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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|cosine]] functions oscillating at different [[Physics:Frequency|frequencies]]. The DCT, first proposed by [[Biography:Nasir Ahmed (engineer)|Nasir Ahmed]] in 1972, is a widely used transformation technique in [[Signal processing|signal processing]] and [[Data compression|data compression]]. It is used in most [[Digital media|digital media]], including digital images (such as [[JPEG]] and HEIF), [[Engineering:Digital video|digital video]] (such as MPEG and H.26x), [[Digital audio|digital audio]] (such as [[Organization:Dolby Digital|Dolby Digital]], [[MP3]] and [[Advanced Audio Coding|AAC]]), [[Engineering:Digital television|digital television]] (such as SDTV, [[Engineering:HDTV|HDTV]] and [[Engineering:Video on demand|VOD]]), digital radio (such as AAC+ and DAB+), and [[Speech coding|speech coding]] (such as [[AAC-LD]], [[Siren (codec)|Siren]] and [[Software:Opus (audio format)|Opus]]). DCTs are also important to numerous other applications in science and engineering, such as [[Digital signal processing|digital signal processing]], [[Engineering:Telecommunication|telecommunication]] devices, reducing network bandwidth usage, and [[Spectral method|spectral method]]s for the numerical solution of partial differential equations.

A DCT is a [[List of Fourier-related transforms|Fourier-related transform]] similar to the [[Discrete Fourier transform|discrete Fourier transform]] (DFT), but using only [[Real number|real number]]s. The DCTs are generally related to [[Fourier series]] coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with [[Even and odd functions|even]] symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input or output data are shifted by half a sample. 

There are eight standard DCT variants, of which four are common.
The most common variant of discrete cosine transform is the type-II DCT, which is often called simply ''the DCT''. This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT, is correspondingly often called simply ''the inverse DCT'' or ''the IDCT''. Two related transforms are the [[Discrete sine transform|discrete sine transform]] (DST), which is equivalent to a DFT of real and odd functions, and the [[Modified discrete cosine transform|modified discrete cosine transform]] (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT to multidimensional signals. A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT. One of these is the integer DCT (IntDCT),<ref name="Stankovic"/> an [[Integer|integer]] approximation of the standard DCT,<ref name="Britanak2010" />{{rp|pages= [https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA141 ix, xiii, 1, 141–304]}} used in several ISO/IEC and ITU-T international standards.<ref name="Stankovic"/><ref name="Britanak2010"/>

DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks.<ref name="Alikhani"/> DCT blocks sizes including 8x8 pixels for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels.<ref name="Stankovic"/><ref name="apple"/> The DCT has a strong ''energy compaction'' property,<ref name="pubDCT"/><ref name="pubRaoYip"/> capable of achieving high quality at high [[Data compression ratio|data compression ratio]]s.<ref name="Barbero"/><ref name="Lea">{{cite journal|last1=Lea|first1=William|date=1994|title=Video on demand: Research Paper 94/68|url=https://researchbriefings.parliament.uk/ResearchBriefing/Summary/RP94-68|journal=House of Commons Library|access-date=20 September 2019}}</ref> However, blocky compression artifacts can appear when heavy DCT compression is applied.

== History ==
The DCT was first conceived by [[Biography:Nasir Ahmed (engineer)|Nasir Ahmed]], T. Natarajan and K. R. Rao while working at [[Organization:Kansas State University|Kansas State University]]. The concept was proposed to the [[Organization:National Science Foundation|National Science Foundation]] in 1972. The DCT was originally intended for [[Image compression|image compression]].<ref name="Ahmed" /><ref name="Stankovic"/> Ahmed developed a practical DCT algorithm with his PhD students T. Raj Natarajan, Wills Dietrich, and Jeremy Fries, and his friend Dr. K. R. Rao at the [[Organization:University of Texas at Arlington|University of Texas at Arlington]] in 1973.<ref name="Ahmed"/> They presented their results in a January 1974 paper, titled ''Discrete Cosine Transform''.<ref name="pubDCT" /><ref name="pubRaoYip" /><ref name="t81">{{cite web|date=September 1992|title=T.81 – Digital compression and coding of continuous-tone still images – Requirements and guidelines|url=https://www.w3.org/Graphics/JPEG/itu-t81.pdf|access-date=12 July 2019|publisher=CCITT}}</ref> It described what is now called the type-II DCT (DCT-II),<ref name="Britanak2010" />{{rp|page = [https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA51 51]}} as well as the type-III inverse DCT (IDCT).<ref name="pubDCT"/>

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=[[Engineering:IEEE Transactions on Communications|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>

The [[Discrete sine transform|discrete sine transform]] (DST) was derived from the DCT, by replacing the [[Neumann boundary condition|Neumann condition]] at ''x=0'' with a Dirichlet condition.<ref name="Britanak2010" />{{rp|pages=[https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA35 35{{hyphen}}36]}} The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao.<ref name="pubDCT"/> A type-I DST (DST-I) was later described by Anil K. Jain in 1976, and a type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.<ref>{{cite journal |last1=Dhamija |first1=Swati |last2=Jain |first2=Priyanka |title=Comparative Analysis for Discrete Sine Transform as a suitable method for noise estimation |journal=IJCSI International Journal of Computer Science |date=September 2011 |volume=8 |issue= 5, No. 3 |pages=162–164 (162) |url=https://www.researchgate.net/publication/267228857 |access-date=4 November 2019}}</ref>

In 1975, John A. Roese and Guner S. Robinson adapted the DCT for inter-frame [[Motion compensation|motion-compensated]] video coding. They experimented with the DCT and the [[Fast Fourier transform|fast Fourier transform]] (FFT), developing inter-frame hybrid coders for both, and found that the DCT is the most efficient due to its reduced complexity, capable of compressing image data down to 0.25-[[Bit|bit]] per [[Pixel|pixel]] for a videotelephone scene with image quality comparable to an [[Intra-frame coding|intra-frame coder]] requiring 2-bit per pixel.<ref>{{cite book |last1=Huang |first1=T. S. |title=Image Sequence Analysis |date=1981 |publisher=Springer Science & Business Media |isbn=9783642870378 |page=29 |url=https://books.google.com/books?id=bAirCAAAQBAJ&pg=PA29}}</ref><ref name="Roese">{{cite journal |last1=Roese |first1=John A. |last2=Robinson |first2=Guner S. |editor-first1=Andrew G. |editor-last1=Tescher |title=Combined Spatial And Temporal Coding Of Digital Image Sequences |journal=Efficient Transmission of Pictorial Information |date=30 October 1975 |volume=0066 |pages=172–181 |doi=10.1117/12.965361 |publisher=International Society for Optics and Photonics|bibcode=1975SPIE...66..172R |s2cid=62725808 }}</ref> In 1979, Anil K. Jain and Jaswant R. Jain further developed motion-compensated DCT video compression,<ref>{{cite book |last1=Cianci |first1=Philip J. |title=High Definition Television: The Creation, Development and Implementation of HDTV Technology |date=2014 |publisher=McFarland |isbn=9780786487974 |page=63 |url=https://books.google.com/books?id=0mbsfr38GTgC&pg=PA63}}</ref><ref name="ITU">{{cite web |title=History of Video Compression |url=https://www.itu.int/wftp3/av-arch/jvt-site/2002_07_Klagenfurt/JVT-D068.doc |website=ITU-T |publisher=Joint Video Team (JVT) of ISO/IEC MPEG & ITU-T VCEG (ISO/IEC JTC1/SC29/WG11 and ITU-T SG16 Q.6) |date=July 2002 |pages=11, 24–9, 33, 40–1, 53–6 |access-date=3 November 2019}}</ref> also called block motion compensation.<ref name="ITU"/> This led to Chen developing a practical video compression algorithm, called motion-compensated DCT or adaptive scene coding, in 1981.<ref name="ITU"/> Motion-compensated DCT later became the standard coding technique for video compression from the late 1980s onwards.<ref name="Ghanbari"/><ref name="Li">{{cite book |last1=Li |first1=Jian Ping |title=Proceedings of the International Computer Conference 2006 on Wavelet Active Media Technology and Information Processing: Chongqing, China, 29-31 August 2006 |date=2006 |publisher=[[Company:World Scientific|World Scientific]] |isbn=9789812709998 |page=847 |url=https://books.google.com/books?id=FZiK3zXdK7sC&pg=PA847}}</ref>

A DCT variant, the [[Modified discrete cosine transform|modified discrete cosine transform]] (MDCT), was developed by John P. Princen, A.W. Johnson and Alan B. Bradley at the [[Organization:University of Surrey|University of Surrey]] in 1987,<ref>{{cite book |last1=Princen |first1=John P. |last2=Johnson |first2=A.W. |last3=Bradley |first3=Alan B. |title=ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing |chapter=Subband/Transform coding using filter bank designs based on time domain aliasing cancellation |date=1987 |volume=12 |pages=2161–2164 |doi=10.1109/ICASSP.1987.1169405|s2cid=58446992 }}</ref> following earlier work by Princen and Bradley in 1986.<ref>{{cite journal|doi=10.1109/TASSP.1986.1164954|title=Analysis/Synthesis filter bank design based on time domain aliasing cancellation|year=1986|last1=Princen|first1=J.|last2=Bradley|first2=A.|journal=IEEE Transactions on Acoustics, Speech, and Signal Processing|volume=34|issue=5|pages=1153–1161}}</ref> The MDCT is used in most modern audio compression formats, such as [[Organization:Dolby Digital|Dolby Digital]] (AC-3),<ref name="Luo"/><ref name="Britanak2011"/> [[MP3]] (which uses a hybrid DCT-[[Fast Fourier transform|FFT]] algorithm),<ref name="Guckert">{{cite web |last1=Guckert |first1=John |title=The Use of FFT and MDCT in MP3 Audio Compression |url=http://www.math.utah.edu/~gustafso/s2012/2270/web-projects/Guckert-audio-compression-svd-mdct-MP3.pdf |website=[[Organization:University of Utah|University of Utah]] |date=Spring 2012 |access-date=14 July 2019}}</ref> [[Advanced Audio Coding]] (AAC),<ref name=brandenburg>{{cite web|url=http://graphics.ethz.ch/teaching/mmcom12/slides/mp3_and_aac_brandenburg.pdf|title=MP3 and AAC Explained|last=Brandenburg|first=Karlheinz|year=1999|url-status=live|archive-url=https://web.archive.org/web/20170213191747/https://graphics.ethz.ch/teaching/mmcom12/slides/mp3_and_aac_brandenburg.pdf|archive-date=2017-02-13}}</ref> and [[Software:Vorbis|Vorbis]] ([[Software:Ogg|Ogg]]).<ref name="vorbis-mdct"/>

Nasir Ahmed also developed a lossless DCT algorithm with Giridhar Mandyam and Neeraj Magotra at the [[Organization:University of New Mexico|University of New Mexico]] in 1995. This allows the DCT technique to be used for [[Lossless compression|lossless compression]] of images. It is a modification of the original DCT algorithm, and incorporates elements of inverse DCT and [[Delta modulation|delta modulation]]. It is a more effective lossless compression algorithm than [[Entropy coding|entropy coding]].<ref>{{cite journal |last1=Mandyam |first1=Giridhar D. |last2=Ahmed |first2=Nasir |last3=Magotra |first3=Neeraj |editor-first1=Arturo A. |editor-first2=Robert J. |editor-first3=Edward J. |editor-last1=Rodriguez |editor-last2=Safranek |editor-last3=Delp |s2cid=13894279 |title=DCT-based scheme for lossless image compression |journal=Digital Video Compression: Algorithms and Technologies 1995 |date=17 April 1995 |volume=2419 |pages=474–478 |doi=10.1117/12.206386 |publisher=International Society for Optics and Photonics|bibcode=1995SPIE.2419..474M }}</ref> Lossless DCT is also known as LDCT.<ref>{{cite book |last1=Komatsu |first1=K. |last2=Sezaki |first2=Kaoru |title=Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181) |chapter=Reversible discrete cosine transform |date=1998 |volume=3 |pages=1769–1772 vol.3 |doi=10.1109/ICASSP.1998.681802 |isbn=0-7803-4428-6 |s2cid=17045923 |chapter-url=https://www.researchgate.net/publication/3747502}}</ref>

==Applications==
The DCT is the most widely used transformation technique in [[Signal processing|signal processing]],<ref name="Muchahary">{{cite book |last1=Muchahary |first1=D. |last2=Mondal |first2=A. J. |last3=Parmar |first3=R. S. |last4=Borah |first4=A. D. |last5=Majumder |first5=A. |title=2015 Fifth International Conference on Communication Systems and Network Technologies |chapter=A Simplified Design Approach for Efficient Computation of DCT |date=2015 |pages=483–487 |doi=10.1109/CSNT.2015.134|isbn=978-1-4799-1797-6 |s2cid=16411333 }}</ref> and by far the most widely used linear transform in [[Data compression|data compression]].<ref>{{cite book |last1=Chen |first1=Wai Kai |title=The Electrical Engineering Handbook |date=2004 |publisher=[[Company:Elsevier|Elsevier]] |isbn=9780080477480 |page=906 |url=https://books.google.com/books?id=qhHsSlazGrQC&pg=PA906}}</ref> Uncompressed [[Digital media|digital media]] as well as [[Lossless compression|lossless compression]] have high [[Computer memory|memory]] and [[Bandwidth (computing)|bandwidth]] requirements, which is significantly reduced by the DCT [[Lossy compression|lossy compression]] technique,<ref name="Barbero">{{cite journal |last1=Barbero |first1=M. |last2=Hofmann |first2=H. |last3=Wells |first3=N. D. |title=DCT source coding and current implementations for HDTV |journal=EBU Technical Review |date=14 November 1991 |issue=251 |pages=22–33 |publisher=European Broadcasting Union |url=https://tech.ebu.ch/publications/trev_251-barbero |access-date=4 November 2019}}</ref><ref name="Lea"/> capable of achieving [[Data compression ratio|data compression ratio]]s from 8:1 to 14:1 for near-studio-quality,<ref name="Barbero"/> up to 100:1 for acceptable-quality content.<ref name="Lea"/> DCT compression standards are used in digital media technologies, such as digital images, digital photos,<ref name="Atlantic">{{cite web |title=What Is a JPEG? The Invisible Object You See Every Day |url=https://www.theatlantic.com/technology/archive/2013/09/what-is-a-jpeg-the-invisible-object-you-see-every-day/279954/ |access-date=13 September 2019 |website=The Atlantic |date=24 September 2013}}</ref><ref name="epfl">{{cite news |last1=Pessina |first1=Laure-Anne |title=JPEG changed our world |url=https://actu.epfl.ch/news/jpeg-changed-our-world/ |access-date=13 September 2019 |work=EPFL News |publisher=[[Organization:École Polytechnique Fédérale de Lausanne|École Polytechnique Fédérale de Lausanne]] |date=12 December 2014}}</ref> [[Engineering:Digital video|digital video]],<ref name="Ghanbari">{{cite book |last1=Ghanbari |first1=Mohammed |title=Standard Codecs: Image Compression to Advanced Video Coding |date=2003 |publisher=[[Organization:Institution of Engineering and Technology|Institution of Engineering and Technology]] |isbn=9780852967102 |pages=1–2 |url=https://books.google.com/books?id=7XuU8T3ooOAC&pg=PA1}}</ref><ref name="Lee1995">{{cite journal|last1=Lee|first1=Ruby Bei-Loh|last2=Beck|first2=John P.|last3=Lamb|first3=Joel|last4=Severson|first4=Kenneth E.|date=April 1995|title=Real-time software MPEG video decoder on multimedia-enhanced PA 7100LC processors|url=https://www.hpl.hp.com/hpjournal/95apr/apr95a7.pdf|journal=Hewlett-Packard Journal|volume=46|issue=2|issn=0018-1153}}</ref> [[Engineering:Streaming media|streaming media]],<ref name="Lee">{{cite book |last1=Lee |first1=Jack |title=Scalable Continuous Media Streaming Systems: Architecture, Design, Analysis and Implementation |date=2005 |publisher={{wipe|John Wiley & Sons}} |isbn=9780470857649 |page=25 |url=https://books.google.com/books?id=7fuvu52cyNEC&pg=PA25}}</ref> [[Engineering:Digital television|digital television]], [[Streaming television|streaming television]], [[Engineering:Video on demand|video on demand]] (VOD),<ref name="Lea"/> [[Engineering:Digital cinema|digital cinema]],<ref name="Luo"/> [[Engineering:High-definition video|high-definition video]] (HD video), and [[Engineering:High-definition television|high-definition television]] (HDTV).<ref name="Barbero"/><ref name="Shishikui">{{cite book |last1=Shishikui |first1=Yoshiaki |last2=Nakanishi |first2=Hiroshi |last3=Imaizumi |first3=Hiroyuki |title=Signal Processing of HDTV |chapter=An HDTV Coding Scheme using Adaptive-Dimension DCT |date=October 26–28, 1993 |pages=611–618 |doi=10.1016/B978-0-444-81844-7.50072-3 |chapter-url=https://books.google.com/books?id=j9XSBQAAQBAJ&pg=PA611 |publisher=[[Company:Elsevier|Elsevier]] |isbn=9781483298511}}</ref>

The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy compression, because it has a strong ''energy compaction'' property.<ref name="pubDCT"/><ref name="pubRaoYip"/> In typical applications, most of the signal information tends to be concentrated in a few low-frequency components of the DCT. For strongly correlated Markov processes, the DCT can approach the compaction efficiency of the Karhunen-Loève transform (which is optimal in the decorrelation sense). As explained below, this stems from the boundary conditions implicit in the cosine functions.

DCTs are widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even and odd boundary conditions at the two ends of the array.

DCTs are closely related to [[Chebyshev polynomials]], and fast DCT algorithms (below) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in [[Clenshaw–Curtis quadrature]].

===General applications===
The DCT is widely used in many applications, which include the following.

{{columns-list|colwidth=50em|
*[[Audio signal processing]] — audio coding, audio data compression (lossy and lossless),<ref name="Ochoa129">{{cite book |last1=Ochoa-Dominguez |first1=Humberto |last2=Rao |first2=K. R. |title=Discrete Cosine Transform, Second Edition |date=2019 |publisher=CRC Press |isbn=9781351396486 |pages=1–3, 129 |url=https://books.google.com/books?id=dVOWDwAAQBAJ}}</ref> [[Surround sound|surround sound]],<ref name="Luo"/> acoustic echo and [[Physics:Adaptive feedback cancellation|feedback cancellation]], [[Social:Phoneme|phoneme]] recognition, time-domain aliasing cancellation (TDAC)<ref name="Ochoa"/>
**[[Digital audio]]<ref name="Stankovic"/>
**Digital radio — Digital Audio Broadcasting (DAB+),<ref name="Britanak"/> [[Engineering:HD Radio|HD Radio]]<ref name="Jones">{{cite book |last1=Jones |first1=Graham A. |last2=Layer |first2=David H. |last3=Osenkowsky |first3=Thomas G. |title=National Association of Broadcasters Engineering Handbook: NAB Engineering Handbook |date=2013 |publisher=Taylor & Francis |isbn=978-1-136-03410-7 |pages=558–9 |url=https://books.google.com/books?id=K9N1TVhf82YC&pg=PA558}}</ref>
**[[Speech processing]] — [[Speech coding|speech coding]]<ref name="Hersent"/><ref name="AppleInsider standards 1"/> [[Speech recognition|speech recognition]], [[Voice activity detection|voice activity detection]] (VAD)<ref name="Ochoa"/>
**[[Engineering:Digital telephony|Digital telephony]] — [[Engineering:Voice over IP|voice over IP]] (VoIP),<ref name="Hersent"/> [[Engineering:Mobile telephony|mobile telephony]], video telephony,<ref name="AppleInsider standards 1"/> teleconferencing, videoconferencing<ref name="Stankovic"/>
*[[Biometrics]] — [[Biology:Fingerprint|fingerprint]] orientation, facial recognition systems, biometric [[Digital watermarking|watermarking]], fingerprint-based biometric watermarking, [[Biology:Palm print|palm print]] identification/recognition<ref name="Ochoa"/>
**[[Face detection]] — [[Facial recognition system|facial recognition]]<ref name="Ochoa"/>
*Computers and the [[Internet]] — the [[World Wide Web]], [[Social:Social media|social media]],<ref name="Atlantic"/><ref name="epfl"/> [[Engineering:Internet video|Internet video]]<ref name="Encodes"/>
**Network bandwidth usage reducation<ref name="Stankovic"/>
*[[Engineering:Consumer electronics|Consumer electronics]]<ref name="Ochoa"/> — [[Engineering:Multimedia|multimedia]] systems,<ref name="Stankovic"/> multimedia [[Engineering:Telecommunication|telecommunication]] devices,<ref name="Stankovic"/> consumer devices<ref name="Encodes"/>
*[[Cryptography]] — [[Encryption|encryption]], steganography, [[Copyright|copyright]] protection<ref name="Ochoa"/>
*[[Data compression]] — [[Transform coding|transform coding]], [[Lossy compression|lossy compression]], [[Lossless compression|lossless compression]]<ref name="Ochoa129"/>
**Encoding operations — [[Quantization (signal processing)|quantization]], perceptual weighting, [[Entropy encoding|entropy encoding]], variable bitrate encoding<ref name="Stankovic"/>
*[[Digital media]]<ref name="Lee"/> — [[Digital distribution|digital distribution]]<ref name="Bitmovin"/>
**[[Engineering:Streaming media|Streaming media]]<ref name="Lee"/> — streaming audio, streaming video, [[Streaming television|streaming television]], video-on-demand (VOD)<ref name="Lea"/>
*Forgery detection<ref name="Ochoa"/>
*Geophysical [[Earth:Transient electromagnetics|transient electromagnetics]] (transient EM)<ref name="Ochoa"/>
*[[Image]]s — artist identification,<ref name="Ochoa">{{cite book |last1=Ochoa-Dominguez |first1=Humberto |last2=Rao |first2=K. R. |title=Discrete Cosine Transform, Second Edition |date=2019 |publisher=CRC Press |isbn=9781351396486 |pages=1–3 |url=https://books.google.com/books?id=dVOWDwAAQBAJ&pg=PA1}}</ref> [[Physics:Focus (optics)|focus]] and [[Physics:Bokeh|blurriness]] measure,<ref name="Ochoa"/> [[Feature extraction|feature extraction]]<ref name="Ochoa"/>
**[[Color]] formatting — formatting [[Physics:Luminance|luminance]] and color differences, color formats (such as YUV444 and YUV411), decoding operations such as the inverse operation between display color formats ([[YIQ]], [[YUV]], RGB)<ref name="Stankovic"/>
**[[Digital imaging]] — [[Digital image|digital image]]s, [[Engineering:Digital camera|digital camera]]s, digital photography,<ref name="Atlantic"/><ref name="epfl"/> [[High-dynamic-range imaging|high-dynamic-range imaging]] (HDR imaging)<ref>{{cite book |last1=Ochoa-Dominguez |first1=Humberto |last2=Rao |first2=K. R. |title=Discrete Cosine Transform, Second Edition |date=2019 |publisher=CRC Press |isbn=9781351396486 |page=186 |url=https://books.google.com/books?id=dVOWDwAAQBAJ&pg=PA186}}</ref>
**[[Image compression]]<ref name="Ochoa"/><ref name="McKernan58"/> — [[Image file format|image file format]]s,<ref name="Baraniuk"/> [[2D-plus-depth|multiview image]] compression, progressive image transmission<ref name="Ochoa"/>
**[[Image processing]] — [[Digital image processing|digital image processing]],<ref name="Stankovic"/> [[Image analysis|image analysis]], [[Content-based image retrieval|content-based image retrieval]], [[Corner detection|corner detection]], directional block-wise [[Sparse approximation|image representation]], [[Edge detection|edge detection]], image enhancement, [[Image fusion|image fusion]], [[Image segmentation|image segmentation]], [[Interpolation|interpolation]], [[Image noise|image noise]] level estimation, mirroring, rotation, [[Philosophy:Just-noticeable difference|just-noticeable distortion]] (JND) profile, spatiotemporal masking effects, [[Foveated imaging|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, [[Image restoration|image restoration]], [[Inpainting|inpainting]], [[Photo recovery|visual recovery]]<ref name="Ochoa"/>
*Medical technology
**[[Biology:Electrocardiography|Electrocardiography]] (ECG) — [[Medicine:Vectorcardiography|vectorcardiography]] (VCG)<ref name="Ochoa"/>
**[[Medicine:Medical imaging|Medical imaging]] — medical image compression, image fusion, watermarking, [[Medicine:Brain tumor|brain tumor]] compression classification<ref name="Ochoa"/>
*[[Pattern recognition]]<ref name="Ochoa"/>
*[[Region of interest]] (ROI) extraction<ref name="Ochoa"/>
*[[Signal processing]] — [[Digital signal processing|digital signal processing]], [[Engineering:Digital signal processor|digital signal processor]]s (DSP), DSP [[Software|software]], [[Engineering:Multiplexing|multiplexing]], signaling, control signals, analog-to-digital conversion (ADC),<ref name="Stankovic"/> compressive sampling, DCT pyramid [[Error concealment|error concealment]], downsampling, [[Upsampling|upsampling]], [[Physics:Signal-to-noise ratio|signal-to-noise ratio]] (SNR) estimation, [[Transmux|transmux]], [[Wiener filter]]<ref name="Ochoa"/>
**Complex cepstrum feature analysis<ref name="Ochoa"/>
**DCT [[Filter (signal processing)|filtering]]<ref name="Ochoa"/>
*Surveillance<ref name="Ochoa"/>
*Vehicular [[Engineering:Event data recorder|event data recorder]] camera<ref name="Ochoa"/>
*[[Engineering:Video|Video]]
**[[Engineering:Digital cinema|Digital cinema]]<ref name="McKernan58">{{cite book |last1=McKernan |first1=Brian |title=Digital cinema: the revolution in cinematography, postproduction, distribution |date=2005 |publisher=McGraw-Hill |isbn=978-0-07-142963-4 |page=58 |url=https://books.google.com/books?id=5vBTAAAAMAAJ |quote=DCT is used in most of the compression systems standardized by the Moving Picture Experts Group (MPEG), is the dominant technology for image compression. In particular, it is the core technology of MPEG-2, the system used for DVDs, digital television broadcasting, that has been used for many of the trials of digital cinema.}}</ref> — [[Engineering:Digital cinematography|digital cinematography]], [[Engineering:Digital movie camera|digital movie camera]]s, [[Engineering:Video editing|video editing]], [[Engineering:Film editing|film editing]],<ref>{{cite book |last1=Ascher |first1=Steven |last2=Pincus |first2=Edward |title=The Filmmaker's Handbook: A Comprehensive Guide for the Digital Age: Fifth Edition |date=2012 |publisher=Penguin |isbn=978-1-101-61380-1 |pages=246–7 |url=https://books.google.com/books?id=zp4KMKwnYVoC&pg=PA246}}</ref><ref>{{cite book |last1=Bertalmio |first1=Marcelo |title=Image Processing for Cinema |date=2014 |publisher=CRC Press |isbn=978-1-4398-9928-1 |page=95 |url=https://books.google.com/books?id=6mnNBQAAQBAJ&pg=PA95}}</ref> [[Organization:Dolby Digital|Dolby Digital]] audio<ref name="Stankovic"/><ref name="Luo"/>
**[[Engineering:Digital television|Digital television]] (DTV)<ref name="Barbero"/> — digital television broadcasting,<ref name="McKernan58"/> [[Engineering:Standard-definition television|standard-definition television]] (SDTV), high-definition TV (HDTV),<ref name="Barbero"/><ref name="Shishikui"/> HDTV [[Engineering:Video decoder|encoder/decoder chips]], ultra HDTV (UHDTV)<ref name="Stankovic"/>
**[[Engineering:Digital video|Digital video]]<ref name="Ghanbari"/><ref name="Lee1995"/> — digital versatile disc (DVD),<ref name="McKernan58"/> [[Engineering:High-definition video|high-definition]] (HD) video<ref name="Barbero"/><ref name="Shishikui"/> 
**Video coding — video compression,<ref name="Stankovic"/> video coding standards,<ref name="Ochoa"/> [[Physics:Motion estimation|motion estimation]], [[Motion compensation|motion compensation]], inter-frame prediction, [[Motion vector|motion vector]]s,<ref name="Stankovic"/> [[Stereoscopic video coding|3D video coding]], local distortion detection probability (LDDP) model, [[Moving object detection|moving object detection]], [[Multiview Video Coding]] (MVC)<ref name="Ochoa"/>
**[[Engineering:Video processing|Video processing]] — [[Motion analysis|motion analysis]], 3D-DCT motion analysis, [[Engineering:Video content analysis|video content analysis]], [[Data extraction|data extraction]],<ref name="Ochoa"/> [[Engineering:Video browsing|video browsing]],<ref>{{cite book |last1=Zhang |first1=HongJiang |chapter=Content-Based Video Browsing And Retrieval |editor-last1=Furht |editor-first1=Borko |title=Handbook of Internet and Multimedia Systems and Applications |date=1998 |publisher=CRC Press |isbn=9780849318580 |pages=[https://archive.org/details/handbookofintern0000unse_a3l0/page/83 83–108 (89)] |chapter-url=https://books.google.com/books?id=5zfC1wI0wzUC&pg=PA89 |url=https://archive.org/details/handbookofintern0000unse_a3l0/page/83 }}</ref> professional [[Engineering:Video production|video production]]<ref name="loc"/>
*Watermarking — [[Digital watermarking|digital watermarking]], image watermarking, video watermarking, 3D video watermarking, [[Digital watermarking#Reversible data hiding|reversible data hiding]], watermarking detection<ref name="Ochoa"/>
*[[Engineering:Wireless|Wireless]] technology
**Mobile devices<ref name="Encodes"/> — mobile phones, smartphones,<ref name="AppleInsider standards 1"/> videophones<ref name="Stankovic"/>
**[[Engineering:Radio frequency|Radio frequency]] (RF) technology — RF engineering, [[Astronomy:Aperture synthesis|aperture]] [[Engineering:Sensor array|arrays]],<ref name="Ochoa"/> [[Beamforming|beamforming]], [[Engineering:Digital electronics|digital]] arithmetic circuits, directional [[Engineering:Sensor|sensing]], space imaging<ref>{{cite journal |last1=Potluri |first1=U. S. |last2=Madanayake |first2=A. |last3=Cintra |first3=R. J. |last4=Bayer |first4=F. M. |last5=Rajapaksha |first5=N. |title=Multiplier-free DCT approximations for RF multi-beam digital aperture-array space imaging and directional sensing |journal=Measurement Science and Technology |date=17 October 2012 |volume=23 |issue=11 |pages=114003 |doi=10.1088/0957-0233/23/11/114003 |s2cid=119888170 |issn=0957-0233}}</ref>
*[[Wireless sensor network]] (WSN) — wireless acoustic sensor networks<ref name="Ochoa"/>
}}

===Visual media standards===
The DCT-II, is an important image compression technique. It is used in image compression standards such as [[JPEG]], and video compression standards such as H.26x, MJPEG, MPEG, [[Engineering:DV (video format)|DV]], [[Software:Theora|Theora]] and [[Daala]]. There, the two-dimensional DCT-II of <math>N \times N</math> blocks are computed and the results are [[Quantization (signal processing)|quantized]] and [[Entropy encoding|entropy coded]]. In this case, <math>N</math> is typically 8 and the DCT-II formula is applied to each row and column of the block. The result is an 8&nbsp;× 8 transform coefficient array in which the <math>(0,0)</math> element (top-left) is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.

The integer DCT, an integer approximation of the DCT,<ref name="Britanak2010"/><ref name="Stankovic"/> is used in [[Advanced Video Coding]] (AVC),<ref name="Wang">{{cite journal |last1=Wang |first1=Hanli |last2=Kwong |first2=S. |last3=Kok |first3=C. |title=Efficient prediction algorithm of integer DCT coefficients for H.264/AVC optimization |journal=IEEE Transactions on Circuits and Systems for Video Technology |date=2006 |volume=16 |issue=4 |pages=547–552 |doi=10.1109/TCSVT.2006.871390|s2cid=2060937 }}</ref><ref name="Stankovic"/> introduced in 2003, and [[High Efficiency Video Coding]] (HEVC),<ref name="apple"/><ref name="Stankovic"/> introduced in 2013. The integer DCT is also used in the High Efficiency Image Format (HEIF), which uses a subset of the HEVC video coding format for coding still images.<ref name="apple"/> AVC uses 4&nbsp;x 4 and 8&nbsp;x 8 blocks. HEVC and HEIF use varied block sizes between 4&nbsp;x 4 and 32&nbsp;x 32 pixels.<ref name="apple"/><ref name="Stankovic"/> {{As of|2019}}, AVC is by far the most commonly used format for the recording, compression and distribution of video content, used by 91% of video developers, followed by HEVC which is used by 43% of developers.<ref name="Bitmovin">{{cite web |url=https://cdn2.hubspot.net/hubfs/3411032/Bitmovin%20Magazine/Video%20Developer%20Report%202019/bitmovin-video-developer-report-2019.pdf |title=Video Developer Report 2019 |website=[[Company:Bitmovin|Bitmovin]] |year=2019 |access-date=5 November 2019}}</ref>

====Image formats====
{| class="wikitable"
|-
! Image compression standard !! Year !! Common applications
|-
| [[JPEG]]<ref name="Stankovic"/> ||1992|| The most widely used image compression standard<ref name="Hudson">{{cite journal |last1=Hudson |first1=Graham |last2=Léger |first2=Alain |last3=Niss |first3=Birger |last4=Sebestyén |first4=István |last5=Vaaben |first5=Jørgen |title=JPEG-1 standard 25 years: past, present, and future reasons for a success |journal=[[Physics:Journal of Electronic Imaging|Journal of Electronic Imaging]] |date=31 August 2018 |volume=27 |issue=4 |pages=1 |doi=10.1117/1.JEI.27.4.040901|doi-access=free }}</ref><ref>{{cite web |title=The JPEG image format explained |url=https://home.bt.com/tech-gadgets/photography/what-is-a-jpeg-11364206889349 |website=BT.com |publisher=[[Company:BT Group|BT Group]] |access-date=5 August 2019 |date=31 May 2018}}</ref> and digital image format.<ref name="Baraniuk">{{cite news |last1=Baraniuk |first1=Chris |title=Copy protections could come to JPegs |url=https://www.bbc.co.uk/news/technology-34538705 |access-date=13 September 2019 |work=BBC News |agency=BBC |date=15 October 2015}}</ref>
|-
| [[JPEG XR]] ||2009|| [[Open XML Paper Specification]]
|-
| [[Software:WebP|WebP]] ||2010|| A graphic format that supports the lossy compression of digital images. Developed by [[Company:Google|Google]].
|-
| High Efficiency Image Format (HEIF) ||2013|| [[Image file format]] based on HEVC compression. It improves compression over JPEG,<ref name="apple">{{cite web |last1=Thomson |first1=Gavin |last2=Shah |first2=Athar |title=Introducing HEIF and HEVC |url=https://devstreaming-cdn.apple.com/videos/wwdc/2017/503i6plfvfi7o3222/503/503_introducing_heif_and_hevc.pdf |publisher=[[Company:Apple Inc.|Apple Inc.]] |year=2017 |access-date=5 August 2019}}</ref> and supports [[Engineering:Animation|animation]] with much more efficient compression than the animated GIF format.<ref>{{cite web |title=HEIF Comparison - High Efficiency Image File Format |url=https://nokiatech.github.io/heif/comparison.html |publisher=Nokia Technologies |access-date=5 August 2019}}</ref>
|-
| [[Better Portable Graphics|BPG]] ||2014||Based on HEVC compression
|-
| [[Software:JPEG XL|JPEG XL]]<ref name="jxl">{{Cite web |url=http://ds.jpeg.org/whitepapers/jpeg-xl-whitepaper.pdf |title=JPEG XL White Paper |last1=Alakuijala | first1=Jyrki |last2=Sneyers |first2=Jon |last3=Versari |first3=Luca |last4=Wassenberg |first4=Jan |access-date=14 Jan 2022 |date=22 January 2021 |website=JPEG Org. |archive-date=2 May 2021 |archive-url=https://web.archive.org/web/20210502025653/http://ds.jpeg.org/whitepapers/jpeg-xl-whitepaper.pdf |url-status=live |quote=Variable-sized DCT (square or rectangular from 2x2 to 256x256) serves as a fast approximation of the optimal decorrelating transform.}}</ref> ||2020|| A royalty-free raster-graphics file format that supports both lossy and lossless compression. 
|}<!--User:Kvng/RTH-->

====Video formats====
{| class="wikitable"
|-
! Video coding standard !! Year !! Common applications
|-
| [[H.261]]<ref name="video-standards">{{cite web|first=Yao|last=Wang|archive-url=https://web.archive.org/web/20130123211453/http://eeweb.poly.edu/~yao/EL6123/coding_standards_pt1.pdf|archive-date=2013-01-23|url=http://eeweb.poly.edu/~yao/EL6123/coding_standards_pt1.pdf|title=Video Coding Standards: Part I|year=2006}}</ref><ref>{{cite web|first=Yao|last=Wang|archive-url=https://web.archive.org/web/20130123211453/http://eeweb.poly.edu/~yao/EL6123/coding_standards_pt2.pdf|archive-date=2013-01-23|url=http://eeweb.poly.edu/~yao/EL6123/coding_standards_pt2.pdf|title=Video Coding Standards: Part II|year=2006}}</ref> ||1988|| First of a family of video coding standards. Used primarily in older video conferencing and video telephone products.
|-
| [[Motion JPEG]] (MJPEG)<ref>{{cite book |last1=Hoffman |first1=Roy |title=Data Compression in Digital Systems |date=2012 |publisher=Springer Science & Business Media |isbn=9781461560319 |page=255 |url=https://books.google.com/books?id=FOfTBwAAQBAJ}}</ref> ||1992|| [[Software:QuickTime|QuickTime]], [[Engineering:Video editing|video editing]], [[Non-linear editing|non-linear editing]], digital cameras
|-
|-
| MPEG-2 Video (H.262)<ref name="Rao"/> ||1995|| Storage and handling of digital images in broadcast applications, [[Engineering:Digital television|digital television]], [[Engineering:HDTV|HDTV]], cable, satellite, high-speed [[Internet]], [[Engineering:DVD|DVD]] video distribution
|-
| [[Engineering:DV (video format)|DV]] ||1995|| Camcorders, [[Engineering:Digital cassettes|digital cassettes]]
|-
| [[H.263]] ([[MPEG-4 Part 2]])<ref name="video-standards"/> ||1996|| Video telephony over public switched telephone network (PSTN), H.320, [[Integrated Services Digital Network]] (ISDN)<ref>{{cite news |last1=Davis |first1=Andrew |title=The H.320 Recommendation Overview |url=https://www.eetimes.com/document.asp?doc_id=1275886 |access-date=7 November 2019 |work=EE Times |date=13 June 1997}}</ref><ref>{{cite book |title=IEEE WESCANEX 97: communications, power, and computing : conference proceedings |date=May 22–23, 1997 |publisher=[[Organization:Institute of Electrical and Electronics Engineers|Institute of Electrical and Electronics Engineers]] |location=University of Manitoba, Winnipeg, Manitoba, Canada |isbn=9780780341470 |page=30 |url=https://books.google.com/books?id=8vhEAQAAIAAJ |quote=H.263 is similar to, but more complex than H.261. It is currently the most widely used international video compression standard for video telephony on ISDN (Integrated Services Digital Network) telephone lines.}}</ref>
|-
| [[Advanced Video Coding]] (AVC / H.264 / [[MPEG-4]])<ref name="Stankovic"/><ref name="Wang"/> ||2003|| Most common HD video recording/compression/distribution format, [[Engineering:Internet video|Internet video]], [[Company:YouTube|YouTube]], Blu-ray Discs, [[Engineering:HDTV|HDTV]] broadcasts, web browsers, [[Streaming television|streaming television]], mobile devices, consumer devices, [[Company:Netflix|Netflix]],<ref name="Encodes">{{cite news |author=Netflix Technology Blog |title=More Efficient Mobile Encodes for Netflix Downloads |url=https://medium.com/netflix-techblog/more-efficient-mobile-encodes-for-netflix-downloads-625d7b082909 |access-date=20 October 2019 |work=Medium.com |publisher=[[Company:Netflix|Netflix]] |date=19 April 2017}}</ref> video telephony, [[Software:FaceTime|FaceTime]]<ref name="AppleInsider standards 1">{{cite web|url=http://www.appleinsider.com/articles/10/06/08/inside_iphone_4_facetime_video_calling.html|date=June 8, 2010|access-date=June 9, 2010|title=Inside iPhone 4: FaceTime video calling|publisher=AppleInsider|author=Daniel Eran Dilger}}</ref>
|-
| [[Software:Theora|Theora]] ||2004|| Internet video, web browsers
|-
| [[VC-1]] ||2006|| [[Software:Windows|Windows]] media, Blu-ray Discs
|-
| [[Apple ProRes]] ||2007|| Professional [[Engineering:Video production|video production]].<ref name="loc">{{cite web |title=Apple ProRes 422 Codec Family |url=http://www.loc.gov/preservation/digital/formats/fdd/fdd000389.shtml |website=Library of Congress |access-date=13 October 2019 |date=17 November 2014}}</ref>
|-
| [[VP9]]||2010|| A video codec developed by [[Company:Google|Google]] used in the [[WebM]] container format with [[HTML5]].
|-
| [[High Efficiency Video Coding]] (HEVC / H.265)<ref name="Stankovic"/><ref name="apple"/> ||2013|| The emerging successor to the H.264/MPEG-4 AVC standard, having substantially improved compression capability.
|-
| [[Daala]] ||2013|| Research video format by Xiph.org.
|-
| [[Software:AV1|AV1]]<ref name="AV1">{{cite web |url=https://aomediacodec.github.io/av1-spec/av1-spec.pdf |title=AV1 Bitstream & Decoding Process Specification |author=Peter de Rivaz |author2=Jack Haughton |date=2018 |publisher=[[Organization:Alliance for Open Media|Alliance for Open Media]] |access-date=2022-01-14}}</ref> ||2018|| An open source format based on VP10 ([[VP9]]'s internal successor), [[Daala]] and [[Thor (video codec)|Thor]]; used by content providers such as [[Company:YouTube|YouTube]]<ref name="YT AV1 Beta Playlist">{{cite web |url=https://www.youtube.com/playlist?list=PLyqf6gJt7KuHBmeVzZteZUlNUQAVLwrZS |title=AV1 Beta Launch Playlist |author=YouTube Developers |website=[[Company:YouTube|YouTube]] |date=15 September 2018 |access-date=14 January 2022 |quote=The first videos to receive YouTube's AV1 transcodes.}}</ref><ref name="YT AV1">{{cite web |url=https://www.ghacks.net/2018/09/13/how-to-enable-av1-support-on-youtube/ |title=How to enable AV1 support on YouTube |last=Brinkmann |first=Martin |date=13 September 2018 |access-date=14 January 2022}}</ref> and [[Company:Netflix|Netflix]].<ref name="Netflix AV1 Android">{{cite web |url=https://netflixtechblog.com/netflix-now-streaming-av1-on-android-d5264a515202 |title=Netflix Now Streaming AV1 on Android |author=Netflix Technology Blog |date=5 February 2020 |access-date=14 January 2022}}</ref><ref name="Netflix AV1 TV">{{cite web |url=https://netflixtechblog.com/bringing-av1-streaming-to-netflix-members-tvs-b7fc88e42320 |title=Bringing AV1 Streaming to Netflix Members' TVs |author=Netflix Technology Blog |date=9 November 2021 |access-date=14 January 2022}}</ref>
|}

===MDCT audio standards===
{{See|Modified discrete cosine transform}}

====General audio====
{| class="wikitable"
|-
! Audio compression standard !! Year
!Common applications
|-
| [[Organization:Dolby Digital|Dolby Digital]] (AC-3)<ref name="Luo">{{cite book |last1=Luo |first1=Fa-Long |title=Mobile Multimedia Broadcasting Standards: Technology and Practice |date=2008 |publisher=Springer Science & Business Media |isbn=9780387782638 |page=590 |url=https://books.google.com/books?id=l6PovWat8SMC&pg=PA590}}</ref><ref name="Britanak2011">{{cite journal |last1=Britanak |first1=V. |title=On Properties, Relations, and Simplified Implementation of Filter Banks in the Dolby Digital (Plus) AC-3 Audio Coding Standards |journal=IEEE Transactions on Audio, Speech, and Language Processing |date=2011 |volume=19 |issue=5 |pages=1231–1241 |doi=10.1109/TASL.2010.2087755|s2cid=897622 }}</ref>
| 1991
| Cinema, [[Engineering:Digital cinema|digital cinema]], [[Engineering:DVD|DVD]], [[Engineering:Blu-ray|Blu-ray]], [[Engineering:Streaming media|streaming media]], video games
|-
| [[Adaptive Transform Acoustic Coding]] (ATRAC)<ref name="Luo"/>
| 1992
| [[MiniDisc]]
|-
| MPEG Layer III (MP3)<ref name="Guckert"/><ref name="Stankovic"/>
| 1993
| [[Digital audio]] distribution, MP3 players, portable media players, [[Engineering:Streaming media|streaming media]]
|-
| [[Perceptual Audio Coder]] (PAC)<ref name="Luo"/>
| 1996
| Digital audio radio service (DARS)
|-
| [[Advanced Audio Coding]] (AAC / [[MP4]] Audio)<ref name=brandenburg/><ref name="Luo"/>
| 1997
| [[Digital audio]] distribution, portable media players, [[Engineering:Streaming media|streaming media]], game consoles, mobile devices, [[Software:IOS|iOS]], [[Software:ITunes|iTunes]], [[Software:Android (operating system)|Android]], [[Engineering:BlackBerry|BlackBerry]]
|-
| [[High-Efficiency Advanced Audio Coding]] (AAC+)<ref>{{cite journal |last1=Herre |first1=J. |last2=Dietz |first2=M. |title=MPEG-4 high-efficiency AAC coding [Standards in a Nutshell] |journal=IEEE Signal Processing Magazine |date=2008 |volume=25 |issue=3 |pages=137–142 |doi=10.1109/MSP.2008.918684|bibcode=2008ISPM...25..137H }}</ref><ref name="Britanak"/>{{rp|page = [https://books.google.com/books?id=cZ4vDwAAQBAJ&pg=PA478 478]}}
| 1997
| Digital radio, digital audio broadcasting (DAB+),<ref name="Britanak">{{cite book |last1=Britanak |first1=Vladimir |last2=Rao |first2=K. R. |title=Cosine-/Sine-Modulated Filter Banks: General Properties, Fast Algorithms and Integer Approximations |date=2017 |publisher=Springer |isbn=9783319610801 |page=478 |url=https://books.google.com/books?id=cZ4vDwAAQBAJ&pg=PA478}}</ref> Digital Radio Mondiale (DRM)
|-
| [[Cook Codec]]
| 1998
| [[RealAudio]]
|-
| [[Windows Media Audio]] (WMA)<ref name="Luo"/>
| 1999
| [[Software:Windows Media|Windows Media]]
|-
| [[Software:Vorbis|Vorbis]]<ref name="vorbis-mdct">{{cite web |author=Xiph.Org Foundation |publisher=Xiph.Org Foundation |url=http://www.xiph.org/vorbis/doc/Vorbis_I_spec.html#x1-50001.1.2 |title=Vorbis I specification - 1.1.2 Classification |date=2009-06-02 |access-date=2009-09-22}}</ref><ref name="Luo"/>
| 2000
| [[Digital audio]] distribution, radio stations, [[Engineering:Streaming media|streaming media]], [[Engineering:Video game|video game]]s, [[Software:Spotify|Spotify]], [[Wikipedia]]
|-
| [[High-Definition Coding]] (HDC)<ref name="Jones"/>
| 2002
| Digital radio, [[Engineering:HD Radio|HD Radio]]
|-
| [[Dynamic Resolution Adaptation]] (DRA)<ref name="Luo"/>
| 2008
| China national audio standard, [[Engineering:China Multimedia Mobile Broadcasting|China Multimedia Mobile Broadcasting]], [[Engineering:DVB-H|DVB-H]]
|-
| [[Software:Opus (audio format)|Opus]]<ref>{{cite conference|last1=Valin|first1=Jean-Marc|last2=Maxwell|first2=Gregory|last3=Terriberry|first3=Timothy B.|last4=Vos|first4=Koen|date=October 2013|title=High-Quality, Low-Delay Music Coding in the Opus Codec|conference=135th AES Convention|publisher=[[Organization:Audio Engineering Society|Audio Engineering Society]]|arxiv=1602.04845}}</ref>
| 2012
| VoIP,<ref name="homepage">{{cite web|url=http://opus-codec.org/|title=Opus Codec|work=Opus|publisher=Xiph.org Foundation|type=Home page|access-date=July 31, 2012}}</ref> mobile telephony, [[Software:WhatsApp|WhatsApp]],<ref name="Register">{{cite news|url=https://www.theregister.co.uk/2015/10/27/whatsapp_forensic_analysis/|title=WhatsApp laid bare: Info-sucking app's innards probed|last1=Leyden|first1=John|date=27 October 2015|work=The Register|access-date=19 October 2019}}</ref><ref name="Hazra">{{cite book|title=Security in Computing and Communications: 5th International Symposium, SSCC 2017|last1=Hazra|first1=Sudip|last2=Mateti|first2=Prabhaker|date=September 13–16, 2017|publisher=Springer|isbn=9789811068980|editor-last1=Thampi|editor-first1=Sabu M.|pages=286–299 (290)|chapter=Challenges in Android Forensics|doi=10.1007/978-981-10-6898-0_24|editor-last2=Pérez|editor-first2=Gregorio Martínez|editor-last3=Westphall|editor-first3=Carlos Becker|editor-last4=Hu|editor-first4=Jiankun|editor-last5=Fan|editor-first5=Chun I.|editor-last6=Mármol|editor-first6=Félix Gómez|chapter-url=https://books.google.com/books?id=1u09DwAAQBAJ&pg=PA290}}</ref><ref name="Srivastava">{{cite book|title=Cyber Security in Parallel and Distributed Computing: Concepts, Techniques, Applications and Case Studies|last1=Srivastava|first1=Saurabh Ranjan|last2=Dube|first2=Sachin|last3=Shrivastaya|first3=Gulshan|last4=Sharma|first4=Kavita|chapter=Smartphone Triggered Security Challenges ‐ Issues, Case Studies and Prevention |date=2019|publisher=John Wiley & Sons|isbn=9781119488057|editor-last1=Le|editor-first1=Dac-Nhuong|pages=187–206 (200)|doi=10.1002/9781119488330.ch12|s2cid=214034702 |editor-last2=Kumar|editor-first2=Raghvendra|editor-last3=Mishra|editor-first3=Brojo Kishore|editor-last4=Chatterjee|editor-first4=Jyotir Moy|editor-last5=Khari|editor-first5=Manju|chapter-url=https://books.google.com/books?id=FzGtDwAAQBAJ&pg=PA200}}</ref> [[Engineering:PlayStation 4|PlayStation 4]]<ref name="playstation">{{cite web|url=https://doc.dl.playstation.net/doc/ps4-oss/|title=Open Source Software used in PlayStation 4|publisher=Sony Interactive Entertainment Inc.|access-date=2017-12-11}}</ref>
|-
| [[Dolby AC-4]]<ref>{{cite web |title=Dolby AC-4: Audio Delivery for Next-Generation Entertainment Services |url=https://www.dolby.com/us/en/technologies/ac-4/Next-Generation-Entertainment-Services.pdf |website=[[Organization:Dolby Laboratories|Dolby Laboratories]] |date=June 2015 |access-date=11 November 2019}}</ref>
| rowspan="2" | 2017
| rowspan="2" | [[Engineering:ATSC 3.0|ATSC 3.0]], [[Engineering:Ultra-high-definition television|ultra-high-definition television]] (UHD TV)
|-
| [[MPEG-H 3D Audio]]<ref>{{cite journal |last1=Bleidt |first1=R. L. |last2=Sen |first2=D. |last3=Niedermeier |first3=A. |last4=Czelhan |first4=B. |last5=Füg |first5=S. |display-authors=0 |title=Development of the MPEG-H TV Audio System for ATSC 3.0 |journal=IEEE Transactions on Broadcasting |date=2017 |volume=63 |issue=1 |pages=202–236 |doi=10.1109/TBC.2017.2661258 |s2cid=30821673 |url=https://www.iis.fraunhofer.de/content/dam/iis/en/doc/ame/Conference-Paper/BleidtR-IEEE-2017-Development-of-MPEG-H-TV-Audio-System-for-ATSC-3-0.pdf}}</ref>
|}

====Speech coding====
{| class="wikitable"
|-
![[Speech coding]] standard !! Year
!Common applications
|-
|[[AAC-LD]] (LD-MDCT)<ref>{{cite conference |last1=Schnell |first1=Markus |last2=Schmidt |first2=Markus |last3=Jander |first3=Manuel |last4=Albert |first4=Tobias |last5=Geiger |first5=Ralf |last6=Ruoppila |first6=Vesa |last7=Ekstrand |first7=Per |last8=Bernhard |first8=Grill |date=October 2008 |title=MPEG-4 Enhanced Low Delay AAC - A New Standard for High Quality Communication |url=https://www.iis.fraunhofer.de/content/dam/iis/de/doc/ame/conference/AES-125-Convention_AAC-ELD-NewStandardForHighQualityCommunication_AES7503.pdf |conference=125th AES Convention |publisher=[[Organization:Audio Engineering Society|Audio Engineering Society]] |access-date=20 October 2019 |website=Fraunhofer IIS}}</ref>
|1999
|[[Engineering:Mobile telephony|Mobile telephony]], voice-over-IP (VoIP), [[Software:IOS|iOS]], [[Software:FaceTime|FaceTime]]<ref name="AppleInsider standards 1"/>
|-
|[[Siren (codec)|Siren]]<ref name="Hersent">{{cite book |last1=Hersent |first1=Olivier |last2=Petit |first2=Jean-Pierre |last3=Gurle |first3=David |title=Beyond VoIP Protocols: Understanding Voice Technology and Networking Techniques for IP Telephony |date=2005 |publisher={{wipe|John Wiley & Sons}} |isbn=9780470023631 |page=55 |url=https://books.google.com/books?id=SMvNToRs-DgC&pg=PA55}}</ref>
|1999
|VoIP, [[Wideband audio|wideband audio]], [[G.722.1]]
|-
|[[G.722.1]]<ref>{{cite conference |last1=Lutzky |first1=Manfred |last2=Schuller |first2=Gerald |last3=Gayer |first3=Marc |last4=Krämer |first4=Ulrich |last5=Wabnik |first5=Stefan |title=A guideline to audio codec delay |url=https://www.iis.fraunhofer.de/content/dam/iis/de/doc/ame/conference/AES-116-Convention_guideline-to-audio-codec-delay_AES116.pdf |website=Fraunhofer IIS |conference=116th AES Convention |publisher=[[Organization:Audio Engineering Society|Audio Engineering Society]] |date=May 2004 |access-date=24 October 2019}}</ref>
|1999
|VoIP, wideband audio, [[G.722]]
|-
|[[G.729.1]]<ref name="Nagireddi">{{cite book|url=https://books.google.com/books?id=5AneeZFE71MC&pg=PA69|title=VoIP Voice and Fax Signal Processing|last1=Nagireddi|first1=Sivannarayana|date=2008|publisher={{wipe|John Wiley & Sons}}|isbn=9780470377864|page=69}}</ref>
|2006
|[[G.729]], VoIP, wideband audio,<ref name="Nagireddi" /> [[Engineering:Mobile telephony|mobile telephony]]
|-
|[[Enhanced Variable Rate Codec B|EVRC-WB]]<ref name="Britanak"/>{{rp|page=[https://books.google.com/books?id=cZ4vDwAAQBAJ&pg=PA31 31], 478]}}
|2007
|[[Wideband audio]]
|-
|[[G.718]]<ref>{{Cite web|url=https://www.itu.int/ITU-T/workprog/wp_item.aspx?isn=344|title=ITU-T Work Programme|website=ITU}}</ref>
|2008
|VoIP, wideband audio, mobile telephony
|-
|[[G.719]]<ref name="Britanak"/>
|2008
|Teleconferencing, videoconferencing, voice mail
|-
|[[Software:CELT|CELT]]<ref name="presentation">{{cite AV media |first=Timothy B.|last=Terriberry |title=Presentation of the CELT codec |url=http://people.xiph.org/~greg/video/linux_conf_au_CELT_2.ogv |time=65 minutes}}, also {{cite web|url=http://www.celt-codec.org/presentations/misc/lca-celt.pdf|title=CELT codec presentation slides}}</ref>
|2011
|VoIP,<ref name="ekiga">{{cite web|url=http://blog.ekiga.net/?p=112|title=Ekiga 3.1.0 available}}</ref><ref name="AutoV5-14">{{Cite web|url=https://signalwire.com/freeswitch|title=☏ FreeSWITCH|website=SignalWire}}</ref> mobile telephony
|-
|[[Enhanced Voice Services]] (EVS)<ref>{{cite web|url=https://www.iis.fraunhofer.de/content/dam/iis/de/doc/ame/wp/FraunhoferIIS_Technical-Paper_EVS.pdf|title=Enhanced Voice Services (EVS) Codec|date=March 2017|publisher=Fraunhofer IIS|access-date=19 October 2019}}</ref>
|2014
|Mobile telephony, VoIP, wideband audio
|}

===MD DCT===

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 Processing. |date=May 1997 |volume=6 |issue=5 |pages=758–763 |doi=10.1109/83.568933|pmid=18282969 |bibcode=1997ITIP....6..758C |hdl=10397/1928 |url=http://www.en.polyu.edu.hk/~wcsiu/paper_store/Journal/1997/1997_J3-IEEE-Chan%26Siu.pdf |citeseerx=10.1.1.516.2824 }}</ref> video coding algorithms,<ref name="app3DCT">{{Citation |first1=J. |last1=Song |first2=Z. |last2=SXiong |first3=X. |last3=Liu |first4=Y. |last4=Liu |title= An algorithm for layered video coding and transmission| journal= Proc. Fourth Int. Conf./Exh. High Performance Comput. Asia-Pacific Region |volume=2 |pages=700–703}}</ref> adaptive video coding<ref name="app4DCT">{{Citation |first1=S.-C |last1=Tai |first2=Y. |last2=Gi |first3=C.-W. |last3=Lin |title= An adaptive 3-D discrete cosine transform coder for medical image compression |journal= IEEE Trans. Inf. Technol. Biomed. |date=September 2000 |volume=4 |issue=3 |pages=259–263 |doi=10.1109/4233.870036|pmid=11026596 |s2cid=18016215 }}</ref> and 3-D Compression.<ref name="app5DCT">{{Citation |first1=B. |last1=Yeo |first2=B. |last2=Liu |title= Volume rendering of DCT-based compressed 3D scalar data |journal=IEEE Trans. Comput. Graphics. |date=May 1995 |volume=1 |pages=29–43 |doi=10.1109/2945.468390}}</ref> Due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using M-D DCTs is rapidly increasing. DCT-IV has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks,<ref>{{cite book| doi=10.1109/ISCAS.2000.856261 | chapter=Perfect reconstruction modulated filter banks with sum of powers-of-two coefficients | title=2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353) | year=2000 | last1=Chan | first1=S.C. | last2=Liu | first2=W. | last3=Ho | first3=K.I. | volume=2 | pages=73–76 | hdl=10722/46174 | isbn=0-7803-5482-6 | s2cid=1757438 }}</ref> lapped orthogonal transform<ref>{{cite journal |last1=Queiroz |first1=R. L. |last2=Nguyen |first2=T. Q. |title=Lapped transforms for efficient transform/subband coding |journal=IEEE Trans. Signal Process. |date=1996 |volume=44 |issue=5 |pages=497–507}}</ref>{{sfn|Malvar|1992}} and cosine-modulated wavelet bases.<ref>{{cite journal |last1=Chan |first1=S. C. |last2=Luo |first2=L. |last3=Ho |first3=K. L. |title=M-Channel compactly supported biorthogonal cosine-modulated wavelet bases |journal=IEEE Trans. Signal Process. |date=1998 |volume=46 |issue=2 |pages=1142–1151|doi=10.1109/78.668566 |bibcode=1998ITSP...46.1142C |hdl=10722/42775 |hdl-access=free }}</ref>

===Digital signal processing===
DCT plays a very important role in [[Digital signal processing|digital signal processing]]. By using the DCT, the signals can be compressed. DCT can be used in [[Biology:Electrocardiography|electrocardiography]] for the compression of ECG signals. DCT2 provides a better compression ratio than DCT.

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, [[Engineering:Multiplexing|multiplexing]], control signals, signaling, and analog-to-digital conversion. DCTs are also commonly used for [[Engineering:High-definition television|high-definition television]] (HDTV) encoder/decoder [[Engineering:Integrated circuit|chips]].<ref name="Stankovic"/>

===Compression artifacts===
{{See|Compression artifact}}

A common issue with DCT compression in [[Digital media|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=[[Company:Academic Press|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]], H.26x and MPEG formats), DCT-based blocky compression artifacts are widespread in [[Digital media|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|data compression ratio]]s.<ref name="Katsaggelos"/> This can also cause the "mosquito noise" effect, commonly found in [[Engineering:Digital video|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|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|digital media]] formats such as [[JPEG]] digital images and [[MP3]] [[Digital audio|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, discrete cosine transforms (DCTs) express a function or a signal in terms of a sum of sinusoids with different frequencies and [[Physics:Amplitude|amplitude]]s. Like the [[Discrete Fourier transform|discrete Fourier transform]] (DFT), a DCT operates on a function at a finite number of discrete data points. The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines (in the form of complex exponentials). However, this visible difference is merely a consequence of a deeper distinction: a DCT implies different boundary conditions from the DFT or other related transforms.

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 [[Sine and cosine transforms|cosine transform]], implies an [[Even and odd functions|even]] extension of the original function.

[[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).

These choices lead to all the standard variations of DCTs and also [[Discrete sine transform|discrete sine transform]]s (DSTs). 
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. 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|partial differential equation]]s by [[Spectral method|spectral method]]s, the boundary conditions are directly specified as a part of the problem being solved. Or, for the [[Modified discrete cosine transform|MDCT]] (based on the type-IV DCT), the boundary conditions are intimately involved in the MDCT's critical property of time-domain aliasing cancellation. In a more subtle fashion, the boundary conditions are responsible for the "energy compactification" properties that make DCTs useful for image and audio compression, because the boundaries affect the rate of convergence of any Fourier-like series.

In particular, it is well known that any [[Classification of discontinuities|discontinuities]] in a function reduce the [[Rate of convergence|rate of convergence]] of the Fourier series, so that more sinusoids are needed to represent the function with a given accuracy. The same principle governs the usefulness of the DFT and other transforms for signal compression; the smoother a function is, the fewer terms in its DFT or DCT are required to represent it accurately, and the more it can be compressed. (Here, we think of the DFT or DCT as approximations for the [[Fourier series]] or cosine series of a function, respectively, in order to talk about its "smoothness".) However, the implicit periodicity of the DFT means that discontinuities usually occur at the boundaries: any random segment of a signal is unlikely to have the same value at both the left and right boundaries. (A similar problem arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.) In contrast, a DCT where ''both'' boundaries are even ''always'' yields a continuous extension at the boundaries (although the [[Slope|slope]] is generally discontinuous). This is why DCTs, and in particular DCTs of types I, II, V, and VI (the types that have two even boundaries) generally perform better for signal compression than DFTs and DSTs. In practice, a type-II DCT is usually preferred for such applications, in part for reasons of computational convenience.

== Formal definition ==
Formally, the discrete cosine transform is a linear, invertible [[Function (mathematics)|function]] <math> f : \R^{N} \to \R^{N} </math> (where <math> \R</math> denotes the set of [[Real number|real number]]s), or equivalently an invertible {{mvar|N}} × {{mvar|N}} [[Square matrix|square matrix]]. There are several variants of the DCT with slightly modified definitions. The {{mvar|N}} real numbers <math>~ x_0,\ \ldots\ x_{N - 1} ~</math> are transformed into the {{mvar|N}} real numbers <math> X_0,\, \ldots,\, X_{N - 1} </math> according to one of the formulas:

=== DCT-I ===
:<math>X_k
 = \frac{1}{2} (x_0 + (-1)^k x_{N-1}) 
 + \sum_{n=1}^{N-2} x_n \cos \left[\, \frac{\pi}{\,N-1\,} \, n \, k \,\right]
 \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\,}\, ,</math> and correspondingly multiply the <math> X_0 </math> and <math> X_{N-1}</math> terms by <math> 1/\sqrt{2\,} \,,</math> which makes the DCT-I matrix [[Orthogonal matrix|orthogonal]], if one further multiplies by an overall scale factor of <math> \sqrt{\tfrac{2}{N-1\,}\,} ,</math> but breaks the direct correspondence with a real-even [[Discrete Fourier transform|DFT]].

The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a [[Discrete Fourier transform|DFT]] of <math> 2(N-1) </math> real numbers with even symmetry. For example, a DCT-I of <math>N = 5 </math> real numbers <math> a\ b\ c\ d\ e </math> is exactly equivalent to a DFT of eight real numbers {{not a typo|<math> a\ b\ c\ d\ e\ d\ c\ b </math>}} (even symmetry), divided by two. (In contrast, DCT types II-IV involve a half-sample shift in the equivalent DFT.)

Note, however, that the DCT-I is not defined for <math> N </math> less than 2, while all other DCT types are defined for any positive <math> N .</math>

Thus, the DCT-I corresponds to the boundary conditions: <math> x_n </math> is even around <math> n = 0 </math> and even around <math> n = N - 1 </math>; similarly for <math> X_k .</math>

=== DCT-II ===
:<math>X_k =
 \sum_{n=0}^{N-1} x_n \cos \left[\, \tfrac{\,\pi\,}{N} \left( n + \tfrac{1}{2} \right) k \, \right]
 \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 "the DCT".<ref name="pubDCT"/><ref name="pubRaoYip"/>

This transform is exactly equivalent (up to an overall scale factor of 2) to a [[Discrete Fourier transform|DFT]] of <math>4N</math> real inputs of even symmetry where the even-indexed elements are zero. That is, it is half of the [[Discrete Fourier transform|DFT]] of the <math>4N</math> inputs <math> y_n ,</math> where <math> y_{2n} = 0 ,</math> <math> y_{2n+1} = x_n </math> for <math> 0 \leq n < N ,</math> <math> y_{2N} = 0 ,</math> and <math> y_{4N-n} = y_n </math> for <math> 0 < n < 2N .</math> DCT-II transformation is also possible using 2{{mvar|N}} signal followed by a multiplication by half shift. This is demonstrated by [[Biography:John Makhoul|Makhoul]].

Some authors further multiply the <math> X_0 </math> term by <math> 1/\sqrt{N\,} \, </math> and multiply the rest of the matrix by an overall scale factor of <math display="inline">\sqrt{{2}/{N}}</math> (see below for the corresponding change in DCT-III). This makes the DCT-II matrix [[Orthogonal matrix|orthogonal]], but breaks the direct correspondence with a real-even [[Discrete Fourier transform|DFT]] of half-shifted input. This is the normalization used by Matlab, for example, see.<ref>{{cite web |url=https://www.mathworks.com/help/signal/ref/dct.html |title=Discrete cosine transform - MATLAB dct |website=www.mathworks.com |access-date=2019-07-11}}</ref> In many applications, such as [[JPEG]], the scaling is arbitrary because scale factors can be combined with a subsequent computational step (e.g. the [[Quantization (signal processing)|quantization]] step in JPEG<ref>{{cite book |isbn=9780442012724 |title=JPEG: Still Image Data Compression Standard |last1=Pennebaker |first1=William B. |last2=Mitchell |first2=Joan L. |date=31 December 1992|publisher=Springer }}</ref>), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications.<ref>{{cite journal |url=https://search.ieice.org/bin/summary.php?id=e71-e_11_1095 |first1=Y. |last1=Arai |first2=T. |last2=Agui |first3=M. |last3=Nakajima |title=A fast DCT-SQ scheme for images |journal=IEICE Transactions |volume=71 |issue=11 |pages= 1095–1097 |year=1988}}</ref><ref>{{cite journal |doi=10.1016/j.sigpro.2008.01.004 |title=Type-II/III DCT/DST algorithms with reduced number of arithmetic operations |year=2008 |last1=Shao |first1=Xuancheng |last2=Johnson |first2=Steven G. |journal=Signal Processing |volume=88 |issue=6 |pages=1553–1564 |arxiv=cs/0703150 |s2cid=986733}}</ref>

The DCT-II implies the boundary conditions: <math> x_n </math> is even around <math> n = -1/2 </math> and even around <math> n = N - 1/2 \,;</math> <math> X_k </math> is even around <math> k = 0 </math> and odd around <math> k = N .</math>

=== DCT-III ===
:<math> X_k =
  \tfrac{1}{2} x_0 +
  \sum_{n=1}^{N-1} x_n \cos \left[\, \tfrac{\,\pi\,}{N} \left( k + \tfrac{1}{2} \right) n \,\right]
  \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 "the inverse DCT" ("IDCT").<ref name="pubRaoYip"/>

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 [[Discrete Fourier transform|DFT]] of half-shifted output.

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>

=== DCT-IV ===
:<math>X_k =
 \sum_{n=0}^{N-1} x_n \cos \left[\, \tfrac{\,\pi\,}{N} \, \left(n + \tfrac{1}{2} \right)\left(k + \tfrac{1}{2} \right) \,\right]
 \qquad \text{ for } k = 0,\ \ldots\ N-1 ~.</math>

The DCT-IV matrix becomes [[Orthogonal matrix|orthogonal]] (and thus, being clearly symmetric, its own inverse) if one further multiplies by an overall scale factor of <math display="inline"> \sqrt{2/N}.</math>

A variant of the DCT-IV, where data from different transforms are ''overlapped'', is called the [[Modified discrete cosine transform|modified discrete cosine transform]] (MDCT).<ref>{{harvnb|Malvar|1992}}</ref>

The DCT-IV implies the boundary conditions: <math> x_n </math> is even around <math>n = -1/2</math> and odd around <math>n = N - 1/2;</math> similarly for <math>X_k.</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 [[Discrete Fourier transform|DFT]]s of even order (regardless of whether <math> N </math> is even or odd), since the corresponding DFT is of length <math> 2(N-1) </math> (for DCT-I) or <math> 4 N </math> (for DCT-II & III) or <math> 8 N </math> (for DCT-IV). The four additional types of discrete cosine transform<ref>{{harvnb|Martucci|1994}}</ref> correspond essentially to real-even DFTs of logically odd order, which have factors of <math> N \pm {1}/{2} </math> in the denominators of the cosine arguments.

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.

(The trivial real-even array, a length-one DFT (odd length) of a single number {{mvar|a}}&nbsp;, corresponds to a DCT-V of length <math> N = 1 .</math>)

== Inverse transforms ==
Using the normalization conventions above, the inverse of DCT-I is DCT-I multiplied by 2/(''N''&nbsp;−&nbsp;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 [[Discrete Fourier transform|DFT]], the normalization factor in front of these transform definitions is merely a convention and differs between treatments. For example, some authors multiply the transforms by <math display="inline">\sqrt{2/N}</math> so that the inverse does not require any additional multiplicative factor. Combined with appropriate factors of {{sqrt|2}} (see above), this can be used to make the transform matrix [[Orthogonal matrix|orthogonal]].

== Multidimensional DCTs ==

Multidimensional variants of the various DCT types follow straightforwardly from the one-dimensional definitions: they are simply a separable product (equivalently, a composition) of DCTs along each dimension.

=== M-D DCT-II ===

For example, a two-dimensional DCT-II of an image or a matrix is simply the one-dimensional DCT-II, from above, performed along the rows and then along the columns (or vice versa). That is, the 2D DCT-II is given by the formula (omitting normalization and other scale factors, as above):

:<math>
\begin{align}
X_{k_1,k_2} &=
 \sum_{n_1=0}^{N_1-1}
\left( \sum_{n_2=0}^{N_2-1}
 x_{n_1,n_2} 
\cos \left[\frac{\pi}{N_2} \left(n_2+\frac{1}{2}\right) k_2 \right]\right)
\cos \left[\frac{\pi}{N_1} \left(n_1+\frac{1}{2}\right) k_1 \right]\\
&= \sum_{n_1=0}^{N_1-1}
 \sum_{n_2=0}^{N_2-1}
 x_{n_1,n_2} 
\cos \left[\frac{\pi}{N_1} \left(n_1+\frac{1}{2}\right) k_1 \right]
\cos \left[\frac{\pi}{N_2} \left(n_2+\frac{1}{2}\right) k_2 \right] .
\end{align}
</math>
:The inverse of a multi-dimensional DCT is just a separable product of the inverses of the corresponding one-dimensional DCTs (see above), e.g. the one-dimensional inverses applied along one dimension at a time in a row-column algorithm.

The ''3-D DCT-II'' is only the extension of ''2-D DCT-II'' in three dimensional space and mathematically can be calculated by the formula

:<math>
X_{k_1,k_2,k_3} =
 \sum_{n_1=0}^{N_1-1}
 \sum_{n_2=0}^{N_2-1}
\sum_{n_3=0}^{N_3-1}
 x_{n_1,n_2,n_3} 
\cos \left[\frac{\pi}{N_1} \left(n_1+\frac{1}{2}\right) k_1 \right]
\cos \left[\frac{\pi}{N_2} \left(n_2+\frac{1}{2}\right) k_2 \right]
\cos \left[\frac{\pi}{N_3} \left(n_3+\frac{1}{2}\right) k_3 \right],\quad 
\text{for } k_i = 0,1,2,\dots,N_i-1.
</math>

The inverse of '''3-D DCT-II''' is '''3-D DCT-III''' and can be computed from the formula given by
:<math>
x_{n_1,n_2,n_3} =
 \sum_{k_1=0}^{N_1-1}
 \sum_{k_2=0}^{N_2-1}
\sum_{k_3=0}^{N_3-1}
 X_{k_1,k_2,k_3} 
\cos \left[\frac{\pi}{N_1} \left(n_1+\frac{1}{2}\right) k_1 \right]
\cos \left[\frac{\pi}{N_2} \left(n_2+\frac{1}{2}\right) k_2 \right]
\cos \left[\frac{\pi}{N_3} \left(n_3+\frac{1}{2}\right) k_3 \right],\quad
\text{for } n_i=0,1,2,\dots,N_i-1.
</math>

Technically, computing a two-, three- (or -multi) dimensional DCT by sequences of one-dimensional DCTs along each dimension is known as a ''row-column'' algorithm. As with [[Fast Fourier transform#Multidimensional FFTs|multidimensional FFT algorithms]], however, there exist other methods to compute the same thing while performing the computations in a different order (i.e. interleaving/combining the algorithms for the different dimensions). Owing to the rapid growth in the applications based on the 3-D DCT, several fast algorithms are developed for the computation of 3-D DCT-II. Vector-Radix algorithms are applied for computing M-D DCT to reduce the computational complexity and to increase the computational speed. To compute 3-D DCT-II efficiently, a fast algorithm, Vector-Radix Decimation in Frequency (VR DIF) algorithm was developed.

====3-D DCT-II VR DIF====
In order to apply the VR DIF algorithm the input data is to be formulated and rearranged as follows.<ref>{{cite journal|doi=10.1049/ip-f-2.1990.0063|title=Direct methods for computing discrete sinusoidal transforms|year=1990|last1=Chan|first1=S.C.|last2=Ho|first2=K.L.|journal=IEE Proceedings F - Radar and Signal Processing|volume=137|issue=6|page=433}}</ref><ref name=":0">{{cite journal|first1=O.|last1=Alshibami|first2=S.|last2=Boussakta|title=Three-dimensional algorithm for the 3-D DCT-III|journal=Proc. Sixth Int. Symp. Commun., Theory Applications|date=July 2001|pages=104–107}}</ref> The transform size ''N × N × N'' is assumed to be&nbsp;2.

[[File:Stages of the 3-D DCT-II VR DIF algorithm.jpg|thumb|The four basic stages of computing 3-D DCT-II using VR DIF Algorithm.|336x336px]]

:<math>
\begin{array}{lcl}\tilde{x}(n_1,n_2,n_3) =x(2n_1,2n_2,2n_3)\\ 
\tilde{x}(n_1,n_2,N-n_3-1)=x(2n_1,2n_2,2n_3+1)\\
\tilde{x}(n_1,N-n_2-1,n_3)=x(2n_1,2n_2+1,2n_3)\\
\tilde{x}(n_1,N-n_2-1,N-n_3-1)=x(2n_1,2n_2+1,2n_3+1)\\
\tilde{x}(N-n_1-1,n_2,n_3)=x(2n_1+1,2n_2,2n_3)\\
\tilde{x}(N-n_1-1,n_2,N-n_3-1)=x(2n_1+1,2n_2,2n_3+1)\\
\tilde{x}(N-n_1-1,N-n_2-1,n_3)=x(2n_1+1,2n_2+1,2n_3)\\
\tilde{x}(N-n_1-1,N-n_2-1,N-n_3-1)=x(2n_1+1,2n_2+1,2n_3+1)\\
\end{array}
</math>
:where <math>0\leq n_1,n_2,n_3 \leq \frac{N}{2} -1</math>

The figure to the adjacent shows the four stages that are involved in calculating 3-D DCT-II using VR DIF algorithm. The first stage is the 3-D reordering using the index mapping illustrated by the above equations. The second stage is the butterfly calculation. Each butterfly calculates eight points together as shown in the figure just below, where <math>c(\varphi_i)=\cos(\varphi_i)</math>.

The original 3-D DCT-II now can be written as

:<math>X(k_1,k_2,k_3)=\sum_{n_1=1}^{N-1}\sum_{n_2=1}^{N-1}\sum_{n_3=1}^{N-1}\tilde{x}(n_1,n_2,n_3) \cos(\varphi k_1)\cos(\varphi k_2)\cos(\varphi k_3)
</math>

where <math>\varphi_i= \frac{\pi}{2N}(4N_i+1),\text{ and } i= 1,2,3.</math>

If the even and the odd parts of <math>k_1,k_2</math> and <math>k_3</math> and are considered, the general formula for the calculation of the 3-D DCT-II can be expressed as [[File:Single butterfly of the 3-D DCT-II VR DIF algorithm.jpg|thumb|The single butterfly stage of VR DIF algorithm.|310x310px]]
:<math>X(k_1,k_2,k_3)=\sum_{n_1=1}^{\tfrac N 2 -1}\sum_{n_2=1}^{\tfrac N 2 -1}\sum_{n_1=1}^{\tfrac N 2 -1}\tilde{x}_{ijl}(n_1,n_2,n_3) \cos(\varphi (2k_1+i)\cos(\varphi (2k_2+j)
\cos(\varphi (2k_3+l))</math>

where

: <math>\tilde{x}_{ijl}(n_1,n_2,n_3)=\tilde{x}(n_1,n_2,n_3)+(-1)^l\tilde{x}\left(n_1,n_2,n_3+\frac{n}{2}\right) </math>

: <math>+(-1)^j\tilde{x}\left(n_1,n_2+\frac{n}{2},n_3\right)+(-1)^{j+l}\tilde{x}\left(n_1,n_2+\frac{n}{2},n_3+\frac{n}{2}\right) </math>

: <math>+(-1)^i\tilde{x}\left(n_1+\frac{n}{2},n_2,n_3\right)+(-1)^{i+j}\tilde{x}\left(n_1+\frac{n}{2}+\frac{n}{2},n_2,n_3\right) </math>

: <math>+(-1)^{i+l}\tilde{x}\left(n_1+\frac{n}{2},n_2,n_3+\frac{n}{3}\right)</math>

: <math>+(-1)^{i+j+l}\tilde{x}\left(n_1+\frac{n}{2},n_2+\frac{n}{2},n_3+\frac{n}{2}\right) \text{ where } i,j,l= 0 \text{ or } 1.</math>

===== Arithmetic complexity =====
The whole 3-D DCT calculation needs <math>~ [\log_2 N] ~</math> stages, and each stage involves <math>~ \tfrac{1}{8}\ N^3 ~</math> butterflies. The whole 3-D DCT requires <math>~ \left[ \tfrac{1}{8}\ N^3 \log_2 N \right] ~</math> butterflies to be computed. Each butterfly requires seven real multiplications (including trivial multiplications) and 24&nbsp;real additions (including trivial additions). Therefore, the total number of real multiplications needed for this stage is <math>~ \left[ \tfrac{7}{8}\ N^3\ \log_2 N \right] ~,</math> and the total number of real additions i.e. including the post-additions (recursive additions) which can be calculated directly after the butterfly stage or after the bit-reverse stage are given by<ref name=":0" /> <math>~ \underbrace{\left[\frac{3}{2}N^3 \log_2N\right]}_\text{Real}+\underbrace{\left[\frac{3}{2}N^3 \log_2N-3N^3+3N^2\right]}_\text{Recursive} = \left[\frac{9}{2}N^3 \log_2N-3N^3+3N^2\right] ~.</math>

The conventional method to calculate MD-DCT-II is using a Row-Column-Frame (RCF) approach which is computationally complex and less productive on most advanced recent hardware platforms. The number of multiplications required to compute VR DIF Algorithm when compared to RCF algorithm are quite a few in number. The number of Multiplications and additions involved in RCF approach are given by <math>~\left[\frac{3}{2}N^3 \log_2 N \right]~</math> and <math>~ \left[\frac{9}{2}N^3 \log_2 N - 3N^3 + 3N^2 \right] ~,</math> respectively. From Table 1, it can be seen that the total number
{| class="wikitable sortable"
|+TABLE 1
Comparison of VR DIF & RCF Algorithms for computing 3D-DCT-II
!Transform Size
!3D VR Mults
!RCF Mults
!3D VR Adds
!RCF Adds
|-
|8 × 8 × 8
|2.625
|4.5
|10.875
|10.875
|-
|16 × 16 × 16
|3.5
|6
|15.188
|15.188
|-
|32 × 32 × 32
|4.375
|7.5
|19.594
|19.594
|-
|64 × 64 × 64
|5.25
|9
|24.047
|24.047
|}
of multiplications associated with the 3-D DCT VR algorithm is less than that associated with the RCF approach by more than 40%. In addition, the RCF approach involves matrix transpose and more indexing and data swapping than the new VR algorithm. This makes the 3-D DCT VR algorithm more efficient and better suited for 3-D applications that involve the 3-D DCT-II such as video compression and other 3-D image processing applications.

The main consideration in choosing a fast algorithm is to avoid computational and structural complexities. As the technology of computers and DSPs advances, the execution time of arithmetic operations (multiplications and additions) is becoming very fast, and regular computational structure becomes the most important factor.<ref>{{cite journal |doi=10.1109/78.827550 |title=On the computation of two-dimensional DCT |year=2000 |last1=Guoan Bi |last2=Gang Li |last3=Kai-Kuang Ma |last4=Tan |first4=T.C. |journal=IEEE Transactions on Signal Processing |volume=48 |issue=4 |pages=1171–1183 |bibcode=2000ITSP...48.1171B}}</ref> Therefore, although the above proposed 3-D VR algorithm does not achieve the theoretical lower bound on the number of multiplications,<ref>{{cite journal |doi=10.1109/18.144722 |title=On the multiplicative complexity of discrete cosine transforms |date=July 1992a |last1=Feig |first1=E. |last2=Winograd |first2=S. |journal=IEEE Transactions on Information Theory |volume=38 |issue=4 |pages=1387–1391}}</ref> it has a simpler computational structure as compared to other 3-D DCT algorithms. It can be implemented in place using a single butterfly and possesses the properties of the [[Cooley–Tukey FFT algorithm]] in 3-D. Hence, the 3-D VR presents a good choice for reducing arithmetic operations in the calculation of the 3-D DCT-II, while keeping the simple structure that characterize butterfly-style [[Cooley–Tukey FFT algorithm]]s.

[[File:DCT-8x8.png|thumb|250px|Two-dimensional DCT frequencies from the JPEG DCT]]
The image to the right shows a combination of horizontal and vertical frequencies for an {{nobr| 8 × 8 }} <math>(~ N_1 = N_2 = 8 ~)</math> two-dimensional DCT. Each step from left to right and top to bottom is an increase in frequency by 1/2 cycle.
For example, moving right one from the top-left square yields a half-cycle increase in the horizontal frequency. Another move to the right yields two half-cycles. A move down yields two half-cycles horizontally and a half-cycle vertically. The source data {{nobr|( 8×8 )}} is transformed to a [[Linear combination|linear combination]] of these 64&nbsp;frequency squares.

=== MD-DCT-IV ===
The M-D DCT-IV is just an extension of 1-D DCT-IV on to {{mvar|M}}&nbsp;dimensional domain. The 2-D DCT-IV of a matrix or an image is given by

:<math> X_{k,\ell} =
 \sum_{n=0}^{N-1} \; \sum_{m=0}^{M-1} \ x_{n,m} \cos\left(\ \frac{\,( 2 m + 1 )( 2 k + 1 )\ \pi \,}{4N} \ \right) \cos\left(\ \frac{\, ( 2n + 1 )( 2 \ell + 1 )\ \pi \,}{4M} \ \right) ~,</math>

: for <math>~~ k = 0,\ 1,\ 2\ \ldots\ N-1 ~~</math> and <math>~~ \ell= 0,\ 1,\ 2,\ \ldots\ M-1 ~.</math>

We can compute the MD DCT-IV using the regular row-column method or we can use the polynomial transform method<ref>{{cite book |last=Nussbaumer |first=H.J. |title=Fast Fourier transform and convolution algorithms |publisher=Springer-Verlag |location=New York |date=1981 |edition=1st }}</ref> for the fast and efficient computation. The main idea of this algorithm is to use the Polynomial Transform to convert the multidimensional DCT into a series of 1-D DCTs directly. MD DCT-IV also has several applications in various fields.

== Computation ==
Although the direct application of these formulas would require <math>~ \mathcal{O}(N^2) ~</math> operations, it is possible to compute the same thing with only <math>~ \mathcal{O}(N \log N ) ~</math> complexity by factorizing the computation similarly to the [[Fast Fourier transform|fast Fourier transform]] (FFT). One can also compute DCTs via FFTs combined with <math>~\mathcal{O}(N)~</math> pre- and post-processing steps. In general, <math>~\mathcal{O}(N \log N )~</math> methods to compute DCTs are known as fast cosine transform (FCT) algorithms.

The most efficient algorithms, in principle, are usually those that are specialized directly for the DCT, as opposed to using an ordinary FFT plus <math>~ \mathcal{O}(N) ~</math> extra operations (see below for an exception). However, even "specialized" DCT algorithms (including all of those that achieve the lowest known arithmetic counts, at least for [[Power of two|power-of-two]] sizes) are typically closely related to FFT algorithms – since DCTs are essentially DFTs of real-even data, one can design a fast DCT algorithm by taking an FFT and eliminating the redundant operations due to this symmetry. This can even be done automatically {{harv|Frigo|Johnson|2005}}. Algorithms based on the [[Cooley–Tukey FFT algorithm]] are most common, but any other FFT algorithm is also applicable. For example, the Winograd FFT algorithm leads to minimal-multiplication algorithms for the DFT, albeit generally at the cost of more additions, and a similar algorithm was proposed by {{harv|Feig|Winograd|1992a}} for the DCT. Because the algorithms for DFTs, DCTs, and similar transforms are all so closely related, any improvement in algorithms for one transform will theoretically lead to immediate gains for the other transforms as well {{harv|Duhamel|Vetterli|1990}}.

While DCT algorithms that employ an unmodified FFT often have some theoretical overhead compared to the best specialized DCT algorithms, the former also have a distinct advantage: Highly optimized FFT programs are widely available. Thus, in practice, it is often easier to obtain high performance for general lengths {{mvar|N}} with FFT-based algorithms.{{efn|
Algorithmic performance on modern hardware is typically not principally determined by simple arithmetic counts, and optimization requires substantial engineering effort to make best use, within its intrinsic limits, of available built-in hardware optimization.
}}
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 [[Software:FFTPACK|FFTPACK]] and [[Software:FFTW|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> butterflies, once the trivial and / or duplicate parts are eliminated and / or merged.
}}
Because the even-indexed elements are zero, this radix-4 step is exactly the same as a split-radix step. If the subsequent size <math>~ N ~</math> real-data FFT is also performed by a real-data [[Split-radix FFT algorithm|split-radix algorithm]] (as in {{harvtxt|Sorensen|Jones|Heideman|Burrus|1987}}), then the resulting algorithm actually matches what was long the lowest published arithmetic count for the power-of-two DCT-II (<math>~ 2 N \log_2 N - N + 2 ~</math> real-arithmetic operations{{efn|
The precise count of real arithmetic operations, and in particular the count of real multiplications, depends somewhat on the scaling of the transform definition. The <math>~ 2 N \log_2 N - N + 2 ~</math> count is for the DCT-II definition shown here; two multiplications can be saved if the transform is scaled by an overall <math>\sqrt2</math> factor. Additional multiplications can be saved if one permits the outputs of the transform to be rescaled individually, as was shown by {{harvtxt|Arai|Agui|Nakajima|1988}} for the size-8 case used in JPEG.
}}).

A recent reduction in the operation count to <math>~ \tfrac{17}{9} N \log_2 N + \mathcal{O}(N)</math> also uses a real-data FFT.<ref>{{cite journal |doi=10.1016/j.sigpro.2008.01.004 |title=Type-II/III DCT/DST algorithms with reduced number of arithmetic operations |journal=Signal Processing |volume=88 |issue=6 |pages=1553–1564 |year=2008 |last1=Shao |first1=Xuancheng |last2=Johnson |first2=Steven G. |arxiv=cs/0703150 |s2cid=986733}}</ref> So, there is nothing intrinsically bad about computing the DCT via an FFT from an arithmetic perspective – it is sometimes merely a question of whether the corresponding FFT algorithm is optimal. (As a practical matter, the function-call overhead in invoking a separate FFT routine might be significant for small <math>~ N ~,</math> but this is an implementation rather than an algorithmic question since it can be solved by unrolling or inlining.)

==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 8x8 grayscale image of capital letter A.
[[File:letter-a-8x8.png|frame|center|Original size, scaled 10x (nearest neighbor), scaled 10x (bilinear).]]

[[File:dct-table.png|frame|center|Basis functions of the discrete cosine transformation with corresponding coefficients (specific for our image).
<br/>DCT of the image =
<math>
\begin{bmatrix}
6.1917 & -0.3411 & 1.2418  &  0.1492  &  0.1583  &  0.2742 &  -0.0724  &  0.0561 \\
0.2205 & 0.0214 & 0.4503  &  0.3947  & -0.7846 &  -0.4391  &  0.1001  & -0.2554 \\
1.0423 & 0.2214 & -1.0017 &  -0.2720  &  0.0789 &  -0.1952  &  0.2801  &  0.4713 \\
-0.2340 & -0.0392 & -0.2617 &  -0.2866 &   0.6351 &   0.3501 &  -0.1433  &  0.3550 \\
0.2750 & 0.0226 & 0.1229  &  0.2183  & -0.2583  & -0.0742  & -0.2042  & -0.5906 \\
0.0653 & 0.0428 & -0.4721 &  -0.2905  &  0.4745  &  0.2875  & -0.0284  & -0.1311 \\
0.3169 & 0.0541 & -0.1033 &  -0.0225  & -0.0056  &  0.1017  & -0.1650 &  -0.1500 \\
-0.2970 & -0.0627 & 0.1960 &   0.0644  & -0.1136 &  -0.1031 &   0.1887  &  0.1444 \\
\end{bmatrix}

</math>.]]

Each basis function is multiplied by its coefficient and then this product is added to the final image.

[[File:idct-animation.gif|frame|center|On the left is the final image. In the middle is the weighted function (multiplied by a coefficient) which is added to the final image. On the right is the current function and corresponding coefficient. Images are scaled (using bilinear interpolation) by factor 10×.]]

==See also==
* [[Discrete wavelet transform]]
* [[JPEG#Discrete cosine transform|JPEG{{nbsp}}{{hyphen}}{{nbsp}}Discrete{{nbsp}}cosine{{nbsp}}transform]]{{nbsp}}{{hyphen}}{{nbsp}}Contains a potentially easier to understand example of DCT transformation
* [[List of Fourier-related transforms]]
* [[Modified discrete cosine transform]]

==Notes==
{{Reflist|group=lower-alpha}}

== References ==
{{reflist|3|refs=

<ref name="Ahmed">{{cite journal |last=Ahmed |first=Nasir |title=How I Came Up With the Discrete Cosine Transform |journal=Digital Signal Processing |date=January 1991 |volume=1 |issue=1 |pages=4–5 |doi=10.1016/1051-2004(91)90086-Z |url=https://www.scribd.com/doc/52879771/DCT-History-How-I-Came-Up-with-the-Discrete-Cosine-Transform}}</ref>

<ref name="Stankovic">{{cite journal | last1 = Stanković | first1 = Radomir S. | last2 = Astola | first2 = Jaakko T. |title = Reminiscences of the Early Work in DCT: Interview with K.R. Rao | journal = Reprints from the Early Days of Information Sciences | publisher = Tampere International Center for Signal Processing | date = 2012 | volume = 60 | url = https://ethw.org/w/images/1/19/Report-60.pdf | access-date = 2021-12-30 | archive-url = https://web.archive.org/web/20211230214050/https://ethw.org/w/images/1/19/Report-60.pdf | archive-date = 2021-12-30 | url-status = live | issn = 1456-2774 | isbn = 978-9521528187 | via = [[Organization:Engineering and Technology History Wiki|ETHW]] | df = dmy-all}}</ref>

<ref name="Britanak2010">{{cite book | last1 = Britanak | first1 = Vladimir | last2 = Yip | first2 = Patrick C. | last3 = Rao | first3 = K. R. |  date = 2006-11-06 | title = Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations | language = en | publisher = [[Company:Academic Press|Academic Press]] | isbn = 978-0123736246 | lccn = 2006931102 | oclc = 220853454 | ol = OL18495589M | s2cid = 118873224 | df = dmy-all}}</ref>

<ref name="pubDCT">{{cite journal | last1 = Ahmed | first1 = Nasir |  last2 = Natarajan | first2 = T. Raj | last3 = Rao | first3 = K.R. | author-link3 = K. R. Rao | date = 1974-01-01 | title = Discrete Cosine Transform | journal = [[IEEE Transactions on Computers]] | language = en | publisher = IEEE Computer Society | volume = C-23 | issue = 1 | pages = 90{{hyphen}}93 | doi = 10.1109/T-C.1974.223784 | eissn = 1557-9956 | issn = 0018-9340 | lccn = 75642478 | oclc = 1799331 | s2cid = 206619973 | df = dmy-all }}
</ref>

<ref name="pubRaoYip">{{cite book | last1 = Rao | first1 = K. Ramamohan |  last2 = Yip | first2 = Patrick C. | date = 1990-09-11 | title = Discrete Cosine Transform: Algorithms, Advantages, Applications | series = Signal, Image and Speech Processing | language = en | publisher = [[Company:Academic Press|Academic Press]] | doi = 10.1016/c2009-0-22279-3 | arxiv = 1109.0337 | isbn = 978-0125802031 | lccn = 89029800 | oclc = 1008648293 | ol = OL2207570M | s2cid = 12270940 | df = dmy-all}}</ref>

}}

==Further reading==
<!-- these references are cited inline above, in Harvard (author, date) style because they pre-date the ref tag, but need to be converted to ref style -->
* {{Cite journal | last1 = Narasimha | first1 = M. | last2 = Peterson | first2 = A. | doi = 10.1109/TCOM.1978.1094144 | title = On the Computation of the Discrete Cosine Transform | journal = IEEE Transactions on Communications | volume = 26 | issue = 6 | pages = 934–936| date=June 1978 }}
* {{Cite journal | last1 = Makhoul | first1 = J. | doi = 10.1109/TASSP.1980.1163351 | title = A fast cosine transform in one and two dimensions | journal = IEEE Transactions on Acoustics, Speech, and Signal Processing | volume = 28 | issue = 1| pages = 27–34 | date=February 1980 }}
* {{Cite journal | last1 = Sorensen | first1 = H. | last2 = Jones | first2 = D. | last3 = Heideman | first3 = M. | last4 = Burrus | first4 = C. | title = Real-valued fast Fourier transform algorithms | doi = 10.1109/TASSP.1987.1165220 | journal = IEEE Transactions on Acoustics, Speech, and Signal Processing | volume = 35 | issue = 6 | pages = 849–863| date=June 1987 | citeseerx = 10.1.1.205.4523 }}
* {{Cite journal |last1=Plonka |first1=G. |last2=Tasche|first2=M. |title=Fast and numerically stable algorithms for discrete cosine transforms |journal=Linear Algebra and Its Applications |volume=394 |issue=1 |pages=309–345 |date=January 2005 |doi=10.1016/j.laa.2004.07.015 |doi-access=free }}
* {{Cite journal | last1 = Duhamel | first1 = P. | last2 = Vetterli | first2 = M. | doi = 10.1016/0165-1684(90)90158-U | title = Fast fourier transforms: A tutorial review and a state of the art | journal = Signal Processing | volume = 19 | issue = 4 | pages = 259–299| date=April 1990 | url = http://infoscience.epfl.ch/record/59946 | type = Submitted manuscript }}
* {{Cite journal | last1 = Ahmed | first1 = N. |  doi = 10.1016/1051-2004(91)90086-Z | title = How I came up with the discrete cosine transform | journal = Digital Signal Processing | volume = 1 | issue = 1| pages = 4–9 | date=January 1991 | url = https://www.scribd.com/doc/52879771/DCT-History-How-I-Came-Up-with-the-Discrete-Cosine-Transform}}
* {{Cite journal | last1 = Feig | first1 = E. | last2 = Winograd | first2 = S. | doi = 10.1109/78.157218 | title = Fast algorithms for the discrete cosine transform | journal = IEEE Transactions on Signal Processing | volume = 40 | issue = 9 | pages = 2174–2193| date=September 1992b | bibcode = 1992ITSP...40.2174F }}
* {{Citation |last1=Malvar |first1=Henrique |title=Signal Processing with Lapped Transforms |publisher=Artech House |location=Boston |year=1992 |isbn=978-0-89006-467-2}}
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==External links==
* Syed Ali Khayam: [https://web.archive.org/web/20150711105353/http://wisnet.seecs.nust.edu.pk/publications/tech_reports/DCT_TR802.pdf The Discrete Cosine Transform (DCT): Theory and Application]
* [http://www.reznik.org/software.html#IDCT Implementation of MPEG integer approximation of 8x8 IDCT (ISO/IEC 23002-2)]
* Matteo Frigo and [[Biography:Steven G. Johnson|Steven G. Johnson]]: ''FFTW'', [http://www.fftw.org/ FFTW Home Page]. A free ([[Software:GNU General Public License|GPL]]) C library that can compute fast DCTs (types I-IV) in one or more dimensions, of arbitrary size.
* 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].
* [http://ltfat.sourceforge.net/ LTFAT] is a free Matlab/Octave toolbox with interfaces to the FFTW implementation of the DCTs and DSTs of type I-IV.

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{{DEFAULTSORT:Discrete Cosine Transform}}
[[Category:Digital signal processing]]
[[Category:Fourier analysis]]
[[Category:Discrete transforms]]
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[[Category:JPEG]]
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{{Sourceattribution|Discrete cosine transform}}