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In mathematics, a Gaussian function (named after Carl Friedrich Gauss) is a function of the form:

for some real constants a > 0, b, and c.

The a is the height of the Gaussian peak, b is the position of the center of the peak and c is related to the FWHM of the peak according to

Gaussian functions with c² = 2 are eigenfunctions of the Fourier transform. This means that the Fourier transform of a Gaussian function, f, is not only another Gaussian function but a scalar multiple of f.

Gaussian functions are among those functions that are elementary but lack elementary antiderivatives. Nonetheless their improper integrals over the whole real line can be evaluated exactly (see Gaussian integral):

Contents 1 Two-dimensional Gaussian function 1.1 Meaning of parameters for the general equation 2 Applications 3 See also 4 External links

A particular example of a two-dimensional Gaussian function is

Here the coefficient A is the amplitude, x_o,y_o is the center and σ_x, σ_y are the x and y spreads of the blob. The figure on the left was created using A = 1, x_o = 0, y_o = 0, σ_x = σ_y = 1.

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

where the matrix

is positive-definite.

Using this formulation, the figure on the left can be created using A = 1, (x_o, y_o) = (0, 0), a = c = 1, b = 0.

For the general form of the equation the coefficient A is the amplitude and (x_o, y_o) is the center of the blob.

If we set

then we rotate the blob by an angle θ. This can be seen in the following examples:

θ = 0

θ = π / 6

θ = π / 3

Using the following MATLAB code one can see the effect of changing the parameters easily

A = 1;
x0 = 0; y0 = 0;
for theta = 0:pi/100:pi
sigma_x = 1;
sigma_y = 2;
a = (cos(theta)/sigma_x)^2 + (sin(theta)/sigma_y)^2;
b = -sin(2*theta)/(sigma_x)^2 + sin(2*theta)/(sigma_y)^2 ;
c = (sin(theta)/sigma_x)^2 + (cos(theta)/sigma_y)^2;

[X, Y] = meshgrid(-5:.1:5, -5:.1:5);
Z = A*exp( - (a*(X-x0).^2 + b*(X-x0).*(Y-y0) + c*(Y-y0).^2)) ;
surf(X,Y,Z);shading interp;view(-36,36);axis equal;drawnow
end

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

Also see multivariate normal distribution.

Layman's explanation

The Gaussian function accounts for distribution anomalies, for example the bombardment of molecules to elicit ions in mass spectometry. The kinetic energy of the molecules before ionization would ideally be zero in reference to the direction of acceleration through the magnetic sector. Though if the respective molecule's kinetic energy was negative prior to the moment of ionization by the transferred kinetic energy of the electron, the resultant velocity at the detector would be slower than predicted. Vice versa for a molecule maintaining a relative positive velocity prior to the moment of ionization experiences a faster velocity than predicted for it's weight. The effects are represented in a bell curve whereby the bulk lie within the predicted domain, and deviations from which diminish exponentially in either direction from that point. The aforementioned accounts for a single prescribed distribution, though the effects may be complexed by multiple overlapping bell curves. For visual, imagine a distributed anomaly falling within a neighboring bell, thereby distorting the seen data.

The integral of the Gaussian function is the error function.

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.
• A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.
• The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals (see also basis set (chemistry)).
• Mathematically, the Gaussian function plays an important role in the definition of the Hermite polynomials.
• Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory.
• Gaussian beams are used in optical and microwave systems,
• Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing -- see the article on scale space representation. Specifically, derivatives of Gaussians are used as a basis for defining a large number of types of visual operations.
• Gaussian functions are used in some types of artificial neural networks

• Lorentzian function
• Multivariate normal distribution

• Mathworld, includes a proof for the relations between c and FWHM