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{{Short description|Technique used in signal processing and data compression}}
A '''discrete cosine transform''' ('''DCT''') expresses a finite sequence of [[data points]] in terms of a sum of [[cosine]] functions oscillating at different [[frequency|frequencies]]. The DCT, first proposed by [[Nasir Ahmed (engineer)|Nasir Ahmed]] in 1972, is a widely used transformation technique in [[signal processing]] and [[data compression]]. It is used in most [[digital media]], including [[digital images]] (such as [[JPEG]] and [[HEIF]]), [[digital video]] (such as [[MPEG]] and {{nowrap|[[H.26x]]}}), [[digital audio]] (such as [[Dolby Digital]], [[MP3]] and [[Advanced Audio Coding|AAC]]), [[digital television]] (such as [[SDTV]], [[HDTV]] and [[Video on demand|VOD]]), [[digital radio]] (such as [[AAC+]] and [[DAB+]]), and [[speech coding]] (such as [[AAC-LD]], [[Siren (codec)|Siren]] and [[Opus (audio format)|Opus]]). DCTs are also important to numerous other applications in [[science and engineering]], such as [[digital signal processing]], [[telecommunication]] devices, reducing [[network bandwidth]] usage, and [[spectral method]]s for the numerical solution of [[partial differential equations]].
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== History ==
The DCT was first conceived by [[Nasir Ahmed (engineer)|Nasir Ahmed
Since its introduction in 1974, there has been significant research on the DCT.<ref name="t81"/> In 1977, Wen-Hsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm.<ref name="A Fast Computational Algorithm for">{{cite journal |last1=Chen |first1=Wen-Hsiung |last2=Smith |first2=C. H. |last3=Fralick |first3=S. C. |title=A Fast Computational Algorithm for the Discrete Cosine Transform |journal=[[IEEE Transactions on Communications]] |date=September 1977 |volume=25 |issue=9 |pages=1004–1009 |doi=10.1109/TCOM.1977.1093941}}</ref><ref name="t81"/> Further developments include a 1978 paper by M. J. Narasimha and A. M. Peterson, and a 1984 paper by B. G. Lee.<ref name="t81"/> These research papers, along with the original 1974 Ahmed paper and the 1977 Chen paper, were cited by the [[Joint Photographic Experts Group]] as the basis for [[JPEG]]'s lossy image compression algorithm in 1992.<ref name="t81"/><ref name="chen">{{cite journal |last1=Smith |first1=C. |last2=Fralick |first2=S. |title=A Fast Computational Algorithm for the Discrete Cosine Transform |journal=IEEE Transactions on Communications |date=1977 |volume=25 |issue=9 |pages=1004–1009 |doi=10.1109/TCOM.1977.1093941 |issn=0090-6778}}</ref>
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**[[Image processing]] — [[digital image processing]],<ref name="Stankovic"/> [[image analysis]], [[content-based image retrieval]], [[corner detection]], directional block-wise [[Sparse approximation|image representation]], [[edge detection]], [[image enhancement]], [[image fusion]], [[image segmentation]], [[interpolation]], [[image noise]] level estimation, mirroring, rotation, [[Just-noticeable difference|just-noticeable distortion]] (JND) profile, [[spatiotemporal]] masking effects, [[foveated imaging]]<ref name="Ochoa"/>
**[[Image quality]] assessment — DCT-based quality degradation metric (DCT QM)<ref name="Ochoa"/>
**[[Image reconstruction]] — directional [[image texture|textures]] auto inspection,
*[[Medical technology]]
**[[Electrocardiography]] (ECG) — [[vectorcardiography]] (VCG)<ref name="Ochoa"/>
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{{See also|ZPEG}}
Multidimensional DCTs (MD DCTs) have several applications, mainly 3-D DCTs such as the 3-D DCT-II, which has several new applications like Hyperspectral Imaging coding systems,<ref name="appDCT">{{Citation |first1=G. P. |last1=Abousleman |first2=M. W. |last2=Marcellin |first3=B. R. |last3=Hunt |title=Compression of hyperspectral imagery using 3-D DCT and hybrid DPCM/DCT |journal=IEEE Trans. Geosci. Remote Sens. |date=January 1995 |volume=33 |issue=1 |pages=26–34 |doi=10.1109/36.368225|bibcode=1995ITGRS..33...26A }}</ref> variable temporal length 3-D DCT coding,<ref name="app2DCT">{{Citation |first1=Y. |last1=Chan |first2=W. |last2=Siu |title= Variable temporal-length 3-D discrete cosine transform coding |journal=IEEE Trans. Image Process
===Digital signal processing===
DCT plays an important role in [[digital signal processing]] specifically [[data compression]]. The DCT is widely implemented in [[digital signal processors]] (DSP), as well as digital signal processing software. Many companies have developed DSPs based on DCT technology. DCTs are widely used for applications such as [[encoding]], decoding, video, audio, [[multiplexing]], control signals, [[signaling]], and [[analog-to-digital conversion]]. DCTs are also commonly used for [[high-definition television]] (HDTV) encoder/decoder [[integrated circuit|chips]].<ref name="Stankovic"/>▼
▲The DCT is widely implemented in [[digital signal processors]] (DSP), as well as digital signal processing software. Many companies have developed DSPs based on DCT technology. DCTs are widely used for applications such as [[encoding]], decoding, video, audio, [[multiplexing]], control signals, [[signaling]], and [[analog-to-digital conversion]]. DCTs are also commonly used for [[high-definition television]] (HDTV) encoder/decoder [[integrated circuit|chips]].<ref name="Stankovic"/>
===Compression artifacts===
A common issue with DCT compression in [[digital media]] are blocky [[compression artifacts]],<ref name="Katsaggelos">{{cite book |last1=Katsaggelos |first1=Aggelos K. |last2=Babacan |first2=S. Derin |last3=Chun-Jen |first3=Tsai |title=The Essential Guide to Image Processing |date=2009 |publisher=[[Academic Press]] |isbn=9780123744579 |pages=349–383|chapter=Chapter 15 - Iterative Image Restoration}}</ref> caused by DCT blocks.<ref name="Alikhani">{{cite web |last1=Alikhani |first1=Darya |title=Beyond resolution: Rosa Menkman's glitch art |url=http://postmatter.merimedia.com/articles/archive-2012-2016/2015/51-rosa-menkman/ |website=POSTmatter |date=April 1, 2015 |access-date=19 October 2019 |archive-date=19 October 2019 |archive-url=https://web.archive.org/web/20191019082218/http://postmatter.merimedia.com/articles/archive-2012-2016/2015/51-rosa-menkman/ |url-status=dead }}</ref>
DCT blocks are often used in [[glitch art]].<ref name="Alikhani"/> The artist [[Rosa Menkman]] makes use of DCT-based compression artifacts in her glitch art,<ref name="Menkman">{{cite book |last1=Menkman |first1=Rosa |title=The Glitch Moment(um) |url=https://networkcultures.org/_uploads/NN%234_RosaMenkman.pdf |publisher=Institute of Network Cultures |isbn=978-90-816021-6-7 |date=October 2011 |access-date=19 October 2019}}</ref> particularly the DCT blocks found in most [[digital media]] formats such as [[JPEG]] digital images and [[MP3]]
▲A common issue with DCT compression in [[digital media]] are blocky [[compression artifacts]],<ref name="Katsaggelos">{{cite book |last1=Katsaggelos |first1=Aggelos K. |last2=Babacan |first2=S. Derin |last3=Chun-Jen |first3=Tsai |title=The Essential Guide to Image Processing |date=2009 |publisher=[[Academic Press]] |isbn=9780123744579 |pages=349–383|chapter=Chapter 15 - Iterative Image Restoration}}</ref> caused by DCT blocks.<ref name="Alikhani">{{cite web |last1=Alikhani |first1=Darya |title=Beyond resolution: Rosa Menkman's glitch art |url=http://postmatter.merimedia.com/articles/archive-2012-2016/2015/51-rosa-menkman/ |website=POSTmatter |date=April 1, 2015 |access-date=19 October 2019 |archive-date=19 October 2019 |archive-url=https://web.archive.org/web/20191019082218/http://postmatter.merimedia.com/articles/archive-2012-2016/2015/51-rosa-menkman/ |url-status=dead }}</ref> The DCT algorithm can cause block-based artifacts when heavy compression is applied. Due to the DCT being used in the majority of digital image and [[video coding standards]] (such as the [[JPEG]], {{nowrap|[[H.26x]]}} and [[MPEG]] formats), DCT-based blocky compression artifacts are widespread in [[digital media]]. In a DCT algorithm, an image (or frame in an image sequence) is divided into square blocks which are processed independently from each other, then the DCT of these blocks is taken, and the resulting DCT coefficients are [[Quantization (signal processing)|quantized]]. This process can cause blocking artifacts, primarily at high [[data compression ratio]]s.<ref name="Katsaggelos"/> This can also cause the "[[mosquito noise]]" effect, commonly found in [[digital video]] (such as the MPEG formats).<ref>{{cite web |title=Mosquito noise |url=https://www.pcmag.com/encyclopedia/term/55914/mosquito-noise |website=[[PC Magazine]] |access-date=19 October 2019}}</ref>
▲DCT blocks are often used in [[glitch art]].<ref name="Alikhani"/> The artist [[Rosa Menkman]] makes use of DCT-based compression artifacts in her glitch art,<ref name="Menkman">{{cite book |last1=Menkman |first1=Rosa |title=The Glitch Moment(um) |url=https://networkcultures.org/_uploads/NN%234_RosaMenkman.pdf |publisher=Institute of Network Cultures |isbn=978-90-816021-6-7 |date=October 2011 |access-date=19 October 2019}}</ref> particularly the DCT blocks found in most [[digital media]] formats such as [[JPEG]] digital images and [[MP3]] [[digital audio]].<ref name="Alikhani"/> Another example is ''Jpegs'' by German photographer [[Thomas Ruff]], which uses intentional [[JPEG]] artifacts as the basis of the picture's style.<ref>{{cite book|chapter=jpegs|first=Thomas|last=Ruff|title=Aperture|date=May 31, 2009|page=132|publisher=Aperture |isbn=9781597110938}}</ref><ref>{{cite web|url=http://jmcolberg.com/weblog/2009/04/review_jpegs_by_thomas_ruff/|title=Review: jpegs by Thomas Ruff|first=Jörg|last=Colberg|date=April 17, 2009}}</ref>
==Informal overview==
Like any Fourier-related transform,
The Fourier-related transforms that operate on a function over a finite [[___domain of a function|___domain]], such as the DFT or DCT or a [[Fourier series]], can be thought of as implicitly defining an ''extension'' of that function outside the ___domain. That is, once you write a function <math>f(x)</math> as a sum of sinusoids, you can evaluate that sum at any <math>x</math>, even for <math>x</math> where the original <math>f(x)</math> was not specified. The DFT, like the Fourier series, implies a [[periodic function|periodic]] extension of the original function. A DCT, like a [[
[[Image:DCT-symmetries.svg|thumb|right|350px|Illustration of the implicit even/odd extensions of DCT input data, for ''N''=11 data points (red dots), for the four most common types of DCT (types I-IV). Note the subtle differences at the interfaces between the data and the extensions: in DCT-II and DCT-IV both the end points are replicated in the extensions but not in DCT-I or DCT-III (and a zero point is inserted at the sign reversal extension in DCT-III).]]
However, because DCTs operate on ''finite'', ''discrete'' sequences, two issues arise that do not apply for the continuous cosine transform. First, one has to specify whether the function is even or odd at ''both'' the left and right boundaries of the ___domain (i.e. the min-''n'' and max-''n'' boundaries in the definitions below, respectively). Second, one has to specify around ''what point'' the function is even or odd. In particular, consider a sequence ''abcd'' of four equally spaced data points, and say that we specify an even ''left'' boundary. There are two sensible possibilities: either the data are even about the sample ''a'', in which case the even extension is ''dcbabcd'', or the data are even about the point ''halfway'' between ''a'' and the previous point, in which case the even extension is ''dcbaabcd'' (''a'' is repeated).
Each boundary can be either even or odd (2 choices per boundary) and can be symmetric about a data point or the point halfway between two data points (2 choices per boundary), for a total of 2 × 2 × 2 × 2 = 16 possibilities. These choices lead to all the standard variations of DCTs and also [[discrete sine transform]]s (DSTs). Half of these possibilities, those where the ''left'' boundary is even, correspond to the 8 types of DCT; the other half are the 8 types of DST.
These different boundary conditions strongly affect the applications of the transform and lead to uniquely useful properties for the various DCT types. Most directly, when using Fourier-related transforms to solve [[partial differential equation]]s by [[spectral method]]s, the boundary conditions are directly specified as a part of the problem being solved. Or, for the
In particular, it is well known that any [[Classification of discontinuities|discontinuities]] in a function reduce the [[rate of convergence]] of the Fourier series
== Formal definition ==
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:<math>X_k
= \frac{1}{2} (x_0 + (-1)^k x_{N-1})
+ \sum_{n=1}^{N-2} x_n \cos \left[\, \
\qquad \text{ for } ~ k = 0,\ \ldots\ N-1 ~.</math>
Some authors further multiply the <math>x_0 </math> and <math> x_{N-1} </math> terms by <math> \sqrt{2\,}\
The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a
Note, however, that the DCT-I is not defined for <math>
Thus, the DCT-I corresponds to the boundary conditions: <math>
=== DCT-II ===
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\qquad \text{ for } ~ k = 0,\ \dots\ N-1 ~.</math>
The DCT-II is probably the most commonly used form, and is often simply referred to as
This transform is exactly equivalent (up to an overall scale factor of 2) to a
Some authors further multiply the <math>
The DCT-II implies the boundary conditions: <math>
=== DCT-III ===
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\qquad \text{ for } ~ k = 0,\ \ldots\ N-1 ~.</math>
Because it is the inverse of DCT-II up to a scale factor (see below), this form is sometimes simply referred to as
Some authors divide the <math>x_0</math> term by <math>\sqrt{2}</math> instead of by 2 (resulting in an overall <math>x_0/\sqrt{2}</math> term) and multiply the resulting matrix by an overall scale factor of <math display="inline"> \sqrt{2/N}</math> (see above for the corresponding change in DCT-II), so that the DCT-II and DCT-III are transposes of one another. This makes the DCT-III matrix [[orthogonal matrix|orthogonal]], but breaks the direct correspondence with a real-even
The DCT-III implies the boundary conditions: <math>x_n</math> is even around <math>n = 0</math> and odd around <math>n = N ;</math> <math>X_k</math> is even around <math>k = -1/2</math> and even around <math>k = N - 1/2.</math>
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A variant of the DCT-IV, where data from different transforms are ''overlapped'', is called the [[modified discrete cosine transform]] (MDCT).<ref>{{harvnb|Malvar|1992}}</ref>
The DCT-IV implies the boundary conditions: <math>
=== DCT V-VIII ===
DCTs of types I–IV treat both boundaries consistently regarding the point of symmetry: they are even/odd around either a data point for both boundaries or halfway between two data points for both boundaries. By contrast, DCTs of types V-VIII imply boundaries that are even/odd around a data point for one boundary and halfway between two data points for the other boundary.
In other words, DCT types I–IV are equivalent to real-even
However, these variants seem to be rarely used in practice. One reason, perhaps, is that [[Fast Fourier transform|FFT]] algorithms for odd-length DFTs are generally more complicated than [[Fast Fourier transform|FFT]] algorithms for even-length DFTs (e.g. the simplest radix-2 algorithms are only for even lengths), and this increased intricacy carries over to the DCTs as described below.
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Using the normalization conventions above, the inverse of DCT-I is DCT-I multiplied by 2/(''N'' − 1). The inverse of DCT-IV is DCT-IV multiplied by 2/''N''. The inverse of DCT-II is DCT-III multiplied by 2/''N'' and vice versa.<ref name="pubRaoYip"/>
Like for the
== Multidimensional DCTs ==
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Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the {{nobr| 8 × 8 }} DCT-II used in [[JPEG]] compression, or the small DCTs (or MDCTs) typically used in audio compression. (Reduced code size may also be a reason to use a specialized DCT for embedded-device applications.)
In fact, even the DCT algorithms using an ordinary FFT are sometimes equivalent to pruning the redundant operations from a larger FFT of real-symmetric data, and they can even be optimal from the perspective of arithmetic counts. For example, a type-II DCT is equivalent to a DFT of size <math>~ 4N ~</math> with real-even symmetry whose even-indexed elements are zero. One of the most common methods for computing this via an FFT (e.g. the method used in [[FFTPACK]] and [[Fastest Fourier Transform in the West|FFTW]]) was described by {{harvtxt|Narasimha|Peterson|1978}} and {{harvtxt|Makhoul|1980}}, and this method in hindsight can be seen as one step of a radix-4 decimation-in-time Cooley–Tukey algorithm applied to the "logical" real-even DFT corresponding to the DCT-II.{{efn|
The radix-4 step reduces the size <math>~ 4N ~</math> DFT to four size <math>~ N ~</math> DFTs of real data, two of which are zero, and two of which are equal to one another by the even symmetry. Hence giving a single size <math>~ N ~</math> FFT of real data plus <math>~ \mathcal{O}(N) ~</math> [[butterfly (FFT algorithm)|butterflies]], once the trivial and / or duplicate parts are eliminated and / or merged.
}}
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==Example of IDCT==
[[File:DCT filter comparison.png|thumb|right|An example showing eight different filters applied to a test image (top left) by multiplying its DCT spectrum (top right) with each filter.]]
Consider this {{resx|8x8}} [[grayscale image]] of capital letter A.
[[File:letter-a-8x8.png|frame|center|Original size, scaled 10x (nearest neighbor), scaled 10x (bilinear).]]
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* Takuya Ooura: General Purpose FFT Package, [http://www.kurims.kyoto-u.ac.jp/~ooura/fft.html FFT Package 1-dim / 2-dim]. Free C & FORTRAN libraries for computing fast DCTs (types II–III) in one, two or three dimensions, power of 2 sizes.
* Tim Kientzle: Fast algorithms for computing the 8-point DCT and IDCT, [http://drdobbs.com/parallel/184410889 Algorithm Alley].
* [
{{Compression Methods|state=expanded}}
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[[Category:Data compression]]
[[Category:Image compression]]
[[Category:H.26x]]
[[Category:JPEG]]
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