Discrete cosine transform

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class="vector-toc-link" href="#Multidimensional_DCT"> <div class="vector-toc-text"> 2.4 Multidimensional DCT </div> </a> <ul id="toc-Multidimensional_DCT-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Digital_signal_processing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Digital_signal_processing"> <div class="vector-toc-text"> 2.5 Digital signal processing </div> </a> <ul id="toc-Digital_signal_processing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compression_artifacts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compression_artifacts"> <div class="vector-toc-text"> 2.6 Compression artifacts </div> </a> <ul id="toc-Compression_artifacts-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Informal_overview" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Informal_overview"> <div class="vector-toc-text"> 3 Informal overview </div> </a> <ul id="toc-Informal_overview-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> 4 Formal definition </div> </a> <button aria-controls="toc-Formal_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Formal definition subsection </button> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> <li id="toc-DCT-I" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#DCT-I"> <div class="vector-toc-text"> 4.1 DCT-I </div> </a> <ul id="toc-DCT-I-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-DCT-II" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#DCT-II"> <div class="vector-toc-text"> 4.2 DCT-II </div> </a> <ul id="toc-DCT-II-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-DCT-III" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#DCT-III"> <div class="vector-toc-text"> 4.3 DCT-III </div> </a> <ul id="toc-DCT-III-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-DCT-IV" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#DCT-IV"> <div class="vector-toc-text"> 4.4 DCT-IV </div> </a> <ul id="toc-DCT-IV-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-DCT_V-VIII" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#DCT_V-VIII"> <div class="vector-toc-text"> 4.5 DCT V-VIII </div> </a> <ul id="toc-DCT_V-VIII-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Inverse_transforms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inverse_transforms"> <div class="vector-toc-text"> 5 Inverse transforms </div> </a> <ul id="toc-Inverse_transforms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multidimensional_DCTs" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Multidimensional_DCTs"> <div class="vector-toc-text"> 6 Multidimensional DCTs </div> </a> <button aria-controls="toc-Multidimensional_DCTs-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Multidimensional DCTs subsection </button> <ul id="toc-Multidimensional_DCTs-sublist" class="vector-toc-list"> <li id="toc-M-D_DCT-II" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#M-D_DCT-II"> <div class="vector-toc-text"> 6.1 M-D DCT-II </div> </a> <ul id="toc-M-D_DCT-II-sublist" class="vector-toc-list"> <li id="toc-3-D_DCT-II_VR_DIF" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#3-D_DCT-II_VR_DIF"> <div class="vector-toc-text"> 6.1.1 3-D DCT-II VR DIF </div> </a> <ul id="toc-3-D_DCT-II_VR_DIF-sublist" class="vector-toc-list"> <li id="toc-Arithmetic_complexity" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Arithmetic_complexity"> <div class="vector-toc-text"> 6.1.1.1 Arithmetic complexity </div> </a> <ul id="toc-Arithmetic_complexity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-MD-DCT-IV" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#MD-DCT-IV"> <div class="vector-toc-text"> 6.2 MD-DCT-IV </div> </a> <ul id="toc-MD-DCT-IV-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> 7 Computation </div> </a> <ul id="toc-Computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_of_IDCT" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Example_of_IDCT"> <div class="vector-toc-text"> 8 Example of IDCT </div> </a> <ul id="toc-Example_of_IDCT-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 10 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 11 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 12 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 13 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" 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Available in 28 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-28" aria-hidden="true" > 28 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_%D8%AC%D9%8A%D8%A8_%D8%A7%D9%84%D8%AA%D9%85%D8%A7%D9%85_%D8%A7%D9%84%D9%85%D8%AA%D9%82%D8%B7%D8%B9" title="تحويل جيب التمام المتقطع – Arabic" lang="ar" hreflang="ar" data-title="تحويل جيب التمام المتقطع" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Transformada_cosinus_discreta" title="Transformada cosinus discreta – Catalan" lang="ca" hreflang="ca" data-title="Transformada cosinus discreta" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diskr%C3%A9tn%C3%AD_kosinov%C3%A1_transformace" title="Diskrétní kosinová transformace – Czech" lang="cs" hreflang="cs" data-title="Diskrétní kosinová transformace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Diskrete_Kosinustransformation" title="Diskrete Kosinustransformation – German" lang="de" hreflang="de" data-title="Diskrete Kosinustransformation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diskreetne_koosinusteisendus" title="Diskreetne koosinusteisendus – Estonian" lang="et" hreflang="et" data-title="Diskreetne koosinusteisendus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Transformada_de_coseno_discreta" title="Transformada de coseno discreta – Spanish" lang="es" hreflang="es" data-title="Transformada de coseno discreta" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kosinuaren_transformatu_diskretu" title="Kosinuaren transformatu diskretu – Basque" lang="eu" hreflang="eu" data-title="Kosinuaren transformatu diskretu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A8%D8%AF%DB%8C%D9%84_%DA%A9%D8%B3%DB%8C%D9%86%D9%88%D8%B3%DB%8C_%DA%AF%D8%B3%D8%B3%D8%AA%D9%87" title="تبدیل کسینوسی گسسته – Persian" lang="fa" hreflang="fa" data-title="تبدیل کسینوسی گسسته" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Transform%C3%A9e_en_cosinus_discr%C3%A8te" title="Transformée en cosinus discrète – French" lang="fr" hreflang="fr" data-title="Transformée en cosinus discrète" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%EC%82%B0_%EC%BD%94%EC%82%AC%EC%9D%B8_%EB%B3%80%ED%99%98" title="이산 코사인 변환 – Korean" lang="ko" hreflang="ko" data-title="이산 코사인 변환" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Trasformata_discreta_del_coseno" title="Trasformata discreta del coseno – Italian" lang="it" hreflang="it" data-title="Trasformata discreta del coseno" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A1%E0%B2%BF%E0%B2%B8%E0%B3%8D%E0%B2%95%E0%B3%8D%E0%B2%B0%E0%B3%80%E0%B2%9F%E0%B3%8D%E2%80%8C_%E0%B2%95%E0%B3%8A%E0%B2%B8%E0%B3%88%E0%B2%A8%E0%B3%8D%E2%80%8C_%E0%B2%9F%E0%B3%8D%E0%B2%B0%E0%B2%BE%E0%B2%A8%E0%B3%8D%E0%B2%B8%E0%B3%8D%E2%80%8C%E0%B2%AB%E0%B2%BE%E0%B2%B0%E0%B3%8D%E0%B2%AE%E0%B3%8D" title="ಡಿಸ್ಕ್ರೀಟ್‌ ಕೊಸೈನ್‌ ಟ್ರಾನ್ಸ್‌ಫಾರ್ಮ್ – Kannada" lang="kn" hreflang="kn" data-title="ಡಿಸ್ಕ್ರೀಟ್‌ ಕೊಸೈನ್‌ ಟ್ರಾನ್ಸ್‌ಫಾರ್ಮ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D1%82%D1%96%D0%BA_%D0%BA%D0%BE%D1%81%D0%B8%D0%BD%D1%83%D1%81%D1%82%D1%8B%D2%9B_%D1%82%D2%AF%D1%80%D0%BB%D0%B5%D0%BD%D0%B4%D1%96%D1%80%D1%83" title="Дискреттік косинустық түрлендіру – Kazakh" lang="kk" hreflang="kk" data-title="Дискреттік косинустық түрлендіру" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Diskreti_kosinuso_transformacija" title="Diskreti kosinuso transformacija – Lithuanian" lang="lt" hreflang="lt" data-title="Diskreti kosinuso transformacija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Diszkr%C3%A9t_koszinusz-transzform%C3%A1ci%C3%B3" title="Diszkrét koszinusz-transzformáció – Hungarian" lang="hu" hreflang="hu" data-title="Diszkrét koszinusz-transzformáció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Discrete_cosinustransformatie" title="Discrete cosinustransformatie – Dutch" lang="nl" hreflang="nl" data-title="Discrete cosinustransformatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9B%A2%E6%95%A3%E3%82%B3%E3%82%B5%E3%82%A4%E3%83%B3%E5%A4%89%E6%8F%9B" title="離散コサイン変換 – Japanese" lang="ja" hreflang="ja" data-title="離散コサイン変換" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dyskretna_transformacja_kosinusowa" title="Dyskretna transformacja kosinusowa – Polish" lang="pl" hreflang="pl" data-title="Dyskretna transformacja kosinusowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Transformada_discreta_de_cosseno" title="Transformada discreta de cosseno – Portuguese" lang="pt" hreflang="pt" data-title="Transformada discreta de cosseno" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Transformata_cosinus_discret%C4%83" title="Transformata cosinus discretă – Romanian" lang="ro" hreflang="ro" data-title="Transformata cosinus discretă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BD%D0%BE%D0%B5_%D0%BA%D0%BE%D1%81%D0%B8%D0%BD%D1%83%D1%81%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5" title="Дискретное косинусное преобразование – Russian" lang="ru" hreflang="ru" data-title="Дискретное косинусное преобразование" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Discrete_cosine_transform" title="Discrete cosine transform – Simple English" lang="en-simple" hreflang="en-simple" data-title="Discrete cosine transform" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BD%D0%B0_%D0%BA%D0%BE%D1%81%D0%B8%D0%BD%D1%83%D1%81_%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Дискретна косинус трансформација – Serbian" lang="sr" hreflang="sr" data-title="Дискретна косинус трансформација" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diskreetti_kosinimuunnos" title="Diskreetti kosinimuunnos – Finnish" lang="fi" hreflang="fi" data-title="Diskreetti kosinimuunnos" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Diskret_cosinustransform" title="Diskret cosinustransform – Swedish" lang="sv" hreflang="sv" data-title="Diskret cosinustransform" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9B%E0%B8%A5%E0%B8%87%E0%B9%82%E0%B8%84%E0%B9%84%E0%B8%8B%E0%B8%99%E0%B9%8C%E0%B9%84%E0%B8%A1%E0%B9%88%E0%B8%95%E0%B9%88%E0%B8%AD%E0%B9%80%E0%B8%99%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%87" title="การแปลงโคไซน์ไม่ต่อเนื่อง – Thai" lang="th" hreflang="th" data-title="การแปลงโคไซน์ไม่ต่อเนื่อง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B5%D1%82%D0%BD%D0%B5_%D0%BA%D0%BE%D1%81%D0%B8%D0%BD%D1%83%D1%81%D0%BD%D0%B5_%D0%BF%D0%B5%D1%80%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Дискретне 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.ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Jagged_85_cleanup plainlinks metadata ambox ambox-content ambox-jagged-85-cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article may misquote or misrepresent many of its sources. Please see the <a href="/wiki/Wikipedia_talk:Requests_for_comment/Jagged_85/Cleanup" title="Wikipedia talk:Requests for comment/Jagged 85/Cleanup">cleanup page</a> for more information. Editors: please remove this warning only after the diffs listed [[Wikipedia talk:Requests for comment/Jagged 85/{{{subpage}}}|here]] have been checked for accuracy. (July 2022)</div></td></tr></tbody></table> A discrete cosine transform (DCT) expresses a finite sequence of <a href="/wiki/Data_points" class="mw-redirect" title="Data points">data points</a> in terms of a sum of <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> functions oscillating at different <a href="/wiki/Frequency" title="Frequency">frequencies</a>. The DCT, first proposed by <a href="/wiki/Nasir_Ahmed_(engineer)" title="Nasir Ahmed (engineer)">Nasir Ahmed</a> in 1972, is a widely used transformation technique in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> and <a href="/wiki/Data_compression" title="Data compression">data compression</a>. It is used in most <a href="/wiki/Digital_media" title="Digital media">digital media</a>, including <a href="/wiki/Digital_images" class="mw-redirect" title="Digital images">digital images</a> (such as <a href="/wiki/JPEG" title="JPEG">JPEG</a> and <a href="/wiki/HEIF" class="mw-redirect" title="HEIF">HEIF</a>), <a href="/wiki/Digital_video" title="Digital video">digital video</a> (such as <a href="/wiki/MPEG" class="mw-redirect" title="MPEG">MPEG</a> and <a href="/wiki/H.26x" class="mw-redirect" title="H.26x">H.26x</a>), <a href="/wiki/Digital_audio" title="Digital audio">digital audio</a> (such as <a href="/wiki/Dolby_Digital" title="Dolby Digital">Dolby Digital</a>, <a href="/wiki/MP3" title="MP3">MP3</a> and <a href="/wiki/Advanced_Audio_Coding" title="Advanced Audio Coding">AAC</a>), <a href="/wiki/Digital_television" title="Digital television">digital television</a> (such as <a href="/wiki/SDTV" class="mw-redirect" title="SDTV">SDTV</a>, <a href="/wiki/HDTV" class="mw-redirect" title="HDTV">HDTV</a> and <a href="/wiki/Video_on_demand" title="Video on demand">VOD</a>), <a href="/wiki/Digital_radio" title="Digital radio">digital radio</a> (such as <a href="/wiki/AAC%2B" class="mw-redirect" title="AAC+">AAC+</a> and <a href="/wiki/DAB%2B" class="mw-redirect" title="DAB+">DAB+</a>), and <a href="/wiki/Speech_coding" title="Speech coding">speech coding</a> (such as <a href="/wiki/AAC-LD" title="AAC-LD">AAC-LD</a>, <a href="/wiki/Siren_(codec)" title="Siren (codec)">Siren</a> and <a href="/wiki/Opus_(audio_format)" title="Opus (audio format)">Opus</a>). DCTs are also important to numerous other applications in <a href="/wiki/Science_and_engineering" class="mw-redirect" title="Science and engineering">science and engineering</a>, such as <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a>, <a href="/wiki/Telecommunication" class="mw-redirect" title="Telecommunication">telecommunication</a> devices, reducing <a href="/wiki/Network_bandwidth" class="mw-redirect" title="Network bandwidth">network bandwidth</a> usage, and <a href="/wiki/Spectral_method" title="Spectral method">spectral methods</a> for the numerical solution of <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>. A DCT is a <a href="/wiki/List_of_Fourier-related_transforms" title="List of Fourier-related transforms">Fourier-related transform</a> similar to the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (DFT), but using only <a href="/wiki/Real_number" title="Real number">real numbers</a>. The DCTs are generally related to <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> 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 <a href="/wiki/Even_and_odd_functions" title="Even and odd functions">even</a> 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 <a href="/wiki/Discrete_sine_transform" title="Discrete sine transform">discrete sine transform</a> (DST), which is equivalent to a DFT of real and <a href="/wiki/Odd_function" class="mw-redirect" title="Odd function">odd functions</a>, and the <a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">modified discrete cosine transform</a> (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),<a href="#cite_note-Stankovic-1">[1]</a> an <a href="/wiki/Integer" title="Integer">integer</a> approximation of the standard DCT,<a href="#cite_note-Britanak2010-2">[2]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA141">ix, xiii, 1, 141–304</a>  used in several <a href="/wiki/ISO/IEC" class="mw-redirect" title="ISO/IEC">ISO/IEC</a> and <a href="/wiki/ITU-T" title="ITU-T">ITU-T</a> international standards.<a href="#cite_note-Stankovic-1">[1]</a><a href="#cite_note-Britanak2010-2">[2]</a> DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks.<a href="#cite_note-Alikhani-3">[3]</a> DCT blocks sizes including 8x8 <a href="/wiki/Pixels" class="mw-redirect" title="Pixels">pixels</a> for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels.<a href="#cite_note-Stankovic-1">[1]</a><a href="#cite_note-apple-4">[4]</a> The DCT has a strong energy compaction property,<a href="#cite_note-pubDCT-5">[5]</a><a href="#cite_note-pubRaoYip-6">[6]</a> capable of achieving high quality at high <a href="/wiki/Data_compression_ratio" title="Data compression ratio">data compression ratios</a>.<a href="#cite_note-Barbero-7">[7]</a><a href="#cite_note-Lea-8">[8]</a> However, blocky <a href="/wiki/Compression_artifacts" class="mw-redirect" title="Compression artifacts">compression artifacts</a> can appear when heavy DCT compression is applied. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=1" title="Edit section: History">edit</a>]</div> The DCT was first conceived by <a href="/wiki/Nasir_Ahmed_(engineer)" title="Nasir Ahmed (engineer)">Nasir Ahmed</a>, T. Natarajan and <a href="/wiki/K._R._Rao" title="K. R. Rao">K. R. Rao</a> while working at <a href="/wiki/Kansas_State_University" title="Kansas State University">Kansas State University</a>. The concept was proposed to the <a href="/wiki/National_Science_Foundation" title="National Science Foundation">National Science Foundation</a> in 1972. The DCT was originally intended for <a href="/wiki/Image_compression" title="Image compression">image compression</a>.<a href="#cite_note-Ahmed-9">[9]</a><a href="#cite_note-Stankovic-1">[1]</a> Ahmed developed a practical DCT algorithm with his PhD students T. Raj Natarajan, Wills Dietrich, and Jeremy Fries, and his friend Dr. <a href="/wiki/K._R._Rao" title="K. R. Rao">K. R. Rao</a> at the <a href="/wiki/University_of_Texas_at_Arlington" title="University of Texas at Arlington">University of Texas at Arlington</a> in 1973.<a href="#cite_note-Ahmed-9">[9]</a> They presented their results in a January 1974 paper, titled Discrete Cosine Transform.<a href="#cite_note-pubDCT-5">[5]</a><a href="#cite_note-pubRaoYip-6">[6]</a><a href="#cite_note-t81-10">[10]</a> It described what is now called the type-II DCT (DCT-II),<a href="#cite_note-Britanak2010-2">[2]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA51">51</a>  as well as the type-III inverse DCT (IDCT).<a href="#cite_note-pubDCT-5">[5]</a> Since its introduction in 1974, there has been significant research on the DCT.<a href="#cite_note-t81-10">[10]</a> In 1977, Wen-Hsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm.<a href="#cite_note-A_Fast_Computational_Algorithm_for-11">[11]</a><a href="#cite_note-t81-10">[10]</a> Further developments include a 1978 paper by M. J. Narasimha and A. M. Peterson, and a 1984 paper by B. G. Lee.<a href="#cite_note-t81-10">[10]</a> These research papers, along with the original 1974 Ahmed paper and the 1977 Chen paper, were cited by the <a href="/wiki/Joint_Photographic_Experts_Group" title="Joint Photographic Experts Group">Joint Photographic Experts Group</a> as the basis for <a href="/wiki/JPEG" title="JPEG">JPEG</a>'s lossy image compression algorithm in 1992.<a href="#cite_note-t81-10">[10]</a><a href="#cite_note-chen-12">[12]</a> The <a href="/wiki/Discrete_sine_transform" title="Discrete sine transform">discrete sine transform</a> (DST) was derived from the DCT, by replacing the <a href="/wiki/Neumann_boundary_condition" title="Neumann boundary condition">Neumann condition</a> at x=0 with a <a href="/wiki/Dirichlet_condition" class="mw-redirect" title="Dirichlet condition">Dirichlet condition</a>.<a href="#cite_note-Britanak2010-2">[2]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA35">35-36</a>  The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao.<a href="#cite_note-pubDCT-5">[5]</a> A type-I DST (DST-I) was later described by <a href="/wiki/Anil_K._Jain_(electrical_engineer,_born_1946)" title="Anil K. Jain (electrical engineer, born 1946)">Anil K. Jain</a> in 1976, and a type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.<a href="#cite_note-13">[13]</a> In 1975, John A. Roese and Guner S. Robinson adapted the DCT for <a href="/wiki/Inter-frame" class="mw-redirect" title="Inter-frame">inter-frame</a> <a href="/wiki/Motion_compensation" title="Motion compensation">motion-compensated</a> <a href="/wiki/Video_coding" class="mw-redirect" title="Video coding">video coding</a>. They experimented with the DCT and the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (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-<a href="/wiki/Bit" title="Bit">bit</a> per <a href="/wiki/Pixel" title="Pixel">pixel</a> for a <a href="/wiki/Videotelephone" class="mw-redirect" title="Videotelephone">videotelephone</a> scene with image quality comparable to an <a href="/wiki/Intra-frame_coding" title="Intra-frame coding">intra-frame coder</a> requiring 2-bit per pixel.<a href="#cite_note-14">[14]</a><a href="#cite_note-Roese-15">[15]</a> In 1979, <a href="/wiki/Anil_K._Jain_(electrical_engineer,_born_1946)" title="Anil K. Jain (electrical engineer, born 1946)">Anil K. Jain</a> and Jaswant R. Jain further developed motion-compensated DCT video compression,<a href="#cite_note-16">[16]</a><a href="#cite_note-ITU-17">[17]</a> also called block motion compensation.<a href="#cite_note-ITU-17">[17]</a> This led to Chen developing a practical video compression algorithm, called motion-compensated DCT or adaptive scene coding, in 1981.<a href="#cite_note-ITU-17">[17]</a> Motion-compensated DCT later became the standard coding technique for video compression from the late 1980s onwards.<a href="#cite_note-Ghanbari-18">[18]</a><a href="#cite_note-Li-19">[19]</a> A DCT variant, the <a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">modified discrete cosine transform</a> (MDCT), was developed by John P. Princen, A.W. Johnson and Alan B. Bradley at the <a href="/wiki/University_of_Surrey" title="University of Surrey">University of Surrey</a> in 1987,<a href="#cite_note-20">[20]</a> following earlier work by Princen and Bradley in 1986.<a href="#cite_note-21">[21]</a> The MDCT is used in most modern <a href="/wiki/Audio_compression_(data)" class="mw-redirect" title="Audio compression (data)">audio compression</a> formats, such as <a href="/wiki/Dolby_Digital" title="Dolby Digital">Dolby Digital</a> (AC-3),<a href="#cite_note-Luo-22">[22]</a><a href="#cite_note-Britanak2011-23">[23]</a> <a href="/wiki/MP3" title="MP3">MP3</a> (which uses a hybrid DCT-<a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a> algorithm),<a href="#cite_note-Guckert-24">[24]</a> <a href="/wiki/Advanced_Audio_Coding" title="Advanced Audio Coding">Advanced Audio Coding</a> (AAC),<a href="#cite_note-brandenburg-25">[25]</a> and <a href="/wiki/Vorbis" title="Vorbis">Vorbis</a> (<a href="/wiki/Ogg" title="Ogg">Ogg</a>).<a href="#cite_note-vorbis-mdct-26">[26]</a> Nasir Ahmed also developed a lossless DCT algorithm with Giridhar Mandyam and Neeraj Magotra at the <a href="/wiki/University_of_New_Mexico" title="University of New Mexico">University of New Mexico</a> in 1995. This allows the DCT technique to be used for <a href="/wiki/Lossless_compression" title="Lossless compression">lossless compression</a> of images. It is a modification of the original DCT algorithm, and incorporates elements of inverse DCT and <a href="/wiki/Delta_modulation" title="Delta modulation">delta modulation</a>. It is a more effective lossless compression algorithm than <a href="/wiki/Entropy_coding" title="Entropy coding">entropy coding</a>.<a href="#cite_note-27">[27]</a> Lossless DCT is also known as LDCT.<a href="#cite_note-28">[28]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=2" title="Edit section: Applications">edit</a>]</div> The DCT is the most widely used transformation technique in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>,<a href="#cite_note-Muchahary-29">[29]</a> and by far the most widely used linear transform in <a href="/wiki/Data_compression" title="Data compression">data compression</a>.<a href="#cite_note-30">[30]</a> Uncompressed <a href="/wiki/Digital_media" title="Digital media">digital media</a> as well as <a href="/wiki/Lossless_compression" title="Lossless compression">lossless compression</a> have high <a href="/wiki/Computer_memory" title="Computer memory">memory</a> and <a href="/wiki/Bandwidth_(computing)" title="Bandwidth (computing)">bandwidth</a> requirements, which is significantly reduced by the DCT <a href="/wiki/Lossy_compression" title="Lossy compression">lossy compression</a> technique,<a href="#cite_note-Barbero-7">[7]</a><a href="#cite_note-Lea-8">[8]</a> capable of achieving <a href="/wiki/Data_compression_ratio" title="Data compression ratio">data compression ratios</a> from 8:1 to 14:1 for near-studio-quality,<a href="#cite_note-Barbero-7">[7]</a> up to 100:1 for acceptable-quality content.<a href="#cite_note-Lea-8">[8]</a> DCT compression standards are used in digital media technologies, such as <a href="/wiki/Digital_images" class="mw-redirect" title="Digital images">digital images</a>, <a href="/wiki/Digital_photo" class="mw-redirect" title="Digital photo">digital photos</a>,<a href="#cite_note-Atlantic-31">[31]</a><a href="#cite_note-epfl-32">[32]</a> <a href="/wiki/Digital_video" title="Digital video">digital video</a>,<a href="#cite_note-Ghanbari-18">[18]</a><a href="#cite_note-Lee1995-33">[33]</a> <a href="/wiki/Streaming_media" title="Streaming media">streaming media</a>,<a href="#cite_note-Lee-34">[34]</a> <a href="/wiki/Digital_television" title="Digital television">digital television</a>, <a href="/wiki/Streaming_television" title="Streaming television">streaming television</a>, <a href="/wiki/Video_on_demand" title="Video on demand">video on demand</a> (VOD),<a href="#cite_note-Lea-8">[8]</a> <a href="/wiki/Digital_cinema" title="Digital cinema">digital cinema</a>,<a href="#cite_note-Luo-22">[22]</a> <a href="/wiki/High-definition_video" title="High-definition video">high-definition video</a> (HD video), and <a href="/wiki/High-definition_television" title="High-definition television">high-definition television</a> (HDTV).<a href="#cite_note-Barbero-7">[7]</a><a href="#cite_note-Shishikui-35">[35]</a> 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.<a href="#cite_note-pubDCT-5">[5]</a><a href="#cite_note-pubRaoYip-6">[6]</a> In typical applications, most of the signal information tends to be concentrated in a few low-frequency components of the DCT. For strongly correlated <a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov processes</a>, the DCT can approach the compaction efficiency of the <a href="/wiki/Karhunen-Lo%C3%A8ve_transform" class="mw-redirect" title="Karhunen-Loève transform">Karhunen-Loève transform</a> (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 <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> by <a href="/wiki/Spectral_methods" class="mw-redirect" title="Spectral methods">spectral methods</a>, 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 <a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a>, and fast DCT algorithms (below) are used in <a href="/wiki/Chebyshev_approximation" class="mw-redirect" title="Chebyshev approximation">Chebyshev approximation</a> of arbitrary functions by series of Chebyshev polynomials, for example in <a href="/wiki/Clenshaw%E2%80%93Curtis_quadrature" title="Clenshaw–Curtis quadrature">Clenshaw–Curtis quadrature</a>. <div class="mw-heading mw-heading3"><h3 id="General_applications">General applications</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=3" title="Edit section: General applications">edit</a>]</div> The DCT is widely used in many applications, which include the following. <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 50em;"> <ul><li><a href="/wiki/Audio_signal_processing" title="Audio signal processing">Audio signal processing</a> — <a href="/wiki/Audio_coding" class="mw-redirect" title="Audio coding">audio coding</a>, <a href="/wiki/Audio_data_compression" class="mw-redirect" title="Audio data compression">audio data compression</a> (lossy and lossless),<a href="#cite_note-Ochoa129-36">[36]</a> <a href="/wiki/Surround_sound" title="Surround sound">surround sound</a>,<a href="#cite_note-Luo-22">[22]</a> <a href="/wiki/Acoustic_echo_cancellation" class="mw-redirect" title="Acoustic echo cancellation">acoustic echo</a> and <a href="/wiki/Adaptive_feedback_cancellation" title="Adaptive feedback cancellation">feedback cancellation</a>, <a href="/wiki/Phoneme" title="Phoneme">phoneme</a> recognition, <a href="/wiki/Time-domain_aliasing_cancellation" class="mw-redirect" title="Time-domain aliasing cancellation">time-domain aliasing cancellation</a> (TDAC)<a href="#cite_note-Ochoa-37">[37]</a> <ul><li><a href="/wiki/Digital_audio" title="Digital audio">Digital audio</a><a href="#cite_note-Stankovic-1">[1]</a></li> <li><a href="/wiki/Digital_radio" title="Digital radio">Digital radio</a> — <a href="/wiki/Digital_Audio_Broadcasting" title="Digital Audio Broadcasting">Digital Audio Broadcasting</a> (DAB+),<a href="#cite_note-Britanak-38">[38]</a> <a href="/wiki/HD_Radio" title="HD Radio">HD Radio</a><a href="#cite_note-Jones-39">[39]</a></li> <li><a href="/wiki/Speech_processing" title="Speech processing">Speech processing</a> — <a href="/wiki/Speech_coding" title="Speech coding">speech coding</a><a href="#cite_note-Hersent-40">[40]</a><a href="#cite_note-AppleInsider_standards_1-41">[41]</a> <a href="/wiki/Speech_recognition" title="Speech recognition">speech recognition</a>, <a href="/wiki/Voice_activity_detection" title="Voice activity detection">voice activity detection</a> (VAD)<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Digital_telephony" class="mw-redirect" title="Digital telephony">Digital telephony</a> — <a href="/wiki/Voice_over_IP" title="Voice over IP">voice over IP</a> (VoIP),<a href="#cite_note-Hersent-40">[40]</a> <a href="/wiki/Mobile_telephony" title="Mobile telephony">mobile telephony</a>, <a href="/wiki/Video_telephony" class="mw-redirect" title="Video telephony">video telephony</a>,<a href="#cite_note-AppleInsider_standards_1-41">[41]</a> <a href="/wiki/Teleconferencing" class="mw-redirect" title="Teleconferencing">teleconferencing</a>, <a href="/wiki/Videoconferencing" class="mw-redirect" title="Videoconferencing">videoconferencing</a><a href="#cite_note-Stankovic-1">[1]</a></li></ul></li> <li><a href="/wiki/Biometrics" title="Biometrics">Biometrics</a> — <a href="/wiki/Fingerprint" title="Fingerprint">fingerprint</a> orientation, <a href="/wiki/Facial_recognition_systems" class="mw-redirect" title="Facial recognition systems">facial recognition systems</a>, biometric <a href="/wiki/Digital_watermarking" title="Digital watermarking">watermarking</a>, fingerprint-based biometric watermarking, <a href="/wiki/Palm_print" title="Palm print">palm print</a> identification/recognition<a href="#cite_note-Ochoa-37">[37]</a> <ul><li><a href="/wiki/Face_detection" title="Face detection">Face detection</a> — <a href="/wiki/Facial_recognition_system" title="Facial recognition system">facial recognition</a><a href="#cite_note-Ochoa-37">[37]</a></li></ul></li> <li><a href="/wiki/Computers" class="mw-redirect" title="Computers">Computers</a> and the <a href="/wiki/Internet" title="Internet">Internet</a> — the <a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a>, <a href="/wiki/Social_media" title="Social media">social media</a>,<a href="#cite_note-Atlantic-31">[31]</a><a href="#cite_note-epfl-32">[32]</a> <a href="/wiki/Internet_video" title="Internet video">Internet video</a><a href="#cite_note-Encodes-42">[42]</a> <ul><li><a href="/wiki/Network_bandwidth" class="mw-redirect" title="Network bandwidth">Network bandwidth</a> usage reducation<a href="#cite_note-Stankovic-1">[1]</a></li></ul></li> <li><a href="/wiki/Consumer_electronics" title="Consumer electronics">Consumer electronics</a><a href="#cite_note-Ochoa-37">[37]</a> — <a href="/wiki/Multimedia" title="Multimedia">multimedia</a> systems,<a href="#cite_note-Stankovic-1">[1]</a> multimedia <a href="/wiki/Telecommunication" class="mw-redirect" title="Telecommunication">telecommunication</a> devices,<a href="#cite_note-Stankovic-1">[1]</a> consumer devices<a href="#cite_note-Encodes-42">[42]</a></li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a> — <a href="/wiki/Encryption" title="Encryption">encryption</a>, <a href="/wiki/Steganography" title="Steganography">steganography</a>, <a href="/wiki/Copyright" title="Copyright">copyright</a> protection<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Data_compression" title="Data compression">Data compression</a> — <a href="/wiki/Transform_coding" title="Transform coding">transform coding</a>, <a href="/wiki/Lossy_compression" title="Lossy compression">lossy compression</a>, <a href="/wiki/Lossless_compression" title="Lossless compression">lossless compression</a><a href="#cite_note-Ochoa129-36">[36]</a> <ul><li><a href="/wiki/Encoding" class="mw-redirect" title="Encoding">Encoding</a> operations — <a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">quantization</a>, perceptual weighting, <a href="/wiki/Entropy_encoding" class="mw-redirect" title="Entropy encoding">entropy encoding</a>, <a href="/wiki/Variable_bitrate_encoding" class="mw-redirect" title="Variable bitrate encoding">variable bitrate encoding</a><a href="#cite_note-Stankovic-1">[1]</a></li></ul></li> <li><a href="/wiki/Digital_media" title="Digital media">Digital media</a><a href="#cite_note-Lee-34">[34]</a> — <a href="/wiki/Digital_distribution" title="Digital distribution">digital distribution</a><a href="#cite_note-Bitmovin-43">[43]</a> <ul><li><a href="/wiki/Streaming_media" title="Streaming media">Streaming media</a><a href="#cite_note-Lee-34">[34]</a> — <a href="/wiki/Streaming_audio" class="mw-redirect" title="Streaming audio">streaming audio</a>, <a href="/wiki/Streaming_video" class="mw-redirect" title="Streaming video">streaming video</a>, <a href="/wiki/Streaming_television" title="Streaming television">streaming television</a>, <a href="/wiki/Video-on-demand" class="mw-redirect" title="Video-on-demand">video-on-demand</a> (VOD)<a href="#cite_note-Lea-8">[8]</a></li></ul></li> <li><a href="/wiki/Forgery_detection" class="mw-redirect" title="Forgery detection">Forgery detection</a><a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Geophysical" class="mw-redirect" title="Geophysical">Geophysical</a> <a href="/wiki/Transient_electromagnetics" title="Transient electromagnetics">transient electromagnetics</a> (transient EM)<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Image" title="Image">Images</a> — <a href="/wiki/Artist" title="Artist">artist</a> identification,<a href="#cite_note-Ochoa-37">[37]</a> <a href="/wiki/Focus_(optics)" title="Focus (optics)">focus</a> and <a href="/wiki/Bokeh" title="Bokeh">blurriness</a> measure,<a href="#cite_note-Ochoa-37">[37]</a> <a href="/wiki/Feature_extraction" class="mw-redirect" title="Feature extraction">feature extraction</a><a href="#cite_note-Ochoa-37">[37]</a> <ul><li><a href="/wiki/Color" title="Color">Color</a> formatting — formatting <a href="/wiki/Luminance" title="Luminance">luminance</a> and color differences, color formats (such as <a href="/wiki/YUV444" class="mw-redirect" title="YUV444">YUV444</a> and <a href="/wiki/YUV411" class="mw-redirect" title="YUV411">YUV411</a>), <a href="/wiki/Encoding" class="mw-redirect" title="Encoding">decoding</a> operations such as the inverse operation between display color formats (<a href="/wiki/YIQ" title="YIQ">YIQ</a>, <a href="/wiki/YUV" class="mw-redirect" title="YUV">YUV</a>, <a href="/wiki/RGB" class="mw-redirect" title="RGB">RGB</a>)<a href="#cite_note-Stankovic-1">[1]</a></li> <li><a href="/wiki/Digital_imaging" title="Digital imaging">Digital imaging</a> — <a href="/wiki/Digital_image" title="Digital image">digital images</a>, <a href="/wiki/Digital_camera" title="Digital camera">digital cameras</a>, <a href="/wiki/Digital_photography" title="Digital photography">digital photography</a>,<a href="#cite_note-Atlantic-31">[31]</a><a href="#cite_note-epfl-32">[32]</a> <a href="/wiki/High-dynamic-range_imaging" class="mw-redirect" title="High-dynamic-range imaging">high-dynamic-range imaging</a> (HDR imaging)<a href="#cite_note-44">[44]</a></li> <li><a href="/wiki/Image_compression" title="Image compression">Image compression</a><a href="#cite_note-Ochoa-37">[37]</a><a href="#cite_note-McKernan58-45">[45]</a> — <a href="/wiki/Image_file_format" title="Image file format">image file formats</a>,<a href="#cite_note-Baraniuk-46">[46]</a> <a href="/wiki/2D-plus-depth" title="2D-plus-depth">multiview image</a> compression, <a href="/wiki/Progressive_JPEG" class="mw-redirect" title="Progressive JPEG">progressive image</a> transmission<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">Image processing</a> — <a href="/wiki/Digital_image_processing" title="Digital image processing">digital image processing</a>,<a href="#cite_note-Stankovic-1">[1]</a> <a href="/wiki/Image_analysis" title="Image analysis">image analysis</a>, <a href="/wiki/Content-based_image_retrieval" title="Content-based image retrieval">content-based image retrieval</a>, <a href="/wiki/Corner_detection" title="Corner detection">corner detection</a>, directional block-wise <a href="/wiki/Sparse_approximation" title="Sparse approximation">image representation</a>, <a href="/wiki/Edge_detection" title="Edge detection">edge detection</a>, <a href="/wiki/Image_enhancement" class="mw-redirect" title="Image enhancement">image enhancement</a>, <a href="/wiki/Image_fusion" title="Image fusion">image fusion</a>, <a href="/wiki/Image_segmentation" title="Image segmentation">image segmentation</a>, <a href="/wiki/Interpolation" title="Interpolation">interpolation</a>, <a href="/wiki/Image_noise" title="Image noise">image noise</a> level estimation, mirroring, rotation, <a href="/wiki/Just-noticeable_difference" title="Just-noticeable difference">just-noticeable distortion</a> (JND) profile, <a href="/wiki/Spatiotemporal" class="mw-redirect" title="Spatiotemporal">spatiotemporal</a> masking effects, <a href="/wiki/Foveated_imaging" title="Foveated imaging">foveated imaging</a><a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Image_quality" title="Image quality">Image quality</a> assessment — DCT-based quality degradation metric (DCT QM)<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Image_reconstruction" class="mw-redirect" title="Image reconstruction">Image reconstruction</a> — directional <a href="/wiki/Image_texture" title="Image texture">textures</a> auto inspection, image restoration, <a href="/wiki/Inpainting" title="Inpainting">inpainting</a>, <a href="/wiki/Photo_recovery" title="Photo recovery">visual recovery</a><a href="#cite_note-Ochoa-37">[37]</a></li></ul></li> <li><a href="/wiki/Medical_technology" class="mw-redirect" title="Medical technology">Medical technology</a> <ul><li><a href="/wiki/Electrocardiography" title="Electrocardiography">Electrocardiography</a> (ECG) — <a href="/wiki/Vectorcardiography" title="Vectorcardiography">vectorcardiography</a> (VCG)<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Medical_imaging" title="Medical imaging">Medical imaging</a> — medical image compression, image fusion, watermarking, <a href="/wiki/Brain_tumor" title="Brain tumor">brain tumor</a> <a href="/wiki/Brain_compression" class="mw-redirect" title="Brain compression">compression</a> classification<a href="#cite_note-Ochoa-37">[37]</a></li></ul></li> <li><a href="/wiki/Pattern_recognition" title="Pattern recognition">Pattern recognition</a><a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Region_of_interest" title="Region of interest">Region of interest</a> (ROI) extraction<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a> — <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a>, <a href="/wiki/Digital_signal_processor" title="Digital signal processor">digital signal processors</a> (DSP), DSP <a href="/wiki/Software" title="Software">software</a>, <a href="/wiki/Multiplexing" title="Multiplexing">multiplexing</a>, <a href="/wiki/Signaling" class="mw-redirect" title="Signaling">signaling</a>, control signals, <a href="/wiki/Analog-to-digital_conversion" class="mw-redirect" title="Analog-to-digital conversion">analog-to-digital conversion</a> (ADC),<a href="#cite_note-Stankovic-1">[1]</a> <a href="/wiki/Compressive_sampling" class="mw-redirect" title="Compressive sampling">compressive sampling</a>, DCT pyramid <a href="/wiki/Error_concealment" title="Error concealment">error concealment</a>, <a href="/wiki/Downsampling" class="mw-redirect" title="Downsampling">downsampling</a>, <a href="/wiki/Upsampling" title="Upsampling">upsampling</a>, <a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a> (SNR) estimation, <a href="/wiki/Transmux" title="Transmux">transmux</a>, <a href="/wiki/Wiener_filter" title="Wiener filter">Wiener filter</a><a href="#cite_note-Ochoa-37">[37]</a> <ul><li><a href="/wiki/Complex_cepstrum" class="mw-redirect" title="Complex cepstrum">Complex cepstrum</a> feature analysis<a href="#cite_note-Ochoa-37">[37]</a></li> <li>DCT <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">filtering</a><a href="#cite_note-Ochoa-37">[37]</a></li></ul></li> <li><a href="/wiki/Surveillance" title="Surveillance">Surveillance</a><a href="#cite_note-Ochoa-37">[37]</a></li> <li>Vehicular <a href="/wiki/Event_data_recorder" title="Event data recorder">event data recorder</a> camera<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Video" title="Video">Video</a> <ul><li><a href="/wiki/Digital_cinema" title="Digital cinema">Digital cinema</a><a href="#cite_note-McKernan58-45">[45]</a> — <a href="/wiki/Digital_cinematography" title="Digital cinematography">digital cinematography</a>, <a href="/wiki/Digital_movie_camera" title="Digital movie camera">digital movie cameras</a>, <a href="/wiki/Video_editing" title="Video editing">video editing</a>, <a href="/wiki/Film_editing" title="Film editing">film editing</a>,<a href="#cite_note-47">[47]</a><a href="#cite_note-48">[48]</a> <a href="/wiki/Dolby_Digital" title="Dolby Digital">Dolby Digital</a> audio<a href="#cite_note-Stankovic-1">[1]</a><a href="#cite_note-Luo-22">[22]</a></li> <li><a href="/wiki/Digital_television" title="Digital television">Digital television</a> (DTV)<a href="#cite_note-Barbero-7">[7]</a> — <a href="/wiki/Digital_television_broadcasting" class="mw-redirect" title="Digital television broadcasting">digital television broadcasting</a>,<a href="#cite_note-McKernan58-45">[45]</a> <a href="/wiki/Standard-definition_television" title="Standard-definition television">standard-definition television</a> (SDTV), <a href="/wiki/High-definition_TV" class="mw-redirect" title="High-definition TV">high-definition TV</a> (HDTV),<a href="#cite_note-Barbero-7">[7]</a><a href="#cite_note-Shishikui-35">[35]</a> HDTV <a href="/wiki/Video_decoder" title="Video decoder">encoder/decoder chips</a>, <a href="/wiki/Ultra_HDTV" class="mw-redirect" title="Ultra HDTV">ultra HDTV</a> (UHDTV)<a href="#cite_note-Stankovic-1">[1]</a></li> <li><a href="/wiki/Digital_video" title="Digital video">Digital video</a><a href="#cite_note-Ghanbari-18">[18]</a><a href="#cite_note-Lee1995-33">[33]</a> — <a href="/wiki/Digital_versatile_disc" class="mw-redirect" title="Digital versatile disc">digital versatile disc</a> (DVD),<a href="#cite_note-McKernan58-45">[45]</a> <a href="/wiki/High-definition_video" title="High-definition video">high-definition</a> (HD) video<a href="#cite_note-Barbero-7">[7]</a><a href="#cite_note-Shishikui-35">[35]</a></li> <li><a href="/wiki/Video_coding" class="mw-redirect" title="Video coding">Video coding</a> — <a href="/wiki/Video_compression" class="mw-redirect" title="Video compression">video compression</a>,<a href="#cite_note-Stankovic-1">[1]</a> <a href="/wiki/Video_coding_standards" class="mw-redirect" title="Video coding standards">video coding standards</a>,<a href="#cite_note-Ochoa-37">[37]</a> <a href="/wiki/Motion_estimation" title="Motion estimation">motion estimation</a>, <a href="/wiki/Motion_compensation" title="Motion compensation">motion compensation</a>, <a href="/wiki/Inter-frame" class="mw-redirect" title="Inter-frame">inter-frame</a> prediction, <a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">motion vectors</a>,<a href="#cite_note-Stankovic-1">[1]</a> <a href="/wiki/Stereoscopic_video_coding" title="Stereoscopic video coding">3D video coding</a>, local distortion detection probability (LDDP) model, <a href="/wiki/Moving_object_detection" title="Moving object detection">moving object detection</a>, <a href="/wiki/Multiview_Video_Coding" title="Multiview Video Coding">Multiview Video Coding</a> (MVC)<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Video_processing" title="Video processing">Video processing</a> — <a href="/wiki/Motion_analysis" title="Motion analysis">motion analysis</a>, 3D-DCT motion analysis, <a href="/wiki/Video_content_analysis" title="Video content analysis">video content analysis</a>, <a href="/wiki/Data_extraction" title="Data extraction">data extraction</a>,<a href="#cite_note-Ochoa-37">[37]</a> <a href="/wiki/Video_browsing" title="Video browsing">video browsing</a>,<a href="#cite_note-49">[49]</a> professional <a href="/wiki/Video_production" title="Video production">video production</a><a href="#cite_note-loc-50">[50]</a></li></ul></li> <li><a href="/wiki/Watermark" title="Watermark">Watermarking</a> — <a href="/wiki/Digital_watermarking" title="Digital watermarking">digital watermarking</a>, <a href="/wiki/Image_watermarking" class="mw-redirect" title="Image watermarking">image watermarking</a>, video watermarking, <a href="/wiki/3D_video" class="mw-redirect" title="3D video">3D video</a> watermarking, <a href="/wiki/Digital_watermarking#Reversible_data_hiding" title="Digital watermarking">reversible data hiding</a>, watermarking detection<a href="#cite_note-Ochoa-37">[37]</a></li> <li><a href="/wiki/Wireless" title="Wireless">Wireless</a> technology <ul><li><a href="/wiki/Mobile_devices" class="mw-redirect" title="Mobile devices">Mobile devices</a><a href="#cite_note-Encodes-42">[42]</a> — <a href="/wiki/Mobile_phones" class="mw-redirect" title="Mobile phones">mobile phones</a>, <a href="/wiki/Smartphones" class="mw-redirect" title="Smartphones">smartphones</a>,<a href="#cite_note-AppleInsider_standards_1-41">[41]</a> <a href="/wiki/Videophones" class="mw-redirect" title="Videophones">videophones</a><a href="#cite_note-Stankovic-1">[1]</a></li> <li><a href="/wiki/Radio_frequency" title="Radio frequency">Radio frequency</a> (RF) technology — <a href="/wiki/RF_engineering" class="mw-redirect" title="RF engineering">RF engineering</a>, <a href="/wiki/Aperture_synthesis" title="Aperture synthesis">aperture</a> <a href="/wiki/Sensor_array" title="Sensor array">arrays</a>,<a href="#cite_note-Ochoa-37">[37]</a> <a href="/wiki/Beamforming" title="Beamforming">beamforming</a>, <a href="/wiki/Digital_electronics" title="Digital electronics">digital</a> <a href="/wiki/Arithmetic_circuit" class="mw-redirect" title="Arithmetic circuit">arithmetic circuits</a>, directional <a href="/wiki/Sensor" title="Sensor">sensing</a>, <a href="/wiki/Astronomical_image_processing" class="mw-redirect" title="Astronomical image processing">space imaging</a><a href="#cite_note-51">[51]</a></li></ul></li> <li><a href="/wiki/Wireless_sensor_network" title="Wireless sensor network">Wireless sensor network</a> (WSN) — wireless <a href="/wiki/Surface_acoustic_wave_sensor" title="Surface acoustic wave sensor">acoustic sensor</a> networks<a href="#cite_note-Ochoa-37">[37]</a></li></ul></div> <div class="mw-heading mw-heading3"><h3 id="Visual_media_standards">Visual media standards</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=4" title="Edit section: Visual media standards">edit</a>]</div> The DCT-II is an important image compression technique. It is used in image compression standards such as <a href="/wiki/JPEG" title="JPEG">JPEG</a>, and <a href="/wiki/Video_compression" class="mw-redirect" title="Video compression">video compression</a> standards such as <a href="/wiki/H.26x" class="mw-redirect" title="H.26x">H.26x</a>, <a href="/wiki/MJPEG" class="mw-redirect" title="MJPEG">MJPEG</a>, <a href="/wiki/MPEG" class="mw-redirect" title="MPEG">MPEG</a>, <a href="/wiki/DV_(video_format)" title="DV (video format)">DV</a>, <a href="/wiki/Theora" title="Theora">Theora</a> and <a href="/wiki/Daala" title="Daala">Daala</a>. There, the two-dimensional DCT-II of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\times N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\times N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a86c5231bb3cbb863d9d428ebe9ac8db8d4ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.968ex; height:2.176ex;" alt="{\displaystyle N\times N}"> blocks are computed and the results are <a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">quantized</a> and <a href="/wiki/Entropy_encoding" class="mw-redirect" title="Entropy encoding">entropy coded</a>. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"> 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,<a href="#cite_note-Britanak2010-2">[2]</a><a href="#cite_note-Stankovic-1">[1]</a> is used in <a href="/wiki/Advanced_Video_Coding" title="Advanced Video Coding">Advanced Video Coding</a> (AVC),<a href="#cite_note-Wang-52">[52]</a><a href="#cite_note-Stankovic-1">[1]</a> introduced in 2003, and <a href="/wiki/High_Efficiency_Video_Coding" title="High Efficiency Video Coding">High Efficiency Video Coding</a> (HEVC),<a href="#cite_note-apple-4">[4]</a><a href="#cite_note-Stankovic-1">[1]</a> introduced in 2013. The integer DCT is also used in the <a href="/wiki/High_Efficiency_Image_Format" class="mw-redirect" title="High Efficiency Image Format">High Efficiency Image Format</a> (HEIF), which uses a subset of the <a href="/wiki/HEVC" class="mw-redirect" title="HEVC">HEVC</a> video coding format for coding still images.<a href="#cite_note-apple-4">[4]</a> 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 <a href="/wiki/Pixels" class="mw-redirect" title="Pixels">pixels</a>.<a href="#cite_note-apple-4">[4]</a><a href="#cite_note-Stankovic-1">[1]</a> As of 2019<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Discrete_cosine_transform&action=edit">[update]</a>, 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.<a href="#cite_note-Bitmovin-43">[43]</a> <div class="mw-heading mw-heading4"><h4 id="Image_formats">Image formats</h4>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=5" title="Edit section: Image formats">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th>Image compression standard</th> <th>Year</th> <th>Common applications </th></tr> <tr> <td><a href="/wiki/JPEG" title="JPEG">JPEG</a><a href="#cite_note-Stankovic-1">[1]</a></td> <td>1992</td> <td>The most widely used image compression standard<a href="#cite_note-Hudson-53">[53]</a><a href="#cite_note-54">[54]</a> and digital <a href="/wiki/Image_format" class="mw-redirect" title="Image format">image format</a>.<a href="#cite_note-Baraniuk-46">[46]</a> </td></tr> <tr> <td><a href="/wiki/JPEG_XR" title="JPEG XR">JPEG XR</a></td> <td>2009</td> <td><a href="/wiki/Open_XML_Paper_Specification" title="Open XML Paper Specification">Open XML Paper Specification</a> </td></tr> <tr> <td><a href="/wiki/WebP" title="WebP">WebP</a></td> <td>2010</td> <td>A graphic format that supports the lossy compression of digital images. Developed by <a href="/wiki/Google" title="Google">Google</a>. </td></tr> <tr> <td><a href="/wiki/High_Efficiency_Image_Format" class="mw-redirect" title="High Efficiency Image Format">High Efficiency Image Format</a> (HEIF)</td> <td>2013</td> <td><a href="/wiki/Image_file_format" title="Image file format">Image file format</a> based on HEVC compression. It improves compression over JPEG,<a href="#cite_note-apple-4">[4]</a> and supports <a href="/wiki/Animation" title="Animation">animation</a> with much more efficient compression than the <a href="/wiki/Animated_GIF" class="mw-redirect" title="Animated GIF">animated GIF</a> format.<a href="#cite_note-55">[55]</a> </td></tr> <tr> <td><a href="/wiki/Better_Portable_Graphics" title="Better Portable Graphics">BPG</a></td> <td>2014</td> <td>Based on HEVC compression </td></tr> <tr> <td><a href="/wiki/JPEG_XL" title="JPEG XL">JPEG XL</a><a href="#cite_note-jxl-56">[56]</a></td> <td>2020</td> <td>A royalty-free raster-graphics file format that supports both lossy and lossless compression. </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Video_formats">Video formats</h4>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=6" title="Edit section: Video formats">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th><a href="/wiki/Video_coding_standard" class="mw-redirect" title="Video coding standard">Video coding standard</a></th> <th>Year</th> <th>Common applications </th></tr> <tr> <td><a href="/wiki/H.261" title="H.261">H.261</a><a href="#cite_note-video-standards-57">[57]</a><a href="#cite_note-58">[58]</a></td> <td>1988</td> <td>First of a family of <a href="/wiki/Video_coding_standards" class="mw-redirect" title="Video coding standards">video coding standards</a>. Used primarily in older <a href="/wiki/Video_conferencing" class="mw-redirect" title="Video conferencing">video conferencing</a> and <a href="/wiki/Video_telephone" class="mw-redirect" title="Video telephone">video telephone</a> products. </td></tr> <tr> <td><a href="/wiki/Motion_JPEG" title="Motion JPEG">Motion JPEG</a> (MJPEG)<a href="#cite_note-59">[59]</a></td> <td>1992</td> <td><a href="/wiki/QuickTime" title="QuickTime">QuickTime</a>, <a href="/wiki/Video_editing" title="Video editing">video editing</a>, <a href="/wiki/Non-linear_editing" title="Non-linear editing">non-linear editing</a>, <a href="/wiki/Digital_cameras" class="mw-redirect" title="Digital cameras">digital cameras</a> </td></tr> <tr> <td><a href="/wiki/MPEG-1" title="MPEG-1">MPEG-1</a> Video<a href="#cite_note-Rao-60">[60]</a></td> <td>1993</td> <td><a href="/wiki/Digital_video" title="Digital video">Digital video</a> distribution on <a href="/wiki/CD" class="mw-redirect" title="CD">CD</a> or <a href="/wiki/Internet_video" title="Internet video">Internet video</a> </td></tr> <tr> <td><a href="/wiki/MPEG-2_Video" class="mw-redirect" title="MPEG-2 Video">MPEG-2 Video</a> (H.262)<a href="#cite_note-Rao-60">[60]</a></td> <td>1995</td> <td>Storage and handling of digital images in broadcast applications, <a href="/wiki/Digital_television" title="Digital television">digital television</a>, <a href="/wiki/HDTV" class="mw-redirect" title="HDTV">HDTV</a>, cable, satellite, high-speed <a href="/wiki/Internet" title="Internet">Internet</a>, <a href="/wiki/DVD" title="DVD">DVD</a> video distribution </td></tr> <tr> <td><a href="/wiki/DV_(video_format)" title="DV (video format)">DV</a></td> <td>1995</td> <td><a href="/wiki/Camcorders" class="mw-redirect" title="Camcorders">Camcorders</a>, <a href="/wiki/Digital_cassettes" title="Digital cassettes">digital cassettes</a> </td></tr> <tr> <td><a href="/wiki/H.263" title="H.263">H.263</a> (<a href="/wiki/MPEG-4_Part_2" title="MPEG-4 Part 2">MPEG-4 Part 2</a>)<a href="#cite_note-video-standards-57">[57]</a></td> <td>1996</td> <td><a href="/wiki/Video_telephony" class="mw-redirect" title="Video telephony">Video telephony</a> over <a href="/wiki/Public_switched_telephone_network" title="Public switched telephone network">public switched telephone network</a> (PSTN), <a href="/wiki/H.320" title="H.320">H.320</a>, <a href="/wiki/Integrated_Services_Digital_Network" class="mw-redirect" title="Integrated Services Digital Network">Integrated Services Digital Network</a> (ISDN)<a href="#cite_note-61">[61]</a><a href="#cite_note-62">[62]</a> </td></tr> <tr> <td><a href="/wiki/Advanced_Video_Coding" title="Advanced Video Coding">Advanced Video Coding</a> (AVC, H.264, <a href="/wiki/MPEG-4" title="MPEG-4">MPEG-4</a>)<a href="#cite_note-Stankovic-1">[1]</a><a href="#cite_note-Wang-52">[52]</a></td> <td>2003</td> <td>Popular <a href="/wiki/HD_video" class="mw-redirect" title="HD video">HD video</a> recording, compression and distribution format, <a href="/wiki/Internet_video" title="Internet video">Internet video</a>, <a href="/wiki/YouTube" title="YouTube">YouTube</a>, <a href="/wiki/Blu-ray_Discs" class="mw-redirect" title="Blu-ray Discs">Blu-ray Discs</a>, <a href="/wiki/HDTV" class="mw-redirect" title="HDTV">HDTV</a> broadcasts, <a href="/wiki/Web_browsers" class="mw-redirect" title="Web browsers">web browsers</a>, <a href="/wiki/Streaming_television" title="Streaming television">streaming television</a>, <a href="/wiki/Mobile_devices" class="mw-redirect" title="Mobile devices">mobile devices</a>, consumer devices, <a href="/wiki/Netflix" title="Netflix">Netflix</a>,<a href="#cite_note-Encodes-42">[42]</a> <a href="/wiki/Video_telephony" class="mw-redirect" title="Video telephony">video telephony</a>, <a href="/wiki/FaceTime" title="FaceTime">FaceTime</a><a href="#cite_note-AppleInsider_standards_1-41">[41]</a> </td></tr> <tr> <td><a href="/wiki/Theora" title="Theora">Theora</a></td> <td>2004</td> <td>Internet video, web browsers </td></tr> <tr> <td><a href="/wiki/VC-1" title="VC-1">VC-1</a></td> <td>2006</td> <td><a href="/wiki/Windows" class="mw-redirect" title="Windows">Windows</a> media, <a href="/wiki/Blu-ray_Disc" class="mw-redirect" title="Blu-ray Disc">Blu-ray Discs</a> </td></tr> <tr> <td><a href="/wiki/Apple_ProRes" title="Apple ProRes">Apple ProRes</a></td> <td>2007</td> <td>Professional video production.<a href="#cite_note-loc-50">[50]</a> </td></tr> <tr> <td><a href="/wiki/VP9" title="VP9">VP9</a></td> <td>2010</td> <td>A video codec developed by <a href="/wiki/Google" title="Google">Google</a> used in the <a href="/wiki/WebM" title="WebM">WebM</a> container format with <a href="/wiki/HTML5" title="HTML5">HTML5</a>. </td></tr> <tr> <td><a href="/wiki/High_Efficiency_Video_Coding" title="High Efficiency Video Coding">High Efficiency Video Coding</a> (HEVC, H.265)<a href="#cite_note-Stankovic-1">[1]</a><a href="#cite_note-apple-4">[4]</a></td> <td>2013</td> <td>Successor to the H.264 standard, having substantially improved compression capability </td></tr> <tr> <td><a href="/wiki/Daala" title="Daala">Daala</a></td> <td>2013</td> <td>Research video format by <a href="/wiki/Xiph.org" class="mw-redirect" title="Xiph.org">Xiph.org</a> </td></tr> <tr> <td><a href="/wiki/AV1" title="AV1">AV1</a><a href="#cite_note-AV1-63">[63]</a></td> <td>2018</td> <td>An open source format based on VP10 (<a href="/wiki/VP9" title="VP9">VP9</a>'s internal successor), <a href="/wiki/Daala" title="Daala">Daala</a> and <a href="/wiki/Thor_(video_codec)" title="Thor (video codec)">Thor</a>; used by content providers such as <a href="/wiki/YouTube" title="YouTube">YouTube</a><a href="#cite_note-YT_AV1_Beta_Playlist-64">[64]</a><a href="#cite_note-YT_AV1-65">[65]</a> and <a href="/wiki/Netflix" title="Netflix">Netflix</a>.<a href="#cite_note-Netflix_AV1_Android-66">[66]</a><a href="#cite_note-Netflix_AV1_TV-67">[67]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="MDCT_audio_standards">MDCT audio standards</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=7" title="Edit section: MDCT audio standards">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">Modified discrete cosine transform</a></div> <div class="mw-heading mw-heading4"><h4 id="General_audio">General audio</h4>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=8" title="Edit section: General audio">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th>Audio compression standard </th> <th>Year </th> <th>Common applications </th></tr> <tr> <td><a href="/wiki/Dolby_Digital" title="Dolby Digital">Dolby Digital</a> (AC-3)<a href="#cite_note-Luo-22">[22]</a><a href="#cite_note-Britanak2011-23">[23]</a> </td> <td>1991 </td> <td><a href="/wiki/Film" title="Film">Cinema</a>, <a href="/wiki/Digital_cinema" title="Digital cinema">digital cinema</a>, <a href="/wiki/DVD" title="DVD">DVD</a>, <a href="/wiki/Blu-ray" title="Blu-ray">Blu-ray</a>, <a href="/wiki/Streaming_media" title="Streaming media">streaming media</a>, <a href="/wiki/Video_games" class="mw-redirect" title="Video games">video games</a> </td></tr> <tr> <td><a href="/wiki/Adaptive_Transform_Acoustic_Coding" class="mw-redirect" title="Adaptive Transform Acoustic Coding">Adaptive Transform Acoustic Coding</a> (ATRAC)<a href="#cite_note-Luo-22">[22]</a> </td> <td>1992 </td> <td><a href="/wiki/MiniDisc" title="MiniDisc">MiniDisc</a> </td></tr> <tr> <td><a href="/wiki/MP3" title="MP3">MP3</a><a href="#cite_note-Guckert-24">[24]</a><a href="#cite_note-Stankovic-1">[1]</a> </td> <td>1993 </td> <td><a href="/wiki/Digital_audio" title="Digital audio">Digital audio</a> distribution, <a href="/wiki/MP3_players" class="mw-redirect" title="MP3 players">MP3 players</a>, <a href="/wiki/Portable_media_players" class="mw-redirect" title="Portable media players">portable media players</a>, <a href="/wiki/Streaming_media" title="Streaming media">streaming media</a> </td></tr> <tr> <td><a href="/wiki/Perceptual_Audio_Coder" title="Perceptual Audio Coder">Perceptual Audio Coder</a> (PAC)<a href="#cite_note-Luo-22">[22]</a> </td> <td>1996 </td> <td><a href="/wiki/Digital_audio_radio_service" title="Digital audio radio service">Digital audio radio service</a> (DARS) </td></tr> <tr> <td><a href="/wiki/Advanced_Audio_Coding" title="Advanced Audio Coding">Advanced Audio Coding</a> (AAC / <a href="/wiki/MP4" class="mw-redirect" title="MP4">MP4</a> Audio)<a href="#cite_note-brandenburg-25">[25]</a><a href="#cite_note-Luo-22">[22]</a> </td> <td>1997 </td> <td><a href="/wiki/Digital_audio" title="Digital audio">Digital audio</a> distribution, <a href="/wiki/Portable_media_players" class="mw-redirect" title="Portable media players">portable media players</a>, <a href="/wiki/Streaming_media" title="Streaming media">streaming media</a>, <a href="/wiki/Game_consoles" class="mw-redirect" title="Game consoles">game consoles</a>, <a href="/wiki/Mobile_devices" class="mw-redirect" title="Mobile devices">mobile devices</a>, <a href="/wiki/IOS" title="IOS">iOS</a>, <a href="/wiki/ITunes" title="ITunes">iTunes</a>, <a href="/wiki/Android_(operating_system)" title="Android (operating system)">Android</a>, <a href="/wiki/BlackBerry" title="BlackBerry">BlackBerry</a> </td></tr> <tr> <td><a href="/wiki/High-Efficiency_Advanced_Audio_Coding" title="High-Efficiency Advanced Audio Coding">High-Efficiency Advanced Audio Coding</a> (AAC+)<a href="#cite_note-Herre-68">[68]</a><a href="#cite_note-Britanak-38">[38]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cZ4vDwAAQBAJ&pg=PA478">478</a>  </td> <td>1997 </td> <td><a href="/wiki/Digital_radio" title="Digital radio">Digital radio</a>, <a href="/wiki/Digital_audio_broadcasting" class="mw-redirect" title="Digital audio broadcasting">digital audio broadcasting</a> (DAB+),<a href="#cite_note-Britanak-38">[38]</a> <a href="/wiki/Digital_Radio_Mondiale" title="Digital Radio Mondiale">Digital Radio Mondiale</a> (DRM) </td></tr> <tr> <td><a href="/wiki/Cook_Codec" title="Cook Codec">Cook Codec</a> </td> <td>1998 </td> <td><a href="/wiki/RealAudio" title="RealAudio">RealAudio</a> </td></tr> <tr> <td><a href="/wiki/Windows_Media_Audio" title="Windows Media Audio">Windows Media Audio</a> (WMA)<a href="#cite_note-Luo-22">[22]</a> </td> <td>1999 </td> <td><a href="/wiki/Windows_Media" title="Windows Media">Windows Media</a> </td></tr> <tr> <td><a href="/wiki/Vorbis" title="Vorbis">Vorbis</a><a href="#cite_note-vorbis-mdct-26">[26]</a><a href="#cite_note-Luo-22">[22]</a> </td> <td>2000 </td> <td><a href="/wiki/Digital_audio" title="Digital audio">Digital audio</a> distribution, <a href="/wiki/Radio_station" class="mw-redirect" title="Radio station">radio stations</a>, <a href="/wiki/Streaming_media" title="Streaming media">streaming media</a>, <a href="/wiki/Video_game" title="Video game">video games</a>, <a href="/wiki/Spotify" title="Spotify">Spotify</a>, <a href="/wiki/Wikipedia" title="Wikipedia">Wikipedia</a> </td></tr> <tr> <td><a href="/wiki/High-Definition_Coding" title="High-Definition Coding">High-Definition Coding</a> (HDC)<a href="#cite_note-Jones-39">[39]</a> </td> <td>2002 </td> <td>Digital radio, <a href="/wiki/HD_Radio" title="HD Radio">HD Radio</a> </td></tr> <tr> <td><a href="/wiki/Dynamic_Resolution_Adaptation" title="Dynamic Resolution Adaptation">Dynamic Resolution Adaptation</a> (DRA)<a href="#cite_note-Luo-22">[22]</a> </td> <td>2008 </td> <td>China national audio standard, <a href="/wiki/China_Multimedia_Mobile_Broadcasting" title="China Multimedia Mobile Broadcasting">China Multimedia Mobile Broadcasting</a>, <a href="/wiki/DVB-H" title="DVB-H">DVB-H</a> </td></tr> <tr> <td><a href="/wiki/Opus_(audio_format)" title="Opus (audio format)">Opus</a><a href="#cite_note-Valin-69">[69]</a> </td> <td>2012 </td> <td>VoIP,<a href="#cite_note-homepage-70">[70]</a> mobile telephony, <a href="/wiki/WhatsApp" title="WhatsApp">WhatsApp</a>,<a href="#cite_note-Register-71">[71]</a><a href="#cite_note-Hazra-72">[72]</a><a href="#cite_note-Srivastava-73">[73]</a> <a href="/wiki/PlayStation_4" title="PlayStation 4">PlayStation 4</a><a href="#cite_note-PlayStation-74">[74]</a> </td></tr> <tr> <td><a href="/wiki/Dolby_AC-4" title="Dolby AC-4">Dolby AC-4</a><a href="#cite_note-Dolby_AC-4-75">[75]</a> </td> <td rowspan="2">2015 </td> <td rowspan="2"><a href="/wiki/ATSC_3.0" title="ATSC 3.0">ATSC 3.0</a>, <a href="/wiki/Ultra-high-definition_television" title="Ultra-high-definition television">ultra-high-definition television</a> (UHD TV) </td></tr> <tr> <td><a href="/wiki/MPEG-H_3D_Audio" title="MPEG-H 3D Audio">MPEG-H 3D Audio</a><a href="#cite_note-Bleidt-76">[76]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Speech_coding">Speech coding</h4>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=9" title="Edit section: Speech coding">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th><a href="/wiki/Speech_coding" title="Speech coding">Speech coding</a> standard</th> <th>Year </th> <th>Common applications </th></tr> <tr> <td><a href="/wiki/AAC-LD" title="AAC-LD">AAC-LD</a> (LD-MDCT)<a href="#cite_note-Schnell-77">[77]</a> </td> <td>1999 </td> <td><a href="/wiki/Mobile_telephony" title="Mobile telephony">Mobile telephony</a>, <a href="/wiki/Voice-over-IP" class="mw-redirect" title="Voice-over-IP">voice-over-IP</a> (VoIP), <a href="/wiki/IOS" title="IOS">iOS</a>, <a href="/wiki/FaceTime" title="FaceTime">FaceTime</a><a href="#cite_note-AppleInsider_standards_1-41">[41]</a> </td></tr> <tr> <td><a href="/wiki/Siren_(codec)" title="Siren (codec)">Siren</a><a href="#cite_note-Hersent-40">[40]</a> </td> <td>1999 </td> <td><a href="/wiki/VoIP" class="mw-redirect" title="VoIP">VoIP</a>, <a href="/wiki/Wideband_audio" title="Wideband audio">wideband audio</a>, <a href="/wiki/G.722.1" title="G.722.1">G.722.1</a> </td></tr> <tr> <td><a href="/wiki/G.722.1" title="G.722.1">G.722.1</a><a href="#cite_note-Lutzky-78">[78]</a> </td> <td>1999 </td> <td>VoIP, wideband audio, <a href="/wiki/G.722" title="G.722">G.722</a> </td></tr> <tr> <td><a href="/wiki/G.729.1" title="G.729.1">G.729.1</a><a href="#cite_note-Nagireddi-79">[79]</a> </td> <td>2006 </td> <td><a href="/wiki/G.729" title="G.729">G.729</a>, VoIP, wideband audio,<a href="#cite_note-Nagireddi-79">[79]</a> <a href="/wiki/Mobile_telephony" title="Mobile telephony">mobile telephony</a> </td></tr> <tr> <td><a href="/wiki/Enhanced_Variable_Rate_Codec_B" title="Enhanced Variable Rate Codec B">EVRC-WB</a><a href="#cite_note-Britanak-38">[38]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cZ4vDwAAQBAJ&pg=PA31">31</a>, 478]  </td> <td>2007 </td> <td><a href="/wiki/Wideband_audio" title="Wideband audio">Wideband audio</a> </td></tr> <tr> <td><a href="/wiki/G.718" title="G.718">G.718</a><a href="#cite_note-ITU-T-80">[80]</a> </td> <td>2008 </td> <td>VoIP, wideband audio, mobile telephony </td></tr> <tr> <td><a href="/wiki/G.719" title="G.719">G.719</a><a href="#cite_note-Britanak-38">[38]</a> </td> <td>2008 </td> <td><a href="/wiki/Teleconferencing" class="mw-redirect" title="Teleconferencing">Teleconferencing</a>, <a href="/wiki/Videoconferencing" class="mw-redirect" title="Videoconferencing">videoconferencing</a>, <a href="/wiki/Voice_mail" class="mw-redirect" title="Voice mail">voice mail</a> </td></tr> <tr> <td><a href="/wiki/CELT" title="CELT">CELT</a><a href="#cite_note-Terriberry-81">[81]</a> </td> <td>2011 </td> <td>VoIP,<a href="#cite_note-ekiga-82">[82]</a><a href="#cite_note-FreeSWITCH-83">[83]</a> mobile telephony </td></tr> <tr> <td><a href="/wiki/Enhanced_Voice_Services" title="Enhanced Voice Services">Enhanced Voice Services</a> (EVS)<a href="#cite_note-EVS-84">[84]</a> </td> <td>2014 </td> <td>Mobile telephony, VoIP, wideband audio </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Multidimensional_DCT">Multidimensional DCT</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=10" title="Edit section: Multidimensional DCT">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/ZPEG" title="ZPEG">ZPEG</a></div> 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,<a href="#cite_note-appDCT-85">[85]</a> variable temporal length 3-D DCT coding,<a href="#cite_note-app2DCT-86">[86]</a> <a href="/wiki/Video_coding" class="mw-redirect" title="Video coding">video coding</a> algorithms,<a href="#cite_note-app3DCT-87">[87]</a> adaptive video coding<a href="#cite_note-app4DCT-88">[88]</a> and 3-D Compression.<a href="#cite_note-app5DCT-89">[89]</a> Due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using MD DCTs is rapidly increasing. <a href="#DCT-IV">DCT-IV</a> has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks,<a href="#cite_note-90">[90]</a> lapped orthogonal transform<a href="#cite_note-91">[91]</a><a href="#cite_note-FOOTNOTEMalvar1992-92">[92]</a> and cosine-modulated wavelet bases.<a href="#cite_note-93">[93]</a> <div class="mw-heading mw-heading3"><h3 id="Digital_signal_processing">Digital signal processing</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=11" title="Edit section: Digital signal processing">edit</a>]</div> DCT plays an important role in <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a> specifically <a href="/wiki/Data_compression" title="Data compression">data compression</a>. The DCT is widely implemented in <a href="/wiki/Digital_signal_processors" class="mw-redirect" title="Digital signal processors">digital signal processors</a> (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 <a href="/wiki/Encoding" class="mw-redirect" title="Encoding">encoding</a>, decoding, video, audio, <a href="/wiki/Multiplexing" title="Multiplexing">multiplexing</a>, control signals, <a href="/wiki/Signaling" class="mw-redirect" title="Signaling">signaling</a>, and <a href="/wiki/Analog-to-digital_conversion" class="mw-redirect" title="Analog-to-digital conversion">analog-to-digital conversion</a>. DCTs are also commonly used for <a href="/wiki/High-definition_television" title="High-definition television">high-definition television</a> (HDTV) encoder/decoder <a href="/wiki/Integrated_circuit" title="Integrated circuit">chips</a>.<a href="#cite_note-Stankovic-1">[1]</a> <div class="mw-heading mw-heading3"><h3 id="Compression_artifacts">Compression artifacts</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=12" title="Edit section: Compression artifacts">edit</a>]</div> A common issue with DCT compression in <a href="/wiki/Digital_media" title="Digital media">digital media</a> are blocky <a href="/wiki/Compression_artifacts" class="mw-redirect" title="Compression artifacts">compression artifacts</a>,<a href="#cite_note-Katsaggelos-94">[94]</a> caused by DCT blocks.<a href="#cite_note-Alikhani-3">[3]</a> 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 blocks is taken within each block and the resulting DCT coefficients are <a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">quantized</a>. This process can cause blocking artifacts, primarily at high <a href="/wiki/Data_compression_ratio" title="Data compression ratio">data compression ratios</a>.<a href="#cite_note-Katsaggelos-94">[94]</a> This can also cause the <a href="/wiki/Mosquito_noise" class="mw-redirect" title="Mosquito noise">mosquito noise</a> effect, commonly found in <a href="/wiki/Digital_video" title="Digital video">digital video</a>.<a href="#cite_note-95">[95]</a> DCT blocks are often used in <a href="/wiki/Glitch_art" title="Glitch art">glitch art</a>.<a href="#cite_note-Alikhani-3">[3]</a> The artist <a href="/wiki/Rosa_Menkman" title="Rosa Menkman">Rosa Menkman</a> makes use of DCT-based compression artifacts in her glitch art,<a href="#cite_note-Menkman-96">[96]</a> particularly the DCT blocks found in most <a href="/wiki/Digital_media" title="Digital media">digital media</a> formats such as <a href="/wiki/JPEG" title="JPEG">JPEG</a> digital images and <a href="/wiki/MP3" title="MP3">MP3</a> audio.<a href="#cite_note-Alikhani-3">[3]</a> Another example is Jpegs by German photographer <a href="/wiki/Thomas_Ruff" title="Thomas Ruff">Thomas Ruff</a>, which uses intentional <a href="/wiki/JPEG" title="JPEG">JPEG</a> artifacts as the basis of the picture's style.<a href="#cite_note-97">[97]</a><a href="#cite_note-98">[98]</a> <div class="mw-heading mw-heading2"><h2 id="Informal_overview">Informal overview</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=13" title="Edit section: Informal overview">edit</a>]</div> Like any Fourier-related transform, DCTs express a function or a signal in terms of a sum of <a href="/wiki/Sinusoid" class="mw-redirect" title="Sinusoid">sinusoids</a> with different <a href="/wiki/Frequencies" class="mw-redirect" title="Frequencies">frequencies</a> and <a href="/wiki/Amplitude" title="Amplitude">amplitudes</a>. Like the DFT, a DCT operates on a function at a finite number of <a href="/wiki/Discrete_signal" class="mw-redirect" title="Discrete signal">discrete data points</a>. 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 <a href="/wiki/Complex_exponential" class="mw-redirect" title="Complex exponential">complex exponentials</a>). However, this visible difference is merely a consequence of a deeper distinction: a DCT implies different <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a> from the DFT or other related transforms. The Fourier-related transforms that operate on a function over a finite <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>, such as the DFT or DCT or a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>, can be thought of as implicitly defining an extension of that function outside the domain. That is, once you write a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> as a sum of sinusoids, you can evaluate that sum at any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, even for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> where the original <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> was not specified. The DFT, like the Fourier series, implies a <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> extension of the original function. A DCT, like a <a href="/wiki/Sine_and_cosine_transforms" title="Sine and cosine transforms">cosine transform</a>, implies an <a href="/wiki/Even_and_odd_functions" title="Even and odd functions">even</a> extension of the original function. <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:DCT-symmetries.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/DCT-symmetries.svg/350px-DCT-symmetries.svg.png" decoding="async" width="350" height="392" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/DCT-symmetries.svg/525px-DCT-symmetries.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/DCT-symmetries.svg/700px-DCT-symmetries.svg.png 2x" data-file-width="580" data-file-height="650" /></a><figcaption>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).</figcaption></figure> 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 <a href="/wiki/Discrete_sine_transform" title="Discrete sine transform">discrete sine transforms</a> (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 <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> by <a href="/wiki/Spectral_method" title="Spectral method">spectral methods</a>, the boundary conditions are directly specified as a part of the problem being solved. Or, for the <a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a> (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 <a href="/wiki/Classification_of_discontinuities" title="Classification of discontinuities">discontinuities</a> in a function reduce the <a href="/wiki/Rate_of_convergence" title="Rate of convergence">rate of convergence</a> 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 <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> or <a href="/wiki/Cosine_series" class="mw-redirect" title="Cosine series">cosine series</a> 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 <a href="/wiki/Slope" title="Slope">slope</a> 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. <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=14" title="Edit section: Formal definition">edit</a>]</div> Formally, the discrete cosine transform is a <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a>, invertible <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{N}\to \mathbb {R} ^{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{N}\to \mathbb {R} ^{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1850d46354a856720b4f8cdc6f913a4fa493a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.569ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} ^{N}\to \mathbb {R} ^{N}}"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> denotes the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>), or equivalently an invertible N × N <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a>. There are several variants of the DCT with slightly modified definitions. The N real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~x_{0},\ \ldots \ x_{N-1}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~x_{0},\ \ldots \ x_{N-1}~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d750bfe1061fa3732b3af6d493820dd219f8513f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.36ex; height:2.009ex;" alt="{\displaystyle ~x_{0},\ \ldots \ x_{N-1}~}"> are transformed into the N real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0},\,\ldots ,\,X_{N-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{0},\,\ldots ,\,X_{N-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b006d2eb2d2a9fa369fb3aa42c6b52399346d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.647ex; height:2.509ex;" alt="{\displaystyle X_{0},\,\ldots ,\,X_{N-1}}"> according to one of the formulas: <div class="mw-heading mw-heading3"><h3 id="DCT-I">DCT-I</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=15" title="Edit section: DCT-I">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}={\frac {1}{2}}(x_{0}+(-1)^{k}x_{N-1})+\sum _{n=1}^{N-2}x_{n}\cos \left[\,{\tfrac {\ \pi }{\,N-1\,}}\,n\,k\,\right]\qquad {\text{ for }}~k=0,\ \ldots \ N-1~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mi>π</mi> </mrow> <mrow> <mspace width="thinmathspace" /> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mrow> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>n</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mtext> </mtext> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}={\frac {1}{2}}(x_{0}+(-1)^{k}x_{N-1})+\sum _{n=1}^{N-2}x_{n}\cos \left[\,{\tfrac {\ \pi }{\,N-1\,}}\,n\,k\,\right]\qquad {\text{ for }}~k=0,\ \ldots \ N-1~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51126bccb0538b10ecd80fb4b40646d24951be39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:78.119ex; height:7.343ex;" alt="{\displaystyle X_{k}={\frac {1}{2}}(x_{0}+(-1)^{k}x_{N-1})+\sum _{n=1}^{N-2}x_{n}\cos \left[\,{\tfrac {\ \pi }{\,N-1\,}}\,n\,k\,\right]\qquad {\text{ for }}~k=0,\ \ldots \ N-1~.}"></dd></dl> Some authors further multiply the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{N-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{N-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7df9ede29e0429afc8d9b7d32c831ab2df6723e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.122ex; height:2.009ex;" alt="{\displaystyle x_{N-1}}"> terms by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\,}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\,}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df644510238514755b939db799739666eb00c53d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.519ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2\,}}\,,}"> and correspondingly multiply the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6381fdad2b9f11954b1fc2db08bbaccf634ededa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{0}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{N-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{N-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd3fd31df96b1fdf8e2a133c4ba717011dc3022" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.716ex; height:2.509ex;" alt="{\displaystyle X_{N-1}}"> terms by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {2\,}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {2\,}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf6a452910f04614c5f2410b5fdc1b9dab1f35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.844ex; height:3.176ex;" alt="{\displaystyle 1/{\sqrt {2\,}}\,,}"> which, if one further multiplies by an overall scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {{\tfrac {2}{N-1\,}}\,}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mrow> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {{\tfrac {2}{N-1\,}}\,}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e4f46bdc1196d00114fcec44ba4e433eb157a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.141ex; height:4.843ex;" alt="{\displaystyle {\sqrt {{\tfrac {2}{N-1\,}}\,}},}">, makes the DCT-I matrix <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> but breaks the direct correspondence with a real-even <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFT</a>. The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFT</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(N-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2(N-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4410f19685c51f4a556471897ccb38286f5f07c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.038ex; height:2.843ex;" alt="{\displaystyle 2(N-1)}"> real numbers with even symmetry. For example, a DCT-I of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea204e60e1d27578912bb557d2859e759f4d67c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=5}"> real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\ b\ c\ d\ e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>b</mi> <mtext> </mtext> <mi>c</mi> <mtext> </mtext> <mi>d</mi> <mtext> </mtext> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ b\ c\ d\ e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a260e0d32521ceb4b64afe560076c896eb5766c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.856ex; height:2.176ex;" alt="{\displaystyle a\ b\ c\ d\ e}"> is exactly equivalent to a DFT of eight real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\ b\ c\ d\ e\ d\ c\ b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>b</mi> <mtext> </mtext> <mi>c</mi> <mtext> </mtext> <mi>d</mi> <mtext> </mtext> <mi>e</mi> <mtext> </mtext> <mi>d</mi> <mtext> </mtext> <mi>c</mi> <mtext> </mtext> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ b\ c\ d\ e\ d\ c\ b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7d02159dc6511f922294354a02574172002e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.818ex; height:2.176ex;" alt="{\displaystyle a\ b\ c\ d\ e\ d\ c\ b}"> (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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> less than 2, while all other DCT types are defined for any positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/356b8b60a047de347b447f2bdafaaccf47502031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.71ex; height:2.176ex;" alt="{\displaystyle N.}"> Thus, the DCT-I corresponds to the boundary conditions: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"> is even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}"> and even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=N-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=N-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41749647b4bc797ccb5bff075fa4730e5944abc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.56ex; height:2.343ex;" alt="{\displaystyle n=N-1}">; similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c04fca5a5d7f8d48f21d5640d286507dccc5dd93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.66ex; height:2.509ex;" alt="{\displaystyle X_{k}.}"> <div class="mw-heading mw-heading3"><h3 id="DCT-II">DCT-II</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=16" title="Edit section: DCT-II">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>π</mi> <mspace width="thinmathspace" /> </mrow> <mi>N</mi> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mi>k</mi> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mtext> </mtext> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6a9bb1a9eedda6874f60046dee69bbd4b94c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.266ex; height:7.343ex;" alt="{\displaystyle 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~.}"></dd></dl> The DCT-II is probably the most commonly used form, and is often simply referred to as "the DCT".<a href="#cite_note-pubDCT-5">[5]</a><a href="#cite_note-pubRaoYip-6">[6]</a> This transform is exactly equivalent (up to an overall scale factor of 2) to a <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFT</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc080b972918855f349631dd1d838c538162366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 4N}"> real inputs of even symmetry where the even-indexed elements are zero. That is, it is half of the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFT</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc080b972918855f349631dd1d838c538162366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 4N}"> inputs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa69de01af153c0d9aa2d8ef09a96c40232a1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.005ex; height:2.009ex;" alt="{\displaystyle y_{n},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{2n}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{2n}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0cb4c0ecf91f06aea4f27a1e94ef6bafc5ce4dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.087ex; height:2.509ex;" alt="{\displaystyle y_{2n}=0,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{2n+1}=x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{2n+1}=x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42ee5bd574a42782c18d2e3079835b3b21e357c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.927ex; height:2.009ex;" alt="{\displaystyle y_{2n+1}=x_{n}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq n<N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>n</mi> <mo><</mo> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq n<N,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a4faf72d7f59cff647c2c6a0105e6f24b1ffdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.465ex; height:2.509ex;" alt="{\displaystyle 0\leq n<N,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{2N}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{2N}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb3a8f4298ada293035b890aac38a11a5545c81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.56ex; height:2.509ex;" alt="{\displaystyle y_{2N}=0,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{4N-n}=y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{4N-n}=y_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba17f6d7126821c452390bb727456802c303170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.374ex; height:2.009ex;" alt="{\displaystyle y_{4N-n}=y_{n}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<n<2N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>n</mi> <mo><</mo> <mn>2</mn> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<n<2N.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/964664557b0bf4f075273b0938c8562b37c16c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.627ex; height:2.176ex;" alt="{\displaystyle 0<n<2N.}"> DCT-II transformation is also possible using 2N signal followed by a multiplication by half shift. This is demonstrated by <a href="/wiki/John_Makhoul" title="John Makhoul">Makhoul</a>. Some authors further multiply the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6381fdad2b9f11954b1fc2db08bbaccf634ededa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{0}}"> term by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {N\,}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>N</mi> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {N\,}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c006fa07dc5191017bd6d1c46947d36bac3aa14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.099ex; height:3.176ex;" alt="{\displaystyle 1/{\sqrt {N\,}}\,}"> and multiply the rest of the matrix by an overall scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {{2}/{N}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {{2}/{N}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b99799742c369e887980ac1ed84dc8425c67378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.712ex; height:3.343ex;" alt="{\textstyle {\sqrt {{2}/{N}}}}"> (see below for the corresponding change in DCT-III). This makes the DCT-II matrix <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a>, but breaks the direct correspondence with a real-even <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFT</a> of half-shifted input. This is the normalization used by <a href="/wiki/Matlab" class="mw-redirect" title="Matlab">Matlab</a>, for example, see.<a href="#cite_note-99">[99]</a> In many applications, such as <a href="/wiki/JPEG" title="JPEG">JPEG</a>, the scaling is arbitrary because scale factors can be combined with a subsequent computational step (e.g. the <a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">quantization</a> step in JPEG<a href="#cite_note-100">[100]</a>), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications.<a href="#cite_note-101">[101]</a><a href="#cite_note-102">[102]</a> The DCT-II implies the boundary conditions: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"> is even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=-1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=-1/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2e0129774c12b9419b3032d59ad929050f5fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.789ex; height:2.843ex;" alt="{\displaystyle n=-1/2}"> and even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=N-1/2\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=N-1/2\,;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a211c793f73a1a9c95ba0e1630111285b26e34f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.919ex; height:2.843ex;" alt="{\displaystyle n=N-1/2\,;}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\displaystyle X_{k}}"> is even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}"> and odd around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=N.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/885cd540dd443e374615746849123c82a453b479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.02ex; height:2.176ex;" alt="{\displaystyle k=N.}"> <div class="mw-heading mw-heading3"><h3 id="DCT-III">DCT-III</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=17" title="Edit section: DCT-III">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>π</mi> <mspace width="thinmathspace" /> </mrow> <mi>N</mi> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mi>n</mi> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mtext> </mtext> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54964bf4edeb811b15c3455a127339bf4f7c87f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.149ex; height:7.343ex;" alt="{\displaystyle 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~.}"></dd></dl> 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").<a href="#cite_note-pubRaoYip-6">[6]</a> Some authors divide the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> term by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"> instead of by 2 (resulting in an overall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}/{\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7aa315bf15ab3c7b0571f2ed9d807711514893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.645ex; height:3.176ex;" alt="{\displaystyle x_{0}/{\sqrt {2}}}"> term) and multiply the resulting matrix by an overall scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {2/N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {2/N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc67f411ae2805d323fa48a91e852adce10c0fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.712ex; height:3.343ex;" alt="{\textstyle {\sqrt {2/N}}}"> (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 <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a>, but breaks the direct correspondence with a real-even <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFT</a> of half-shifted output. The DCT-III implies the boundary conditions: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"> is even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}"> and odd around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=N;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>N</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=N;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32aae191706a06c08fb75a3727ecc0297257692b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.204ex; height:2.509ex;" alt="{\displaystyle n=N;}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\displaystyle X_{k}}"> is even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=-1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=-1/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c85787e5beba35dd2ae145d7aa5a8a1758d01d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.605ex; height:2.843ex;" alt="{\displaystyle k=-1/2}"> and even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=N-1/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=N-1/2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd7c65518ba86721acbba155a2b877b00168142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.348ex; height:2.843ex;" alt="{\displaystyle k=N-1/2.}"> <div class="mw-heading mw-heading3"><h3 id="DCT-IV">DCT-IV</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=18" title="Edit section: DCT-IV">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>π</mi> <mspace width="thinmathspace" /> </mrow> <mi>N</mi> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mtext> </mtext> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd046a94a30671a18f12d188598bcfb96e797eae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:65.701ex; height:7.343ex;" alt="{\displaystyle 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~.}"></dd></dl> The DCT-IV matrix becomes <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> (and thus, being clearly symmetric, its own inverse) if one further multiplies by an overall scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {2/N}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {2/N}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed8f5aa98dce26668299e9cb96bf42ab0fec18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.359ex; height:3.343ex;" alt="{\textstyle {\sqrt {2/N}}.}"> A variant of the DCT-IV, where data from different transforms are overlapped, is called the <a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">modified discrete cosine transform</a> (MDCT).<a href="#cite_note-103">[103]</a> The DCT-IV implies the boundary conditions: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"> is even around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=-1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=-1/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2e0129774c12b9419b3032d59ad929050f5fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.789ex; height:2.843ex;" alt="{\displaystyle n=-1/2}"> and odd around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=N-1/2;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=N-1/2;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d84fd329c2f42cc7f34fd3a6e4553da6d1f3b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.531ex; height:2.843ex;" alt="{\displaystyle n=N-1/2;}"> similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c04fca5a5d7f8d48f21d5640d286507dccc5dd93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.66ex; height:2.509ex;" alt="{\displaystyle X_{k}.}"> <div class="mw-heading mw-heading3"><h3 id="DCT_V-VIII">DCT V-VIII</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=19" title="Edit section: DCT V-VIII">edit</a>]</div> 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 <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFTs</a> of even order (regardless of whether <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> is even or odd), since the corresponding DFT is of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(N-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2(N-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4410f19685c51f4a556471897ccb38286f5f07c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.038ex; height:2.843ex;" alt="{\displaystyle 2(N-1)}"> (for DCT-I) or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc080b972918855f349631dd1d838c538162366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 4N}"> (for DCT-II & III) or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06b40cbc3ff3084650894d4577be045dd193c4a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 8N}"> (for DCT-IV). The four additional types of discrete cosine transform<a href="#cite_note-104">[104]</a> correspond essentially to real-even DFTs of logically odd order, which have factors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\pm {1}/{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\pm {1}/{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/144aacae5fc6adf3fc9eb60ed2df3ebe2161f4b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.391ex; height:2.843ex;" alt="{\displaystyle N\pm {1}/{2}}"> in the denominators of the cosine arguments. However, these variants seem to be rarely used in practice. One reason, perhaps, is that <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a> algorithms for odd-length DFTs are generally more complicated than <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a> 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 a , corresponds to a DCT-V of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606bc237d1a1ee6bf610c9c2730a9dcf2a2bb001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.971ex; height:2.176ex;" alt="{\displaystyle N=1.}">) <div class="mw-heading mw-heading2"><h2 id="Inverse_transforms">Inverse transforms</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=20" title="Edit section: Inverse transforms">edit</a>]</div> Using the normalization conventions above, the inverse of DCT-I is DCT-I multiplied by 2/(N − 1). The inverse of DCT-IV is DCT-IV multiplied by 2/N. The inverse of DCT-II is DCT-III multiplied by 2/N and vice versa.<a href="#cite_note-pubRaoYip-6">[6]</a> Like for the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">DFT</a>, 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {2/N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {2/N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc67f411ae2805d323fa48a91e852adce10c0fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.712ex; height:3.343ex;" alt="{\textstyle {\sqrt {2/N}}}"> so that the inverse does not require any additional multiplicative factor. Combined with appropriate factors of √2 (see above), this can be used to make the transform matrix <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a>. <div class="mw-heading mw-heading2"><h2 id="Multidimensional_DCTs">Multidimensional DCTs</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=21" title="Edit section: Multidimensional DCTs">edit</a>]</div> 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. <div class="mw-heading mw-heading3"><h3 id="M-D_DCT-II">M-D DCT-II</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=22" title="Edit section: M-D DCT-II">edit</a>]</div> 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): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b130f0c2ff55fcf6a7c26247d018e159abe7f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:75.371ex; height:15.676ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}"></dd> <dd>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.</dd></dl> 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 <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83b579efc3f88bb1ea1ea9f9e7ce0b2cc938a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:133.337ex; height:7.843ex;" alt="{\displaystyle 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.}"></dd></dl> The inverse of 3-D DCT-II is 3-D DCT-III and can be computed from the formula given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e084f4d392271aad0753fc1e18370df38d57c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:133.521ex; height:7.843ex;" alt="{\displaystyle 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.}"></dd></dl> 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 <a href="/wiki/Fast_Fourier_transform#Multidimensional_FFTs" title="Fast Fourier transform">multidimensional FFT algorithms</a>, 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. <div class="mw-heading mw-heading4"><h4 id="3-D_DCT-II_VR_DIF">3-D DCT-II VR DIF</h4>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=23" title="Edit section: 3-D DCT-II VR DIF">edit</a>]</div> In order to apply the VR DIF algorithm the input data is to be formulated and rearranged as follows.<a href="#cite_note-105">[105]</a><a href="#cite_note-:0-106">[106]</a> The transform size N × N × N is assumed to be 2. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Stages_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Stages_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg/246px-Stages_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg" decoding="async" width="246" height="336" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Stages_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg/369px-Stages_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Stages_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg/493px-Stages_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg 2x" data-file-width="654" data-file-height="892" /></a><figcaption>The four basic stages of computing 3-D DCT-II using VR DIF Algorithm.</figcaption></figure> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>N</mi> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7794e26b232b1ebc17d08b9c9737732fe19bfe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:71.174ex; height:26.509ex;" alt="{\displaystyle {\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}}}"></dd> <dd>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq n_{1},n_{2},n_{3}\leq {\frac {N}{2}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq n_{1},n_{2},n_{3}\leq {\frac {N}{2}}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8de53f73f29cd25062e7a41dcd19e00662404bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.677ex; height:5.176ex;" alt="{\displaystyle 0\leq n_{1},n_{2},n_{3}\leq {\frac {N}{2}}-1}"></dd></dl> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\varphi _{i})=\cos(\varphi _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(\varphi _{i})=\cos(\varphi _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5dbeed0f0eb13e29237ff617b07f495d9fa0b28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.475ex; height:2.843ex;" alt="{\displaystyle c(\varphi _{i})=\cos(\varphi _{i})}">. The original 3-D DCT-II now can be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed57cfa5040af2521d3a7f2396a235690a614ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:68.501ex; height:7.676ex;" alt="{\displaystyle 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})}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{i}={\frac {\pi }{2N}}(4N_{i}+1),{\text{ and }}i=1,2,3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>N</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{i}={\frac {\pi }{2N}}(4N_{i}+1),{\text{ and }}i=1,2,3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7bdd6a16d0148ac7391571fee3f843c2574478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.167ex; height:4.676ex;" alt="{\displaystyle \varphi _{i}={\frac {\pi }{2N}}(4N_{i}+1),{\text{ and }}i=1,2,3.}"> If the even and the odd parts of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1},k_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{1},k_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38464b8cbb84b71e60d41873e4bfe788027f3cf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.565ex; height:2.509ex;" alt="{\displaystyle k_{1},k_{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d32e1c66b85257bfd6ad8be93186742d71a804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{3}}"> and are considered, the general formula for the calculation of the 3-D DCT-II can be expressed as <figure typeof="mw:File/Thumb"><a href="/wiki/File:Single_butterfly_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Single_butterfly_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg/310px-Single_butterfly_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg" decoding="async" width="310" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Single_butterfly_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg/465px-Single_butterfly_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Single_butterfly_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg/620px-Single_butterfly_of_the_3-D_DCT-II_VR_DIF_algorithm.jpg 2x" data-file-width="1811" data-file-height="864" /></a><figcaption>The single butterfly stage of VR DIF algorithm.</figcaption></figure> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 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))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fad790076764fe61641954a1130bd19ecfedf34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:89.982ex; height:9.343ex;" alt="{\displaystyle 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))}"></dd></dl> where <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d171604dd04fc955373cc315919399393e28317d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.494ex; height:4.843ex;" alt="{\displaystyle {\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)}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +(-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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>l</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +(-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)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39845a0b04f45d2437904528644bb3da933a7867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.824ex; height:4.843ex;" alt="{\displaystyle +(-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)}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +(-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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +(-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)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469baaa623b197c66106234a88e5a0de7389792c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.791ex; height:4.843ex;" alt="{\displaystyle +(-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)}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +(-1)^{i+l}{\tilde {x}}\left(n_{1}+{\frac {n}{2}},n_{2},n_{3}+{\frac {n}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>l</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +(-1)^{i+l}{\tilde {x}}\left(n_{1}+{\frac {n}{2}},n_{2},n_{3}+{\frac {n}{3}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88d2cc01816c1dd9fbbd6d1cce7c014265d0638d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.206ex; height:4.843ex;" alt="{\displaystyle +(-1)^{i+l}{\tilde {x}}\left(n_{1}+{\frac {n}{2}},n_{2},n_{3}+{\frac {n}{3}}\right)}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +(-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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>+</mo> <mi>l</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +(-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.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0762a6b2ecaa4e5dc1b87e4b34b95f59c832fff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:61.556ex; height:4.843ex;" alt="{\displaystyle +(-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.}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Arithmetic_complexity">Arithmetic complexity</h5>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=24" title="Edit section: Arithmetic complexity">edit</a>]</div> The whole 3-D DCT calculation needs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~[\log _{2}N]~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">[</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo stretchy="false">]</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~[\log _{2}N]~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d8bb5d31213b1734af79717a89534491e1f3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.932ex; height:2.843ex;" alt="{\displaystyle ~[\log _{2}N]~}"> stages, and each stage involves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\tfrac {1}{8}}\ N^{3}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~{\tfrac {1}{8}}\ N^{3}~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c7db17eb75a30c803b114125e2d7864cb1d24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.577ex; height:3.676ex;" alt="{\displaystyle ~{\tfrac {1}{8}}\ N^{3}~}"> butterflies. The whole 3-D DCT requires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\left[{\tfrac {1}{8}}\ N^{3}\log _{2}N\right]~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\left[{\tfrac {1}{8}}\ N^{3}\log _{2}N\right]~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91efa02764854739533932d0917df6fbb8faad42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.41ex; height:4.843ex;" alt="{\displaystyle ~\left[{\tfrac {1}{8}}\ N^{3}\log _{2}N\right]~}"> butterflies to be computed. Each butterfly requires seven real multiplications (including trivial multiplications) and 24 real additions (including trivial additions). Therefore, the total number of real multiplications needed for this stage is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\left[{\tfrac {7}{8}}\ N^{3}\ \log _{2}N\right]~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mtext> </mtext> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\left[{\tfrac {7}{8}}\ N^{3}\ \log _{2}N\right]~,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eee6265f015b96f01166b9015dbe8228d41b6f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.638ex; height:4.843ex;" alt="{\displaystyle ~\left[{\tfrac {7}{8}}\ N^{3}\ \log _{2}N\right]~,}"> 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<a href="#cite_note-:0-106">[106]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\underbrace {\left[{\frac {3}{2}}N^{3}\log _{2}N\right]} _{\text{Real}}+\underbrace {\left[{\frac {3}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]} _{\text{Recursive}}=\left[{\frac {9}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> </mrow> <mo>]</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Real</mtext> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo>−</mo> <mn>3</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Recursive</mtext> </mrow> </munder> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo>−</mo> <mn>3</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\underbrace {\left[{\frac {3}{2}}N^{3}\log _{2}N\right]} _{\text{Real}}+\underbrace {\left[{\frac {3}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]} _{\text{Recursive}}=\left[{\frac {9}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced5ebac308145cfd6efa3c1b191749e4ae23f0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:80.338ex; height:9.676ex;" alt="{\displaystyle ~\underbrace {\left[{\frac {3}{2}}N^{3}\log _{2}N\right]} _{\text{Real}}+\underbrace {\left[{\frac {3}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]} _{\text{Recursive}}=\left[{\frac {9}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]~.}"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\left[{\frac {3}{2}}N^{3}\log _{2}N\right]~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\left[{\frac {3}{2}}N^{3}\log _{2}N\right]~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35aa8c853807e7646c3ab9aaaa804f9e92f519d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.43ex; height:6.176ex;" alt="{\displaystyle ~\left[{\frac {3}{2}}N^{3}\log _{2}N\right]~}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\left[{\frac {9}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo>−</mo> <mn>3</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\left[{\frac {9}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]~,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fcd1a48d8b3795e4c1f5cec76e8642e096d084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.437ex; height:6.176ex;" alt="{\displaystyle ~\left[{\frac {9}{2}}N^{3}\log _{2}N-3N^{3}+3N^{2}\right]~,}"> respectively. From Table 1, it can be seen that the total number <table class="wikitable sortable"> <caption>TABLE 1 Comparison of VR DIF & RCF Algorithms for computing 3D-DCT-II </caption> <tbody><tr> <th>Transform Size </th> <th>3D VR Mults </th> <th>RCF Mults </th> <th>3D VR Adds </th> <th>RCF Adds </th></tr> <tr> <td>8 × 8 × 8 </td> <td>2.625 </td> <td>4.5 </td> <td>10.875 </td> <td>10.875 </td></tr> <tr> <td>16 × 16 × 16 </td> <td>3.5 </td> <td>6 </td> <td>15.188 </td> <td>15.188 </td></tr> <tr> <td>32 × 32 × 32 </td> <td>4.375 </td> <td>7.5 </td> <td>19.594 </td> <td>19.594 </td></tr> <tr> <td>64 × 64 × 64 </td> <td>5.25 </td> <td>9 </td> <td>24.047 </td> <td>24.047 </td></tr></tbody></table> 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.<a href="#cite_note-107">[107]</a> Therefore, although the above proposed 3-D VR algorithm does not achieve the theoretical lower bound on the number of multiplications,<a href="#cite_note-108">[108]</a> 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 <a href="/wiki/Cooley%E2%80%93Tukey_FFT_algorithm" title="Cooley–Tukey FFT algorithm">Cooley–Tukey FFT algorithm</a> 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 <a href="/wiki/Cooley%E2%80%93Tukey_FFT_algorithm" title="Cooley–Tukey FFT algorithm">Cooley–Tukey FFT algorithms</a>. <figure typeof="mw:File/Thumb"><a href="/wiki/File:DCT-8x8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/DCT-8x8.png/250px-DCT-8x8.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/DCT-8x8.png/375px-DCT-8x8.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/24/DCT-8x8.png 2x" data-file-width="438" data-file-height="438" /></a><figcaption>Two-dimensional DCT frequencies from the <a href="/wiki/JPEG#Discrete_cosine_transform" title="JPEG">JPEG DCT</a></figcaption></figure> The image to the right shows a combination of horizontal and vertical frequencies for an 8 × 8 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (~N_{1}=N_{2}=8~)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mtext> </mtext> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>8</mn> <mtext> </mtext> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (~N_{1}=N_{2}=8~)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9285dfe5c6ba6839a02429645ebd85156e077c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.171ex; height:2.843ex;" alt="{\displaystyle (~N_{1}=N_{2}=8~)}"> 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 ( 8×8 ) is transformed to a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of these 64 frequency squares. <div class="mw-heading mw-heading3"><h3 id="MD-DCT-IV">MD-DCT-IV</h3>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=25" title="Edit section: MD-DCT-IV">edit</a>]</div> The M-D DCT-IV is just an extension of 1-D DCT-IV on to M dimensional domain. The 2-D DCT-IV of a matrix or an image is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k,\ell }=\sum _{n=0}^{N-1}\;\sum _{m=0}^{M-1}\ x_{n,m}\cos \left(\ {\frac {\,(2m+1)(2k+1)\ \pi \,}{4N}}\ \right)\cos \left(\ {\frac {\,(2n+1)(2\ell +1)\ \pi \,}{4M}}\ \right)~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mspace width="thickmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>π</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mn>4</mn> <mi>N</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>ℓ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>π</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mn>4</mn> <mi>M</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k,\ell }=\sum _{n=0}^{N-1}\;\sum _{m=0}^{M-1}\ x_{n,m}\cos \left(\ {\frac {\,(2m+1)(2k+1)\ \pi \,}{4N}}\ \right)\cos \left(\ {\frac {\,(2n+1)(2\ell +1)\ \pi \,}{4M}}\ \right)~,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f69415ad62be9a5d95a7b59fbb7568a1dfcfd68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:79.039ex; height:7.343ex;" alt="{\displaystyle X_{k,\ell }=\sum _{n=0}^{N-1}\;\sum _{m=0}^{M-1}\ x_{n,m}\cos \left(\ {\frac {\,(2m+1)(2k+1)\ \pi \,}{4N}}\ \right)\cos \left(\ {\frac {\,(2n+1)(2\ell +1)\ \pi \,}{4M}}\ \right)~,}"></dd></dl> <dl><dd>for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~k=0,\ 1,\ 2\ \ldots \ N-1~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mo>…</mo> <mtext> </mtext> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~k=0,\ 1,\ 2\ \ldots \ N-1~~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc16a175673f1702ff43940a201124ae0b734949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.074ex; height:2.509ex;" alt="{\displaystyle ~~k=0,\ 1,\ 2\ \ldots \ N-1~~}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~~\ell =0,\ 1,\ 2,\ \ldots \ M-1~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mtext> </mtext> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~~\ell =0,\ 1,\ 2,\ \ldots \ M-1~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d7e531e7909ae7996c6d7918bb35c0fe8b38b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.311ex; height:2.509ex;" alt="{\displaystyle ~~\ell =0,\ 1,\ 2,\ \ldots \ M-1~.}"></dd></dl> We can compute the MD DCT-IV using the regular row-column method or we can use the polynomial transform method<a href="#cite_note-109">[109]</a> 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. <div class="mw-heading mw-heading2"><h2 id="Computation">Computation</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=26" title="Edit section: Computation">edit</a>]</div> Although the direct application of these formulas would require <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\mathcal {O}}(N^{2})~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~{\mathcal {O}}(N^{2})~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df84243526b91109c3fe5a41dea855f8bf63af03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.998ex; height:3.176ex;" alt="{\displaystyle ~{\mathcal {O}}(N^{2})~}"> operations, it is possible to compute the same thing with only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\mathcal {O}}(N\log N)~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~{\mathcal {O}}(N\log N)~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab22ab2eef7e7838d517dbeb16c8115c3fd9715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.694ex; height:2.843ex;" alt="{\displaystyle ~{\mathcal {O}}(N\log N)~}"> complexity by factorizing the computation similarly to the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT). One can also compute DCTs via FFTs combined with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\mathcal {O}}(N)~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~{\mathcal {O}}(N)~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11453790a547932579d2d1d6d0797ec38920e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle ~{\mathcal {O}}(N)~}"> pre- and post-processing steps. In general, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\mathcal {O}}(N\log N)~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~{\mathcal {O}}(N\log N)~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab22ab2eef7e7838d517dbeb16c8115c3fd9715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.694ex; height:2.843ex;" alt="{\displaystyle ~{\mathcal {O}}(N\log N)~}"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\mathcal {O}}(N)~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~{\mathcal {O}}(N)~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11453790a547932579d2d1d6d0797ec38920e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle ~{\mathcal {O}}(N)~}"> 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 <a href="/wiki/Power_of_two" title="Power of two">power-of-two</a> 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 (<a href="#CITEREFFrigoJohnson2005">Frigo & Johnson 2005</a>). Algorithms based on the <a href="/wiki/Cooley%E2%80%93Tukey_FFT_algorithm" title="Cooley–Tukey FFT algorithm">Cooley–Tukey FFT algorithm</a> are most common, but any other FFT algorithm is also applicable. For example, the <a href="/w/index.php?title=Winograd_FFT_algorithm&action=edit&redlink=1" class="new" title="Winograd FFT algorithm (page does not exist)">Winograd FFT algorithm</a> leads to minimal-multiplication algorithms for the DFT, albeit generally at the cost of more additions, and a similar algorithm was proposed by (<a href="#CITEREFFeigWinograd1992a">Feig & Winograd 1992a</a>) 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 (<a href="#CITEREFDuhamelVetterli1990">Duhamel & Vetterli 1990</a>). 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 N with FFT-based algorithms.<a href="#cite_note-110">[a]</a> Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the 8 × 8 DCT-II used in <a href="/wiki/JPEG" title="JPEG">JPEG</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~4N~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>4</mn> <mi>N</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~4N~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab67f680d80d426fdeeb9d5827fccd3a4e09735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.387ex; height:2.176ex;" alt="{\displaystyle ~4N~}"> 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 <a href="/wiki/FFTPACK" title="FFTPACK">FFTPACK</a> and <a href="/wiki/FFTW" title="FFTW">FFTW</a>) was described by <a href="#CITEREFNarasimhaPeterson1978">Narasimha & Peterson (1978)</a> and <a href="#CITEREFMakhoul1980">Makhoul (1980)</a>, 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.<a href="#cite_note-111">[b]</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~N~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>N</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~N~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9025f4623f468c4be704a2634cffd698a87fd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.225ex; height:2.176ex;" alt="{\displaystyle ~N~}"> real-data FFT is also performed by a real-data <a href="/wiki/Split-radix_FFT_algorithm" title="Split-radix FFT algorithm">split-radix algorithm</a> (as in <a href="#CITEREFSorensenJonesHeidemanBurrus1987">Sorensen et al. (1987)</a>), then the resulting algorithm actually matches what was long the lowest published arithmetic count for the power-of-two DCT-II (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~2N\log _{2}N-N+2~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>2</mn> <mi>N</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo>−</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~2N\log _{2}N-N+2~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424899e2a755cf09985e0f8385be4374a5ef3bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.158ex; height:2.676ex;" alt="{\displaystyle ~2N\log _{2}N-N+2~}"> real-arithmetic operations<a href="#cite_note-112">[c]</a>). A recent reduction in the operation count to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\tfrac {17}{9}}N\log _{2}N+{\mathcal {O}}(N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>17</mn> <mn>9</mn> </mfrac> </mstyle> </mrow> <mi>N</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~{\tfrac {17}{9}}N\log _{2}N+{\mathcal {O}}(N)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7adfa7941847bff3b93739db511b3f21205073b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.551ex; height:3.843ex;" alt="{\displaystyle ~{\tfrac {17}{9}}N\log _{2}N+{\mathcal {O}}(N)}"> also uses a real-data FFT.<a href="#cite_note-113">[110]</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~N~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>N</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~N~,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7471f2b90757f67d798f526321e3e5e4eb4e217" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.872ex; height:2.509ex;" alt="{\displaystyle ~N~,}"> but this is an implementation rather than an algorithmic question since it can be solved by unrolling or inlining.) <div class="mw-heading mw-heading2"><h2 id="Example_of_IDCT">Example of IDCT</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=27" title="Edit section: Example of IDCT">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:DCT_filter_comparison.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/DCT_filter_comparison.png/220px-DCT_filter_comparison.png" decoding="async" width="220" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/DCT_filter_comparison.png/330px-DCT_filter_comparison.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/DCT_filter_comparison.png/440px-DCT_filter_comparison.png 2x" data-file-width="538" data-file-height="709" /></a><figcaption>An example showing eight different filters applied to a test image (top left) by multiplying its DCT spectrum (top right) with each filter.</figcaption></figure> Consider this 8x8 grayscale image of capital letter A. <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Letter-a-8x8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/1a/Letter-a-8x8.png" decoding="async" width="240" height="81" class="mw-file-element" data-file-width="240" data-file-height="81" /></a><figcaption>Original size, scaled 10x (nearest neighbor), scaled 10x (bilinear).</figcaption></figure> <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Dct-table.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/63/Dct-table.png" decoding="async" width="684" height="684" class="mw-file-element" data-file-width="684" data-file-height="684" /></a><figcaption>Basis functions of the discrete cosine transformation with corresponding coefficients (specific for our image). DCT of the image = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>6.1917</mn> </mtd> <mtd> <mo>−</mo> <mn>0.3411</mn> </mtd> <mtd> <mn>1.2418</mn> </mtd> <mtd> <mn>0.1492</mn> </mtd> <mtd> <mn>0.1583</mn> </mtd> <mtd> <mn>0.2742</mn> </mtd> <mtd> <mo>−</mo> <mn>0.0724</mn> </mtd> <mtd> <mn>0.0561</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.2205</mn> </mtd> <mtd> <mn>0.0214</mn> </mtd> <mtd> <mn>0.4503</mn> </mtd> <mtd> <mn>0.3947</mn> </mtd> <mtd> <mo>−</mo> <mn>0.7846</mn> </mtd> <mtd> <mo>−</mo> <mn>0.4391</mn> </mtd> <mtd> <mn>0.1001</mn> </mtd> <mtd> <mo>−</mo> <mn>0.2554</mn> </mtd> </mtr> <mtr> <mtd> <mn>1.0423</mn> </mtd> <mtd> <mn>0.2214</mn> </mtd> <mtd> <mo>−</mo> <mn>1.0017</mn> </mtd> <mtd> <mo>−</mo> <mn>0.2720</mn> </mtd> <mtd> <mn>0.0789</mn> </mtd> <mtd> <mo>−</mo> <mn>0.1952</mn> </mtd> <mtd> <mn>0.2801</mn> </mtd> <mtd> <mn>0.4713</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>0.2340</mn> </mtd> <mtd> <mo>−</mo> <mn>0.0392</mn> </mtd> <mtd> <mo>−</mo> <mn>0.2617</mn> </mtd> <mtd> <mo>−</mo> <mn>0.2866</mn> </mtd> <mtd> <mn>0.6351</mn> </mtd> <mtd> <mn>0.3501</mn> </mtd> <mtd> <mo>−</mo> <mn>0.1433</mn> </mtd> <mtd> <mn>0.3550</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.2750</mn> </mtd> <mtd> <mn>0.0226</mn> </mtd> <mtd> <mn>0.1229</mn> </mtd> <mtd> <mn>0.2183</mn> </mtd> <mtd> <mo>−</mo> <mn>0.2583</mn> </mtd> <mtd> <mo>−</mo> <mn>0.0742</mn> </mtd> <mtd> <mo>−</mo> <mn>0.2042</mn> </mtd> <mtd> <mo>−</mo> <mn>0.5906</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.0653</mn> </mtd> <mtd> <mn>0.0428</mn> </mtd> <mtd> <mo>−</mo> <mn>0.4721</mn> </mtd> <mtd> <mo>−</mo> <mn>0.2905</mn> </mtd> <mtd> <mn>0.4745</mn> </mtd> <mtd> <mn>0.2875</mn> </mtd> <mtd> <mo>−</mo> <mn>0.0284</mn> </mtd> <mtd> <mo>−</mo> <mn>0.1311</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.3169</mn> </mtd> <mtd> <mn>0.0541</mn> </mtd> <mtd> <mo>−</mo> <mn>0.1033</mn> </mtd> <mtd> <mo>−</mo> <mn>0.0225</mn> </mtd> <mtd> <mo>−</mo> <mn>0.0056</mn> </mtd> <mtd> <mn>0.1017</mn> </mtd> <mtd> <mo>−</mo> <mn>0.1650</mn> </mtd> <mtd> <mo>−</mo> <mn>0.1500</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>0.2970</mn> </mtd> <mtd> <mo>−</mo> <mn>0.0627</mn> </mtd> <mtd> <mn>0.1960</mn> </mtd> <mtd> <mn>0.0644</mn> </mtd> <mtd> <mo>−</mo> <mn>0.1136</mn> </mtd> <mtd> <mo>−</mo> <mn>0.1031</mn> </mtd> <mtd> <mn>0.1887</mn> </mtd> <mtd> <mn>0.1444</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbbd724a246196dad6f8fb89243fee203908c887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:86.248ex; height:25.509ex;" alt="{\displaystyle {\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}}}">.</figcaption></figure> Each basis function is multiplied by its coefficient and then this product is added to the final image. <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Idct-animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/Idct-animation.gif" decoding="async" width="241" height="81" class="mw-file-element" data-file-width="241" data-file-height="81" /></a><figcaption>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×.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=28" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">Discrete wavelet transform</a></li> <li><a href="/wiki/JPEG#Discrete_cosine_transform" title="JPEG">JPEG - Discrete cosine transform</a> - Contains a potentially easier to understand example of DCT transformation</li> <li><a href="/wiki/List_of_Fourier-related_transforms" title="List of Fourier-related transforms">List of Fourier-related transforms</a></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">Modified discrete cosine transform</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=29" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-110"><a href="#cite_ref-110">^</a> 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. </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> The radix-4 step reduces the size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~4N~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>4</mn> <mi>N</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~4N~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab67f680d80d426fdeeb9d5827fccd3a4e09735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.387ex; height:2.176ex;" alt="{\displaystyle ~4N~}"> DFT to four size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~N~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>N</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~N~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9025f4623f468c4be704a2634cffd698a87fd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.225ex; height:2.176ex;" alt="{\displaystyle ~N~}"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~N~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>N</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~N~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9025f4623f468c4be704a2634cffd698a87fd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.225ex; height:2.176ex;" alt="{\displaystyle ~N~}"> FFT of real data plus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\mathcal {O}}(N)~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~{\mathcal {O}}(N)~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11453790a547932579d2d1d6d0797ec38920e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle ~{\mathcal {O}}(N)~}"> <a href="/wiki/Butterfly_(FFT_algorithm)" class="mw-redirect" title="Butterfly (FFT algorithm)">butterflies</a>, once the trivial and / or duplicate parts are eliminated and / or merged. </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~2N\log _{2}N-N+2~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>2</mn> <mi>N</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo>−</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~2N\log _{2}N-N+2~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424899e2a755cf09985e0f8385be4374a5ef3bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.158ex; height:2.676ex;" alt="{\displaystyle ~2N\log _{2}N-N+2~}"> count is for the DCT-II definition shown here; two multiplications can be saved if the transform is scaled by an overall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"> factor. Additional multiplications can be saved if one permits the outputs of the transform to be rescaled individually, as was shown by <a href="#CITEREFAraiAguiNakajima1988">Arai, Agui & Nakajima (1988)</a> for the size-8 case used in JPEG. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=30" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Stankovic-1">^ <a href="#cite_ref-Stankovic_1-0">a</a> <a href="#cite_ref-Stankovic_1-1">b</a> <a href="#cite_ref-Stankovic_1-2">c</a> <a href="#cite_ref-Stankovic_1-3">d</a> <a href="#cite_ref-Stankovic_1-4">e</a> <a href="#cite_ref-Stankovic_1-5">f</a> <a href="#cite_ref-Stankovic_1-6">g</a> <a href="#cite_ref-Stankovic_1-7">h</a> <a href="#cite_ref-Stankovic_1-8">i</a> <a href="#cite_ref-Stankovic_1-9">j</a> <a href="#cite_ref-Stankovic_1-10">k</a> <a href="#cite_ref-Stankovic_1-11">l</a> <a href="#cite_ref-Stankovic_1-12">m</a> <a href="#cite_ref-Stankovic_1-13">n</a> <a href="#cite_ref-Stankovic_1-14">o</a> <a href="#cite_ref-Stankovic_1-15">p</a> <a href="#cite_ref-Stankovic_1-16">q</a> <a href="#cite_ref-Stankovic_1-17">r</a> <a href="#cite_ref-Stankovic_1-18">s</a> <a href="#cite_ref-Stankovic_1-19">t</a> <a href="#cite_ref-Stankovic_1-20">u</a> <a href="#cite_ref-Stankovic_1-21">v</a> <a href="#cite_ref-Stankovic_1-22">w</a> <a href="#cite_ref-Stankovic_1-23">x</a> <a href="#cite_ref-Stankovic_1-24">y</a> <a href="#cite_ref-Stankovic_1-25">z</a> <a href="#cite_ref-Stankovic_1-26">aa</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFStankovićAstola2012" class="citation journal cs1">Stanković, Radomir S.; Astola, Jaakko T. 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May 22–23, 1997. p. 30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780780341470" title="Special:BookSources/9780780341470"><bdi>9780780341470</bdi></a>. <q>H.263 is similar to, but more complex than H.261. 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Aperture. p. 132. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781597110938" title="Special:BookSources/9781597110938"><bdi>9781597110938</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=jpegs&rft.btitle=Aperture&rft.pages=132&rft.pub=Aperture&rft.date=2009-05-31&rft.isbn=9781597110938&rft.aulast=Ruff&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFColberg2009" class="citation web cs1">Colberg, Jörg (April 17, 2009). <a rel="nofollow" class="external text" href="http://jmcolberg.com/weblog/2009/04/review_jpegs_by_thomas_ruff/">"Review: jpegs by Thomas Ruff"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Review%3A+jpegs+by+Thomas+Ruff&rft.date=2009-04-17&rft.aulast=Colberg&rft.aufirst=J%C3%B6rg&rft_id=http%3A%2F%2Fjmcolberg.com%2Fweblog%2F2009%2F04%2Freview_jpegs_by_thomas_ruff%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathworks.com/help/signal/ref/dct.html">"Discrete cosine transform - MATLAB dct"</a>. www.mathworks.com. Retrieved 2019-07-11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathworks.com&rft.atitle=Discrete+cosine+transform+-+MATLAB+dct&rft_id=https%3A%2F%2Fwww.mathworks.com%2Fhelp%2Fsignal%2Fref%2Fdct.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPennebakerMitchell1992" class="citation book cs1">Pennebaker, William B.; Mitchell, Joan L. (31 December 1992). JPEG: Still Image Data Compression Standard. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780442012724" title="Special:BookSources/9780442012724"><bdi>9780442012724</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=JPEG%3A+Still+Image+Data+Compression+Standard&rft.pub=Springer&rft.date=1992-12-31&rft.isbn=9780442012724&rft.aulast=Pennebaker&rft.aufirst=William+B.&rft.au=Mitchell%2C+Joan+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAraiAguiNakajima1988" class="citation journal cs1">Arai, Y.; Agui, T.; Nakajima, M. (1988). <a rel="nofollow" class="external text" href="https://search.ieice.org/bin/summary.php?id=e71-e_11_1095">"A fast DCT-SQ scheme for images"</a>. 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Signal Processing. 88 (6): 1553–1564. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs/0703150">cs/0703150</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008SigPr..88.1553S">2008SigPr..88.1553S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.sigpro.2008.01.004">10.1016/j.sigpro.2008.01.004</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:986733">986733</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Signal+Processing&rft.atitle=Type-II%2FIII+DCT%2FDST+algorithms+with+reduced+number+of+arithmetic+operations&rft.volume=88&rft.issue=6&rft.pages=1553-1564&rft.date=2008&rft_id=info%3Aarxiv%2Fcs%2F0703150&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A986733%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.sigpro.2008.01.004&rft_id=info%3Abibcode%2F2008SigPr..88.1553S&rft.aulast=Shao&rft.aufirst=Xuancheng&rft.au=Johnson%2C+Steven+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> <a href="#CITEREFMalvar1992">Malvar 1992</a> </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> <a href="#CITEREFMartucci1994">Martucci 1994</a> </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChanHo1990" class="citation journal cs1">Chan, S.C.; Ho, K.L. 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IEE Proceedings F - Radar and Signal Processing. 137 (6): 433. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fip-f-2.1990.0063">10.1049/ip-f-2.1990.0063</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEE+Proceedings+F+-+Radar+and+Signal+Processing&rft.atitle=Direct+methods+for+computing+discrete+sinusoidal+transforms&rft.volume=137&rft.issue=6&rft.pages=433&rft.date=1990&rft_id=info%3Adoi%2F10.1049%2Fip-f-2.1990.0063&rft.aulast=Chan&rft.aufirst=S.C.&rft.au=Ho%2C+K.L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-:0-106">^ <a href="#cite_ref-:0_106-0">a</a> <a href="#cite_ref-:0_106-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlshibamiBoussakta2001" class="citation journal cs1">Alshibami, O.; Boussakta, S. (July 2001). "Three-dimensional algorithm for the 3-D DCT-III". Proc. Sixth Int. Symp. Commun., Theory Applications: 104–107.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Sixth+Int.+Symp.+Commun.%2C+Theory+Applications&rft.atitle=Three-dimensional+algorithm+for+the+3-D+DCT-III&rft.pages=104-107&rft.date=2001-07&rft.aulast=Alshibami&rft.aufirst=O.&rft.au=Boussakta%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuoan_BiGang_LiKai-Kuang_MaTan2000" class="citation journal cs1">Guoan Bi; Gang Li; Kai-Kuang Ma; Tan, T.C. (2000). "On the computation of two-dimensional DCT". IEEE Transactions on Signal Processing. 48 (4): 1171–1183. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000ITSP...48.1171B">2000ITSP...48.1171B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F78.827550">10.1109/78.827550</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Signal+Processing&rft.atitle=On+the+computation+of+two-dimensional+DCT&rft.volume=48&rft.issue=4&rft.pages=1171-1183&rft.date=2000&rft_id=info%3Adoi%2F10.1109%2F78.827550&rft_id=info%3Abibcode%2F2000ITSP...48.1171B&rft.au=Guoan+Bi&rft.au=Gang+Li&rft.au=Kai-Kuang+Ma&rft.au=Tan%2C+T.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeigWinograd1992a" class="citation journal cs1">Feig, E.; Winograd, S. 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IEEE Transactions on Information Theory. 38 (4): 1387–1391. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F18.144722">10.1109/18.144722</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Information+Theory&rft.atitle=On+the+multiplicative+complexity+of+discrete+cosine+transforms&rft.volume=38&rft.issue=4&rft.pages=1387-1391&rft.date=1992-07&rft_id=info%3Adoi%2F10.1109%2F18.144722&rft.aulast=Feig&rft.aufirst=E.&rft.au=Winograd%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNussbaumer1981" class="citation book cs1">Nussbaumer, H.J. (1981). 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Signal Processing. 88 (6): 1553–1564. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs/0703150">cs/0703150</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008SigPr..88.1553S">2008SigPr..88.1553S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.sigpro.2008.01.004">10.1016/j.sigpro.2008.01.004</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:986733">986733</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Signal+Processing&rft.atitle=Type-II%2FIII+DCT%2FDST+algorithms+with+reduced+number+of+arithmetic+operations&rft.volume=88&rft.issue=6&rft.pages=1553-1564&rft.date=2008&rft_id=info%3Aarxiv%2Fcs%2F0703150&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A986733%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.sigpro.2008.01.004&rft_id=info%3Abibcode%2F2008SigPr..88.1553S&rft.aulast=Shao&rft.aufirst=Xuancheng&rft.au=Johnson%2C+Steven+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Discrete_cosine_transform&action=edit&section=31" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNarasimhaPeterson1978" class="citation journal cs1">Narasimha, M.; Peterson, A. (June 1978). "On the Computation of the Discrete Cosine Transform". IEEE Transactions on Communications. 26 (6): 934–936. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCOM.1978.1094144">10.1109/TCOM.1978.1094144</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Communications&rft.atitle=On+the+Computation+of+the+Discrete+Cosine+Transform&rft.volume=26&rft.issue=6&rft.pages=934-936&rft.date=1978-06&rft_id=info%3Adoi%2F10.1109%2FTCOM.1978.1094144&rft.aulast=Narasimha&rft.aufirst=M.&rft.au=Peterson%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMakhoul1980" class="citation journal cs1">Makhoul, J. (February 1980). "A fast cosine transform in one and two dimensions". IEEE Transactions on Acoustics, Speech, and Signal Processing. 28 (1): 27–34. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTASSP.1980.1163351">10.1109/TASSP.1980.1163351</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Acoustics%2C+Speech%2C+and+Signal+Processing&rft.atitle=A+fast+cosine+transform+in+one+and+two+dimensions&rft.volume=28&rft.issue=1&rft.pages=27-34&rft.date=1980-02&rft_id=info%3Adoi%2F10.1109%2FTASSP.1980.1163351&rft.aulast=Makhoul&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSorensenJonesHeidemanBurrus1987" class="citation journal cs1">Sorensen, H.; Jones, D.; Heideman, M.; Burrus, C. (June 1987). "Real-valued fast Fourier transform algorithms". IEEE Transactions on Acoustics, Speech, and Signal Processing. 35 (6): 849–863. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.205.4523">10.1.1.205.4523</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTASSP.1987.1165220">10.1109/TASSP.1987.1165220</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Acoustics%2C+Speech%2C+and+Signal+Processing&rft.atitle=Real-valued+fast+Fourier+transform+algorithms&rft.volume=35&rft.issue=6&rft.pages=849-863&rft.date=1987-06&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.205.4523%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1109%2FTASSP.1987.1165220&rft.aulast=Sorensen&rft.aufirst=H.&rft.au=Jones%2C+D.&rft.au=Heideman%2C+M.&rft.au=Burrus%2C+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+cosine+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlonkaTasche2005" class="citation journal cs1"><a href="/wiki/Gerlind_Plonka" title="Gerlind Plonka">Plonka, G.</a>; Tasche, M. 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Johnson">Steven G. Johnson</a>: FFTW, <a rel="nofollow" class="external text" href="http://www.fftw.org/">FFTW Home Page</a>. A free (<a href="/wiki/GNU_General_Public_License" title="GNU General Public License">GPL</a>) C library that can compute fast DCTs (types I-IV) in one or more dimensions, of arbitrary size.</li> <li>Takuya Ooura: General Purpose FFT Package, <a rel="nofollow" class="external text" href="http://www.kurims.kyoto-u.ac.jp/~ooura/fft.html">FFT Package 1-dim / 2-dim</a>. Free C & FORTRAN libraries for computing fast DCTs (types II–III) in one, two or three dimensions, power of 2 sizes.</li> <li>Tim Kientzle: Fast algorithms for computing the 8-point DCT and IDCT, <a rel="nofollow" class="external text" href="http://drdobbs.com/parallel/184410889">Algorithm Alley</a>.</li> <li><a rel="nofollow" class="external text" href="http://ltfat.sourceforge.net/">LTFAT</a> is a free Matlab/Octave toolbox with interfaces to the FFTW implementation of the DCTs and DSTs of type I-IV.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output 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methods"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Compression_methods" title="Special:EditPage/Template:Compression methods"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_compression_methods" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_compression" title="Data compression">Data compression</a> methods</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lossless_compression" title="Lossless compression">Lossless</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Entropy_coding" title="Entropy coding">Entropy type</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_coding" title="Adaptive coding">Adaptive coding</a></li> <li><a href="/wiki/Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li> <li><a href="/wiki/Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric numeral systems</a></li> <li><a href="/wiki/Golomb_coding" title="Golomb coding">Golomb</a></li> <li><a href="/wiki/Huffman_coding" title="Huffman coding">Huffman</a> <ul><li><a href="/wiki/Adaptive_Huffman_coding" title="Adaptive Huffman coding">Adaptive</a></li> <li><a href="/wiki/Canonical_Huffman_code" title="Canonical Huffman code">Canonical</a></li> <li><a href="/wiki/Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li> <li><a href="/wiki/Range_coding" title="Range coding">Range</a></li> <li><a href="/wiki/Shannon_coding" title="Shannon coding">Shannon</a></li> <li><a href="/wiki/Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a></li> <li><a href="/wiki/Shannon%E2%80%93Fano%E2%80%93Elias_coding" title="Shannon–Fano–Elias coding">Shannon–Fano–Elias</a></li> <li><a href="/wiki/Tunstall_coding" title="Tunstall coding">Tunstall</a></li> <li><a href="/wiki/Unary_coding" title="Unary coding">Unary</a></li> <li><a href="/wiki/Universal_code_(data_compression)" title="Universal code (data compression)">Universal</a> <ul><li><a href="/wiki/Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a></li> <li><a href="/wiki/Fibonacci_coding" title="Fibonacci coding">Fibonacci</a></li> <li><a href="/wiki/Elias_gamma_coding" title="Elias gamma coding">Gamma</a></li> <li><a href="/wiki/Levenshtein_coding" title="Levenshtein coding">Levenshtein</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Dictionary_coder" title="Dictionary coder">Dictionary type</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Byte_pair_encoding" title="Byte pair encoding">Byte pair encoding</a></li> <li><a href="/wiki/LZ77_and_LZ78" title="LZ77 and LZ78">Lempel–Ziv</a> <ul><li><a href="/wiki/842_(compression_algorithm)" title="842 (compression algorithm)">842</a></li> <li><a href="/wiki/LZ4_(compression_algorithm)" title="LZ4 (compression algorithm)">LZ4</a></li> <li><a href="/wiki/LZJB" class="mw-redirect" title="LZJB">LZJB</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Oberhumer" title="Lempel–Ziv–Oberhumer">LZO</a></li> <li><a href="/wiki/LZRW" title="LZRW">LZRW</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Storer%E2%80%93Szymanski" title="Lempel–Ziv–Storer–Szymanski">LZSS</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Welch" title="Lempel–Ziv–Welch">LZW</a></li> <li><a href="/wiki/LZWL" title="LZWL">LZWL</a></li> <li><a href="/wiki/Snappy_(compression)" title="Snappy (compression)">Snappy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Other types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burrows%E2%80%93Wheeler_transform" title="Burrows–Wheeler transform">BWT</a></li> <li><a href="/wiki/Context_tree_weighting" title="Context tree weighting">CTW</a></li> <li><a href="/wiki/Context_mixing" title="Context mixing">CM</a></li> <li><a href="/wiki/Delta_encoding" title="Delta encoding">Delta</a> <ul><li><a href="/wiki/Incremental_encoding" title="Incremental encoding">Incremental</a></li></ul></li> <li><a href="/wiki/Dynamic_Markov_compression" title="Dynamic Markov compression">DMC</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li> <li><a href="/wiki/Grammar-based_code" title="Grammar-based code">Grammar</a> <ul><li><a href="/wiki/Re-Pair" title="Re-Pair">Re-Pair</a></li> <li><a href="/wiki/Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a></li></ul></li> <li><a class="mw-selflink selflink">LDCT</a></li> <li><a href="/wiki/Move-to-front_transform" title="Move-to-front transform">MTF</a></li> <li><a href="/wiki/PAQ" title="PAQ">PAQ</a></li> <li><a href="/wiki/Prediction_by_partial_matching" title="Prediction by partial matching">PPM</a></li> <li><a href="/wiki/Run-length_encoding" title="Run-length encoding">RLE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Hybrid</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li>LZ77 + Huffman <ul><li><a href="/wiki/Deflate" title="Deflate">Deflate</a></li> <li><a href="/wiki/LZX" title="LZX">LZX</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Stac" title="Lempel–Ziv–Stac">LZS</a></li></ul></li> <li>LZ77 + ANS <ul><li><a href="/wiki/LZFSE" title="LZFSE">LZFSE</a></li></ul></li> <li>LZ77 + Huffman + ANS <ul><li><a href="/wiki/Zstd" title="Zstd">Zstandard</a></li></ul></li> <li>LZ77 + Huffman + context <ul><li><a href="/wiki/Brotli" title="Brotli">Brotli</a></li></ul></li> <li>LZSS + Huffman <ul><li><a href="/wiki/LHA_(file_format)" title="LHA (file format)">LHA/LZH</a></li></ul></li> <li>LZ77 + Range <ul><li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Markov_chain_algorithm" title="Lempel–Ziv–Markov chain algorithm">LZMA</a></li> <li>LZHAM</li></ul></li> <li>RLE + BWT + MTF + Huffman <ul><li><a href="/wiki/Bzip2" title="Bzip2">bzip2</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lossy_compression" title="Lossy compression">Lossy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Transform_coding" title="Transform coding">Transform type</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Discrete cosine transform</a> <ul><li><a class="mw-selflink selflink">DCT</a></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li></ul></li> <li><a href="/wiki/Discrete_sine_transform" title="Discrete sine transform">DST</a></li> <li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li> <li><a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Predictive type</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a> <ul><li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Log_area_ratio" title="Log area ratio">LAR</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li>Motion <ul><li><a href="/wiki/Motion_compensation" title="Motion compensation">Compensation</a></li> <li><a href="/wiki/Motion_estimation" title="Motion estimation">Estimation</a></li> <li><a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li> <li><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression#Audio" title="Data compression">Audio</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_rate" title="Bit rate">Bit rate</a> <ul><li><a href="/wiki/Average_bitrate" title="Average bitrate">ABR</a></li> <li><a href="/wiki/Constant_bitrate" title="Constant bitrate">CBR</a></li> <li><a href="/wiki/Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li> <li><a href="/wiki/Companding" title="Companding">Companding</a></li> <li><a href="/wiki/Convolution" title="Convolution">Convolution</a></li> <li><a href="/wiki/Dynamic_range" title="Dynamic range">Dynamic range</a></li> <li><a href="/wiki/Latency_(audio)" title="Latency (audio)">Latency</a></li> <li><a href="/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon theorem</a></li> <li><a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></li> <li><a href="/wiki/Silence_compression" title="Silence compression">Silence compression</a></li> <li><a href="/wiki/Sound_quality" title="Sound quality">Sound quality</a></li> <li><a href="/wiki/Speech_coding" title="Speech coding">Speech coding</a></li> <li><a href="/wiki/Sub-band_coding" title="Sub-band coding">Sub-band coding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Audio_codec" title="Audio codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/A-law_algorithm" title="A-law algorithm">A-law</a></li> <li><a href="/wiki/%CE%9C-law_algorithm" title="Μ-law algorithm">μ-law</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a> <ul><li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li> <li><a href="/wiki/Delta_modulation" title="Delta modulation">DM</a></li></ul></li> <li><a href="/wiki/Fourier_transform" title="Fourier transform">FT</a> <ul><li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Log_area_ratio" title="Log area ratio">LAR</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li> <li><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Image_compression" title="Image compression">Image</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a></li> <li><a href="/wiki/Coding_tree_unit" title="Coding tree unit">Coding tree unit</a></li> <li><a href="/wiki/Color_space" title="Color space">Color space</a></li> <li><a href="/wiki/Compression_artifact" title="Compression artifact">Compression artifact</a></li> <li><a href="/wiki/Image_resolution" title="Image resolution">Image resolution</a></li> <li><a href="/wiki/Macroblock" title="Macroblock">Macroblock</a></li> <li><a href="/wiki/Pixel" title="Pixel">Pixel</a></li> <li><a href="/wiki/Peak_signal-to-noise_ratio" title="Peak signal-to-noise ratio">PSNR</a></li> <li><a href="/wiki/Quantization_(image_processing)" title="Quantization (image processing)">Quantization</a></li> <li><a href="/wiki/Standard_test_image" title="Standard test image">Standard test image</a></li> <li><a href="/wiki/Texture_compression" title="Texture compression">Texture compression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chain_code" title="Chain code">Chain code</a></li> <li><a class="mw-selflink selflink">DCT</a></li> <li><a href="/wiki/Deflate" title="Deflate">Deflate</a></li> <li><a href="/wiki/Fractal_compression" title="Fractal compression">Fractal</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">KLT</a></li> <li><a href="/wiki/Pyramid_(image_processing)" title="Pyramid (image processing)">LP</a></li> <li><a href="/wiki/Run-length_encoding" title="Run-length encoding">RLE</a></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li> <li><a href="/wiki/Embedded_zerotrees_of_wavelet_transforms" title="Embedded zerotrees of wavelet transforms">EZW</a></li> <li><a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression#Video" title="Data compression">Video</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_rate" title="Bit rate">Bit rate</a> <ul><li><a href="/wiki/Average_bitrate" title="Average bitrate">ABR</a></li> <li><a href="/wiki/Constant_bitrate" title="Constant bitrate">CBR</a></li> <li><a href="/wiki/Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li> <li><a href="/wiki/Display_resolution" title="Display resolution">Display resolution</a></li> <li><a href="/wiki/Film_frame" title="Film frame">Frame</a></li> <li><a href="/wiki/Frame_rate" title="Frame rate">Frame rate</a></li> <li><a href="/wiki/Video_compression_picture_types" title="Video compression picture types">Frame types</a></li> <li><a href="/wiki/Interlaced_video" title="Interlaced video">Interlace</a></li> <li><a href="/wiki/Video#Characteristics_of_video_streams" title="Video">Video characteristics</a></li> <li><a href="/wiki/Video_quality" title="Video quality">Video quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Video_codec" title="Video codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">DCT</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li> <li><a href="/wiki/Deblocking_filter" title="Deblocking filter">Deblocking filter</a></li> <li><a href="/wiki/Lapped_transform" title="Lapped transform">Lapped transform</a></li> <li>Motion <ul><li><a href="/wiki/Motion_compensation" title="Motion compensation">Compensation</a></li> <li><a href="/wiki/Motion_estimation" title="Motion estimation">Estimation</a></li> <li><a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_theory" title="Information theory">Theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Compressed_data_structure" title="Compressed data structure">Compressed data structures</a> <ul><li><a href="/wiki/Compressed_suffix_array" title="Compressed suffix array">Compressed suffix array</a></li> <li><a href="/wiki/FM-index" title="FM-index">FM-index</a></li></ul></li> <li><a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a> <ul><li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">Timeline</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Prefix_code" title="Prefix code">Prefix code</a></li> <li><a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li> <li><a href="/wiki/Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion</a></li> <li><a href="/wiki/Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li> <li><a href="/wiki/Data_compression_symmetry" title="Data compression symmetry">Symmetry</a></li> <li><a href="/wiki/Smallest_grammar_problem" title="Smallest grammar problem">Smallest grammar problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Community</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hutter_Prize" title="Hutter Prize">Hutter Prize</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mark_Adler" title="Mark Adler">Mark Adler</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/23px-Symbol_template_class_pink.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/31px-Symbol_template_class_pink.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Template:Compression_formats" title="Template:Compression formats">Compression formats</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/23px-Symbol_template_class_pink.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/31px-Symbol_template_class_pink.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Template:Compression_software" title="Template:Compression software">Compression software</a> (<a href="/wiki/Codec" title="Codec">codecs</a>)</li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Multimedia_compression_and_container_formats" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Compression_formats" title="Template:Compression formats"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Compression_formats" title="Template talk:Compression formats"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Compression_formats" title="Special:EditPage/Template:Compression formats"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Multimedia_compression_and_container_formats" style="font-size:114%;margin:0 4em"><a href="/wiki/Multimedia" title="Multimedia">Multimedia</a> <a href="/wiki/Data_compression" title="Data compression">compression</a> and <a href="/wiki/Container_format_(computing)" class="mw-redirect" title="Container format (computing)">container</a> formats</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Video_coding_format" title="Video coding format">Video compression</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">ISO</a>, <a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">IEC</a>, <a href="/wiki/Moving_Picture_Experts_Group" title="Moving Picture Experts Group">MPEG</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/DV_(video_format)" title="DV (video format)">DV</a></li> <li><a href="/wiki/Motion_JPEG" title="Motion JPEG">MJPEG</a></li> <li><a href="/wiki/Motion_JPEG_2000" title="Motion JPEG 2000">Motion JPEG 2000</a></li> <li><a href="/wiki/MPEG-1" title="MPEG-1">MPEG-1</a></li> <li><a href="/wiki/MPEG-2" title="MPEG-2">MPEG-2</a> <ul><li><a href="/wiki/H.262/MPEG-2_Part_2" title="H.262/MPEG-2 Part 2">Part 2</a></li></ul></li> <li><a href="/wiki/MPEG-4" title="MPEG-4">MPEG-4</a> <ul><li><a href="/wiki/MPEG-4_Part_2" title="MPEG-4 Part 2">Part 2 / ASP</a></li> <li><a href="/wiki/H.264/MPEG-4_AVC" class="mw-redirect" title="H.264/MPEG-4 AVC">Part 10 / AVC</a></li> <li><a href="/wiki/MPEG-4_IVC" class="mw-redirect" title="MPEG-4 IVC">Part 33 / IVC</a></li></ul></li> <li><a href="/wiki/MPEG-H" title="MPEG-H">MPEG-H</a> <ul><li><a href="/wiki/High_Efficiency_Video_Coding" title="High Efficiency Video Coding">Part 2 / HEVC</a></li></ul></li> <li><a href="/w/index.php?title=MPEG-I&action=edit&redlink=1" class="new" title="MPEG-I (page does not exist)">MPEG-I</a> <ul><li><a href="/wiki/Versatile_Video_Coding" title="Versatile Video Coding">Part 3 / VVC</a></li></ul></li> <li><a href="/wiki/MPEG-5" class="mw-redirect" title="MPEG-5">MPEG-5</a> <ul><li><a href="/wiki/Essential_Video_Coding" title="Essential Video Coding">Part 1 / EVC</a></li> <li><a href="/wiki/LCEVC" title="LCEVC">Part 2 / LCEVC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/ITU-T" title="ITU-T">ITU-T</a>, <a href="/wiki/Video_Coding_Experts_Group" title="Video Coding Experts Group">VCEG</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/H.120" title="H.120">H.120</a></li> <li><a href="/wiki/H.261" title="H.261">H.261</a></li> <li><a href="/wiki/H.262/MPEG-2_Part_2" title="H.262/MPEG-2 Part 2">H.262</a></li> <li><a href="/wiki/H.263" title="H.263">H.263</a></li> <li><a href="/wiki/Advanced_Video_Coding" title="Advanced Video Coding">H.264 / AVC</a></li> <li><a href="/wiki/High_Efficiency_Video_Coding" title="High Efficiency Video Coding">H.265 / HEVC</a></li> <li><a href="/wiki/Versatile_Video_Coding" title="Versatile Video Coding">H.266 / VVC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/Society_of_Motion_Picture_and_Television_Engineers" title="Society of Motion Picture and Television Engineers">SMPTE</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/VC-1" title="VC-1">VC-1</a></li> <li><a href="/wiki/Dirac_(video_compression_format)" title="Dirac (video compression format)">VC-2</a></li> <li><a href="/wiki/Avid_DNxHD" title="Avid DNxHD">VC-3</a></li> <li><a href="/wiki/CineForm" title="CineForm">VC-5</a></li> <li><a href="/wiki/VC-6" title="VC-6">VC-6</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/On2_Technologies" title="On2 Technologies">TrueMotion</a> and AOMedia</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/On2_Technologies#TrueMotion_S" title="On2 Technologies">TrueMotion S</a></li> <li><a href="/wiki/VP3" title="VP3">VP3</a></li> <li><a href="/wiki/VP6" title="VP6">VP6</a></li> <li><a href="/wiki/VP7" class="mw-redirect" title="VP7">VP7</a></li> <li><a href="/wiki/VP8" title="VP8">VP8</a></li> <li><a href="/wiki/VP9" title="VP9">VP9</a></li> <li><a href="/wiki/AV1" title="AV1">AV1</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Chinese Standard</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Audio_Video_Standard#First_generation" title="Audio Video Standard">AVS1 P2/AVS+</a>(GB/T 20090.2/16)</li> <li><a href="/wiki/Audio_Video_Standard#Second_generation" title="Audio Video Standard">AVS2 P2</a>(GB/T 33475.2,GY/T 299.1) <ul><li>HDR Vivid(GY/T 358)</li></ul></li> <li>AVS3 P2(GY/T 368)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Apple_Video" title="Apple Video">Apple Video</a></li> <li><a href="/wiki/Audio_Video_Standard" title="Audio Video Standard">AVS</a></li> <li><a href="/wiki/Bink_Video" title="Bink Video">Bink</a></li> <li><a href="/wiki/Cinepak" title="Cinepak">Cinepak</a></li> <li><a href="/wiki/Daala" title="Daala">Daala</a></li> <li><a href="/wiki/Digital_Video_Interactive" title="Digital Video Interactive">DVI</a></li> <li><a href="/wiki/FFV1" title="FFV1">FFV1</a></li> <li><a href="/wiki/Huffyuv" title="Huffyuv">Huffyuv</a></li> <li><a href="/wiki/Indeo" title="Indeo">Indeo</a></li> <li><a href="/wiki/Lagarith" title="Lagarith">Lagarith</a></li> <li><a href="/wiki/Microsoft_Video_1" title="Microsoft Video 1">Microsoft Video 1</a></li> <li><a href="/wiki/MSU_Lossless_Video_Codec" title="MSU Lossless Video Codec">MSU Lossless</a></li> <li><a href="/wiki/OMS_Video" title="OMS Video">OMS Video</a></li> <li><a href="/wiki/Pixlet" title="Pixlet">Pixlet</a></li> <li><a href="/wiki/Apple_ProRes" title="Apple ProRes">ProRes</a> <ul><li><a href="/wiki/ProRes_422" class="mw-redirect" title="ProRes 422">422</a></li> <li><a href="/wiki/ProRes_4444" class="mw-redirect" title="ProRes 4444">4444</a></li></ul></li> <li>QuickTime <ul><li><a href="/wiki/QuickTime_Animation" title="QuickTime Animation">Animation</a></li> <li><a href="/wiki/QuickTime_Graphics" title="QuickTime Graphics">Graphics</a></li></ul></li> <li><a href="/wiki/RealVideo" title="RealVideo">RealVideo</a></li> <li><a href="/wiki/RTVideo" title="RTVideo">RTVideo</a></li> <li><a href="/wiki/SheerVideo" title="SheerVideo">SheerVideo</a></li> <li><a href="/wiki/Smacker_video" title="Smacker video">Smacker</a></li> <li><a href="/wiki/Sorenson_Media" title="Sorenson Media">Sorenson Video/Spark</a></li> <li><a href="/wiki/Theora" title="Theora">Theora</a></li> <li><a href="/wiki/Thor_(video_codec)" title="Thor (video codec)">Thor</a></li> <li><a href="/wiki/Ut_Video_Codec_Suite" title="Ut Video Codec Suite">Ut</a></li> <li><a href="/wiki/Windows_Media_Video" title="Windows Media Video">WMV</a></li> <li><a href="/wiki/RatDVD" title="RatDVD">XEB</a></li> <li><a href="/wiki/YULS" title="YULS">YULS</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Audio_coding_format" title="Audio coding format">Audio compression</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">ISO</a>, <a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">IEC</a>, <a href="/wiki/Moving_Picture_Experts_Group" title="Moving Picture Experts Group">MPEG</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/MPEG-1_Audio_Layer_II" title="MPEG-1 Audio Layer II">MPEG-1 Layer II</a> <ul><li><a href="/wiki/MPEG_Multichannel" title="MPEG Multichannel">Multichannel</a></li></ul></li> <li><a href="/wiki/MPEG-1_Audio_Layer_I" title="MPEG-1 Audio Layer I">MPEG-1 Layer I</a></li> <li><a href="/wiki/MP3" title="MP3">MPEG-1 Layer III (MP3)</a></li> <li><a href="/wiki/Advanced_Audio_Coding" title="Advanced Audio Coding">AAC</a> <ul><li><a href="/wiki/High-Efficiency_Advanced_Audio_Coding" title="High-Efficiency Advanced Audio Coding">HE-AAC</a></li> <li><a href="/wiki/AAC-LD" title="AAC-LD">AAC-LD</a></li></ul></li> <li><a href="/wiki/MPEG_Surround" title="MPEG Surround">MPEG Surround</a></li> <li><a href="/wiki/Audio_Lossless_Coding" title="Audio Lossless Coding">MPEG-4 ALS</a></li> <li><a href="/wiki/MPEG-4_SLS" title="MPEG-4 SLS">MPEG-4 SLS</a></li> <li><a href="/wiki/Super_Audio_CD#DST" title="Super Audio CD">MPEG-4 DST</a></li> <li><a href="/wiki/Harmonic_Vector_Excitation_Coding" title="Harmonic Vector Excitation Coding">MPEG-4 HVXC</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">MPEG-4 CELP</a></li> <li><a href="/wiki/Unified_Speech_and_Audio_Coding" title="Unified Speech and Audio Coding">MPEG-D USAC</a></li> <li><a href="/wiki/MPEG-H_3D_Audio" title="MPEG-H 3D Audio">MPEG-H 3D Audio</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/ITU-T" title="ITU-T">ITU-T</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G.711" title="G.711">G.711</a> <ul><li><a href="/wiki/A-law_algorithm" title="A-law algorithm">A-law</a></li> <li><a href="/wiki/%CE%9C-law_algorithm" title="Μ-law algorithm">µ-law</a></li></ul></li> <li><a href="/wiki/G.718" title="G.718">G.718</a></li> <li><a href="/wiki/G.719" title="G.719">G.719</a></li> <li><a href="/wiki/G.722" title="G.722">G.722</a></li> <li><a href="/wiki/G.722.1" title="G.722.1">G.722.1</a></li> <li><a href="/wiki/Adaptive_Multi-Rate_Wideband" title="Adaptive Multi-Rate Wideband">G.722.2</a></li> <li><a href="/wiki/G.723" title="G.723">G.723</a></li> <li><a href="/wiki/G.723.1" title="G.723.1">G.723.1</a></li> <li><a href="/wiki/G.726" title="G.726">G.726</a></li> <li><a href="/wiki/G.728" title="G.728">G.728</a></li> <li><a href="/wiki/G.729" title="G.729">G.729</a></li> <li><a href="/wiki/G.729.1" title="G.729.1">G.729.1</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/Internet_Engineering_Task_Force" title="Internet Engineering Task Force">IETF</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Opus_(audio_format)" title="Opus (audio format)">Opus</a></li> <li><a href="/wiki/Internet_Low_Bitrate_Codec" title="Internet Low Bitrate Codec">iLBC</a></li> <li><a href="/wiki/Speex" title="Speex">Speex</a></li> <li><a href="/wiki/Vorbis" title="Vorbis">Vorbis</a></li> <li><a href="/wiki/FLAC" title="FLAC">FLAC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/3GPP" title="3GPP">3GPP</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_Multi-Rate_audio_codec" title="Adaptive Multi-Rate audio codec">AMR</a></li> <li><a href="/wiki/Adaptive_Multi-Rate_Wideband" title="Adaptive Multi-Rate Wideband">AMR-WB</a></li> <li><a href="/wiki/Extended_Adaptive_Multi-Rate_%E2%80%93_Wideband" title="Extended Adaptive Multi-Rate – Wideband">AMR-WB+</a></li> <li><a href="/wiki/Enhanced_Variable_Rate_Codec" title="Enhanced Variable Rate Codec">EVRC</a></li> <li><a href="/wiki/Enhanced_Variable_Rate_Codec_B" title="Enhanced Variable Rate Codec B">EVRC-B</a></li> <li><a href="/wiki/Enhanced_Voice_Services" title="Enhanced Voice Services">EVS</a></li> <li><a href="/wiki/Half_Rate" title="Half Rate">GSM-HR</a></li> <li><a href="/wiki/Full_Rate" title="Full Rate">GSM-FR</a></li> <li><a href="/wiki/Enhanced_full_rate" title="Enhanced full rate">GSM-EFR</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/ETSI" class="mw-redirect" title="ETSI">ETSI</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dolby_Digital" title="Dolby Digital">AC-3</a></li> <li><a href="/wiki/Dolby_AC-4" title="Dolby AC-4">AC-4</a></li> <li><a href="/wiki/DTS_(sound_system)" class="mw-redirect" title="DTS (sound system)">DTS</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/Bluetooth_Special_Interest_Group" title="Bluetooth Special Interest Group">Bluetooth SIG</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/SBC_(codec)" title="SBC (codec)">SBC</a></li> <li><a href="/wiki/LC3_(codec)" title="LC3 (codec)">LC3</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Chinese Standard</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Audio_Video_Standard#First_generation" title="Audio Video Standard">AVS1 P10</a>(GB/T 20090.10)</li> <li><a href="/wiki/Audio_Video_Standard#Second_generation" title="Audio Video Standard">AVS2 P3</a>(GB/T 33475.3) <ul><li><a href="/w/index.php?title=Audio_Vivid&action=edit&redlink=1" class="new" title="Audio Vivid (page does not exist)">Audio Vivid</a>(GY/T 363)</li></ul></li> <li><a href="/wiki/Dynamic_Resolution_Adaptation" title="Dynamic Resolution Adaptation">DRA</a>(GB/T 22726)</li> <li><a href="/wiki/L2HC" title="L2HC">L2HC</a></li> <li>ExAC(SJ/T 11299.4)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Apple_Lossless_Audio_Codec" title="Apple Lossless Audio Codec">ALAC</a></li> <li><a href="/wiki/Asao_(codec)" title="Asao (codec)">Asao</a></li> <li><a href="/wiki/Adaptive_Transform_Acoustic_Coding" class="mw-redirect" title="Adaptive Transform Acoustic Coding">ATRAC</a></li> <li><a href="/wiki/CELT" title="CELT">CELT</a></li> <li><a href="/wiki/Codec_2" title="Codec 2">Codec 2</a></li> <li><a href="/wiki/Internet_Speech_Audio_Codec" title="Internet Speech Audio Codec">iSAC</a></li> <li><a href="/wiki/Lyra_(codec)" title="Lyra (codec)">Lyra</a></li> <li><a href="/wiki/Mixed-excitation_linear_prediction" title="Mixed-excitation linear prediction">MELP</a></li> <li><a href="/wiki/Monkey%27s_Audio" title="Monkey's Audio">Monkey's Audio</a></li> <li><a href="/wiki/MT9" title="MT9">MT9</a></li> <li><a href="/wiki/Musepack" title="Musepack">Musepack</a></li> <li><a href="/wiki/OptimFROG" title="OptimFROG">OptimFROG</a></li> <li><a href="/wiki/Original_Sound_Quality" title="Original Sound Quality">OSQ</a></li> <li><a href="/wiki/Qualcomm_code-excited_linear_prediction" title="Qualcomm code-excited linear prediction">QCELP</a></li> <li><a href="/wiki/Relaxed_code-excited_linear_prediction" title="Relaxed code-excited linear prediction">RCELP</a></li> <li><a href="/wiki/RealAudio" title="RealAudio">RealAudio</a></li> <li><a href="/wiki/RTAudio" title="RTAudio">RTAudio</a></li> <li><a href="/wiki/Avid_Audio#Sound_Designer_file_formats" title="Avid Audio">SD2</a></li> <li><a href="/wiki/Shorten_file_format" class="mw-redirect" title="Shorten file format">SHN</a></li> <li><a href="/wiki/SILK" title="SILK">SILK</a></li> <li><a href="/wiki/Siren_(codec)" title="Siren (codec)">Siren</a></li> <li><a href="/wiki/Selectable_Mode_Vocoder" title="Selectable Mode Vocoder">SMV</a></li> <li><a href="/wiki/SVOPC" title="SVOPC">SVOPC</a></li> <li>TTA <ul><li>True Audio</li></ul></li> <li><a href="/wiki/TwinVQ" title="TwinVQ">TwinVQ</a></li> <li><a href="/wiki/Variable-Rate_Multimode_Wideband" title="Variable-Rate Multimode Wideband">VMR-WB</a></li> <li><a href="/wiki/Vector_sum_excited_linear_prediction" title="Vector sum excited linear prediction">VSELP</a></li> <li><a href="/wiki/WavPack" title="WavPack">WavPack</a></li> <li><a href="/wiki/Windows_Media_Audio" title="Windows Media Audio">WMA</a></li> <li><a href="/wiki/Master_Quality_Authenticated" title="Master Quality Authenticated">MQA</a></li> <li><a href="/wiki/AptX" title="AptX">aptX</a></li> <li><a href="/wiki/AptX#aptX_HD" title="AptX">aptX HD</a></li> <li><a href="/wiki/AptX#aptX_Low_Latency" title="AptX">aptX Low Latency</a></li> <li><a href="/wiki/AptX#aptX_Adaptive" title="AptX">aptX Adaptive</a></li> <li><a href="/wiki/LDAC_(codec)" title="LDAC (codec)">LDAC</a></li> <li><a href="/wiki/LHDC_(codec)" title="LHDC (codec)">LHDC</a></li> <li><a href="/wiki/LHDC_(codec)#LLAC" title="LHDC (codec)">LLAC</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Image_compression" title="Image compression">Image compression</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">IEC</a>, <a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">ISO</a>, <a href="/wiki/Internet_Engineering_Task_Force" title="Internet Engineering Task Force">IETF</a>, <a href="/wiki/World_Wide_Web_Consortium" title="World Wide Web Consortium">W3C</a>, <a href="/wiki/ITU-T" title="ITU-T">ITU-T</a>, <a href="/wiki/Joint_Photographic_Experts_Group" title="Joint Photographic Experts Group">JPEG</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Group_4_compression" title="Group 4 compression">CCITT Group 4</a></li> <li><a href="/wiki/GIF" title="GIF">GIF</a></li> <li><a href="/wiki/High_Efficiency_Image_File_Format#HEIC:_HEVC_in_HEIF" title="High Efficiency Image File Format">HEIC / HEIF</a></li> <li><a href="/wiki/High_Efficiency_Video_Coding#Main_Still_Picture" title="High Efficiency Video Coding">HEVC</a></li> <li><a href="/wiki/JBIG" title="JBIG">JBIG</a></li> <li><a href="/wiki/JBIG2" title="JBIG2">JBIG2</a></li> <li><a href="/wiki/JPEG" title="JPEG">JPEG</a></li> <li><a href="/wiki/JPEG_2000" title="JPEG 2000">JPEG 2000</a></li> <li><a href="/wiki/JPEG-LS" class="mw-redirect" title="JPEG-LS">JPEG-LS</a></li> <li><a href="/wiki/JPEG_XL" title="JPEG XL">JPEG XL</a></li> <li><a href="/wiki/JPEG_XR" title="JPEG XR">JPEG XR</a></li> <li><a href="/wiki/JPEG_XS" title="JPEG XS">JPEG XS</a></li> <li><a href="/wiki/JPEG_XT" title="JPEG XT">JPEG XT</a></li> <li><a href="/wiki/Portable_Network_Graphics" class="mw-redirect" title="Portable Network Graphics">PNG</a></li> <li><a href="/wiki/TIFF" title="TIFF">TIFF</a></li> <li><a href="/wiki/TIFF/EP" title="TIFF/EP">TIFF/EP</a></li> <li><a href="/wiki/TIFF/IT" class="mw-redirect" title="TIFF/IT">TIFF/IT</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/APNG" title="APNG">APNG</a></li> <li><a href="/wiki/AV1" title="AV1">AV1</a></li> <li><a href="/wiki/AVIF" title="AVIF">AVIF</a></li> <li><a href="/wiki/Better_Portable_Graphics" title="Better Portable Graphics">BPG</a></li> <li><a href="/wiki/DjVu" title="DjVu">DjVu</a></li> <li><a href="/wiki/OpenEXR" title="OpenEXR">EXR</a></li> <li><a href="/wiki/Free_Lossless_Image_Format" title="Free Lossless Image Format">FLIF</a></li> <li><a href="/wiki/ICER_(file_format)" title="ICER (file format)">ICER</a></li> <li><a href="/wiki/Multiple-image_Network_Graphics" title="Multiple-image Network Graphics">MNG</a></li> <li><a href="/wiki/Progressive_Graphics_File" title="Progressive Graphics File">PGF</a></li> <li><a href="/wiki/QOI_(image_format)" title="QOI (image format)">QOI</a></li> <li><a href="/wiki/QuickTime_VR" title="QuickTime VR">QTVR</a></li> <li><a href="/wiki/Wireless_Application_Protocol_Bitmap_Format" title="Wireless Application Protocol Bitmap Format">WBMP</a></li> <li><a href="/wiki/WebP" title="WebP">WebP</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digital_container_format" class="mw-redirect" title="Digital container format">Containers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">ISO</a>, <a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">IEC</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/MPEG_elementary_stream" title="MPEG elementary stream">MPEG-ES</a> <ul><li><a href="/wiki/Packetized_elementary_stream" title="Packetized elementary stream">MPEG-PES</a></li></ul></li> <li><a href="/wiki/MPEG_program_stream" title="MPEG program stream">MPEG-PS</a></li> <li><a href="/wiki/MPEG_transport_stream" title="MPEG transport stream">MPEG-TS</a></li> <li><a href="/wiki/ISO/IEC_base_media_file_format" class="mw-redirect" title="ISO/IEC base media file format">ISO/IEC base media file format</a></li> <li><a href="/wiki/MPEG-4_Part_14" class="mw-redirect" title="MPEG-4 Part 14">MPEG-4 Part 14</a> (MP4)</li> <li><a href="/wiki/Motion_JPEG_2000" title="Motion JPEG 2000">Motion JPEG 2000</a></li> <li><a href="/wiki/MPEG-21" title="MPEG-21">MPEG-21 Part 9</a></li> <li><a href="/wiki/MPEG_media_transport" title="MPEG media transport">MPEG media transport</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/ITU-T" title="ITU-T">ITU-T</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/MPEG-2#Systems" title="MPEG-2">H.222.0</a></li> <li><a href="/wiki/Motion_JPEG_2000" title="Motion JPEG 2000">T.802</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/Internet_Engineering_Task_Force" title="Internet Engineering Task Force">IETF</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Real-time_Transport_Protocol" title="Real-time Transport Protocol">RTP</a></li> <li><a href="/wiki/Ogg" title="Ogg">Ogg</a></li> <li><a href="/wiki/Matroska" title="Matroska">Matroska</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/Society_of_Motion_Picture_and_Television_Engineers" title="Society of Motion Picture and Television Engineers">SMPTE</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_Exchange_Format" title="General Exchange Format">GXF</a></li> <li><a href="/wiki/Material_Exchange_Format" title="Material Exchange Format">MXF</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/3GP_and_3G2" title="3GP and 3G2">3GP and 3G2</a></li> <li><a href="/wiki/AMV_video_format" title="AMV video format">AMV</a></li> <li><a href="/wiki/Advanced_Systems_Format" title="Advanced Systems Format">ASF</a></li> <li><a href="/wiki/Audio_Interchange_File_Format" title="Audio Interchange File Format">AIFF</a></li> <li><a href="/wiki/Audio_Video_Interleave" title="Audio Video Interleave">AVI</a></li> <li><a href="/wiki/Au_file_format" title="Au file format">AU</a></li> <li><a href="/wiki/Better_Portable_Graphics" title="Better Portable Graphics">BPG</a></li> <li><a href="/wiki/Bink_Video" title="Bink Video">Bink</a> <ul><li><a href="/wiki/Smacker_video" title="Smacker video">Smacker</a></li></ul></li> <li><a href="/wiki/BMP_file_format" title="BMP file format">BMP</a></li> <li><a href="/wiki/DivX#DivX_Media_Format_(DMF)" title="DivX">DivX Media Format</a></li> <li><a href="/wiki/Enhanced_VOB" title="Enhanced VOB">EVO</a></li> <li><a href="/wiki/Flash_Video" title="Flash Video">Flash Video</a></li> <li><a href="/wiki/High_Efficiency_Image_File_Format" title="High Efficiency Image File Format">HEIF</a></li> <li><a href="/wiki/Interchange_File_Format" title="Interchange File Format">IFF</a></li> <li><a href="/wiki/.m2ts" title=".m2ts">M2TS</a></li> <li><a href="/wiki/Matroska" title="Matroska">Matroska</a> <ul><li><a href="/wiki/WebM" title="WebM">WebM</a></li></ul></li> <li><a href="/wiki/QuickTime_File_Format" title="QuickTime File Format">QuickTime File Format</a></li> <li><a href="/wiki/RatDVD" title="RatDVD">RatDVD</a></li> <li><a href="/wiki/RealMedia" title="RealMedia">RealMedia</a></li> <li><a href="/wiki/Resource_Interchange_File_Format" title="Resource Interchange File Format">RIFF</a> <ul><li><a href="/wiki/WAV" title="WAV">WAV</a></li></ul></li> <li><a href="/wiki/MOD_and_TOD" title="MOD and TOD">MOD and TOD</a></li> <li><a href="/wiki/VOB" title="VOB">VOB, IFO and BUP</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Collaborations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/NETVC" title="NETVC">NETVC</a></li> <li><a href="/wiki/MPEG_LA" title="MPEG LA">MPEG LA</a></li> <li><a href="/wiki/Alliance_for_Open_Media" title="Alliance for Open Media">Alliance for Open Media</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression" title="Data compression">Methods</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Entropy_encoding" class="mw-redirect" title="Entropy encoding">Entropy</a> <ul><li><a href="/wiki/Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li> <li><a href="/wiki/Huffman_coding" title="Huffman coding">Huffman</a></li> <li><a href="/wiki/Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li><a href="/wiki/Lossless_compression" title="Lossless compression">Lossless</a></li> <li><a href="/wiki/Lossy_compression" title="Lossy compression">Lossy</a></li> <li><a href="/wiki/LZ77_and_LZ78" title="LZ77 and LZ78">LZ</a> <ul><li><a href="/wiki/DEFLATE" class="mw-redirect" title="DEFLATE">DEFLATE</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Welch" title="Lempel–Ziv–Welch">LZW</a></li></ul></li> <li><a href="/wiki/Pulse-code_modulation" title="Pulse-code modulation">PCM</a> <ul><li><a href="/wiki/A-law_algorithm" title="A-law algorithm">A-law</a></li> <li><a href="/wiki/%CE%9C-law_algorithm" title="Μ-law algorithm">µ-law</a></li> <li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li></ul></li> <li><a href="/wiki/Transform_coding" title="Transform coding">Transforms</a> <ul><li><a class="mw-selflink selflink">DCT</a></li> <li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Comparison_of_audio_coding_formats" title="Comparison of audio coding formats">Comparison of audio coding formats</a></li> <li><a href="/wiki/Comparison_of_video_codecs" title="Comparison of video codecs">Comparison of video codecs</a></li> <li><a href="/wiki/List_of_codecs" title="List of codecs">List of codecs</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>See <a href="/wiki/Template:Compression_methods" title="Template:Compression methods">Compression methods</a> for techniques and <a href="/wiki/Template:Compression_software" title="Template:Compression software">Compression software</a> for codecs</div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Digital_signal_processing" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Digital_signal_processing" title="Template:Digital signal processing"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Digital_signal_processing" title="Template talk:Digital signal processing"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Digital_signal_processing" title="Special:EditPage/Template:Digital signal processing"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Digital_signal_processing" style="font-size:114%;margin:0 4em"><a href="/wiki/Digital_signal_processing" title="Digital signal processing">Digital signal processing</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Detection_theory" title="Detection theory">Detection theory</a></li> <li><a href="/wiki/Discrete_time_and_continuous_time" title="Discrete time and continuous time">Discrete signal</a></li> <li><a href="/wiki/Estimation_theory" title="Estimation theory">Estimation theory</a></li> <li><a href="/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sub-fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Audio_signal_processing" title="Audio signal processing">Audio signal processing</a></li> <li><a href="/wiki/Digital_image_processing" title="Digital image processing">Digital image processing</a></li> <li><a href="/wiki/Speech_processing" title="Speech processing">Speech processing</a></li> <li><a href="/wiki/Statistical_signal_processing" class="mw-redirect" title="Statistical signal processing">Statistical signal processing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Techniques</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-transform" title="Z-transform">Z-transform</a> <ul><li><a href="/wiki/Advanced_z-transform" title="Advanced z-transform">Advanced z-transform</a></li> <li><a href="/wiki/Matched_Z-transform_method" title="Matched Z-transform method">Matched Z-transform method</a></li></ul></li> <li><a href="/wiki/Bilinear_transform" title="Bilinear transform">Bilinear transform</a></li> <li><a href="/wiki/Constant-Q_transform" title="Constant-Q transform">Constant-Q transform</a></li> <li><a class="mw-selflink selflink">Discrete cosine transform</a> (DCT)</li> <li><a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a> (DFT)</li> <li><a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT)</li> <li><a href="/wiki/Impulse_invariance" title="Impulse invariance">Impulse invariance</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a></li> <li><a href="/wiki/Post%27s_inversion_formula" class="mw-redirect" title="Post's inversion formula">Post's inversion formula</a></li> <li><a href="/wiki/Starred_transform" title="Starred transform">Starred transform</a></li> <li><a href="/wiki/Zak_transform" title="Zak transform">Zak transform</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aliasing" title="Aliasing">Aliasing</a></li> <li><a href="/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">Anti-aliasing filter</a></li> <li><a href="/wiki/Downsampling_(signal_processing)" title="Downsampling (signal processing)">Downsampling</a></li> <li><a href="/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a> / <a href="/wiki/Nyquist_frequency" title="Nyquist frequency">frequency</a></li> <li><a href="/wiki/Oversampling" title="Oversampling">Oversampling</a></li> <li><a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li> <li><a href="/wiki/Sampling_rate" class="mw-redirect" title="Sampling rate">Sampling rate</a></li> <li><a href="/wiki/Undersampling" title="Undersampling">Undersampling</a></li> <li><a href="/wiki/Upsampling" title="Upsampling">Upsampling</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Telecommunications" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Telecommunications" title="Template:Telecommunications"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Telecommunications" title="Template talk:Telecommunications"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Telecommunications" title="Special:EditPage/Template:Telecommunications"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Telecommunications" style="font-size:114%;margin:0 4em"><a href="/wiki/Telecommunications" title="Telecommunications">Telecommunications</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_telecommunication" title="History of telecommunication">History</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beacon#For_defensive_communications" title="Beacon">Beacon</a></li> <li><a href="/wiki/History_of_broadcasting" title="History of broadcasting">Broadcasting</a></li> <li><a href="/wiki/Cable_protection_system" title="Cable protection system">Cable protection system</a></li> <li><a href="/wiki/Cable_television" title="Cable television">Cable TV</a></li> <li><a href="/wiki/Communications_satellite#History" title="Communications satellite">Communications satellite</a></li> <li><a href="/wiki/Computer_network#History" title="Computer network">Computer network</a></li> <li><a href="/wiki/Data_compression" title="Data compression">Data compression</a> <ul><li><a href="/wiki/Audio_coding_format" title="Audio coding format">audio</a></li> <li><a class="mw-selflink selflink">DCT</a></li> <li><a href="/wiki/Image_compression" title="Image compression">image</a></li> <li><a href="/wiki/Video_coding_format" title="Video coding format">video</a></li></ul></li> <li><a href="/wiki/Digital_media" title="Digital media">Digital media</a> <ul><li><a href="/wiki/Internet_video" title="Internet video">Internet video</a></li> <li><a href="/wiki/Online_video_platform" title="Online video platform">online video platform</a></li> <li><a href="/wiki/Social_media" title="Social media">social media</a></li> <li><a href="/wiki/Streaming_media" title="Streaming media">streaming</a></li></ul></li> <li><a href="/wiki/Drums_in_communication" title="Drums in communication">Drums</a></li> <li><a href="/wiki/Edholm%27s_law" title="Edholm's law">Edholm's law</a></li> <li><a href="/wiki/Electrical_telegraph#History" title="Electrical telegraph">Electrical telegraph</a></li> <li><a href="/wiki/Fax#History" title="Fax">Fax</a></li> <li><a href="/wiki/Heliograph#History" title="Heliograph">Heliographs</a></li> <li><a href="/wiki/Hydraulic_telegraph#Greek_hydraulic_semaphore_system" title="Hydraulic telegraph">Hydraulic telegraph</a></li> <li><a href="/wiki/Information_Age" title="Information Age">Information Age</a></li> <li><a href="/wiki/Information_revolution" class="mw-redirect" title="Information revolution">Information revolution</a></li> <li><a href="/wiki/History_of_the_Internet" title="History of the Internet">Internet</a></li> <li><a href="/wiki/Mass_media#History" title="Mass media">Mass media</a></li> <li><a href="/wiki/History_of_mobile_phones" title="History of mobile phones">Mobile phone</a> <ul><li><a href="/wiki/Smartphone" title="Smartphone">Smartphone</a></li></ul></li> <li><a href="/wiki/Optical_communication" title="Optical communication">Optical telecommunication</a></li> <li><a href="/wiki/Optical_telegraph" title="Optical telegraph">Optical telegraphy</a></li> <li><a href="/wiki/Pager" title="Pager">Pager</a></li> <li><a href="/wiki/Photophone" title="Photophone">Photophone</a></li> <li><a href="/wiki/History_of_prepaid_mobile_phones" title="History of prepaid mobile phones">Prepaid mobile phone</a></li> <li><a href="/wiki/History_of_radio" title="History of radio">Radio</a></li> <li><a href="/wiki/Radiotelephone" title="Radiotelephone">Radiotelephone</a></li> <li><a href="/wiki/Communications_satellite" title="Communications satellite">Satellite communications</a></li> <li><a href="/wiki/Semaphore" title="Semaphore">Semaphore</a> <ul><li><a href="/wiki/Phryctoria" title="Phryctoria">Phryctoria</a></li></ul></li> <li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a> <ul><li><a href="/wiki/Semiconductor_device" title="Semiconductor device">device</a></li> <li><a href="/wiki/MOSFET" title="MOSFET">MOSFET</a></li> <li><a href="/wiki/History_of_the_transistor" title="History of the transistor">transistor</a></li></ul></li> <li><a href="/wiki/Smoke_signal" title="Smoke signal">Smoke signals</a></li> <li><a href="/wiki/History_of_telecommunication" title="History of telecommunication">Telecommunications history</a></li> <li><a href="/wiki/Telautograph" title="Telautograph">Telautograph</a></li> <li><a href="/wiki/Telegraphy" title="Telegraphy">Telegraphy</a></li> <li><a href="/wiki/Teleprinter" title="Teleprinter">Teleprinter</a> (teletype)</li> <li><a href="/wiki/History_of_the_telephone" title="History of the telephone">Telephone</a></li> <li><a href="/wiki/The_Telephone_Cases" title="The Telephone Cases">The Telephone Cases</a></li> <li><a href="/wiki/History_of_television" title="History of television">Television</a> <ul><li><a href="/wiki/Digital_television" title="Digital television">digital</a></li> <li><a href="/wiki/Streaming_television" title="Streaming television">streaming</a></li></ul></li> <li><a href="/wiki/Submarine_communications_cable#Early_history:_telegraph_and_coaxial_cables" title="Submarine communications cable">Undersea telegraph line</a></li> <li><a href="/wiki/History_of_videotelephony" title="History of videotelephony">Videotelephony</a></li> <li><a href="/wiki/Whistled_language" title="Whistled language">Whistled language</a></li> <li><a href="/wiki/Wireless_revolution" class="mw-redirect" title="Wireless revolution">Wireless revolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Pioneers</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nasir_Ahmed_(engineer)" title="Nasir Ahmed (engineer)">Nasir Ahmed</a></li> <li><a href="/wiki/Edwin_Howard_Armstrong" title="Edwin Howard Armstrong">Edwin Howard Armstrong</a></li> <li><a href="/wiki/Mohamed_M._Atalla" title="Mohamed M. Atalla">Mohamed M. Atalla</a></li> <li><a href="/wiki/John_Logie_Baird" title="John Logie Baird">John Logie Baird</a></li> <li><a href="/wiki/Paul_Baran" title="Paul Baran">Paul Baran</a></li> <li><a href="/wiki/John_Bardeen" title="John Bardeen">John Bardeen</a></li> <li><a href="/wiki/Alexander_Graham_Bell" title="Alexander Graham Bell">Alexander Graham Bell</a></li> <li><a href="/wiki/Emile_Berliner" title="Emile Berliner">Emile Berliner</a></li> <li><a href="/wiki/Tim_Berners-Lee" title="Tim Berners-Lee">Tim Berners-Lee</a></li> <li><a href="/wiki/Francis_Blake_(inventor)" title="Francis Blake (inventor)">Francis Blake</a></li> <li><a href="/wiki/Jagadish_Chandra_Bose" title="Jagadish Chandra Bose">Jagadish Chandra Bose</a></li> <li><a href="/wiki/Charles_Bourseul" title="Charles Bourseul">Charles Bourseul</a></li> <li><a href="/wiki/Walter_Houser_Brattain" title="Walter Houser Brattain">Walter Houser Brattain</a></li> <li><a href="/wiki/Vint_Cerf" title="Vint Cerf">Vint Cerf</a></li> <li><a href="/wiki/Claude_Chappe" title="Claude Chappe">Claude Chappe</a></li> <li><a href="/wiki/Yogen_Dalal" class="mw-redirect" title="Yogen Dalal">Yogen Dalal</a></li> <li><a href="/wiki/Daniel_Davis_Jr." title="Daniel Davis Jr.">Daniel Davis Jr.</a></li> <li><a href="/wiki/Donald_Davies" title="Donald Davies">Donald Davies</a></li> <li><a href="/wiki/Amos_Dolbear" title="Amos Dolbear">Amos Dolbear</a></li> <li><a href="/wiki/Thomas_Edison" title="Thomas Edison">Thomas Edison</a></li> <li><a href="/wiki/Lee_de_Forest" title="Lee de Forest">Lee de Forest</a></li> <li><a href="/wiki/Philo_Farnsworth" title="Philo Farnsworth">Philo Farnsworth</a></li> <li><a href="/wiki/Reginald_Fessenden" title="Reginald Fessenden">Reginald Fessenden</a></li> <li><a href="/wiki/Elisha_Gray" title="Elisha Gray">Elisha Gray</a></li> <li><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a></li> <li><a href="/wiki/Erna_Schneider_Hoover" title="Erna Schneider Hoover">Erna Schneider Hoover</a></li> <li><a href="/wiki/Harold_Hopkins_(physicist)" title="Harold Hopkins (physicist)">Harold Hopkins</a></li> <li><a href="/wiki/Gardiner_Greene_Hubbard" title="Gardiner Greene Hubbard">Gardiner Greene Hubbard</a></li> <li><a href="/wiki/List_of_Internet_pioneers" title="List of Internet pioneers">Internet pioneers</a></li> <li><a href="/wiki/Bob_Kahn" class="mw-redirect" title="Bob Kahn">Bob Kahn</a></li> <li><a href="/wiki/Dawon_Kahng" title="Dawon Kahng">Dawon Kahng</a></li> <li><a href="/wiki/Charles_K._Kao" title="Charles K. Kao">Charles K. Kao</a></li> <li><a href="/wiki/Narinder_Singh_Kapany" title="Narinder Singh Kapany">Narinder Singh Kapany</a></li> <li><a href="/wiki/Hedy_Lamarr" title="Hedy Lamarr">Hedy Lamarr</a></li> <li><a href="/wiki/Roberto_Landell_de_Moura" title="Roberto Landell de Moura">Roberto Landell de Moura</a></li> <li><a href="/wiki/Innocenzo_Manzetti" title="Innocenzo Manzetti">Innocenzo Manzetti</a></li> <li><a href="/wiki/Guglielmo_Marconi" title="Guglielmo Marconi">Guglielmo Marconi</a></li> <li><a href="/wiki/Robert_Metcalfe" title="Robert Metcalfe">Robert Metcalfe</a></li> <li><a href="/wiki/Antonio_Meucci" title="Antonio Meucci">Antonio Meucci</a></li> <li><a href="/wiki/Samuel_Morse" title="Samuel Morse">Samuel Morse</a></li> <li><a href="/wiki/Jun-ichi_Nishizawa" title="Jun-ichi Nishizawa">Jun-ichi Nishizawa</a></li> <li><a href="/wiki/Charles_Grafton_Page" title="Charles Grafton Page">Charles Grafton Page</a></li> <li><a href="/wiki/Radia_Perlman" title="Radia Perlman">Radia Perlman</a></li> <li><a href="/wiki/Alexander_Stepanovich_Popov" class="mw-redirect" title="Alexander Stepanovich Popov">Alexander Stepanovich Popov</a></li> <li><a href="/wiki/Tivadar_Pusk%C3%A1s" title="Tivadar Puskás">Tivadar Puskás</a></li> <li><a href="/wiki/Johann_Philipp_Reis" title="Johann Philipp Reis">Johann Philipp Reis</a></li> <li><a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li> <li><a href="/wiki/Almon_Brown_Strowger" title="Almon Brown Strowger">Almon Brown Strowger</a></li> <li><a href="/wiki/Henry_Sutton_(inventor)" title="Henry Sutton (inventor)">Henry Sutton</a></li> <li><a href="/wiki/Charles_Sumner_Tainter" title="Charles Sumner Tainter">Charles Sumner Tainter</a></li> <li><a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Nikola Tesla</a></li> <li><a href="/wiki/Camille_Tissot" title="Camille Tissot">Camille Tissot</a></li> <li><a href="/wiki/Alfred_Vail" title="Alfred Vail">Alfred Vail</a></li> <li><a href="/wiki/Thomas_A._Watson" title="Thomas A. Watson">Thomas A. Watson</a></li> <li><a href="/wiki/Charles_Wheatstone" title="Charles Wheatstone">Charles Wheatstone</a></li> <li><a href="/wiki/Vladimir_K._Zworykin" title="Vladimir K. Zworykin">Vladimir K. Zworykin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Transmission_medium" title="Transmission medium">Transmission media</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coaxial_cable" title="Coaxial cable">Coaxial cable</a></li> <li><a href="/wiki/Fiber-optic_communication" title="Fiber-optic communication">Fiber-optic communication</a> <ul><li><a href="/wiki/Optical_fiber" title="Optical fiber">optical fiber</a></li></ul></li> <li><a href="/wiki/Free-space_optical_communication" title="Free-space optical communication">Free-space optical communication</a></li> <li><a href="/wiki/Molecular_communication" title="Molecular communication">Molecular communication</a></li> <li><a href="/wiki/Radio_wave" title="Radio wave">Radio waves</a> <ul><li><a href="/wiki/Wireless" title="Wireless">wireless</a></li></ul></li> <li><a href="/wiki/Transmission_line" title="Transmission line">Transmission line</a> <ul><li><a href="/wiki/Telecommunication_circuit" title="Telecommunication circuit">telecommunication circuit</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Network_topology" title="Network topology">Network topology</a> and switching</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bandwidth_(computing)" title="Bandwidth (computing)">Bandwidth</a></li> <li><a href="/wiki/Telecommunications_link" title="Telecommunications link">Links</a></li> <li><a href="/wiki/Node_(networking)" title="Node (networking)">Nodes</a> <ul><li><a href="/wiki/Terminal_(telecommunication)" title="Terminal (telecommunication)">terminal</a></li></ul></li> <li><a href="/wiki/Network_switch" title="Network switch">Network switching</a> <ul><li><a href="/wiki/Circuit_switching" title="Circuit switching">circuit</a></li> <li><a href="/wiki/Packet_switching" title="Packet switching">packet</a></li></ul></li> <li><a href="/wiki/Telephone_exchange" title="Telephone exchange">Telephone exchange</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multiplexing" title="Multiplexing">Multiplexing</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Space-division_multiple_access" title="Space-division multiple access">Space-division</a></li> <li><a href="/wiki/Frequency-division_multiplexing" title="Frequency-division multiplexing">Frequency-division</a></li> <li><a href="/wiki/Time-division_multiplexing" title="Time-division multiplexing">Time-division</a></li> <li><a href="/wiki/Polarization-division_multiplexing" title="Polarization-division multiplexing">Polarization-division</a></li> <li><a href="/wiki/Orbital_angular_momentum_multiplexing" title="Orbital angular momentum multiplexing">Orbital angular-momentum</a></li> <li><a href="/wiki/Code-division_multiple_access" title="Code-division multiple access">Code-division</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Communication_protocol" title="Communication protocol">Communication protocol</a></li> <li><a href="/wiki/Computer_network" title="Computer network">Computer network</a></li> <li><a href="/wiki/Data_communication" title="Data communication">Data transmission</a></li> <li><a href="/wiki/Store_and_forward" title="Store and forward">Store and forward</a></li> <li><a href="/wiki/Telecommunications_equipment" title="Telecommunications equipment">Telecommunications equipment</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Telecommunications_network" title="Telecommunications network">Types of network</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cellular_network" title="Cellular network">Cellular network</a></li> <li><a href="/wiki/Ethernet" title="Ethernet">Ethernet</a></li> <li><a href="/wiki/Integrated_Services_Digital_Network" class="mw-redirect" title="Integrated Services Digital Network">ISDN</a></li> <li><a href="/wiki/Local_area_network" title="Local area network">LAN</a></li> <li><a href="/wiki/Mobile_telephony" title="Mobile telephony">Mobile</a></li> <li><a href="/wiki/Next-generation_network" title="Next-generation network">NGN</a></li> <li><a href="/wiki/Public_switched_telephone_network" title="Public switched telephone network">Public Switched Telephone</a></li> <li><a href="/wiki/Radio_network" title="Radio network">Radio</a></li> <li><a href="/wiki/Television_broadcasting" class="mw-redirect" title="Television broadcasting">Television</a></li> <li><a href="/wiki/Telex" title="Telex">Telex</a></li> <li><a href="/wiki/UUCP" title="UUCP">UUCP</a></li> <li><a href="/wiki/Wide_area_network" title="Wide area network">WAN</a></li> <li><a href="/wiki/Wireless_network" title="Wireless network">Wireless network</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Telecommunications_network" title="Telecommunications network">Notable networks</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ARPANET" title="ARPANET">ARPANET</a></li> <li><a href="/wiki/BITNET" title="BITNET">BITNET</a></li> <li><a href="/wiki/CYCLADES" title="CYCLADES">CYCLADES</a></li> <li><a href="/wiki/FidoNet" title="FidoNet">FidoNet</a></li> <li><a href="/wiki/Internet" title="Internet">Internet</a></li> <li><a href="/wiki/Internet2" title="Internet2">Internet2</a></li> <li><a href="/wiki/JANET" title="JANET">JANET</a></li> <li><a href="/wiki/NPL_network" title="NPL network">NPL network</a></li> <li><a href="/wiki/Toasternet" title="Toasternet">Toasternet</a></li> <li><a href="/wiki/Usenet" title="Usenet">Usenet</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Locations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category:Telecommunications_in_Africa" title="Category:Telecommunications in Africa">Africa</a></li> <li>Americas <ul><li><a href="/wiki/Category:Telecommunications_in_North_America" title="Category:Telecommunications in North America">North</a></li> <li><a href="/wiki/Category:Telecommunications_in_South_America" title="Category:Telecommunications in South America">South</a></li></ul></li> <li><a href="/wiki/Category:Communications_in_Antarctica" title="Category:Communications in Antarctica">Antarctica</a></li> <li><a href="/wiki/Category:Telecommunications_in_Asia" title="Category:Telecommunications in Asia">Asia</a></li> <li><a href="/wiki/Category:Telecommunications_in_Europe" title="Category:Telecommunications in Europe">Europe</a></li> <li><a href="/wiki/Category:Telecommunications_in_Oceania" title="Category:Telecommunications in Oceania">Oceania</a></li> <li>(<a href="/wiki/List_of_telecommunications_regulatory_bodies" title="List of telecommunications regulatory bodies">Global telecommunications regulation bodies</a>)</li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Telecom-icon.svg/16px-Telecom-icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Telecom-icon.svg/24px-Telecom-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Telecom-icon.svg/32px-Telecom-icon.svg.png 2x" data-file-width="500" data-file-height="500" /> <a href="/wiki/Portal:Telecommunication" title="Portal:Telecommunication">Telecommunication portal</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Telecommunications" title="Category:Telecommunications">Category</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Outline_of_telecommunication" title="Outline of telecommunication">Outline</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /> <a 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alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b7f745dd4-qtmrx","wgBackendResponseTime":180,"wgPageParseReport":{"limitreport":{"cputime":"2.203","walltime":"2.530","ppvisitednodes":{"value":11114,"limit":1000000},"postexpandincludesize":{"value":452480,"limit":2097152},"templateargumentsize":{"value":32399,"limit":2097152},"expansiondepth":{"value":15,"limit":100},"expensivefunctioncount":{"value":6,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":506100,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1886.920 1 -total"," 50.31% 949.311 2 Template:Reflist"," 23.05% 434.846 39 Template:Cite_journal"," 11.13% 210.096 35 Template:Cite_book"," 9.83% 185.442 13 Template:Navbox"," 8.08% 152.373 32 Template:Cite_web"," 7.12% 134.416 1 Template:Short_description"," 6.41% 120.914 1 Template:Compression_Methods"," 5.98% 112.906 1 Template:Sfn"," 5.26% 99.183 2 Template:Pagetype"]},"scribunto":{"limitreport-timeusage":{"value":"1.249","limit":"10.000"},"limitreport-memusage":{"value":8682985,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAbouslemanMarcellinHunt1995\"] = 1,\n [\"CITEREFAhmed1991\"] = 2,\n [\"CITEREFAhmedNatarajanRao1974\"] = 1,\n [\"CITEREFAlakuijalaSneyersVersariWassenberg2021\"] = 1,\n [\"CITEREFAlikhani2015\"] = 1,\n [\"CITEREFAlshibamiBoussakta2001\"] = 1,\n [\"CITEREFAraiAguiNakajima1988\"] = 1,\n [\"CITEREFAscherPincus2012\"] = 1,\n [\"CITEREFBaraniuk2015\"] = 1,\n [\"CITEREFBarberoHofmannWells1991\"] = 1,\n [\"CITEREFBertalmio2014\"] = 1,\n [\"CITEREFBleidtSenNiedermeierCzelhan2017\"] = 1,\n [\"CITEREFBoussaktaAlshibami2004\"] = 1,\n [\"CITEREFBrandenburg1999\"] = 1,\n [\"CITEREFBrinkmann2018\"] = 1,\n [\"CITEREFBritanak2011\"] = 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[\"CITEREFJonesLayerOsenkowsky2013\"] = 1,\n [\"CITEREFKatsaggelosBabacanChun-Jen2009\"] = 1,\n [\"CITEREFKomatsuSezaki1998\"] = 1,\n [\"CITEREFLea1994\"] = 1,\n [\"CITEREFLee2005\"] = 1,\n [\"CITEREFLeeBeckLambSeverson1995\"] = 1,\n [\"CITEREFLeyden2015\"] = 1,\n [\"CITEREFLi2006\"] = 1,\n [\"CITEREFLuo2008\"] = 1,\n [\"CITEREFLutzkySchullerGayerKrämer2004\"] = 1,\n [\"CITEREFMakhoul1980\"] = 1,\n [\"CITEREFMalvar1992\"] = 1,\n [\"CITEREFMandyamAhmedMagotra1995\"] = 1,\n [\"CITEREFMartucci1994\"] = 1,\n [\"CITEREFMcKernan2005\"] = 1,\n [\"CITEREFMenkman2011\"] = 1,\n [\"CITEREFMuchaharyMondalParmarBorah2015\"] = 1,\n [\"CITEREFNagireddi2008\"] = 1,\n [\"CITEREFNarasimhaPeterson1978\"] = 1,\n [\"CITEREFNetflix_Technology_Blog2017\"] = 1,\n [\"CITEREFNetflix_Technology_Blog2020\"] = 1,\n [\"CITEREFNetflix_Technology_Blog2021\"] = 1,\n [\"CITEREFNussbaumer1981\"] = 1,\n [\"CITEREFOchoa-DominguezRao2019\"] = 3,\n [\"CITEREFOppenheimSchaferBuck1999\"] = 1,\n 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Discrete cosine transform - Wikipedia