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Gaussian function

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Template:Short description Template:Redirect Template:More citations needed In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form <math display="block">f(x) = \exp (-x^2)</math> and with parametric extension <math display="block">f(x) = a \exp\left( -\frac{(x - b)^2}{2c^2} \right)</math> for arbitrary real constants Template:Mvar, Template:Mvar and non-zero Template:Mvar. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter Template:Mvar is the height of the curve's peak, Template:Mvar is the position of the center of the peak, and Template:Mvar (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 Template:Math and variance Template:Math. In this case, the Gaussian is of the form<ref>Template:Cite book</ref>

<math display="block">g(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( -\frac{1}{2} \frac{(x - \mu)^2}{\sigma^2} \right).</math>

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

PropertiesEdit

Gaussian functions arise by composing the exponential function with a concave quadratic function:<math display="block">f(x) = \exp(\alpha x^2 + \beta x + \gamma),</math>where

  • <math>\alpha = -1/2c^2,</math>
  • <math>\beta = b/c^2,</math>
  • <math>\gamma = \ln a-(b^2 / 2c^2).</math>

(Note: <math>a = 1/(\sigma\sqrt{2\pi}) </math> in <math> \ln a </math>, not to be confused with <math>\alpha = -1/2c^2</math>)

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

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

<math display="block">\text{FWHM} = 2 \sqrt{2 \ln 2}\,c \approx 2.35482\,c.</math>

The function may then be expressed in terms of the FWHM, represented by Template:Mvar: <math display="block">f(x) = a e^{-4 (\ln 2) (x - b)^2 / w^2}.</math>

Alternatively, the parameter Template:Mvar can be interpreted by saying that the two inflection points of the function occur at Template:Math.

The full width at tenth of maximum (FWTM) for a Gaussian could be of interest and is <math display="block">\text{FWTM} = 2 \sqrt{2 \ln 10}\,c \approx 4.29193\,c.</math>

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

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

<math display="block">\int e^{-x^2} \,dx = \frac{\sqrt\pi}{2} \operatorname{erf} x + C.</math>

Nonetheless, their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral <math display="block">\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi},</math> and one obtains <math display="block">\int_{-\infty}^\infty a e^{-(x - b)^2 / (2c^2)} \,dx = ac \cdot \sqrt{2\pi}.</math>

File:Normal Distribution PDF.svg
Normalized Gaussian curves with expected value Template:Mvar and variance Template:Math. The corresponding parameters are <math display="inline">a = \tfrac{1}{\sigma\sqrt{2\pi}}</math>, Template:Math and Template:Math.

This integral is 1 if and only if <math display="inline">a = \tfrac{1}{c\sqrt{2\pi}}</math> (the normalizing constant), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value Template:Math and variance Template:Math: <math display="block">g(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(\frac{-(x - \mu)^2}{2\sigma^2} \right).</math>

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: <math>c^2 = c_1^2 + c_2^2</math>. 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.<ref>Template:Cite journal</ref>

Taking the Fourier transform (unitary, angular-frequency convention) of a Gaussian function with parameters Template:Math, Template:Math and Template:Math yields another Gaussian function, with parameters <math>c</math>, Template:Math and <math>1/c</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> So in particular the Gaussian functions with Template:Math and <math>c = 1</math> are kept fixed by the Fourier transform (they are eigenfunctions 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 interestingTemplate:Clarify identity from the Poisson summation formula: <math display="block">\sum_{k\in\Z} \exp\left(-\pi \cdot \left(\frac{k}{c}\right)^2\right) = c \cdot \sum_{k\in\Z} \exp\left(-\pi \cdot (kc)^2\right).</math>

Integral of a Gaussian functionEdit

The integral of an arbitrary Gaussian function is<math display="block">\int_{-\infty}^\infty a\,e^{-(x - b)^2/2c^2}\,dx = \ a \, |c| \, \sqrt{2\pi}.</math>

An alternative form is<math display="block">\int_{-\infty}^\infty k\,e^{-f x^2 + g x + h}\,dx = \int_{-\infty}^\infty k\,e^{-f \big(x - g/(2f)\big)^2 + g^2/(4f) + h}\,dx = k\,\sqrt{\frac{\pi}{f}}\,\exp\left(\frac{g^2}{4f} + h\right),</math> where f must be strictly positive for the integral to converge.

Relation to standard Gaussian integralEdit

The integral <math display="block">\int_{-\infty}^\infty ae^{-(x - b)^2/2c^2}\,dx</math> for some real constants a, b and c > 0 can be calculated by putting it into the form of a Gaussian integral. First, the constant a can simply be factored out of the integral. Next, the variable of integration is changed from x to Template:Math: <math display="block">a\int_{-\infty}^\infty e^{-y^2/2c^2}\,dy,</math> and then to <math>z = y/\sqrt{2 c^2}</math>: <math display="block">a\sqrt{2 c^2} \int_{-\infty}^\infty e^{-z^2}\,dz.</math>

Then, using the Gaussian integral identity <math display="block">\int_{-\infty}^\infty e^{-z^2}\,dz = \sqrt{\pi},</math>

we have <math display="block">\int_{-\infty}^\infty ae^{-(x-b)^2/2c^2}\,dx = a\sqrt{2\pi c^2}.</math>

Two-dimensional Gaussian functionEdit

File:Gaussian 2d surface.png
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 sets of the Gaussian will always be ellipses.

A particular example of a two-dimensional Gaussian function is <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, x0, y0 is the center, and σx, σy are the x and y spreads of the blob. The figure on the right was created using A = 1, x0 = 0, y0 = 0, σx = σy = 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.

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

Meaning of parameters for the general equationEdit

For the general form of the equation the coefficient A is the height of the peak and Template:Math 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>{{#invoke:citation/CS1|citation |CitationClass=web }}</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:

File:Gaussian 2d 0 degrees.png
<math>\theta = 0</math>
File:Gaussian 2d 30 degrees.png
<math>\theta = -\pi/6</math>
File:Gaussian 2d 60 degrees.png
<math>\theta = -\pi/3</math>

Using the following 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 functionEdit

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 Template:Mvar: <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>{{#invoke:citation/CS1|citation |CitationClass=web }}</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>

Multi-dimensional Gaussian functionEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In an <math>n</math>-dimensional space a Gaussian function can be defined as <math display="block">f(x) = \exp(-x^\mathsf{T} C x),</math> where <math>x = \begin{bmatrix} x_1 & \cdots & x_n\end{bmatrix}</math> is a column of <math>n</math> coordinates, <math>C</math> is a positive-definite <math>n \times n</math> matrix, and <math>{}^\mathsf{T}</math> denotes matrix transposition.

The integral of this Gaussian function over the whole <math>n</math>-dimensional space is given as <math display="block">\int_{\R^n} \exp(-x^\mathsf{T} C x) \, dx = \sqrt{\frac{\pi^n}{\det C}}.</math>

It can be easily calculated by diagonalizing the matrix <math>C</math> and changing the integration variables to the eigenvectors of <math>C</math>.

More generally a shifted Gaussian function is defined as <math display="block">f(x) = \exp(-x^\mathsf{T} C x + s^\mathsf{T} x),</math> where <math>s = \begin{bmatrix} s_1 & \cdots & s_n\end{bmatrix}</math> is the shift vector and the matrix <math>C</math> can be assumed to be symmetric, <math>C^\mathsf{T} = C</math>, and positive-definite. The following integrals with this function can be calculated with the same technique: <math display="block">\int_{\R^n} e^{-x^\mathsf{T} C x + v^\mathsf{T}x} \, dx = \sqrt{\frac{\pi^n}{\det{C}}} \exp\left(\frac{1}{4} v^\mathsf{T} C^{-1} v\right) \equiv \mathcal{M}.</math> <math display="block">\int_{\mathbb{R}^n} e^{- x^\mathsf{T} C x + v^\mathsf{T} x} (a^\mathsf{T} x) \, dx = (a^T u) \cdot \mathcal{M}, \text{ where } u = \frac{1}{2} C^{-1} v.</math> <math display="block">\int_{\mathbb{R}^n} e^{- x^\mathsf{T} C x + v^\mathsf{T} x} (x^\mathsf{T} D x) \, dx = \left( u^\mathsf{T} D u + \frac{1}{2} \operatorname{tr} (D C^{-1}) \right) \cdot \mathcal{M}.</math> <math display="block">\begin{align} & \int_{\mathbb{R}^n} e^{- x^\mathsf{T} C' x + s'^\mathsf{T} x} \left( -\frac{\partial}{\partial x} \Lambda \frac{\partial}{\partial x} \right) e^{-x^\mathsf{T} C x + s^\mathsf{T} x} \, dx \\ & \qquad = \left( 2 \operatorname{tr}(C' \Lambda C B^{- 1}) + 4 u^\mathsf{T} C' \Lambda C u - 2 u^\mathsf{T} (C' \Lambda s + C \Lambda s') + s'^\mathsf{T} \Lambda s \right) \cdot \mathcal{M}, \end{align}</math> where <math display="inline">u = \frac{1}{2} B^{- 1} v,\ v = s + s',\ B = C + C'.</math>

Estimation of parametersEdit

Template:See also

A number of fields such as stellar photometry, Gaussian beam characterization, and emission/absorption line spectroscopy work with sampled Gaussian functions and need to accurately estimate the height, position, and width parameters of the function. There are three unknown parameters for a 1D Gaussian function (a, b, c) and five for a 2D Gaussian function <math>(A; x_0,y_0; \sigma_X,\sigma_Y)</math>.

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.<ref name="Caruana Searle Heller Shupack 1986 pp. 1162–1167">Template:Cite journal</ref><ref name="Guo">Hongwei Guo, "A simple algorithm for fitting a Gaussian function," IEEE Sign. Proc. Mag. 28(9): 134-137 (2011).</ref> 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.<ref name="Guo" /> 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.

Parameter precisionEdit

Once one has an algorithm for estimating the Gaussian function parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter (i.e., the variance of the estimated height, position, and width of the function). One can also use Cramér–Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data.<ref name="Hagen1">N. Hagen, M. Kupinski, and E. L. Dereniak, "Gaussian profile estimation in one dimension," Appl. Opt. 46:5374–5383 (2007)</ref><ref name="Hagen2">N. Hagen and E. L. Dereniak, "Gaussian profile estimation in two dimensions," Appl. Opt. 47:6842–6851 (2008)</ref>

  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).

When these assumptions are satisfied, the following covariance matrix K applies for the 1D profile parameters <math>a</math>, <math>b</math>, and <math>c</math> under i.i.d. Gaussian noise and under Poisson noise:<ref name="Hagen1" /> <math display="block"> \mathbf{K}_{\text{Gauss}} = \frac{\sigma^2}{\sqrt{\pi} \delta_X Q^2} \begin{pmatrix} \frac{3}{2c} &0 &\frac{-1}{a} \\ 0 &\frac{2c}{a^2} &0 \\ \frac{-1}{a} &0 &\frac{2c}{a^2} \end{pmatrix} \ , \qquad \mathbf{K}_\text{Poiss} = \frac{1}{\sqrt{2 \pi}} \begin{pmatrix} \frac{3a}{2c} &0 &-\frac{1}{2} \\ 0 &\frac{c}{a} &0 \\ -\frac{1}{2} &0 &\frac{c}{2a} \end{pmatrix} \ ,</math> where <math>\delta_X</math> is the width of the pixels used to sample the function, <math>Q</math> is the quantum efficiency of the detector, and <math>\sigma</math> indicates the standard deviation of the measurement noise. Thus, the individual variances for the parameters are, in the Gaussian noise case, <math display="block">\begin{align} \operatorname{var} (a) &= \frac{3 \sigma^2}{2 \sqrt{\pi} \, \delta_X Q^2 c} \\ \operatorname{var} (b) &= \frac{2 \sigma^2 c}{\delta_X \sqrt{\pi} \, Q^2 a^2} \\ \operatorname{var} (c) &= \frac{2 \sigma^2 c}{\delta_X \sqrt{\pi} \, Q^2 a^2} \end{align}</math>

and in the Poisson noise case, <math display="block">\begin{align} \operatorname{var} (a) &= \frac{3a}{2 \sqrt{2 \pi} \, c} \\ \operatorname{var} (b) &= \frac{c}{\sqrt{2 \pi} \, a} \\ \operatorname{var} (c) &= \frac{c}{2 \sqrt{2 \pi} \, a}. \end{align} </math>

For the 2D profile parameters giving the amplitude <math>A</math>, position <math>(x_0,y_0)</math>, and width <math>(\sigma_X,\sigma_Y)</math> of the profile, the following covariance matrices apply:<ref name="Hagen2" />

<math display="block">\begin{align} \mathbf{K}_\text{Gauss} = \frac{\sigma^2}{\pi \delta_X \delta_Y Q^2} & \begin{pmatrix} \frac{2}{\sigma_X \sigma_Y} &0 &0 &\frac{-1}{A \sigma_Y} &\frac{-1}{A \sigma_X} \\ 

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}

\end{pmatrix} \\[6pt] \mathbf{K}_{\operatorname{Poisson}} = \frac{1}{2 \pi} & \begin{pmatrix} 

\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}.

\end{align}</math> where the individual parameter variances are given by the diagonal elements of the covariance matrix.

Discrete GaussianEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:Discrete Gaussian kernel.svg
The discrete Gaussian kernel (solid), compared with the sampled Gaussian kernel (dashed) for scales <math>t = 0.5,1,2,4.</math>

One may ask for a discrete analog to the Gaussian; this is necessary in discrete applications, particularly digital signal processing. A simple answer is to sample the continuous Gaussian, yielding the sampled Gaussian kernel. However, this discrete function does not have the discrete analogs of the properties of the continuous function, and can lead to undesired effects, as described in the article scale space implementation.

An alternative approach is to use the discrete Gaussian kernel:<ref name="lin90">Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234–254.</ref> <math display="block">T(n, t) = e^{-t} I_n(t)</math> where <math>I_n(t)</math> denotes the modified Bessel functions of integer order.

This is the discrete analog of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation.<ref name="lin90"/><ref>Campbell, J, 2007, The SMM model as a boundary value problem using the discrete diffusion equation, Theor Popul Biol. 2007 Dec;72(4):539–46.</ref>

ApplicationsEdit

Gaussian functions appear in many contexts in the natural sciences, the social sciences, mathematics, and engineering. Some examples include:

See alsoEdit

ReferencesEdit

Template:Reflist

Further readingEdit

External linksEdit

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