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A '''discrete cosine transform''' ('''DCT''') expresses a finite sequence of [[data points]] in terms of a sum of [[cosine]] functions oscillating at different [[frequency|frequencies]]. The DCT, first proposed by [[Nasir Ahmed (engineer)|Nasir Ahmed]] in 1972, is a widely used transformation technique in [[signal processing]] and [[data compression]]. It is used in most [[digital media]], including [[digital images]] (such as [[JPEG]] and [[HEIF]]), [[digital video]] (such as [[MPEG]] and {{nowrap|[[H.26x]]}}), [[digital audio]] (such as [[Dolby Digital]], [[MP3]] and [[Advanced Audio Coding|AAC]]), [[digital television]] (such as [[SDTV]], [[HDTV]] and [[Video on demand|VOD]]), [[digital radio]] (such as [[AAC+]] and [[DAB+]]), and [[speech coding]] (such as [[AAC-LD]], [[Siren (codec)|Siren]] and [[Opus (audio format)|Opus]]). DCTs are also important to numerous other applications in [[science and engineering]], such as [[digital signal processing]], [[telecommunication]] devices, reducing [[network bandwidth]] usage, and [[spectral method]]s for the numerical solution of [[partial differential equations]].
A DCT is a [[List of Fourier-related transforms|Fourier-related transform]] similar to the [[discrete Fourier transform]] (DFT), but using only [[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.
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===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 {{nowrap|[[H.26x]]}}, [[MJPEG]], [[MPEG]], [[DV (video format)|DV]], [[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 × 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 {{nowrap|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 x 4 and 8 x 8 blocks. HEVC and HEIF use varied block sizes between 4 x 4 and 32 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=[[Bitmovin]] |year=2019 |access-date=5 November 2019}}</ref>
====Image formats====
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| [[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.
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====Video formats====
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! [[Video coding standard]] !! Year !! Common applications
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| {{nowrap|[[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.
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| [[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|| [[QuickTime]], [[video editing]], [[non-linear editing]], [[digital cameras]]
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| [[MPEG-1]] Video<ref name="Rao">{{cite book | last1 = Rao | first1 = K.R. | author-link1 = K. R. Rao | last2 = Hwang | first2 = J. J. | date = 1996-07-18 | title = Techniques and Standards for Image, Video, and Audio Coding | language = en | publisher = Prentice Hall | at = JPEG: Chapter 8; {{nowrap|H.261}}: Chapter 9; MPEG-1: Chapter 10; MPEG-2: Chapter 11 | isbn = 978-0133099072 | lccn = 96015550 | oclc = 34617596 | ol = OL978319M | s2cid = 56983045 | df = dmy-all}}</ref> ||1993|| [[Digital video]] distribution on [[CD]] or [[Internet video]]
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| [[MPEG-2 Video]] ({{nowrap|H.262}})<ref name="Rao"/> ||1995|| Storage and handling of digital images in broadcast applications, [[digital television]], [[HDTV]], cable, satellite, high-speed [[Internet]], [[DVD]] video distribution
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| [[DV (video format)|DV]] ||1995|| [[Camcorders]], [[digital cassettes]]
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| [[H.263]] ([[MPEG-4 Part 2]])<ref name="video-standards"/> ||1996|| [[Video telephony]] over [[public switched telephone network]] (PSTN), {{nowrap|[[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=[[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={{nowrap|H.263}} is similar to, but more complex than {{nowrap|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>
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| [[Advanced Video Coding]] (AVC,
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| [[Theora]] ||2004|| Internet video, web browsers
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| [[VC-1]] ||2006|| [[Windows]] media, [[Blu-ray Disc]]s
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| [[Apple ProRes]] ||2007|| Professional
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| [[VP9]]||2010|| A video codec developed by [[Google]] used in the [[WebM]] container format with [[HTML5]].
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| [[High Efficiency Video Coding]] (HEVC,
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| [[Daala]] ||2013|| Research video format by [[Xiph.org]]
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| [[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=[[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 [[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=[[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 [[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>
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===MDCT audio standards===
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{{See|Compression artifact}}
A common issue with DCT compression in [[digital media]] are blocky [[compression artifacts]],<ref name="Katsaggelos">{{cite book |last1=Katsaggelos |first1=Aggelos K. |last2=Babacan |first2=S. Derin |last3=Chun-Jen |first3=Tsai |title=The Essential Guide to Image Processing |date=2009 |publisher=[[Academic Press]] |isbn=9780123744579 |pages=349–383|chapter=Chapter 15 - Iterative Image Restoration}}</ref> caused by DCT blocks.<ref name="Alikhani">{{cite web |last1=Alikhani |first1=Darya |title=Beyond resolution: Rosa Menkman's glitch art |url=http://postmatter.merimedia.com/articles/archive-2012-2016/2015/51-rosa-menkman/ |website=POSTmatter |date=April 1, 2015 |access-date=19 October 2019 |archive-date=19 October 2019 |archive-url=https://web.archive.org/web/20191019082218/http://postmatter.merimedia.com/articles/archive-2012-2016/2015/51-rosa-menkman/ |url-status=dead }}</ref> The DCT algorithm can cause block-based artifacts when heavy compression is applied. Due to the DCT being used in the majority of digital image and [[video coding standards]] (such as the [[JPEG]], {{nowrap|[[H.26x]]}} and [[MPEG]] formats), DCT-based blocky compression artifacts are widespread in [[digital media]]. In a DCT algorithm, an image (or frame in an image sequence) is divided into square blocks which are processed independently from each other, then the DCT of these blocks is taken, and the resulting DCT coefficients are [[Quantization (signal processing)|quantized]]. This process can cause blocking artifacts, primarily at high [[data compression ratio]]s.<ref name="Katsaggelos"/> This can also cause the "[[mosquito noise]]" effect, commonly found in [[digital video]] (such as the MPEG formats).<ref>{{cite web |title=Mosquito noise |url=https://www.pcmag.com/encyclopedia/term/55914/mosquito-noise |website=[[PC Magazine]] |access-date=19 October 2019}}</ref>
DCT blocks are often used in [[glitch art]].<ref name="Alikhani"/> The artist [[Rosa Menkman]] makes use of DCT-based compression artifacts in her glitch art,<ref name="Menkman">{{cite book |last1=Menkman |first1=Rosa |title=The Glitch Moment(um) |url=https://networkcultures.org/_uploads/NN%234_RosaMenkman.pdf |publisher=Institute of Network Cultures |isbn=978-90-816021-6-7 |date=October 2011 |access-date=19 October 2019}}</ref> particularly the DCT blocks found in most [[digital media]] formats such as [[JPEG]] digital images and [[MP3]] [[digital audio]].<ref name="Alikhani"/> Another example is ''Jpegs'' by German photographer [[Thomas Ruff]], which uses intentional [[JPEG]] artifacts as the basis of the picture's style.<ref>{{cite book|chapter=jpegs|first=Thomas|last=Ruff|title=Aperture|date=May 31, 2009|page=132|publisher=Aperture |isbn=9781597110938}}</ref><ref>{{cite web|url=http://jmcolberg.com/weblog/2009/04/review_jpegs_by_thomas_ruff/|title=Review: jpegs by Thomas Ruff|first=Jörg|last=Colberg|date=April 17, 2009}}</ref>
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