Editing Convolution (section)

== Applications ==
[[File:Halftone, Gaussian Blur.jpg|thumb|right|[[Gaussian blur]] can be used to obtain a smooth grayscale digital image of a [[halftone]] print.]]

Convolution and related operations are found in many applications in science, engineering and mathematics.
* [[Convolutional neural network]]s apply multiple cascaded ''convolution'' kernels with applications in [[machine vision]] and [[artificial intelligence]].<ref>{{Cite journal|last1=Zhang|first1=Yingjie|last2=Soon|first2=Hong Geok|last3=Ye|first3=Dongsen|last4=Fuh|first4=Jerry Ying Hsi|last5=Zhu|first5=Kunpeng|date=September 2020|title=Powder-Bed Fusion Process Monitoring by Machine Vision With Hybrid Convolutional Neural Networks|url=https://ieeexplore.ieee.org/document/8913613|journal=IEEE Transactions on Industrial Informatics|volume=16|issue=9|pages=5769–5779|doi=10.1109/TII.2019.2956078|s2cid=213010088|issn=1941-0050}}</ref><ref>{{Cite journal|last1=Chervyakov|first1=N.I.|last2=Lyakhov|first2=P.A.|last3=Deryabin|first3=M.A.|last4=Nagornov|first4=N.N.|last5=Valueva|first5=M.V.|last6=Valuev|first6=G.V.|date=September 2020|title=Residue Number System-Based Solution for Reducing the Hardware Cost of a Convolutional Neural Network|url=https://linkinghub.elsevier.com/retrieve/pii/S092523122030583X|journal=Neurocomputing|language=en|volume=407|pages=439–453|doi=10.1016/j.neucom.2020.04.018|s2cid=219470398|quote=Convolutional neural networks represent deep learning architectures that are currently used in a wide range of applications, including computer vision, speech recognition, time series analysis in finance, and many others.}}</ref> Though these are actually '''cross-correlations''' rather than convolutions in most cases.<ref>{{Cite journal|last=Atlas, Homma, and Marks|title=An Artificial Neural Network for Spatio-Temporal Bipolar Patterns: Application to Phoneme Classification|url=https://papers.nips.cc/paper/1987/file/98f13708210194c475687be6106a3b84-Paper.pdf |archive-url=https://web.archive.org/web/20210414091306/https://papers.nips.cc/paper/1987/file/98f13708210194c475687be6106a3b84-Paper.pdf |archive-date=2021-04-14 |url-status=live|journal=Neural Information Processing Systems (NIPS 1987)|volume=1}}</ref>
* In non-[[artificial neural network|neural-network]]-based [[image processing]]
** In [[digital image processing]] convolutional filtering plays an important role in many important [[algorithm]]s in [[edge detection]] and related processes (see [[Kernel (image processing)]])
** In [[optics]], an out-of-focus photograph is a convolution of the sharp image with a lens function. The photographic term for this is [[bokeh]].
** In image processing applications such as adding blurring.
* In digital data processing
** In [[analytical chemistry]], [[Savitzky–Golay smoothing filter]]s are used for the analysis of spectroscopic data. They can improve [[signal-to-noise ratio]] with minimal distortion of the spectra<!--<ref name=Scafer2011>{{cite journal|last=Schafer|first=Ronald W.|title= What is a Savitzky-Golay Filter? &#91;Lecture Notes&#93;|journal= IEEE Signal Processing Magazine|date=July 2011|volume= 28|issue= 4|pages= 111–117|doi=10.1109/MSP.2011.941097|url=https://www-inst.eecs.berkeley.edu/~ee123/fa12/docs/SGFilter.pdf |archive-url=https://web.archive.org/web/20140124083227/https://www-inst.eecs.berkeley.edu/~ee123/fa12/docs/SGFilter.pdf |archive-date=2014-01-24 |url-status=live|access-date=18 May 2013}}</ref>-->
** In [[statistics]], a weighted [[moving average]] is a convolution.
* In [[acoustics]], [[reverberation]] is the convolution of the original sound with [[echo (phenomenon)|echo]]es from objects surrounding the sound source.
** In digital signal processing, convolution is used to map the [[impulse response]] of a real room on a digital audio signal.
** In [[electronic music]] convolution is the imposition of a [[Spectrum|spectral]] or rhythmic structure on a sound. Often this envelope or structure is taken from another sound. The convolution of two signals is the filtering of one through the other.<ref>Zölzer, Udo, ed. (2002). ''DAFX:Digital Audio Effects'', p.48–49. {{isbn|0471490784}}.</ref>
* In [[electrical engineering]], the convolution of one function (the [[Signal (electrical engineering)|input signal]]) with a second function (the impulse response) gives the output of a [[linear time-invariant system]] (LTI). At any given moment, the output is an accumulated effect of all the prior values of the input function, with the most recent values typically having the most influence (expressed as a multiplicative factor). The impulse response function provides that factor as a function of the elapsed time since each input value occurred.
* In [[physics]], wherever there is a [[linear system]] with a "[[superposition principle]]", a convolution operation makes an appearance. For instance, in [[spectroscopy]] line broadening due to the Doppler effect on its own gives a [[Normal distribution|Gaussian]] [[spectral line shape]] and collision broadening alone gives a [[Cauchy distribution|Lorentzian]] line shape. When both effects are operative, the line shape is a convolution of Gaussian and Lorentzian, a [[Voigt function]].
** In [[Time-resolved spectroscopy#Time-resolved fluorescence spectroscopy|time-resolved fluorescence spectroscopy]], the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse.
** In [[computational fluid dynamics]], the [[large eddy simulation]] (LES) [[turbulence model]] uses the convolution operation to lower the range of length scales necessary in computation thereby reducing computational cost.
* In [[probability theory]], the [[probability distribution]] of the sum of two [[independent (probability)|independent]] [[random variable]]s is the convolution of their individual distributions.
** In [[kernel density estimation]], a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian.{{sfn|Diggle|1985}}
* In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a [[convolution-superposition algorithm]].{{Clarify|date=May 2013}}
* In structural reliability, the reliability index can be defined based on the convolution theorem.
** The definition of reliability index for limit state functions with nonnormal distributions can be established corresponding to the [[joint distribution function]]. In fact, the joint distribution function can be obtained using the convolution theory.{{sfn|Ghasemi|Nowak|2017}}
* In [[Smoothed-particle hydrodynamics]], simulations of fluid dynamics are calculated using particles, each with surrounding kernels. For any given particle <math>i</math>, some physical quantity <math>A_i</math> is calculated as a convolution of <math>A_j</math> with a weighting function, where <math>j</math> denotes the neighbors of particle <math>i</math>: those that are located within its kernel. The convolution is approximated as a summation over each neighbor.<ref name="1992ARA&A..30..543M">{{cite journal |last1=Monaghan |first1=J. J. |title=Smoothed particle hydrodynamics |journal=Annual Review of Astronomy and Astrophysics |date=1992 |volume=30 |pages=543–547 |doi=10.1146/annurev.aa.30.090192.002551 |bibcode=1992ARA&A..30..543M |url=https://ui.adsabs.harvard.edu/abs/1992ARA&A..30..543M |access-date=16 February 2021 |ref=1992ARA&A..30..543M}}</ref>
* In [[Fractional calculus]] convolution is instrumental in various definitions of fractional integral and fractional derivative.