Editing Gaussian function (section)

== Two-dimensional Gaussian function ==
[[File:Gaussian 2d surface.png|thumb|3d plot of a Gaussian function with a two-dimensional domain]]
Base form:
<math display="block">f(x,y) = \exp(-x^2-y^2)</math>

In two dimensions, the power to which ''e'' is raised in the Gaussian function is any negative-definite quadratic form.  Consequently, the [[level set]]s of the Gaussian will always be ellipses.

A particular example of a two-dimensional Gaussian function is
<!-- This makes the formula consistent with the 1d formula above -->
<math display="block">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).</math>

Here the coefficient ''A'' is the amplitude, ''x''<sub>0</sub>,&nbsp;''y''<sub>0</sub> is the center, and ''σ''<sub>''x''</sub>,&nbsp;''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>0</sub> = 0, ''y''<sub>0</sub> = 0, ''σ''<sub>''x''</sub> = ''σ''<sub>''y''</sub> = 1.

The volume under the Gaussian function is given by
<math display="block">V = \int_{-\infty}^\infty \int_{-\infty}^\infty f(x, y)\,dx \,dy = 2 \pi A \sigma_X \sigma_Y.</math>

In general, a two-dimensional elliptical Gaussian function is expressed as
<math display="block">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),</math>
where the matrix
<math display="block">\begin{bmatrix} a & b \\ b & c \end{bmatrix}</math>
is [[positive-definite matrix|positive-definite]].

Using this formulation, the figure on the right can be created using {{math|1=''A'' = 1}}, {{math|1=(''x''<sub>0</sub>, ''y''<sub>0</sub>) = (0, 0)}}, {{math|1=''a'' = ''c'' = 1/2}}, {{math|1=''b'' = 0}}.

=== Meaning of parameters for the general equation ===
For the general form of the equation the coefficient ''A'' is the height of the peak and {{math|(''x''<sub>0</sub>, ''y''<sub>0</sub>)}} is the center of the blob.

If we set
<math display="block">
\begin{align}
 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{align}
</math>then we rotate the blob by a positive, counter-clockwise angle <math>\theta</math> (for negative, clockwise rotation, invert the signs in the ''b'' coefficient).<ref>{{cite web |last1=Nawri |first1=Nikolai |title=Berechnung von Kovarianzellipsen |url=http://imkbemu.physik.uni-karlsruhe.de/~eisatlas/covariance_ellipses.pdf |access-date=14 August 2019 |url-status=dead |archive-url=https://web.archive.org/web/20190814081830/http://imkbemu.physik.uni-karlsruhe.de/~eisatlas/covariance_ellipses.pdf |archive-date=2019-08-14}}</ref> 

To get back the coefficients <math>\theta</math>, <math>\sigma_X</math> and <math>\sigma_Y</math> from <math>a</math>, <math>b</math> and <math>c</math> use

<math display="block">\begin{align}
\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 + 2 b \cdot \cos\theta\sin\theta + c \cdot \sin^2\theta)}, \\
\sigma_Y^2 &= \frac{1}{2 (a \cdot \sin^2\theta - 2 b \cdot \cos\theta\sin\theta + c \cdot \cos^2\theta)}.
\end{align}</math>

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

{|
| [[Image:Gaussian 2d 0 degrees.png|thumb|200px|<math>\theta = 0</math>]]
| [[Image:Gaussian 2d 30 degrees.png|thumb|200px|<math>\theta = -\pi/6</math>]]
| [[Image:Gaussian 2d 60 degrees.png|thumb|200px|<math>\theta = -\pi/3</math>]]
|}

Using the following [[GNU Octave|Octave]] code, one can easily see the effect of changing the parameters:

<syntaxhighlight lang="octave">
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
</syntaxhighlight>

Such functions are often used in [[image processing]] and in computational models of [[visual system]] function—see the articles on [[scale space]] and [[affine shape adaptation]].

Also see [[multivariate normal distribution]].

=== Higher-order Gaussian or super-Gaussian function or generalized Gaussian function ===
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>P</math>:
<math display="block">f(x) = A \exp\left(-\left(\frac{(x - x_0)^2}{2\sigma_X^2}\right)^P\right).</math>

This function is known as a super-Gaussian function and is often used for Gaussian beam formulation.<ref>Parent, A., M. Morin, and P. Lavigne. "Propagation of super-Gaussian field distributions". ''[[Optical and Quantum Electronics]]'' 24.9 (1992): S1071–S1079.</ref> This function may also be expressed in terms of the [[full width at half maximum]] (FWHM), represented by {{mvar|w}}:
<math display="block">f(x) = A \exp\left(-\ln 2\left(4\frac{(x - x_0)^2}{w^2}\right)^P\right).</math>

In a two-dimensional formulation, a Gaussian function along <math>x</math> and <math>y</math> can be combined<ref>{{Cite web |url=http://www.aor.com/anonymous/pub/commands.pdf |title=GLAD optical software commands manual, Entry on GAUSSIAN command |date=2016-12-15 |website=Applied Optics Research}}</ref> with potentially different <math>P_X</math> and <math>P_Y</math> to form a rectangular Gaussian distribution:
<math display="block">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).</math>
or an elliptical Gaussian distribution:
<math display="block">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)</math>