Gaussian function

 ( Exponentials ) 

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form $$f(x)\; =\; \backslash exp\; (-x^2)$$ and with parametric extension $$f(x)\; =\; a\; \backslash exp\backslash left(\; -\backslash frac\{(x\; -\; b)^2\}\{2c^2\}\; \backslash right)$$ for arbitrary real number constants , and non-zero . It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, is the position of the center of the peak, and (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".

Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value and variance . In this case, the Gaussian is of the form

$$g(x)\; =\; \backslash frac\{1\}\{\backslash sigma\backslash sqrt\{2\backslash pi\}\}\; \backslash exp\backslash left(\; -\backslash frac\{1\}\{2\}\; \backslash frac\{(x\; -\; \backslash mu)^2\}\{\backslash sigma^2\}\; \backslash right).$$

Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define , in image processing where two-dimensional Gaussians are used for , and in mathematics to solve and diffusion equations and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basis sets.

• $\backslash alpha\; =\; -1/2c^2,$
• $\backslash beta\; =\; b/c^2,$
• $\backslash gamma\; =\; \backslash ln\; a-(b^2\; /\; 2c^2).$

The Gaussian functions are thus those functions whose logarithm is a concave quadratic function.

The parameter is related to the full width at half maximum (FWHM) of the peak according to

$$\backslash text\{FWHM\}\; =\; 2\; \backslash sqrt\{2\; \backslash ln\; 2\}\backslash ,c\; \backslash approx\; 2.35482\backslash ,c.$$

The function may then be expressed in terms of the FWHM, represented by : $$f(x)\; =\; a\; e^\{-4\; (\backslash ln\; 2)\; (x\; -\; b)^2\; /\; w^2\}.$$

Alternatively, the parameter can be interpreted by saying that the two of the function occur at .

The full width at tenth of maximum (FWTM) for a Gaussian could be of interest and is $$\backslash text\{FWTM\}\; =\; 2\; \backslash sqrt\{2\; \backslash ln\; 10\}\backslash ,c\; \backslash approx\; 4.29193\backslash ,c.$$

Gaussian functions are analytic, and their limit as is 0 (for the above case of ).

Gaussian functions are among those functions that are elementary but lack elementary ; the integral of the Gaussian function is the error function:

$$\backslash int\; e^\{-x^2\}\; \backslash ,dx\; =\; \backslash frac\{\backslash sqrt\backslash pi\}\{2\}\; \backslash operatorname\{erf\}\; x\; +\; C.$$

Nonetheless, their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral $$\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-x^2\}\; \backslash ,dx\; =\; \backslash sqrt\{\backslash pi\},$$ and one obtains $$\backslash int\_\{-\backslash infty\}^\backslash infty\; a\; e^\{-(x\; -\; b)^2\; /\; (2c^2)\}\; \backslash ,dx\; =\; ac\; \backslash cdot\; \backslash sqrt\{2\backslash pi\}.$$

This integral is 1 if and only if $a\; =\; \backslash tfrac\{1\}\{c\backslash sqrt\{2\backslash pi\}\}$ (the normalizing constant), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value and variance : $$g(x)\; =\; \backslash frac\{1\}\{\backslash sigma\backslash sqrt\{2\backslash pi\}\}\; \backslash exp\backslash left(\backslash frac\{-(x\; -\; \backslash mu)^2\}\{2\backslash sigma^2\}\; \backslash right).$$

These Gaussians are plotted in the accompanying figure.

The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: $c^2\; =\; c\_1^2\; +\; c\_2^2$. The product of two Gaussian probability density functions (PDFs), though, is not in general a Gaussian PDF.

The Fourier uncertainty principle becomes an equality if and only if (modulated) Gaussian functions are considered.

Taking the Fourier transform (unitary, angular-frequency convention) of a Gaussian function with parameters , and yields another Gaussian function, with parameters $c$, and $1/c$. So in particular the Gaussian functions with and $c\; =\; 1$ are kept fixed by the Fourier transform (they are of the Fourier transform with eigenvalue 1).

A physical realization is that of the diffraction pattern: for example, a photographic slide whose transmittance 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 Poisson summation formula: $$\backslash sum\_\{k\backslash in\backslash Z\}\; \backslash exp\backslash left(-\backslash pi\; \backslash cdot\; \backslash left(\backslash frac\{k\}\{c\}\backslash right)^2\backslash right)\; =\; c\; \backslash cdot\; \backslash sum\_\{k\backslash in\backslash Z\}\; \backslash exp\backslash left(-\backslash pi\; \backslash cdot\; (kc)^2\backslash right).$$

An alternative form is$$\backslash int\_\{-\backslash infty\}^\backslash infty\; k\backslash ,e^\{-f\; x^2\; +\; g\; x\; +\; h\}\backslash ,dx\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; k\backslash ,e^\{-f\; \backslash big(x\; -\; g/(2f)\backslash big)^2\; +\; g^2/(4f)\; +\; h\}\backslash ,dx\; =\; k\backslash ,\backslash sqrt\{\backslash frac\{\backslash pi\}\{f\}\}\backslash ,\backslash exp\backslash left(\backslash frac\{g^2\}\{4f\}\; +\; h\backslash right),$$ where f must be strictly positive for the integral to converge.

Then, using the Gaussian integral identity $$\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-z^2\}\backslash ,dz\; =\; \backslash sqrt\{\backslash pi\},$$

we have $$\backslash int\_\{-\backslash infty\}^\backslash infty\; ae^\{-(x-b)^2/2c^2\}\backslash ,dx\; =\; a\backslash sqrt\{2\backslash pi\; c^2\}.$$

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

A particular example of a two-dimensional Gaussian function is

$$f(x,y)\; =\; A\; \backslash exp\backslash left(-\backslash left(\backslash frac\{(x\; -\; x\_0)^2\}\{2\backslash sigma\_X^2\}\; +\; \backslash frac\{(y\; -\; y\_0)^2\}\{2\backslash sigma\_Y^2\}\; \backslash right)\backslash right).$$

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

The volume under the Gaussian function is given by $$V\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f(x,\; y)\backslash ,dx\; \backslash ,dy\; =\; 2\; \backslash pi\; A\; \backslash sigma\_X\; \backslash sigma\_Y.$$

In general, a two-dimensional elliptical Gaussian function is expressed as $$f(x,\; y)\; =\; A\; \backslash exp\backslash Big(-\backslash big(a(x\; -\; x\_0)^2\; +\; 2b(x\; -\; x\_0)(y\; -\; y\_0)\; +\; c(y\; -\; y\_0)^2\; \backslash big)\backslash Big),$$ where the matrix is positive-definite.

Using this formulation, the figure on the right can be created using , , , .

If we set $$\backslash begin\{align\}$$

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},
     

To get back the coefficients $\backslash theta$, $\backslash sigma\_X$ and $\backslash sigma\_Y$ from $a$, $b$ and $c$ use

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

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

A = 1; x0 = 0; y0 = 0;

sigma_X = 1; sigma_Y = 2;

X, = 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
     

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.

This function is known as a super-Gaussian function and is often used for Gaussian beam formulation.Parent, A., M. Morin, and P. Lavigne. "Propagation of super-Gaussian field distributions". Optical and Quantum Electronics 24.9 (1992): S1071–S1079. This function may also be expressed in terms of the full width at half maximum (FWHM), represented by : $$f(x)\; =\; A\; \backslash exp\backslash left(-\backslash ln\; 2\backslash left(4\backslash frac\{(x\; -\; x\_0)^2\}\{w^2\}\backslash right)^P\backslash right).$$

In a two-dimensional formulation, a Gaussian function along $x$ and $y$ can be combined with potentially different $P\_X$ and $P\_Y$ to form a rectangular Gaussian distribution: $$f(x,\; y)\; =\; A\; \backslash exp\backslash left(-\backslash left(\backslash frac\{(x\; -\; x\_0)^2\}\{2\backslash sigma\_X^2\}\backslash right)^\{P\_X\}\; -\; \backslash left(\backslash frac\{(y\; -\; y\_0)^2\}\{2\backslash sigma\_Y^2\}\backslash right)^\{P\_Y\}\backslash right).$$ or an elliptical Gaussian distribution: $$f(x\; ,\; y)\; =\; A\; \backslash exp\backslash left(-\backslash left(\backslash frac\{(x\; -\; x\_0)^2\}\{2\backslash sigma\_X^2\}\; +\; \backslash frac\{(y\; -\; y\_0)^2\}\{2\backslash sigma\_Y^2\}\backslash right)^P\backslash right)$$

The integral of this Gaussian function over the whole $n$-dimensional space is given as $$\backslash int\_\{\backslash R^n\}\; \backslash exp(-x^\backslash mathsf\{T\}\; C\; x)\; \backslash ,\; dx\; =\; \backslash sqrt\{\backslash frac\{\backslash pi^n\}\{\backslash det\; C\}\}.$$

It can be easily calculated by diagonalizing the matrix $C$ and changing the integration variables to the eigenvectors of $C$.

More generally a shifted Gaussian function is defined as $$f(x)\; =\; \backslash exp(-x^\backslash mathsf\{T\}\; C\; x\; +\; s^\backslash mathsf\{T\}\; x),$$ where is the shift vector and the matrix $C$ can be assumed to be symmetric, $C^\backslash mathsf\{T\}\; =\; C$, and positive-definite. The following integrals with this function can be calculated with the same technique: $$\backslash int\_\{\backslash R^n\}\; e^\{-x^\backslash mathsf\{T\}\; C\; x\; +\; v^\backslash mathsf\{T\}x\}\; \backslash ,\; dx\; =\; \backslash sqrt\{\backslash frac\{\backslash pi^n\}\{\backslash det\{C\}\}\}\; \backslash exp\backslash left(\backslash frac\{1\}\{4\}\; v^\backslash mathsf\{T\}\; C^\{-1\}\; v\backslash right)\; \backslash equiv\; \backslash mathcal\{M\}.$$ $$\backslash int\_\{\backslash mathbb\{R\}^n\}\; e^\{-\; x^\backslash mathsf\{T\}\; C\; x\; +\; v^\backslash mathsf\{T\}\; x\}\; (a^\backslash mathsf\{T\}\; x)\; \backslash ,\; dx\; =\; (a^T\; u)\; \backslash cdot\; \backslash mathcal\{M\},\; \backslash text\{\; where\; \}\; u\; =\; \backslash frac\{1\}\{2\}\; C^\{-1\}\; v.$$ $$\backslash int\_\{\backslash mathbb\{R\}^n\}\; e^\{-\; x^\backslash mathsf\{T\}\; C\; x\; +\; v^\backslash mathsf\{T\}\; x\}\; (x^\backslash mathsf\{T\}\; D\; x)\; \backslash ,\; dx\; =\; \backslash left(\; u^\backslash mathsf\{T\}\; D\; u\; +\; \backslash frac\{1\}\{2\}\; \backslash operatorname\{tr\}\; (D\; C^\{-1\})\; \backslash right)\; \backslash cdot\; \backslash mathcal\{M\}.$$ where $u\; =\; \backslash frac\{1\}\{2\}\; B^\{-\; 1\}\; v,\backslash \; v\; =\; s\; +\; s\text{'},\backslash \; B\; =\; C\; +\; C\text{'}.$

The most common method for estimating the Gaussian parameters is to take the logarithm of the data and fit a parabola to the resulting data set. Hongwei Guo, "A simple algorithm for fitting a Gaussian function," IEEE Sign. Proc. Mag. 28(9): 134-137 (2011). While this provides a simple curve fitting 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 weighted least squares 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 iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perform non-linear regression directly on the data, without involving the logarithmic data transformation; for more options, see probability distribution fitting.

1. The noise in the measured profile is either i.i.d. Gaussian, or the noise is Poisson-distributed.
2. The spacing between each sampling (i.e. the distance between pixels measuring the data) is uniform.
3. 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.
4. 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).

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}
     

\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}.