Gaussian Curve

Gaussian function

Gaussian curves parametrised by expected value and variance (see normal distribution)
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):

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Two-dimensional Gaussian function
A particular example of a two-dimensional Gaussian function is
Here the coefficient A is the amplitude, xo,yo is the center and σx, σy are the x and y spreads of the blob. The figure on the left was created using A = 1, xo = 0, yo = 0, σx = σy = 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, (xo, yo) = (0, 0), a = c = 1, b = 0.
Meaning of parameters for the general equation
For the general form of the equation the coefficient A is the amplitude and (xo, yo) 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.
Applications
The antiderivative 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

Dr Abdul Aziz Awan