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: Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... This article is about mathematics. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually Gauß, Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

for some real constants a > 0, b, and c. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

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 A full width at half maximum (FWHM) 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. ...

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. In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...

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): In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by... The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...

Two-dimensional Gaussian function

2-d Gaussian curve

A particular example of a two-dimensional Gaussian function is Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1200 × 900 pixel, file size: 69 KB, MIME type: image/png) Isometric plot of a two dimensional gaussian created by Kaushik Ghose using MATLAB. The code is [X, Y] = meshgrid(-3:.1:3, -3... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1200 × 900 pixel, file size: 69 KB, MIME type: image/png) Isometric plot of a two dimensional gaussian created by Kaushik Ghose using MATLAB. The code is [X, Y] = meshgrid(-3:.1:3, -3...

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. Look up blob in Wiktionary, the free dictionary. ...

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

where the matrix

is positive-definite. In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...

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.

Meaning of parameters for the general equation

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 Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1200 × 900 pixel, file size: 54 KB, MIME type: image/png) Created by Kaushik Ghose using the MATLAB commands A = 1; x0 = 0; y0 = 0; theta = 0; sigma_x = 1; sigma_y = 2; a = (cos(theta... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1200 × 900 pixel, file size: 54 KB, MIME type: image/png) Created by Kaushik Ghose using the MATLAB commands A = 1; x0 = 0; y0 = 0; theta = 0; sigma_x = 1; sigma_y = 2; a = (cos(theta... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1200 × 900 pixel, file size: 53 KB, MIME type: image/png) Created by Kaushik Ghose using the MATLAB commands A = 1; x0 = 0; y0 = 0; theta = pi/6; sigma_x = 1; sigma_y = 2; a = (cos... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1200 × 900 pixel, file size: 53 KB, MIME type: image/png) Created by Kaushik Ghose using the MATLAB commands A = 1; x0 = 0; y0 = 0; theta = pi/6; sigma_x = 1; sigma_y = 2; a = (cos... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1200 × 900 pixel, file size: 52 KB, MIME type: image/png) Created by Kaushik Ghose using the MATLAB commands A = 1; x0 = 0; y0 = 0; theta = 0; sigma_x = 1; sigma_y = 2; a = (cos(theta... Image File history File links Size of this preview: 800 × 600 pixel Image in higher resolution (1200 × 900 pixel, file size: 52 KB, MIME type: image/png) Created by Kaushik Ghose using the MATLAB commands A = 1; x0 = 0; y0 = 0; theta = 0; sigma_x = 1; sigma_y = 2; a = (cos(theta... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ...

 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. UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system. ... The visual system is the part of the nervous system which allows organisms to see. ... The Scale space theory is a framework for multi-scale signal representation. ... Affine shape adaptation is a methodology for iteratively adapting the shape of the smoothing kernels in an affine group of smoothing kernels to the local image structure in neighbourhood region of a specific image point. ...

Also see multivariate normal distribution. In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian 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 transfered 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.

Applications

The integral of the Gaussian function is the error function. This article is about the concept of integrals in calculus. ... Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ...

Gaussian functions appear in many contexts in the natural sciences, the social sciences, mathematics, and engineering. Some examples include: The term natural science as the way in which different fields of study are defined is determined as much by historical convention as by the present day meaning of the words. ... The social sciences are a group of academic disciplines that study human aspects of the world. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ...

• 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

This article is about the field of statistics. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... In physics, the ground state of a quantum mechanical system is its lowest-energy state. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... In chemistry, a molecular orbital is a region in which an electron may be found in a molecule. ... Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to complement the information obtained by actual chemical experiments, predict hitherto unobserved chemical phenomena, and solve related problems. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In molecular physics, Gaussian orbitals are a mathematical technique used for the computation of electron orbitals in molecules. ... In modern computational chemistry, quantum chemical calculations are typically performed within a finite set of basis functions. ... In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... Quantum field theory (QFT) is the quantum theory of fields. ... In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity (irradiance) distributions are described by Gaussian functions. ... Computer vision is the science and technology of machines that see. ... UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... The Scale space theory is a framework for multi-scale signal representation. ... A neural network is an interconnected group of neurons. ...

See also

• Lorentzian function
• Multivariate normal distribution

... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ...

External links

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