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# Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: for arbitrary real constants, and. [1]

91 relations: Airy disk, Analytic function, Antiderivative, Artificial neural network, Basis set (chemistry), Bell, Bessel function, Carl Friedrich Gauss, Cauchy distribution, Central limit theorem, Computational chemistry, Computer vision, Concave function, Convolution, Cramér–Rao bound, Derivative, Diffusion, Diffusion equation, Digital image processing, Digital signal processing, Dirac delta function, Discrete Fourier transform, Discretization, Eigenfunction, Elementary function, Engineering, Error function, Expected value, Exponential function, Fluorescence microscope, Full width at half maximum, Function (mathematics), Gaussian beam, Gaussian blur, Gaussian filter, Gaussian integral, Gaussian orbital, Geostatistics, GNU Octave, Graph of a function, Green's function, Ground state, Heat equation, Heat kernel, Hermite polynomials, Independent and identically distributed random variables, Inflection point, Integral, Iteratively reweighted least squares, Least squares, ... Expand index (41 more) »

## Airy disk

In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light.

## Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

## Antiderivative

In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function.

## Artificial neural network

Artificial neural networks (ANNs) or connectionist systems are computing systems vaguely inspired by the biological neural networks that constitute animal brains.

## Basis set (chemistry)

A basis set in theoretical and computational chemistry is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer.

## Bell

A bell is a directly struck idiophone percussion instrument.

## Bessel function

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions of Bessel's differential equation for an arbitrary complex number, the order of the Bessel function.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Cauchy distribution

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution.

## Central limit theorem

In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed.

## Computational chemistry

Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems.

## Computer vision

Computer vision is a field that deals with how computers can be made for gaining high-level understanding from digital images or videos.

## Concave function

In mathematics, a concave function is the negative of a convex function.

## Convolution

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

## Cramér–Rao bound

In estimation theory and statistics, the Cramér–Rao bound (CRB), Cramér–Rao lower bound (CRLB), Cramér–Rao inequality, Frechet–Darmois–Cramér–Rao inequality, or information inequality expresses a lower bound on the variance of unbiased estimators of a deterministic (fixed, though unknown) parameter.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Diffusion

Diffusion is the net movement of molecules or atoms from a region of high concentration (or high chemical potential) to a region of low concentration (or low chemical potential) as a result of random motion of the molecules or atoms.

## Diffusion equation

The diffusion equation is a partial differential equation.

## Digital image processing

In computer science, Digital image processing is the use of computer algorithms to perform image processing on digital images.

## Digital signal processing

Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations.

## Dirac delta function

In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.

## Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency.

## Discretization

In mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts.

## Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

## Elementary function

In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations, exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of ''n''th roots).

## Engineering

Engineering is the creative application of science, mathematical methods, and empirical evidence to the innovation, design, construction, operation and maintenance of structures, machines, materials, devices, systems, processes, and organizations.

## Error function

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion.

## Expected value

In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents.

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Fluorescence microscope

A fluorescence microscope is an optical microscope that uses fluorescence and phosphorescence instead of, or in addition to, reflection and absorption to study properties of organic or inorganic substances.

## Full width at half maximum

Full width at half maximum (FWHM) is an expression of the extent of 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.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Gaussian beam

In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function; this also implies a Gaussian intensity (irradiance) profile.

## Gaussian blur

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

## Gaussian filter

In electronics and signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it).

## Gaussian integral

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.

## Gaussian orbital

In computational chemistry and molecular physics, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are functions used as atomic orbitals in the LCAO method for the representation of electron orbitals in molecules and numerous properties that depend on these.

## Geostatistics

Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets.

## GNU Octave

GNU Octave is software featuring a high-level programming language, primarily intended for numerical computations.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Green's function

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential equation defined on a domain, with specified initial conditions or boundary conditions.

## Ground state

The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system.

## Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

## Heat kernel

In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions.

## Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.

## Independent and identically distributed random variables

In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d. or iid or IID) if each random variable has the same probability distribution as the others and all are mutually independent.

## Inflection point

In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuously differentiable plane curve at which the curve crosses its tangent, that is, the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Iteratively reweighted least squares

The method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems with objective functions of the form: by an iterative method in which each step involves solving a weighted least squares problem of the form:C.

## Least squares

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns.

## Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

## List of logarithmic identities

In mathematics, there are many logarithmic identities.

## Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.

## Molecular orbital

In chemistry, a molecular orbital (MO) is a mathematical function describing the wave-like behavior of an electron in a molecule.

## Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions.

## Natural science

Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on empirical evidence from observation and experimentation.

## Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.

## Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

## Periodic summation

In signal processing, any periodic function, s_P(t) with period P, can be represented by a summation of an infinite number of instances of an aperiodic function, s(t), that are offset by integer multiples of P. This representation is called periodic summation: When s_P(t) is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform, S(f) \ \stackrel \ \mathcal\, at intervals of 1/P.

## Photometry (astronomy)

Photometry is a technique of astronomy concerned with measuring the flux, or intensity of an astronomical object's electromagnetic radiation.

## Point source

A point source is a single identifiable localised source of something.

## Poisson distribution

In probability theory and statistics, the Poisson distribution (in English often rendered), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event.

## Poisson summation formula

In 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.

## Positive-definite matrix

In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.

## Probability density function

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

## Probability distribution

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

## Probability theory

Probability theory is the branch of mathematics concerned with probability.

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree.

## Quantum field theory

In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.

## Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator.

## Radial basis function kernel

In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms.

## Random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Reversal film

In photography, reversal film is a type of photographic film that produces a positive image on a transparent base.

## Root mean square

In statistics and its applications, the root mean square (abbreviated RMS or rms) is defined as the square root of the mean square (the arithmetic mean of the squares of a set of numbers).

## Scale space

Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision.

## Scale space implementation

The linear scale-space representation of an N-dimensional continuous signal, is obtained by convolving fC with an N-dimensional Gaussian kernel: In other words: However, for implementation, this definition is impractical, since it is continuous.

## Signal processing

Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.

## Social science

Social science is a major category of academic disciplines, concerned with society and the relationships among individuals within a society.

## Standard deviation

In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.

## Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

## Transmittance

Transmittance of the surface of a material is its effectiveness in transmitting radiant energy.

## Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

## Vacuum state

In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy.

## Variance

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.

## Visual system

The visual system is the part of the central nervous system which gives organisms the ability to process visual detail, as well as enabling the formation of several non-image photo response functions.

## Wave function

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.

## Weierstrass transform

In mathematics, the Weierstrass transform of a function, named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of, weighted with a Gaussian centered at x.

## References

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