In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
Gaussian function (named after
Johann Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
) is a
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of the form:
for some
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
constants
a,
b,
c > 0, and
e ≈ 2.718281828 (
Euler's numberThe mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
).
The
graphIn mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
of a Gaussian is a characteristic symmetric "bell curve" shape that quickly falls off towards plus/minus infinity. The parameter
a is the height of the curve's peak,
b is the position of the centre of the peak, and
c controls the width of the "bell".
Gaussian functions are widely used in
statisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
where they describe the
normal distributions, in
signal processingSignal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
where they serve to define
Gaussian filterIn electronics and signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function. Gaussian filters are designed to give no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian...
s, in
image processingIn electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
where twodimensional Gaussians are used for
Gaussian blurA Gaussian blur is the result of blurring an image by a Gaussian function. It is a widely used effect in graphics software, typically to reduce image noise and reduce detail...
s, and in
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
where they are used to solve
heat equationThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
s and
diffusion equations and to define the
Weierstrass transformIn mathematics, the Weierstrass transform of a function f : R → R, named after Karl Weierstrass, is the function F defined by...
.
Properties
Gaussian functions arise by applying the
exponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
to a general
quadratic functionA quadratic function, in mathematics, is a polynomial function of the formf=ax^2+bx+c,\quad a \ne 0.The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the yaxis....
. The Gaussian functions are thus those functions whose
logarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
is a quadratic function.
The parameter
c is related to the
full width at half maximumFull width at half maximum is an expression of the extent of a function, given by the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value....
(FWHM) of the peak according to

Alternatively, the parameter
c can be interpreted by saying that the two
inflection pointIn differential calculus, an inflection point, point of inflection, or inflection is a point on a curve at which the curvature or concavity changes sign. The curve changes from being concave upwards to concave downwards , or vice versa...
s of the function occur at
x =
b −
c and
x =
b +
c.
The full width at tenth of maximum FWTM for a Gaussian could be of interest and is
 .
Gaussian functions are
analyticIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
, and their
limitIn mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
as
x → ∞ is 0.
Gaussian functions are among those functions that are
elementaryIn mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations...
but lack elementary
antiderivativeIn calculus, an "antiderivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
s; the
integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the Gaussian function is the
error functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
. Nonetheless their improper integrals over the whole real line can be evaluated exactly, using the
Gaussian integralThe Gaussian integral, also known as the EulerPoisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...
and one obtains
This integral is 1 if and only if
a = 1/(
c√(2π)), and in this case the Gaussian is the
probability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of a
normally distributed random variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
with
expected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
μ =
b and
varianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
σ
^{2} =
c^{2}. These Gaussians are graphed in the accompanying figure.
Gaussian functions centered at zero minimize the Fourier uncertainty principle.
The product of two Gaussian functions is a Gaussian, and the
convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to crosscorrelation...
of two Gaussian functions is again a Gaussian, with
.
Taking the Fourier transform (unitary, angular frequency convention) of a Gaussian function with parameters
a,
b = 0 and
c yields another Gaussian function, with parameters
ac,
b = 0 and 1/
c. So in particular the Gaussian functions with
b = 0 and
c = 1 are kept fixed by the Fourier transform (they are
eigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any nonzero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of the Fourier transform with eigenvalue 1).
The fact that the Gaussian function is an eigenfunction of the Continuous Fourier transform
allows to derive the following interesting identity from the
Poisson summation formulaIn mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples...
:
Twodimensional Gaussian function
In two dimensions, the power to which e is raised in the Gaussian function may be any bivariate conic section, including circles, ellipses, and hyperbolas. Depending on which curve is used, the resulting Gaussian will have level sets that are circles, ellipses, or even hyperbolas.
A particular example of a twodimensional 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 right was created using
A = 1,
x_{o} = 0,
y_{o} = 0, σ
_{x} = σ
_{y} = 1.
In general, a twodimensional elliptical Gaussian function is expressed as
where the matrix
is
positivedefiniteIn linear algebra, a positivedefinite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positivedefinite symmetric bilinear form ....
.
Using this formulation, the figure on the right can be created using
A = 1, (
x_{o},
y_{o}) = (0, 0),
a =
c = 1/2,
b = 0.
Meaning of parameters for the general equation
For the general form of the equation the coefficient
A is the height of the peak and (
x_{o},
y_{o}) is the center of the blob.
If we set
then we rotate the blob by a clockwise angle
(for counterclockwise rotation invert the signs in the b coefficient). This can be seen in the following examples:
Using the following
OctaveGNU Octave is a highlevel language, primarily intended for numerical computations. It provides a convenient commandline interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...
code one can easily see the effect of changing the parameters