Editing Discrete cosine transform (section)

== History ==
The DCT was first conceived by [[Nasir Ahmed (engineer)|Nasir Ahmed]] while working at [[Kansas State University]]. The concept was proposed to the [[National Science Foundation]] in 1972. The DCT was originally intended for [[image compression]].<ref name="Ahmed" /><ref name="Stankovic"/> Ahmed developed a practical DCT algorithm with his PhD students T. Raj Natarajan and [[K. R. Rao]] at the [[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=[[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]] (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 (electrical engineer, born 1946)|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]] (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]] per [[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 (electrical engineer, born 1946)|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=[[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]] (MDCT), was developed by John P. Princen, A.W. Johnson and Alan B. Bradley at the [[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 (data)|audio compression]] formats, such as [[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=[[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 [[Vorbis]] ([[Ogg]]).<ref name="vorbis-mdct"/>

Nasir Ahmed also developed a lossless DCT algorithm with Giridhar Mandyam and Neeraj Magotra at the [[University of New Mexico]] in 1995. This allows the DCT technique to be used for [[lossless compression]] of images. It is a modification of the original DCT algorithm, and incorporates elements of inverse DCT and [[delta modulation]]. It is a more effective lossless compression algorithm than [[entropy coding]].<ref>{{cite journal |last1=Mandyam |first1=Giridhar D. |last2=Ahmed |first2=Nasir |author1-link=N. Ahmed |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>