gaussian function. The convolution of a function with a Gaussian is also known as a <a href="Weierstrass_transform" title="Weierstrass transform">Weierstrass transform</a>.</li>

<li>A Gaussian function is the <a href="Wave_function" title="Wave function">wave function</a> of the <a href="Ground_state" title="Ground state">ground state</a> of the <a href="Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a>.</li>

<li>The <a href="Molecular_orbital" title="Molecular orbital">molecular orbitals</a> used in <a href="Computational_chemistry" title="Computational chemistry">computational chemistry</a> can be <a href="Linear_combination" title="Linear combination">linear combinations</a> of Gaussian functions called <a href="Gaussian_orbital" title="Gaussian orbital">Gaussian orbitals</a> (see also <a href="Basis_set_(chemistry)" title="Basis set (chemistry)">basis set (chemistry)</a>).</li>

<li>Mathematically, the <a href="Derivative" title="Derivative">derivatives</a> of the Gaussian function can be represented using <a href="Hermite_functions" class="mw-redirect" title="Hermite functions">Hermite functions</a>. For unit variance, the n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th <a href="Hermite_polynomial" class="mw-redirect" title="Hermite polynomial">Hermite polynomial</a>, up to scale.</li>

<li>Consequently, Gaussian functions are also associated with the <a href="Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum state</a> in <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>.</li>

<li><a href="Gaussian_beam" title="Gaussian beam">Gaussian beams</a> are used in optical systems, microwave systems and lasers.</li>

<li>In <a href="Scale_space" title="Scale space">scale space</a> representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in <a href="Computer_vision" title="Computer vision">computer vision</a> and <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a>. Specifically, derivatives of Gaussians (<a href="Hermite_functions" class="mw-redirect" title="Hermite functions">Hermite functions</a>) are used as a basis for defining a large number of types of visual operations.</li>

<li>Gaussian functions are used to define some types of <a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">artificial neural networks</a>.</li>

<li>In <a href="Fluorescence_microscopy" class="mw-redirect" title="Fluorescence microscopy">fluorescence microscopy</a> a 2D Gaussian function is used to approximate the <a href="Airy_disk" title="Airy disk">Airy disk</a>, describing the intensity distribution produced by a <a href="Point_source" title="Point source">point source</a>.</li>

<li>In <a href="Signal_processing" title="Signal processing">signal processing</a> they serve to define <a href="Gaussian_filter" title="Gaussian filter">Gaussian filters</a>, such as in <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a> where 2D Gaussians are used for <a href="Gaussian_blur" title="Gaussian blur">Gaussian blurs</a>. In <a href="Digital_signal_processing" title="Digital signal processing">digital signal processing</a>, one uses a <a href="Discrete_Gaussian_kernel" class="mw-redirect" title="Discrete Gaussian kernel">discrete Gaussian kernel</a>, which may be approximated by the <a href="Binomial_coefficient" title="Binomial coefficient">Binomial coefficient</a><a href="#cite_note-13">[13]</a> or sampling a Gaussian.</li>

<li>In <a href="Geostatistics" title="Geostatistics">geostatistics</a> 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.<a href="#cite_note-14">[14]</a></li></ul>

<ul><li><a href="Bell-shaped_function" title="Bell-shaped function">Bell-shaped function</a></li>

<li><a href="Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a></li>

<li><a href="Normal_distribution" title="Normal distribution">Normal distribution</a></li>

<li><a href="Radial_basis_function_kernel" title="Radial basis function kernel">Radial basis function kernel</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFSquires2001" class="citation book cs1">Squires, G. L. (2001-08-30). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/product/identifier/9781139164498/type/book">Practical Physics</a> (4 ed.). Cambridge University Press. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fcbo9781139164498">10.1017/cbo9781139164498</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-77940-1</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFFollandSitaram1997" class="citation journal cs1">Folland, Gerald B.; Sitaram, Alladi (1997). "The uncertainty principle: A mathematical survey". The Journal of Fourier Analysis and Applications. 3 (3): 207–238. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997JFAA....3..207F">1997JFAA....3..207F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02649110">10.1007/BF02649110</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1069-5869">1069-5869</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/FourierTransformGaussian.html">"Fourier Transform – Gaussian"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>. Retrieved 19 December 2013.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFNawri" class="citation web cs1">Nawri, Nikolai. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190814081830/http://imkbemu.physik.uni-karlsruhe.de/~eisatlas/covariance_ellipses.pdf">"Berechnung von Kovarianzellipsen"</a> (PDF). Archived from <a rel="nofollow" class="external text" href="http://imkbemu.physik.uni-karlsruhe.de/~eisatlas/covariance_ellipses.pdf">the original</a> (PDF) on 2019-08-14. Retrieved 14 August 2019.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Parent, A., M. Morin, and P. Lavigne. "Propagation of super-Gaussian field distributions". <a href="Optical_and_Quantum_Electronics" title="Optical and Quantum Electronics">Optical and Quantum Electronics</a> 24.9 (1992): S1071–S1079.

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFHaberman2013" class="citation book cs1">Haberman, Richard (2013). "10.3.3 Inverse Fourier transform of a Gaussian". Applied Partial Differential Equations. Boston: PEARSON. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-321-79705-6</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/GaussianFunction.html">Mathworld, includes a proof for the relations between c and FWHM</a></li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath045/kmath045.htm">"Integrating The Bell Curve"</a>. MathPages.com.</cite></li>

<li><a rel="nofollow" class="external text" href="https://github.com/frecker/gaussian-distribution/">Haskell, Erlang and Perl implementation of Gaussian distribution</a></li>

<li><a class="external text external" href="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)</a></li>

<li><a rel="nofollow" class="external text" href="https://github.com/dwaithe/generalMacros/tree/master/gaussian_fitting">Code for fitting Gaussians in ImageJ and Fiji.</a></li></ul></div><div><div class="zim-footer">

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