Gaussian noise, named after
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, is a term from
signal processing theory denoting a kind of
signal noise that has a
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 ...
(pdf) equal to that of the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
(which is also known as the
Gaussian distribution).
In other words, the values that the noise can take are Gaussian-distributed.
The probability density function
of a Gaussian random variable
is given by:
:
where
represents the grey level,
the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
grey value and
its
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
.
A special case is ''White Gaussian noise'', in which the values at any pair of times are
identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
and
statistically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...
(and hence
uncorrelated
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
). In
communication channel
A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for informa ...
testing and modelling, Gaussian noise is used as additive
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
to generate
additive white Gaussian noise.
In
telecommunications
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
and
computer networking
A computer network is a set of computers sharing resources located on or provided by network nodes. The computers use common communication protocols over digital interconnections to communicate with each other. These interconnections are ma ...
, communication channels can be affected by
wideband
In communications, a system is wideband when the message bandwidth significantly exceeds the coherence bandwidth of the Channel (communications), channel. Some communication links have such a high Bit rate, data rate that they are forced to use a ...
Gaussian noise coming from many natural sources, such as the thermal vibrations of atoms in conductors (referred to as thermal noise or
Johnson–Nyquist noise
Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens rega ...
),
shot noise,
black-body radiation
Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spect ...
from the earth and other warm objects, and from celestial sources such as the Sun.
Gaussian noise in digital images
Principal sources of Gaussian noise in
digital images arise during acquisition e.g.
sensor noise
Image noise is random variation of brightness or color information in images, and is usually an aspect of electronic noise. It can be produced by the image sensor and circuitry of a scanner or digital camera. Image noise can also originate in ...
caused by poor illumination and/or high temperature, and/or transmission e.g.
electronic circuit noise.
In
digital image processing
Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
Gaussian noise can be reduced using a
spatial filter
A spatial filter is an optical device which uses the principles of Fourier optics to alter the structure of a beam of light or other electromagnetic radiation, typically coherent laser light. Spatial filtering is commonly used to "clean up" the o ...
, though when smoothing an image, an undesirable outcome may result in the blurring of fine-scaled image edges and details because they also correspond to blocked high frequencies. Conventional spatial filtering techniques for
noise removal include: mean (
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
) filtering,
median filter
The median filter is a non-linear digital filtering technique, often used to remove noise from an image or signal. Such noise reduction is a typical pre-processing step to improve the results of later processing (for example, edge detection on an ...
ing and
Gaussian smoothing
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).
It is a widely used effect in graphics software, ...
.
See also
*
Gaussian process
*
Gaussian smoothing
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).
It is a widely used effect in graphics software, ...
References
{{DEFAULTSORT:Gaussian Noise
Stochastic processes
Normal distribution
Acoustics