Gaussian noise

2025-07-19T16:35:53Z

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{{Short description|Type of noise in signal processing}}
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In [[signal processing]] theory, '''Gaussian noise''', named after [[Carl Friedrich Gauss]], is a kind of [[Noise (spectral phenomenon) |signal noise]] that has a [[probability density function]] (pdf) equal to that of the [[normal distribution]] (which is also known as the Gaussian distribution).<ref name="Barbu" /><ref name="Handbook"/> In other words, the values that the noise can take are Gaussian-distributed.

The probability density function <math>p</math> of a Gaussian random variable <math>z</math> is given by:

: <math>p(z) = \frac 1 {\sigma\sqrt{2\pi}} e^{-\frac{(z-\mu)^2}{2\sigma^2} }</math>

where <math>z</math> represents the grey level, <math>\mu</math> the [[mean]] grey value and <math>\sigma</math> its [[standard deviation]].<ref name="Basel" />

A special case is ''white Gaussian noise'', in which the values at any pair of times are [[iid|identically distributed]] and [[statistically independent]] (and hence [[uncorrelated]]). In [[communication channel]] testing and modelling, Gaussian noise is used as additive [[white noise]] to generate [[additive white Gaussian noise]].

In [[telecommunications]] and [[computer networking]], communication channels can be affected by [[wideband]] 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]]), [[shot noise]], [[black-body radiation]] 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 image]]s arise during acquisition e.g. [[sensor noise]] caused by poor illumination and/or high temperature, and/or transmission e.g. [[Circuit noise level |electronic circuit noise]].<ref name="Basel" /> In [[digital image processing]] Gaussian noise can be reduced using a [[spatial filter]], 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 reduction |noise removal]] include: mean ([[convolution]]) filtering, [[median filter]]ing and [[Gaussian smoothing]].<ref name="Barbu" /><ref name="HIPR2" />

== See also ==

* [[Gaussian process]]
* [[Gaussian smoothing]]

== References ==

{{Reflist|refs=
<ref name="Handbook">{{cite web |url=https://www.sfu.ca/sonic-studio/handbook/Gaussian_Noise.html |title=Handbook for Acoustic Ecology |edition=Second |year=1999 |publisher=Cambridge Street Publishing |editor=Barry Truax |access-date=2012-08-05 |archive-url=https://web.archive.org/web/20171010053540/http://www.sfu.ca/sonic-studio/handbook/Gaussian_Noise.html |archive-date=2017-10-10 |url-status=dead }}</ref>
<ref name="HIPR2">{{cite web | url=http://homepages.inf.ed.ac.uk/rbf/HIPR2/noise.htm | title=Image Synthesis — Noise Generation | access-date=11 October 2013 |author1=Robert Fisher |author2=Simon Perkins |author3=Ashley Walker |author4=Erik Wolfart }}</ref>
<ref name="Basel">{{cite web | url=http://miac.unibas.ch/SIP/06-Restoration.html | title=Image Restoration: Introduction to Signal and Image Processing | publisher=MIAC, University of Basel | date=2012-04-24 | access-date=11 October 2013 | author=Philippe Cattin | archive-date=2016-09-18 | archive-url=https://web.archive.org/web/20160918164948/https://miac.unibas.ch/SIP/06-Restoration.html | url-status=dead }}</ref>
<ref name="Barbu">{{cite journal | title=Variational Image Denoising Approach with Diffusion Porous Media Flow | author=Tudor Barbu | journal=Abstract and Applied Analysis | year=2013 | volume=2013 | pages=8 | doi=10.1155/2013/856876| doi-access=free }}</ref>
}}
{{Noise}}

{{DEFAULTSORT:Gaussian Noise}}
[[Category:Stochastic processes]]
[[Category:Normal distribution]]
[[Category:Acoustics]]

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Gaussian noise