Editing Gaussian function (section)

=== 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]].