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aria-controls="toc-Wavelet_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Wavelet theory subsection</span> </button> <ul id="toc-Wavelet_theory-sublist" class="vector-toc-list"> <li id="toc-Continuous_wavelet_transforms_(continuous_shift_and_scale_parameters)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuous_wavelet_transforms_(continuous_shift_and_scale_parameters)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Continuous wavelet transforms (continuous shift and scale parameters)</span> </div> </a> <ul id="toc-Continuous_wavelet_transforms_(continuous_shift_and_scale_parameters)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_wavelet_transforms_(discrete_shift_and_scale_parameters,_continuous_in_time)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_wavelet_transforms_(discrete_shift_and_scale_parameters,_continuous_in_time)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Discrete wavelet transforms (discrete shift and scale parameters, continuous in time)</span> </div> </a> <ul id="toc-Discrete_wavelet_transforms_(discrete_shift_and_scale_parameters,_continuous_in_time)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiresolution_based_discrete_wavelet_transforms_(continuous_in_time)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiresolution_based_discrete_wavelet_transforms_(continuous_in_time)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Multiresolution based discrete wavelet transforms (continuous in time)</span> </div> </a> <ul id="toc-Multiresolution_based_discrete_wavelet_transforms_(continuous_in_time)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time-causal_wavelets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time-causal_wavelets"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Time-causal wavelets</span> </div> </a> <ul id="toc-Time-causal_wavelets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mother_wavelet" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mother_wavelet"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mother wavelet</span> </div> </a> <ul id="toc-Mother_wavelet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparisons_with_Fourier_transform_(continuous-time)" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Comparisons_with_Fourier_transform_(continuous-time)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Comparisons with Fourier transform (continuous-time)</span> </div> </a> <ul id="toc-Comparisons_with_Fourier_transform_(continuous-time)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_of_a_wavelet" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_of_a_wavelet"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Definition of a wavelet</span> </div> </a> <button aria-controls="toc-Definition_of_a_wavelet-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition of a wavelet subsection</span> </button> <ul id="toc-Definition_of_a_wavelet-sublist" class="vector-toc-list"> <li id="toc-Scaling_filter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaling_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Scaling filter</span> </div> </a> <ul id="toc-Scaling_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scaling_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaling_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Scaling function</span> </div> </a> <ul id="toc-Scaling_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wavelet_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wavelet_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Wavelet function</span> </div> </a> <ul id="toc-Wavelet_function-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>History</span> </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Timeline" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Timeline"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Timeline</span> </div> </a> <ul id="toc-Timeline-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Wavelet_transforms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Wavelet_transforms"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Wavelet transforms</span> </div> </a> <button aria-controls="toc-Wavelet_transforms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Wavelet transforms subsection</span> </button> <ul id="toc-Wavelet_transforms-sublist" class="vector-toc-list"> <li id="toc-Generalized_transforms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_transforms"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Generalized transforms</span> </div> </a> <ul id="toc-Generalized_transforms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-As_a_representation_of_a_signal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_representation_of_a_signal"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>As a representation of a signal</span> </div> </a> <ul id="toc-As_a_representation_of_a_signal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wavelet_denoising" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wavelet_denoising"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Wavelet denoising</span> </div> </a> <ul id="toc-Wavelet_denoising-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiscale_climate_network" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiscale_climate_network"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Multiscale climate network</span> </div> </a> <ul id="toc-Multiscale_climate_network-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-List_of_wavelets" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#List_of_wavelets"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>List of wavelets</span> </div> </a> <button aria-controls="toc-List_of_wavelets-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle List of wavelets subsection</span> </button> <ul id="toc-List_of_wavelets-sublist" class="vector-toc-list"> <li id="toc-Discrete_wavelets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_wavelets"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Discrete wavelets</span> </div> </a> <ul id="toc-Discrete_wavelets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuous_wavelets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuous_wavelets"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Continuous wavelets</span> </div> </a> <ul id="toc-Continuous_wavelets-sublist" class="vector-toc-list"> <li id="toc-Real-valued" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Real-valued"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2.1</span> <span>Real-valued</span> </div> </a> <ul id="toc-Real-valued-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex-valued" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Complex-valued"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2.2</span> <span>Complex-valued</span> </div> </a> <ul id="toc-Complex-valued-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </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"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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"> <span class="vector-toc-numb">11</span> <span>References</span> </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"> <span class="vector-toc-numb">12</span> <span>Further reading</span> </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"> <span class="vector-toc-numb">13</span> <span>External links</span> </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" title="Table of Contents" > <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" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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Available in 26 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-26" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">26 languages</span> </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/%D9%85%D9%88%D9%8A%D8%AC%D8%A9_(%D8%AF%D8%A7%D9%84%D8%A9)" title="مويجة (دالة) – Arabic" lang="ar" hreflang="ar" data-title="مويجة (دالة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Veyvlet" title="Veyvlet – Azerbaijani" lang="az" hreflang="az" data-title="Veyvlet" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D0%B5%D0%B9%D0%B2%D0%BB%D0%B5%D1%82" title="Вейвлет – Belarusian" lang="be" hreflang="be" data-title="Вейвлет" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A3%D0%B5%D0%B9%D0%B2%D0%BB%D0%B5%D1%82" title="Уейвлет – Bulgarian" lang="bg" hreflang="bg" data-title="Уейвлет" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ondeta" title="Ondeta – Catalan" lang="ca" hreflang="ca" data-title="Ondeta" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vlnka" title="Vlnka – Czech" lang="cs" hreflang="cs" data-title="Vlnka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Wavelet" title="Wavelet – German" lang="de" hreflang="de" data-title="Wavelet" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lainik" title="Lainik – Estonian" lang="et" hreflang="et" data-title="Lainik" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ond%C3%ADcula" title="Ondícula – Spanish" lang="es" hreflang="es" data-title="Ondícula" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ondosimila%C4%B5o" title="Ondosimilaĵo – Esperanto" lang="eo" hreflang="eo" data-title="Ondosimilaĵo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D8%AC%DA%A9" title="موجک – Persian" lang="fa" hreflang="fa" data-title="موجک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ondelette" title="Ondelette – French" lang="fr" hreflang="fr" data-title="Ondelette" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%A8%EC%9D%B4%EB%B8%94%EB%A6%BF" title="웨이블릿 – Korean" lang="ko" hreflang="ko" data-title="웨이블릿" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Wavelet" title="Wavelet – Indonesian" lang="id" hreflang="id" data-title="Wavelet" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Wavelet" title="Wavelet – Italian" lang="it" hreflang="it" data-title="Wavelet" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Ondl%C3%A8t" title="Ondlèt – Haitian Creole" lang="ht" hreflang="ht" data-title="Ondlèt" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vilnel%C4%97_(matematika)" title="Vilnelė (matematika) – Lithuanian" lang="lt" hreflang="lt" data-title="Vilnelė (matematika)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wavelet" title="Wavelet – Dutch" lang="nl" hreflang="nl" data-title="Wavelet" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A6%E3%82%A7%E3%83%BC%E3%83%96%E3%83%AC%E3%83%83%E3%83%88" title="ウェーブレット – Japanese" lang="ja" hreflang="ja" data-title="ウェーブレット" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Falki" title="Falki – Polish" lang="pl" hreflang="pl" data-title="Falki" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Wavelet" title="Wavelet – Portuguese" lang="pt" hreflang="pt" data-title="Wavelet" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D0%B9%D0%B2%D0%BB%D0%B5%D1%82" title="Вейвлет – Russian" lang="ru" hreflang="ru" data-title="Вейвлет" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Wavelet" title="Wavelet – Simple English" lang="en-simple" hreflang="en-simple" data-title="Wavelet" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Wavelet" title="Wavelet – Swedish" lang="sv" hreflang="sv" data-title="Wavelet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B5%D0%B9%D0%B2%D0%BB%D0%B5%D1%82" title="Вейвлет – Ukrainian" lang="uk" hreflang="uk" data-title="Вейвлет" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B0%8F%E6%B3%A2%E5%88%86%E6%9E%90" title="小波分析 – Chinese" lang="zh" hreflang="zh" data-title="小波分析" data-language-autonym="中文" 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vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Function for integral Fourier-like transform</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">For the concept in physics, see <a href="/wiki/Wave_packet" title="Wave packet">Wave packet</a>.</div> <p>A <b>wavelet</b> is a <a href="/wiki/Wave" title="Wave">wave</a>-like <a href="/wiki/Oscillation" title="Oscillation">oscillation</a> with an <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Seismic_Wavelet.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Seismic_Wavelet.svg/220px-Seismic_Wavelet.svg.png" decoding="async" width="220" height="374" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Seismic_Wavelet.svg/330px-Seismic_Wavelet.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Seismic_Wavelet.svg/440px-Seismic_Wavelet.svg.png 2x" data-file-width="418" data-file-height="710" /></a><figcaption>Seismic wavelet</figcaption></figure> <p>For example, a wavelet could be created to have a frequency of <a href="/wiki/Middle_C" class="mw-redirect" title="Middle C">middle C</a> and a short duration of roughly one tenth of a second. If this wavelet were to be <a href="/wiki/Convolution" title="Convolution">convolved</a> with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. <a href="/wiki/Correlation" title="Correlation">Correlation</a> is at the core of many practical wavelet applications. </p><p>As a mathematical tool, wavelets can be used to extract information from many kinds of data, including <a href="/wiki/Audio_signal" title="Audio signal">audio signals</a> and images. Sets of wavelets are needed to analyze data fully. "Complementary" wavelets decompose a signal without gaps or overlaps so that the decomposition process is mathematically reversible. Thus, sets of complementary wavelets are useful in <a href="/wiki/Wavelet_compression" class="mw-redirect" title="Wavelet compression">wavelet-based compression</a>/decompression algorithms, where it is desirable to recover the original information with minimal loss. </p><p>In formal terms, this representation is a <a href="/wiki/Wavelet_series" class="mw-redirect" title="Wavelet series">wavelet series</a> representation of a <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable function</a> with respect to either a <a href="/wiki/Complete_orthogonal_system#Incomplete_orthogonal_sets" class="mw-redirect" title="Complete orthogonal system">complete</a>, <a href="/wiki/Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a> set of <a href="/wiki/Basis_function" title="Basis function">basis functions</a>, or an <a href="/wiki/Complete_orthogonal_system#Incomplete_orthogonal_sets" class="mw-redirect" title="Complete orthogonal system">overcomplete</a> set or <a href="/wiki/Frame_of_a_vector_space" class="mw-redirect" title="Frame of a vector space">frame of a vector space</a>, for the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> of square-integrable functions. This is accomplished through <a href="/wiki/Coherent_states_in_mathematical_physics#The_group-theoretical_approach" title="Coherent states in mathematical physics">coherent states</a>. </p><p>In <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a>, the diffraction phenomenon is described by the <a href="/wiki/Huygens%E2%80%93Fresnel_principle" title="Huygens–Fresnel principle">Huygens–Fresnel principle</a> that treats each point in a propagating <a href="/wiki/Wavefront" title="Wavefront">wavefront</a> as a collection of individual spherical wavelets.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The characteristic bending pattern is most pronounced when a wave from a <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">coherent</a> source (such as a laser) encounters a slit/aperture that is comparable in size to its <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>. This is due to the addition, or <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a>, of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of different lengths to the registering surface. Multiple, <a href="/wiki/Diffraction_grating" title="Diffraction grating">closely spaced openings</a> (e.g., a <a href="/wiki/Diffraction_grating" title="Diffraction grating">diffraction grating</a>), can result in a complex pattern of varying intensity. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=1" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word <i>wavelet</i> has been used for decades in digital signal processing and exploration geophysics.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The equivalent <a href="/wiki/French_language" title="French language">French</a> word <i>ondelette</i> meaning "small wave" was used by <a href="/wiki/Jean_Morlet" title="Jean Morlet">Jean Morlet</a> and <a href="/wiki/Alex_Grossmann" title="Alex Grossmann">Alex Grossmann</a> in the early 1980s. </p> <div class="mw-heading mw-heading2"><h2 id="Wavelet_theory">Wavelet theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=2" title="Edit section: Wavelet theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .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-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Wavelet" title="Special:EditPage/Wavelet">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">November 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of <a href="/wiki/Time-frequency_representation" class="mw-redirect" title="Time-frequency representation">time-frequency representation</a> for <a href="/wiki/Continuous-time" class="mw-redirect" title="Continuous-time">continuous-time</a> (analog) signals and so are related to <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Discrete wavelet transform (continuous in time) of a <a href="/wiki/Discrete-time" class="mw-redirect" title="Discrete-time">discrete-time</a> (sampled) signal by using <a href="/wiki/Discrete-time" class="mw-redirect" title="Discrete-time">discrete-time</a> <a href="/wiki/Filterbank" class="mw-redirect" title="Filterbank">filterbanks</a> of dyadic (octave band) configuration is a wavelet approximation to that signal. The coefficients of such a filter bank are called the shift and scaling coefficients in wavelets nomenclature. These filterbanks may contain either <a href="/wiki/Finite_impulse_response" title="Finite impulse response">finite impulse response</a> (FIR) or <a href="/wiki/Infinite_impulse_response" title="Infinite impulse response">infinite impulse response</a> (IIR) filters. The wavelets forming a <a href="/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">continuous wavelet transform</a> (CWT) are subject to the <a href="/wiki/Fourier_uncertainty_principle" class="mw-redirect" title="Fourier uncertainty principle">uncertainty principle</a> of Fourier analysis respective sampling theory:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event. The product of the uncertainties of time and frequency response scale has a lower bound. Thus, in the <a href="/wiki/Scaleogram" class="mw-redirect" title="Scaleogram">scaleogram</a> of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Wavelet transforms are broadly divided into three classes: continuous, discrete and multiresolution-based. </p> <div class="mw-heading mw-heading3"><h3 id="Continuous_wavelet_transforms_(continuous_shift_and_scale_parameters)"><span id="Continuous_wavelet_transforms_.28continuous_shift_and_scale_parameters.29"></span>Continuous wavelet transforms (continuous shift and scale parameters)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=3" title="Edit section: Continuous wavelet transforms (continuous shift and scale parameters)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">continuous wavelet transforms</a>, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of the <a href="/wiki/Lp_space" title="Lp space"><i>L<sup>p</sup></i></a> <a href="/wiki/Function_space" title="Function space">function space</a> <i>L</i><sup>2</sup>(<b>R</b>) ). For instance the signal may be represented on every frequency band of the form [<i>f</i>, 2<i>f</i>] for all positive frequencies <i>f</i> > 0. Then, the original signal can be reconstructed by a suitable integration over all the resulting frequency components. </p><p>The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at scale 1. This subspace in turn is in most situations generated by the shifts of one generating function ψ in <i>L</i><sup>2</sup>(<b>R</b>), the <i>mother wavelet</i>. For the example of the scale one frequency band [1, 2] this function is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)=2\,\operatorname {sinc} (2t)-\,\operatorname {sinc} (t)={\frac {\sin(2\pi t)-\sin(\pi t)}{\pi t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mi>sinc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mspace width="thinmathspace"></mspace> <mi>sinc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>π</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)=2\,\operatorname {sinc} (2t)-\,\operatorname {sinc} (t)={\frac {\sin(2\pi t)-\sin(\pi t)}{\pi t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4909e949fc4835aa914a814ab62dc12dbd534249" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.272ex; height:5.676ex;" alt="{\displaystyle \psi (t)=2\,\operatorname {sinc} (2t)-\,\operatorname {sinc} (t)={\frac {\sin(2\pi t)-\sin(\pi t)}{\pi t}}}" /></span> with the (normalized) <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a>. That, Meyer's, and two other examples of mother wavelets are: </p> <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:MeyerMathematica.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/MeyerMathematica.svg/360px-MeyerMathematica.svg.png" decoding="async" width="360" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/MeyerMathematica.svg/540px-MeyerMathematica.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/MeyerMathematica.svg/720px-MeyerMathematica.svg.png 2x" data-file-width="450" data-file-height="278" /></a><figcaption><a href="/wiki/Meyer_wavelet" title="Meyer wavelet">Meyer</a></figcaption></figure> </td></tr></tbody></table> <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:MorletWaveletMathematica.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/MorletWaveletMathematica.svg/360px-MorletWaveletMathematica.svg.png" decoding="async" width="360" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/MorletWaveletMathematica.svg/540px-MorletWaveletMathematica.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/MorletWaveletMathematica.svg/720px-MorletWaveletMathematica.svg.png 2x" data-file-width="450" data-file-height="278" /></a><figcaption><a href="/wiki/Morlet_wavelet" title="Morlet wavelet">Morlet</a></figcaption></figure> </td></tr></tbody></table> <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:MexicanHatMathematica.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/MexicanHatMathematica.svg/360px-MexicanHatMathematica.svg.png" decoding="async" width="360" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/MexicanHatMathematica.svg/540px-MexicanHatMathematica.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/MexicanHatMathematica.svg/720px-MexicanHatMathematica.svg.png 2x" data-file-width="450" data-file-height="278" /></a><figcaption><a href="/wiki/Mexican_hat_wavelet" class="mw-redirect" title="Mexican hat wavelet">Mexican hat</a></figcaption></figure> </td></tr></tbody></table> <p>The subspace of scale <i>a</i> or frequency band [1/<i>a</i>, 2/<i>a</i>] is generated by the functions (sometimes called <i>child wavelets</i>) <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{a,b}(t)={\frac {1}{\sqrt {a}}}\psi \left({\frac {t-b}{a}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>a</mi> </msqrt> </mfrac> </mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{a,b}(t)={\frac {1}{\sqrt {a}}}\psi \left({\frac {t-b}{a}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb9609cba079252c40b394503e8140b75ca8dd6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.396ex; height:6.509ex;" alt="{\displaystyle \psi _{a,b}(t)={\frac {1}{\sqrt {a}}}\psi \left({\frac {t-b}{a}}\right),}" /></span> where <i>a</i> is positive and defines the scale and <i>b</i> is any real number and defines the shift. The pair (<i>a</i>, <i>b</i>) defines a point in the right halfplane <b>R</b><sub>+</sub> × <b>R</b>. </p><p>The projection of a function <i>x</i> onto the subspace of scale <i>a</i> then has the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{a}(t)=\int _{\mathbb {R} }WT_{\psi }\{x\}(a,b)\cdot \psi _{a,b}(t)\,db}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>W</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{a}(t)=\int _{\mathbb {R} }WT_{\psi }\{x\}(a,b)\cdot \psi _{a,b}(t)\,db}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba4a582c4d9576690997d4a71c49d8ddd93bfcd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.804ex; height:5.676ex;" alt="{\displaystyle x_{a}(t)=\int _{\mathbb {R} }WT_{\psi }\{x\}(a,b)\cdot \psi _{a,b}(t)\,db}" /></span> with <i>wavelet coefficients</i> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle WT_{\psi }\{x\}(a,b)=\langle x,\psi _{a,b}\rangle =\int _{\mathbb {R} }x(t){\psi _{a,b}(t)}\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle WT_{\psi }\{x\}(a,b)=\langle x,\psi _{a,b}\rangle =\int _{\mathbb {R} }x(t){\psi _{a,b}(t)}\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/335bbef207f8e12056b76b6ee0d02a2e63d1dfc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.561ex; height:5.676ex;" alt="{\displaystyle WT_{\psi }\{x\}(a,b)=\langle x,\psi _{a,b}\rangle =\int _{\mathbb {R} }x(t){\psi _{a,b}(t)}\,dt.}" /></span> </p><p>For the analysis of the signal <i>x</i>, one can assemble the wavelet coefficients into a <a href="/wiki/Scaleogram" class="mw-redirect" title="Scaleogram">scaleogram</a> of the signal. </p><p>See a list of some <a href="/wiki/Continuous_wavelets" class="mw-redirect" title="Continuous wavelets">Continuous wavelets</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Discrete_wavelet_transforms_(discrete_shift_and_scale_parameters,_continuous_in_time)"><span id="Discrete_wavelet_transforms_.28discrete_shift_and_scale_parameters.2C_continuous_in_time.29"></span>Discrete wavelet transforms (discrete shift and scale parameters, continuous in time)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=4" title="Edit section: Discrete wavelet transforms (discrete shift and scale parameters, continuous in time)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is computationally impossible to analyze a signal using all wavelet coefficients, so one may wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding wavelet coefficients. One such system is the <a href="/wiki/Affine_transformation" title="Affine transformation">affine</a> system for some real parameters <i>a</i> > 1, <i>b</i> > 0. The corresponding discrete subset of the halfplane consists of all the points (<i>a<sup>m</sup></i>, <i>nb a<sup>m</sup></i>) with <i>m</i>, <i>n</i> in <b>Z</b>. The corresponding <i>child wavelets</i> are now given as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{m,n}(t)={\frac {1}{\sqrt {a^{m}}}}\psi \left({\frac {t-nba^{m}}{a^{m}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </msqrt> </mfrac> </mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>b</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m,n}(t)={\frac {1}{\sqrt {a^{m}}}}\psi \left({\frac {t-nba^{m}}{a^{m}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a298d5f6cb489d1b3d126752c9bbc8a0d0c6536" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.225ex; height:6.509ex;" alt="{\displaystyle \psi _{m,n}(t)={\frac {1}{\sqrt {a^{m}}}}\psi \left({\frac {t-nba^{m}}{a^{m}}}\right).}" /></span> </p><p>A sufficient condition for the reconstruction of any signal <i>x</i> of finite energy by the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=\sum _{m\in \mathbb {Z} }\sum _{n\in \mathbb {Z} }\langle x,\,\psi _{m,n}\rangle \cdot \psi _{m,n}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>⋅</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=\sum _{m\in \mathbb {Z} }\sum _{n\in \mathbb {Z} }\langle x,\,\psi _{m,n}\rangle \cdot \psi _{m,n}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5a8259343b96f673e7904a8717d436e0048b69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.606ex; height:5.676ex;" alt="{\displaystyle x(t)=\sum _{m\in \mathbb {Z} }\sum _{n\in \mathbb {Z} }\langle x,\,\psi _{m,n}\rangle \cdot \psi _{m,n}(t)}" /></span> is that the functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\psi _{m,n}:m,n\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\psi _{m,n}:m,n\in \mathbb {Z} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9170fcd8bfec8e7145b39b53849ead6d0a630c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.754ex; height:3.009ex;" alt="{\displaystyle \{\psi _{m,n}:m,n\in \mathbb {Z} \}}" /></span> form an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of <i>L</i><sup>2</sup>(<b>R</b>). </p> <div class="mw-heading mw-heading3"><h3 id="Multiresolution_based_discrete_wavelet_transforms_(continuous_in_time)"><span id="Multiresolution_based_discrete_wavelet_transforms_.28continuous_in_time.29"></span>Multiresolution based discrete wavelet transforms (continuous in time)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=5" title="Edit section: Multiresolution based discrete wavelet transforms (continuous in time)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Daubechies4-functions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Daubechies4-functions.svg/220px-Daubechies4-functions.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Daubechies4-functions.svg/330px-Daubechies4-functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Daubechies4-functions.svg/440px-Daubechies4-functions.svg.png 2x" data-file-width="1000" data-file-height="750" /></a><figcaption>D4 wavelet</figcaption></figure> <p>In any discretised wavelet transform, there are only a finite number of wavelet coefficients for each bounded rectangular region in the upper halfplane. Still, each coefficient requires the evaluation of an integral. In special situations this numerical complexity can be avoided if the scaled and shifted wavelets form a <a href="/wiki/Multiresolution_analysis" title="Multiresolution analysis">multiresolution analysis</a>. This means that there has to exist an <a href="/wiki/Auxiliary_function" title="Auxiliary function">auxiliary function</a>, the <i>father wavelet</i> φ in <i>L</i><sup>2</sup>(<b>R</b>), and that <i>a</i> is an integer. A typical choice is <i>a</i> = 2 and <i>b</i> = 1. The most famous pair of father and mother wavelets is the <a href="/wiki/Daubechies_wavelets" class="mw-redirect" title="Daubechies wavelets">Daubechies</a> 4-tap wavelet. Note that not every orthonormal discrete wavelet basis can be associated to a multiresolution analysis; for example, the Journe wavelet admits no multiresolution analysis.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>From the mother and father wavelets one constructs the subspaces <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}=\operatorname {span} (\phi _{m,n}:n\in \mathbb {Z} ),{\text{ where }}\phi _{m,n}(t)=2^{-m/2}\phi (2^{-m}t-n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}=\operatorname {span} (\phi _{m,n}:n\in \mathbb {Z} ),{\text{ where }}\phi _{m,n}(t)=2^{-m/2}\phi (2^{-m}t-n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8115653a7cfbb5a936c72560e9592565d4f768a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.368ex; height:3.509ex;" alt="{\displaystyle V_{m}=\operatorname {span} (\phi _{m,n}:n\in \mathbb {Z} ),{\text{ where }}\phi _{m,n}(t)=2^{-m/2}\phi (2^{-m}t-n)}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{m}=\operatorname {span} (\psi _{m,n}:n\in \mathbb {Z} ),{\text{ where }}\psi _{m,n}(t)=2^{-m/2}\psi (2^{-m}t-n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{m}=\operatorname {span} (\psi _{m,n}:n\in \mathbb {Z} ),{\text{ where }}\psi _{m,n}(t)=2^{-m/2}\psi (2^{-m}t-n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03270d84a865fef1d99d97a486741843c7232d07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.236ex; height:3.509ex;" alt="{\displaystyle W_{m}=\operatorname {span} (\psi _{m,n}:n\in \mathbb {Z} ),{\text{ where }}\psi _{m,n}(t)=2^{-m/2}\psi (2^{-m}t-n).}" /></span> The father wavelet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}" /></span> keeps the time domain properties, while the mother wavelets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}" /></span> keeps the frequency domain properties. </p><p>From these it is required that the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}\subset \dots \subset V_{1}\subset V_{0}\subset V_{-1}\subset V_{-2}\subset \dots \subset L^{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⊂</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}\subset \dots \subset V_{1}\subset V_{0}\subset V_{-1}\subset V_{-2}\subset \dots \subset L^{2}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feaff7db46551c53b98997568eaf98c7c90571d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.942ex; height:3.176ex;" alt="{\displaystyle \{0\}\subset \dots \subset V_{1}\subset V_{0}\subset V_{-1}\subset V_{-2}\subset \dots \subset L^{2}(\mathbb {R} )}" /></span> forms a <a href="/wiki/Multiresolution_analysis" title="Multiresolution analysis">multiresolution analysis</a> of <i>L<sup>2</sup></i> and that the subspaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dots ,W_{1},W_{0},W_{-1},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mo>,</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dots ,W_{1},W_{0},W_{-1},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716fde9365cf372e6a39b87b3cd21936b78ac330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.992ex; height:2.509ex;" alt="{\displaystyle \dots ,W_{1},W_{0},W_{-1},\dots }" /></span> are the orthogonal "differences" of the above sequence, that is, <i>W<sub>m</sub></i> is the orthogonal complement of <i>V<sub>m</sub></i> inside the subspace <i>V</i><sub><i>m</i>−1</sub>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{m}\oplus W_{m}=V_{m-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⊕</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{m}\oplus W_{m}=V_{m-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234a0e26d0f624a2d1032fc58f3c226cc51680d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.615ex; height:2.509ex;" alt="{\displaystyle V_{m}\oplus W_{m}=V_{m-1}.}" /></span> </p><p>In analogy to the <a href="/wiki/Sampling_theorem" class="mw-redirect" title="Sampling theorem">sampling theorem</a> one may conclude that the space <i>V<sub>m</sub></i> with sampling distance 2<sup><i>m</i></sup> more or less covers the frequency baseband from 0 to 1/2<sup><i>m</i>-1</sup>. As orthogonal complement, <i>W<sub>m</sub></i> roughly covers the band [1/2<sup><i>m</i>−1</sup>, 1/2<sup><i>m</i></sup>]. </p><p>From those inclusions and orthogonality relations, especially <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}\oplus W_{0}=V_{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{0}\oplus W_{0}=V_{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9456e9ba4b78f577e1adf5f172d0a6c2bcedf7f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.284ex; height:2.509ex;" alt="{\displaystyle V_{0}\oplus W_{0}=V_{-1}}" /></span>, follows the existence of sequences <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=\{h_{n}\}_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=\{h_{n}\}_{n\in \mathbb {Z} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f885cc68854f9fc2b4c682002fb59898b56fe6c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.731ex; height:2.843ex;" alt="{\displaystyle h=\{h_{n}\}_{n\in \mathbb {Z} }}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=\{g_{n}\}_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=\{g_{n}\}_{n\in \mathbb {Z} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e78fa249165b4ea5597a6109e2d471c2260cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.278ex; height:2.843ex;" alt="{\displaystyle g=\{g_{n}\}_{n\in \mathbb {Z} }}" /></span> that satisfy the identities <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{n}=\langle \phi _{0,0},\,\phi _{-1,n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{n}=\langle \phi _{0,0},\,\phi _{-1,n}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/131334c85cd599462bd188acb1bb0b9f3d157f5a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.537ex; height:3.009ex;" alt="{\displaystyle g_{n}=\langle \phi _{0,0},\,\phi _{-1,n}\rangle }" /></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi (t)={\sqrt {2}}\sum _{n\in \mathbb {Z} }g_{n}\phi (2t-n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi (t)={\sqrt {2}}\sum _{n\in \mathbb {Z} }g_{n}\phi (2t-n),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c403a635913af00cc4b3043258c950c38af2d3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.276ex; height:3.343ex;" alt="{\textstyle \phi (t)={\sqrt {2}}\sum _{n\in \mathbb {Z} }g_{n}\phi (2t-n),}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{n}=\langle \psi _{0,0},\,\phi _{-1,n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{n}=\langle \psi _{0,0},\,\phi _{-1,n}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f55b2224a0fa52ab699e041751ad1b0b9aee0ad2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.895ex; height:3.009ex;" alt="{\displaystyle h_{n}=\langle \psi _{0,0},\,\phi _{-1,n}\rangle }" /></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \psi (t)={\sqrt {2}}\sum _{n\in \mathbb {Z} }h_{n}\phi (2t-n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi (t)={\sqrt {2}}\sum _{n\in \mathbb {Z} }h_{n}\phi (2t-n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52a1b479ae7404e6b7c9b681dc26a055edd47809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.634ex; height:3.343ex;" alt="{\textstyle \psi (t)={\sqrt {2}}\sum _{n\in \mathbb {Z} }h_{n}\phi (2t-n).}" /></span> The second identity of the first pair is a <a href="/wiki/Refinable_function" title="Refinable function">refinement equation</a> for the father wavelet φ. Both pairs of identities form the basis for the algorithm of the <a href="/wiki/Fast_wavelet_transform" title="Fast wavelet transform">fast wavelet transform</a>. </p><p>From the multiresolution analysis derives the orthogonal decomposition of the space <i>L</i><sup>2</sup> as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}=V_{j_{0}}\oplus W_{j_{0}}\oplus W_{j_{0}-1}\oplus W_{j_{0}-2}\oplus W_{j_{0}-3}\oplus \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>⊕</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>⊕</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>⊕</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}=V_{j_{0}}\oplus W_{j_{0}}\oplus W_{j_{0}-1}\oplus W_{j_{0}-2}\oplus W_{j_{0}-3}\oplus \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94f04c429db3826b6920e348c973ac447d31f13e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.799ex; height:3.343ex;" alt="{\displaystyle L^{2}=V_{j_{0}}\oplus W_{j_{0}}\oplus W_{j_{0}-1}\oplus W_{j_{0}-2}\oplus W_{j_{0}-3}\oplus \cdots }" /></span> For any signal or function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\in L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\in L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d05d95bcd961252d7f0cfac47934d676ba2b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.977ex; height:2.676ex;" alt="{\displaystyle S\in L^{2}}" /></span> this gives a representation in basis functions of the corresponding subspaces as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\sum _{k}c_{j_{0},k}\phi _{j_{0},k}+\sum _{j\leq j_{0}}\sum _{k}d_{j,k}\psi _{j,k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≤</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\sum _{k}c_{j_{0},k}\phi _{j_{0},k}+\sum _{j\leq j_{0}}\sum _{k}d_{j,k}\psi _{j,k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72093c237edecafba07e23ccff1878c21dbc8799" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.446ex; height:6.009ex;" alt="{\displaystyle S=\sum _{k}c_{j_{0},k}\phi _{j_{0},k}+\sum _{j\leq j_{0}}\sum _{k}d_{j,k}\psi _{j,k}}" /></span> where the coefficients are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{j_{0},k}=\langle S,\phi _{j_{0},k}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mo>,</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{j_{0},k}=\langle S,\phi _{j_{0},k}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1659e7e5542113955d08ee42d954d0c087fe3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.943ex; height:3.009ex;" alt="{\displaystyle c_{j_{0},k}=\langle S,\phi _{j_{0},k}\rangle }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{j,k}=\langle S,\psi _{j,k}\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{j,k}=\langle S,\psi _{j,k}\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d79376940ba8e9ed8c8bb99681bfb0c809bdbb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.257ex; height:3.009ex;" alt="{\displaystyle d_{j,k}=\langle S,\psi _{j,k}\rangle .}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Time-causal_wavelets">Time-causal wavelets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=6" title="Edit section: Time-causal wavelets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For processing temporal signals in real time, it is essential that the wavelet filters do not access signal values from the future as well as that minimal temporal latencies can be obtained. Time-causal wavelets representations have been developed by Szu et al <sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and Lindeberg,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> with the latter method also involving a memory-efficient time-recursive implementation. </p> <div class="mw-heading mw-heading2"><h2 id="Mother_wavelet">Mother wavelet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=7" title="Edit section: Mother wavelet"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For practical applications, and for efficiency reasons, one prefers continuously differentiable functions with compact support as mother (prototype) wavelet (functions). However, to satisfy analytical requirements (in the continuous WT) and in general for theoretical reasons, one chooses the wavelet functions from a subspace of the <a href="/wiki/Lp_space" title="Lp space">space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mathbb {R} )\cap L^{2}(\mathbb {R} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>∩</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(\mathbb {R} )\cap L^{2}(\mathbb {R} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/661260ace6ba96f511e8861802b5148465ce93e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.478ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mathbb {R} )\cap L^{2}(\mathbb {R} ).}" /></span> This is the space of <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measurable</a> functions that are both <a href="/wiki/Absolutely_integrable_function" title="Absolutely integrable function">absolutely integrable</a> and <a href="/wiki/Square-integrable_function" title="Square-integrable function">square integrable</a> in the sense that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|\,dt<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|\,dt<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3e7307dbbf3dcf6cba29d5a0e787e301c049a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.154ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|\,dt<\infty }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fda903b393e402664740b9da0f13c90517b6c76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.855ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt<\infty .}" /></span> </p><p>Being in this space ensures that one can formulate the conditions of zero mean and square norm one: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\psi (t)\,dt=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\psi (t)\,dt=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b16a6dad09944cb11020c716d927494b5459e6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.699ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\psi (t)\,dt=0}" /></span> is the condition for zero mean, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76e6ae9181c8b4f8a94eed13447234699a627f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.047ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt=1}" /></span> is the condition for square norm one. </p><p>For <i>ψ</i> to be a wavelet for the <a href="/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">continuous wavelet transform</a> (see there for exact statement), the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform. </p><p>For the <a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">discrete wavelet transform</a>, one needs at least the condition that the <a href="/wiki/Wavelet_series" class="mw-redirect" title="Wavelet series">wavelet series</a> is a representation of the identity in the <a href="/wiki/Lp_space" title="Lp space">space</a> <i>L</i><sup>2</sup>(<b>R</b>). Most constructions of discrete WT make use of the <a href="/wiki/Multiresolution_analysis" title="Multiresolution analysis">multiresolution analysis</a>, which defines the wavelet by a scaling function. This scaling function itself is a solution to a functional equation. </p><p>In most situations it is useful to restrict ψ to be a continuous function with a higher number <i>M</i> of vanishing moments, i.e. for all integer <i>m</i> < <i>M</i> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }t^{m}\,\psi (t)\,dt=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }t^{m}\,\psi (t)\,dt=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aadc4f5a510fffc207571637aa1f4cc37313c95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.248ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }t^{m}\,\psi (t)\,dt=0.}" /></span> </p><p>The mother wavelet is scaled (or dilated) by a factor of <i>a</i> and translated (or shifted) by a factor of <i>b</i> to give (under Morlet's original formulation): </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{a,b}(t)={1 \over {\sqrt {a}}}\psi \left({t-b \over a}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{a,b}(t)={1 \over {\sqrt {a}}}\psi \left({t-b \over a}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08851076ca67aa5b4240058e7f1a73ca42dbfc9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.396ex; height:6.509ex;" alt="{\displaystyle \psi _{a,b}(t)={1 \over {\sqrt {a}}}\psi \left({t-b \over a}\right).}" /></span> </p><p>For the continuous WT, the pair (<i>a</i>,<i>b</i>) varies over the full half-plane <b>R</b><sub>+</sub> × <b>R</b>; for the discrete WT this pair varies over a discrete subset of it, which is also called <i>affine group</i>. </p><p>These functions are often incorrectly referred to as the basis functions of the (continuous) transform. In fact, as in the continuous Fourier transform, there is no basis in the continuous wavelet transform. Time-frequency interpretation uses a subtly different formulation (after Delprat). </p><p>Restriction： </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {a}}}\int _{-\infty }^{\infty }\varphi _{a1,b1}(t)\varphi \left({\frac {t-b}{a}}\right)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>a</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {a}}}\int _{-\infty }^{\infty }\varphi _{a1,b1}(t)\varphi \left({\frac {t-b}{a}}\right)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0218fedf5722e3e52d4e46447de213683357dc81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.972ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {a}}}\int _{-\infty }^{\infty }\varphi _{a1,b1}(t)\varphi \left({\frac {t-b}{a}}\right)\,dt}" /></span> when <span class="texhtml"><i>a</i><sub>1</sub> = <i>a</i></span> and <span class="texhtml"><i>b</i><sub>1</sub> = <i>b</i></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc70cdfb4274988ac81c279cccf1eb47f3a20e6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.457ex; height:2.843ex;" alt="{\displaystyle \Psi (t)}" /></span> has a finite time interval</li></ol> <div class="mw-heading mw-heading2"><h2 id="Comparisons_with_Fourier_transform_(continuous-time)"><span id="Comparisons_with_Fourier_transform_.28continuous-time.29"></span>Comparisons with Fourier transform (continuous-time)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=8" title="Edit section: Comparisons with Fourier transform (continuous-time)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The wavelet transform is often compared with the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, in which signals are represented as a sum of sinusoids. In fact, the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> can be viewed as a special case of the continuous wavelet transform with the choice of the mother wavelet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)=e^{-2\pi it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)=e^{-2\pi it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd1ab56467663e778df17de5abfcf15caab1c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.78ex; height:3.176ex;" alt="{\displaystyle \psi (t)=e^{-2\pi it}}" /></span>. The main difference in general is that wavelets are localized in both time and frequency whereas the standard <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> is only localized in <a href="/wiki/Frequency" title="Frequency">frequency</a>. The <a href="/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a> (STFT) is similar to the wavelet transform, in that it is also time and frequency localized, but there are issues with the frequency/time resolution trade-off. </p><p>In particular, assuming a rectangular window region, one may think of the STFT as a transform with a slightly different kernel <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)=g(t-u)e^{-2\pi it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)=g(t-u)e^{-2\pi it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd2a3f04daf8b3ddf89205bd4d2481ac6cf458c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.715ex; height:3.176ex;" alt="{\displaystyle \psi (t)=g(t-u)e^{-2\pi it}}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t-u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e64cefea3f37a8ed645af1e6648ee9721c6ccb9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.935ex; height:2.843ex;" alt="{\displaystyle g(t-u)}" /></span> can often be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {rect} \left({\frac {t-u}{\Delta _{t}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mi>u</mi> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {rect} \left({\frac {t-u}{\Delta _{t}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6ee2e6b151fea07533f86e7e4356dbe9b4dec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.305ex; height:4.843ex;" alt="{\textstyle \operatorname {rect} \left({\frac {t-u}{\Delta _{t}}}\right)}" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc65d833175da758154af3fc70cb2db400cae7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.762ex; height:2.509ex;" alt="{\displaystyle \Delta _{t}}" /></span> and <i>u</i> respectively denote the length and temporal offset of the windowing function. Using <a href="/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a>, one may define the wavelet's energy as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|{\hat {\psi }}(\omega )|^{2}\,d\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|{\hat {\psi }}(\omega )|^{2}\,d\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5089d15477e02494b805b5c159396dd710f61425" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.557ex; height:6.009ex;" alt="{\displaystyle E=\int _{-\infty }^{\infty }|\psi (t)|^{2}\,dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|{\hat {\psi }}(\omega )|^{2}\,d\omega }" /></span> From this, the square of the temporal support of the window offset by time <i>u</i> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{u}^{2}={\frac {1}{E}}\int |t-u|^{2}|\psi (t)|^{2}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo>−</mo> <mi>u</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{u}^{2}={\frac {1}{E}}\int |t-u|^{2}|\psi (t)|^{2}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6cb1f43fbd50f539759d86f6d6c83dce63ecb3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.488ex; height:5.676ex;" alt="{\displaystyle \sigma _{u}^{2}={\frac {1}{E}}\int |t-u|^{2}|\psi (t)|^{2}\,dt}" /></span> </p><p>and the square of the spectral support of the window acting on a frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}_{\xi }^{2}={\frac {1}{2\pi E}}\int |\omega -\xi |^{2}|{\hat {\psi }}(\omega )|^{2}\,d\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ω</mi> <mo>−</mo> <mi>ξ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}_{\xi }^{2}={\frac {1}{2\pi E}}\int |\omega -\xi |^{2}|{\hat {\psi }}(\omega )|^{2}\,d\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f747e363b146b81e38e071594c044f8d585059" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.468ex; height:5.676ex;" alt="{\displaystyle {\hat {\sigma }}_{\xi }^{2}={\frac {1}{2\pi E}}\int |\omega -\xi |^{2}|{\hat {\psi }}(\omega )|^{2}\,d\omega }" /></span> </p><p>Multiplication with a rectangular window in the time domain corresponds to convolution with a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} (\Delta _{t}\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sinc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (\Delta _{t}\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f838eeaac8ab4779400d06bf366ab393e2c7e958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.905ex; height:2.843ex;" alt="{\displaystyle \operatorname {sinc} (\Delta _{t}\omega )}" /></span> function in the frequency domain, resulting in spurious <a href="/wiki/Ringing_artifacts" title="Ringing artifacts">ringing artifacts</a> for short/localized temporal windows. With the continuous-time Fourier transform, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{t}\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{t}\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c7c43d97cbfc7784bbbdafea4a5e2c012cc623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.7ex; height:2.509ex;" alt="{\displaystyle \Delta _{t}\to \infty }" /></span> and this convolution is with a delta function in Fourier space, resulting in the true Fourier transform of the signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" /></span>. The window function may be some other <a href="/wiki/Apodizing" class="mw-redirect" title="Apodizing">apodizing filter</a>, such as a <a href="/wiki/Gaussian_filter" title="Gaussian filter">Gaussian</a>. The choice of windowing function will affect the approximation error relative to the true Fourier transform. </p><p>A given resolution cell's time-bandwidth product may not be exceeded with the STFT. All STFT basis elements maintain a uniform spectral and temporal support for all temporal shifts or offsets, thereby attaining an equal resolution in time for lower and higher frequencies. The resolution is purely determined by the sampling width. </p><p> In contrast, the wavelet transform's <a href="/wiki/Multiresolution_analysis" title="Multiresolution analysis">multiresolutional</a> properties enables large temporal supports for lower frequencies while maintaining short temporal widths for higher frequencies by the scaling properties of the wavelet transform. This property extends conventional time-frequency analysis into time-scale analysis.<sup id="cite_ref-FOOTNOTEMallat2009Chpt._7_12-0" class="reference"><a href="#cite_note-FOOTNOTEMallat2009Chpt._7-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Time_frequency_atom_resolution.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Time_frequency_atom_resolution.png/220px-Time_frequency_atom_resolution.png" decoding="async" width="220" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Time_frequency_atom_resolution.png/330px-Time_frequency_atom_resolution.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Time_frequency_atom_resolution.png/440px-Time_frequency_atom_resolution.png 2x" data-file-width="922" data-file-height="476" /></a><figcaption>STFT time-frequency atoms (left) and DWT time-scale atoms (right). The time-frequency atoms are four different basis functions used for the STFT (i.e. <b>four separate Fourier transforms required</b>). The time-scale atoms of the DWT achieve small temporal widths for high frequencies and good temporal widths for low frequencies with a <b>single</b> transform basis set.</figcaption></figure> <p>The discrete wavelet transform is less computationally <a href="/wiki/Complexity" title="Complexity">complex</a>, taking <a href="/wiki/Big_O_notation" title="Big O notation">O(<i>N</i>)</a> time as compared to O(<i>N</i> log <i>N</i>) for the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT). This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT which uses the same basis functions as the discrete Fourier transform (DFT).<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This complexity only applies when the filter size has no relation to the signal size. A wavelet without <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact support</a> such as the <a href="/wiki/Shannon_wavelet" title="Shannon wavelet">Shannon wavelet</a> would require O(<i>N</i><sup>2</sup>). (For instance, a logarithmic Fourier Transform also exists with O(<i>N</i>) complexity, but the original signal must be sampled logarithmically in time, which is only useful for certain types of signals.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>) </p> <div class="mw-heading mw-heading2"><h2 id="Definition_of_a_wavelet">Definition of a wavelet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=9" title="Edit section: Definition of a wavelet"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A wavelet (or a wavelet family) can be defined in various ways: </p> <div class="mw-heading mw-heading3"><h3 id="Scaling_filter">Scaling filter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=10" title="Edit section: Scaling filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An orthogonal wavelet is entirely defined by the scaling filter – a low-pass <a href="/wiki/Finite_impulse_response" title="Finite impulse response">finite impulse response</a> (FIR) filter of length 2<i>N</i> and sum 1. In <a href="/wiki/Biorthogonal_system" title="Biorthogonal system">biorthogonal</a> wavelets, separate decomposition and reconstruction filters are defined. </p><p>For analysis with orthogonal wavelets the high pass filter is calculated as the <a href="/wiki/Quadrature_mirror_filter" title="Quadrature mirror filter">quadrature mirror filter</a> of the low pass, and reconstruction filters are the time reverse of the decomposition filters. </p><p>Daubechies and Symlet wavelets can be defined by the scaling filter. </p> <div class="mw-heading mw-heading3"><h3 id="Scaling_function">Scaling function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=11" title="Edit section: Scaling function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wavelets are defined by the wavelet function ψ(<i>t</i>) (i.e. the mother wavelet) and scaling function φ(<i>t</i>) (also called father wavelet) in the time domain. </p><p>The wavelet function is in effect a band-pass filter and scaling that for each level halves its bandwidth. This creates the problem that in order to cover the entire spectrum, an infinite number of levels would be required. The scaling function filters the lowest level of the transform and ensures all the spectrum is covered. See<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> for a detailed explanation. </p><p>For a wavelet with compact support, φ(<i>t</i>) can be considered finite in length and is equivalent to the scaling filter <i>g</i>. </p><p>Meyer wavelets can be defined by scaling functions </p> <div class="mw-heading mw-heading3"><h3 id="Wavelet_function">Wavelet function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=12" title="Edit section: Wavelet function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The wavelet only has a time domain representation as the wavelet function ψ(<i>t</i>). </p><p>For instance, <a href="/wiki/Mexican_hat_wavelet" class="mw-redirect" title="Mexican hat wavelet">Mexican hat wavelets</a> can be defined by a wavelet function. See a list of a few <a href="/wiki/Continuous_wavelets" class="mw-redirect" title="Continuous wavelets">continuous wavelets</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=13" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The development of wavelets can be linked to several separate trains of thought, starting with <a href="/wiki/Alfr%C3%A9d_Haar" title="Alfréd Haar">Alfréd Haar</a>'s work in the early 20th century. Later work by <a href="/wiki/Dennis_Gabor" title="Dennis Gabor">Dennis Gabor</a> yielded <a href="/wiki/Gabor_atom" title="Gabor atom">Gabor atoms</a> (1946), which are constructed similarly to wavelets, and applied to similar purposes. </p><p>Notable contributions to wavelet theory since then can be attributed to <a href="/wiki/George_Zweig" title="George Zweig">George Zweig</a>’s discovery of the <a href="/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">continuous wavelet transform</a> (CWT) in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound),<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Pierre Goupillaud, <a href="/wiki/Alex_Grossmann" title="Alex Grossmann">Alex Grossmann</a> and <a href="/wiki/Jean_Morlet" title="Jean Morlet">Jean Morlet</a>'s formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work on <a href="/wiki/Str%C3%B6mberg_wavelet" title="Strömberg wavelet">discrete wavelets</a> (1983), the Le Gall–Tabatabai (LGT) 5/3-taps non-orthogonal filter bank with linear phase (1988),<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Ingrid_Daubechies" title="Ingrid Daubechies">Ingrid Daubechies</a>' orthogonal wavelets with compact support (1988), <a href="/wiki/Stephane_Mallat" class="mw-redirect" title="Stephane Mallat">Stéphane Mallat</a>'s non-orthogonal multiresolution framework (1989), <a href="/wiki/Ali_Akansu" title="Ali Akansu">Ali Akansu</a>'s <a href="/wiki/Binomial_QMF" title="Binomial QMF">binomial QMF</a> (1990), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland's <a href="/wiki/Harmonic_wavelet_transform" title="Harmonic wavelet transform">harmonic wavelet transform</a> (1993), and <a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">set partitioning in hierarchical trees</a> (SPIHT) developed by Amir Said with William A. Pearlman in 1996.<sup id="cite_ref-Said_20-0" class="reference"><a href="#cite_note-Said-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/JPEG_2000" title="JPEG 2000">JPEG 2000</a> standard was developed from 1997 to 2000 by a <a href="/wiki/Joint_Photographic_Experts_Group" title="Joint Photographic Experts Group">Joint Photographic Experts Group</a> (JPEG) committee chaired by Touradj Ebrahimi (later the JPEG president).<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In contrast to the DCT algorithm used by the original <a href="/wiki/JPEG" title="JPEG">JPEG</a> format, JPEG 2000 instead uses <a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">discrete wavelet transform</a> (DWT) algorithms. It uses the <a href="/wiki/Cohen-Daubechies-Feauveau_wavelet" class="mw-redirect" title="Cohen-Daubechies-Feauveau wavelet">CDF</a> 9/7 wavelet transform (developed by Ingrid Daubechies in 1992) for its <a href="/wiki/Lossy_compression" title="Lossy compression">lossy compression</a> algorithm, and the Le Gall–Tabatabai (LGT) 5/3 discrete-time filter bank (developed by Didier Le Gall and Ali J. Tabatabai in 1988) for its <a href="/wiki/Lossless_compression" title="Lossless compression">lossless compression</a> algorithm.<sup id="cite_ref-Unser_22-0" class="reference"><a href="#cite_note-Unser-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="/wiki/JPEG_2000" title="JPEG 2000">JPEG 2000</a> technology, which includes the <a href="/wiki/Motion_JPEG_2000" title="Motion JPEG 2000">Motion JPEG 2000</a> extension, was selected as the <a href="/wiki/Video_coding_standard" class="mw-redirect" title="Video coding standard">video coding standard</a> for <a href="/wiki/Digital_cinema" title="Digital cinema">digital cinema</a> in 2004.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Timeline">Timeline</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=14" title="Edit section: Timeline"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>First wavelet (<a href="/wiki/Haar_Wavelet" class="mw-redirect" title="Haar Wavelet">Haar's wavelet</a>) by <a href="/wiki/Alfr%C3%A9d_Haar" title="Alfréd Haar">Alfréd Haar</a> (1909)</li> <li>Since the 1970s: <a href="/wiki/George_Zweig" title="George Zweig">George Zweig</a>, <a href="/wiki/Jean_Morlet" title="Jean Morlet">Jean Morlet</a>, <a href="/wiki/Alex_Grossmann" title="Alex Grossmann">Alex Grossmann</a></li> <li>Since the 1980s: <a href="/wiki/Yves_Meyer" title="Yves Meyer">Yves Meyer</a>, Didier Le Gall, Ali J. Tabatabai, <a href="/wiki/St%C3%A9phane_Mallat" title="Stéphane Mallat">Stéphane Mallat</a>, <a href="/wiki/Ingrid_Daubechies" title="Ingrid Daubechies">Ingrid Daubechies</a>, <a href="/wiki/Ronald_Coifman" title="Ronald Coifman">Ronald Coifman</a>, <a href="/wiki/Ali_Akansu" title="Ali Akansu">Ali Akansu</a>, <a href="/wiki/Victor_Wickerhauser" title="Victor Wickerhauser">Victor Wickerhauser</a></li> <li>Since the 1990s: Nathalie Delprat, Newland, Amir Said, William A. Pearlman, Touradj Ebrahimi, <a href="/wiki/JPEG_2000" title="JPEG 2000">JPEG 2000</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Wavelet_transforms">Wavelet transforms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=15" title="Edit section: Wavelet transforms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet transform</a></div> <p>A wavelet is a mathematical function used to divide a given function or <a href="/wiki/Continuous_signal" class="mw-redirect" title="Continuous signal">continuous-time signal</a> into different scale components. Usually one can assign a frequency range to each scale component. Each scale component can then be studied with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">scaled</a> and <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translated</a> copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transforms</a> for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-<a href="/wiki/Periodic_function" title="Periodic function">periodic</a> and/or non-<a href="/wiki/Stationary_process" title="Stationary process">stationary</a> signals. </p><p>Wavelet transforms are classified into <a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">discrete wavelet transforms</a> (DWTs) and <a href="/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">continuous wavelet transforms</a> (CWTs). Note that both DWT and CWT are continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid. </p><p>There are a large number of wavelet transforms each suitable for different applications. For a full list see <a href="/wiki/List_of_wavelet-related_transforms" title="List of wavelet-related transforms">list of wavelet-related transforms</a> but the common ones are listed below: </p> <ul><li><a href="/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">Continuous wavelet transform</a> (CWT)</li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">Discrete wavelet transform</a> (DWT)</li> <li><a href="/wiki/Fast_wavelet_transform" title="Fast wavelet transform">Fast wavelet transform</a> (FWT)</li> <li><a href="/wiki/Lifting_scheme" title="Lifting scheme">Lifting scheme</a> and <a href="/wiki/Generalized_lifting" class="mw-redirect" title="Generalized lifting">generalized lifting scheme</a></li> <li><a href="/wiki/Wavelet_packet_decomposition" title="Wavelet packet decomposition">Wavelet packet decomposition</a> (WPD)</li> <li><a href="/wiki/Stationary_wavelet_transform" title="Stationary wavelet transform">Stationary wavelet transform</a> (SWT)</li> <li><a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">Fractional Fourier transform</a> (FRFT)</li> <li><a href="/wiki/Fractional_wavelet_transform" title="Fractional wavelet transform">Fractional wavelet transform</a> (FRWT)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Generalized_transforms">Generalized transforms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=16" title="Edit section: Generalized transforms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are a number of generalized transforms of which the wavelet transform is a special case. For example, Yosef Joseph Segman introduced scale into the <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg group</a>, giving rise to a continuous transform space that is a function of time, scale, and frequency. The CWT is a two-dimensional slice through the resulting 3d time-scale-frequency volume. </p><p>Another example of a generalized transform is the <a href="/wiki/Chirplet_transform" title="Chirplet transform">chirplet transform</a> in which the CWT is also a two dimensional slice through the chirplet transform. </p><p>An important application area for generalized transforms involves systems in which high frequency resolution is crucial. For example, <a href="/wiki/Darkfield_microscope" class="mw-redirect" title="Darkfield microscope">darkfield</a> electron optical transforms intermediate between direct and <a href="/wiki/Reciprocal_space" class="mw-redirect" title="Reciprocal space">reciprocal space</a> have been widely used in the <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> of atom clustering, i.e. in the study of <a href="/wiki/Crystal" title="Crystal">crystals</a> and <a href="/wiki/Crystal_defect" class="mw-redirect" title="Crystal defect">crystal defects</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Now that <a href="/wiki/Transmission_electron_microscope" class="mw-redirect" title="Transmission electron microscope">transmission electron microscopes</a> are capable of providing digital images with picometer-scale information on atomic periodicity in <a href="/wiki/Nanostructure" title="Nanostructure">nanostructure</a> of all sorts, the range of <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>/<a href="/wiki/Metrology" title="Metrology">metrology</a><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> applications for intermediate transforms with high frequency resolution (like brushlets<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> and ridgelets<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup>) is growing rapidly. </p><p>Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform in the fractional Fourier transform domains. This transform is capable of providing the time- and fractional-domain information simultaneously and representing signals in the time-fractional-frequency plane.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=17" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Generally, an approximation to DWT is used for <a href="/wiki/Data_compression" title="Data compression">data compression</a> if a signal is already sampled, and the CWT for <a href="/wiki/Signal_analysis" class="mw-redirect" title="Signal analysis">signal analysis</a>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Thus, DWT approximation is commonly used in engineering and computer science,<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> and the CWT in scientific research.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Like some other transforms, wavelet transforms can be used to transform data, then encode the transformed data, resulting in effective compression. For example, <a href="/wiki/JPEG_2000" title="JPEG 2000">JPEG 2000</a> is an image compression standard that uses biorthogonal wavelets. This means that although the frame is overcomplete, it is a <i>tight frame</i> (see types of <a href="/wiki/Frame_of_a_vector_space" class="mw-redirect" title="Frame of a vector space">frames of a vector space</a>), and the same frame functions (except for conjugation in the case of complex wavelets) are used for both analysis and synthesis, i.e., in both the forward and inverse transform. For details see <a href="/wiki/Wavelet_compression" class="mw-redirect" title="Wavelet compression">wavelet compression</a>. </p><p>A related use is for smoothing/denoising data based on wavelet coefficient thresholding, also called wavelet shrinkage. By adaptively thresholding the wavelet coefficients that correspond to undesired frequency components smoothing and/or denoising operations can be performed. </p><p>Wavelet transforms are also starting to be used for communication applications. Wavelet <a href="/wiki/OFDM" class="mw-redirect" title="OFDM">OFDM</a> is the basic modulation scheme used in <a href="/wiki/HD-PLC" class="mw-redirect" title="HD-PLC">HD-PLC</a> (a <a href="/wiki/Power_line_communication" class="mw-redirect" title="Power line communication">power line communications</a> technology developed by <a href="/wiki/Panasonic" title="Panasonic">Panasonic</a>), and in one of the optional modes included in the <a href="/wiki/IEEE_1901" title="IEEE 1901">IEEE 1901</a> standard. Wavelet OFDM can achieve deeper notches than traditional <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a> OFDM, and wavelet OFDM does not require a guard interval (which usually represents significant overhead in FFT OFDM systems).<sup id="cite_ref-galli_35-0" class="reference"><a href="#cite_note-galli-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="As_a_representation_of_a_signal">As a representation of a signal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=18" title="Edit section: As a representation of a signal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Often, signals can be represented well as a sum of sinusoids. However, consider a non-continuous signal with an abrupt discontinuity; this signal can still be represented as a sum of sinusoids, but requires an infinite number, which is an observation known as <a href="/wiki/Gibbs_phenomenon" title="Gibbs phenomenon">Gibbs phenomenon</a>. This, then, requires an infinite number of Fourier coefficients, which is not practical for many applications, such as compression. Wavelets are more useful for describing these signals with discontinuities because of their time-localized behavior (both Fourier and wavelet transforms are frequency-localized, but wavelets have an additional time-localization property). Because of this, many types of signals in practice may be non-sparse in the Fourier domain, but very sparse in the wavelet domain. This is particularly useful in signal reconstruction, especially in the recently popular field of <a href="/wiki/Compressed_sensing" title="Compressed sensing">compressed sensing</a>. (Note that the <a href="/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a> (STFT) is also localized in time and frequency, but there are often problems with the frequency-time resolution trade-off. Wavelets are better signal representations because of <a href="/wiki/Multiresolution_analysis" title="Multiresolution analysis">multiresolution analysis</a>.) </p><p>This motivates why wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional <a href="/wiki/Fourier_Transform" class="mw-redirect" title="Fourier Transform">Fourier transform</a>. Many areas of physics have seen this paradigm shift, including <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a>, <a href="/wiki/Chaos_theory" title="Chaos theory">chaos theory</a>,<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Ab_initio" title="Ab initio">ab initio</a> calculations, <a href="/wiki/Astrophysics" title="Astrophysics">astrophysics</a>, <a href="/wiki/Gravitational_wave" title="Gravitational wave">gravitational wave</a> transient data analysis,<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Density_matrix" title="Density matrix">density-matrix</a> localisation, <a href="/wiki/Seismology" title="Seismology">seismology</a>, <a href="/wiki/Optics" title="Optics">optics</a>, <a href="/wiki/Turbulence" title="Turbulence">turbulence</a> and <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. This change has also occurred in <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, <a href="/wiki/Electroencephalography" title="Electroencephalography">EEG</a>, <a href="/wiki/Electromyography" title="Electromyography">EMG</a>,<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Electrocardiography" title="Electrocardiography">ECG</a> analyses, <a href="/wiki/Neural_oscillation" title="Neural oscillation">brain rhythms</a>, <a href="/wiki/DNA" title="DNA">DNA</a> analysis, <a href="/wiki/Protein" title="Protein">protein</a> analysis, <a href="/wiki/Climatology" title="Climatology">climatology</a>, human sexual response analysis,<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> general <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Speech_recognition" title="Speech recognition">speech recognition</a>, acoustics, vibration signals,<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, <a href="/wiki/Multifractal_analysis" class="mw-redirect" title="Multifractal analysis">multifractal analysis</a>, and <a href="/wiki/Sparse_coding" class="mw-redirect" title="Sparse coding">sparse coding</a>. In <a href="/wiki/Computer_vision" title="Computer vision">computer vision</a> and <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, the notion of <a href="/wiki/Scale_space" title="Scale space">scale space</a> representation and Gaussian derivative operators is regarded as a canonical multi-scale representation. </p> <div class="mw-heading mw-heading3"><h3 id="Wavelet_denoising">Wavelet denoising</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=19" title="Edit section: Wavelet denoising"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wavelet_denoising.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Wavelet_denoising.svg/220px-Wavelet_denoising.svg.png" decoding="async" width="220" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Wavelet_denoising.svg/330px-Wavelet_denoising.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Wavelet_denoising.svg/440px-Wavelet_denoising.svg.png 2x" data-file-width="220" data-file-height="170" /></a><figcaption>Signal denoising by wavelet transform thresholding</figcaption></figure> <p>Suppose we measure a noisy signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=s+v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>s</mi> <mo>+</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=s+v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdcf0d15efb7906c154c9381045331256f42d7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.487ex; height:2.176ex;" alt="{\displaystyle x=s+v}" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /></span> represents the signal and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" /></span> represents the noise. Assume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /></span> has a sparse representation in a certain wavelet basis, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\ \sim \ {\mathcal {N}}(0,\,\sigma ^{2}I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mtext> </mtext> <mo>∼</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\ \sim \ {\mathcal {N}}(0,\,\sigma ^{2}I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063c25269473aca08aef4cabad52e6db2ddad45e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.612ex; height:3.176ex;" alt="{\displaystyle v\ \sim \ {\mathcal {N}}(0,\,\sigma ^{2}I)}" /></span> </p><p>Let the wavelet transform of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span> be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=W^{T}x=W^{T}s+W^{T}v=p+z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>x</mi> <mo>=</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>s</mi> <mo>+</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>v</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=W^{T}x=W^{T}s+W^{T}v=p+z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874b4f46c1bf435f84e30aca4a139026e562d41b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.628ex; height:3.009ex;" alt="{\displaystyle y=W^{T}x=W^{T}s+W^{T}v=p+z}" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=W^{T}s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=W^{T}s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7e393ecbb8b148513368964f74446f5a034808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.345ex; height:3.009ex;" alt="{\displaystyle p=W^{T}s}" /></span> is the wavelet transform of the signal component and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=W^{T}v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=W^{T}v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765701eac628252f4c7c54d66cd608031e657e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.211ex; height:2.676ex;" alt="{\displaystyle z=W^{T}v}" /></span> is the wavelet transform of the noise component. </p><p>Most elements in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> are 0 or close to 0, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\ \sim \ \ {\mathcal {N}}(0,\,\sigma ^{2}I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mtext> </mtext> <mo>∼</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\ \sim \ \ {\mathcal {N}}(0,\,\sigma ^{2}I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adebd614784b55a2bd0287a9c0c1113430f976e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.153ex; height:3.176ex;" alt="{\displaystyle z\ \sim \ \ {\mathcal {N}}(0,\,\sigma ^{2}I)}" /></span> </p><p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" /></span> is orthogonal, the estimation problem amounts to recovery of a signal in iid <a href="/wiki/Gaussian_noise" title="Gaussian noise">Gaussian noise</a>. As <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> is sparse, one method is to apply a Gaussian mixture model for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span>. </p><p>Assume a prior <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\ \sim \ a{\mathcal {N}}(0,\,\sigma _{1}^{2})+(1-a){\mathcal {N}}(0,\,\sigma _{2}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mtext> </mtext> <mo>∼</mo> <mtext> </mtext> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\ \sim \ a{\mathcal {N}}(0,\,\sigma _{1}^{2})+(1-a){\mathcal {N}}(0,\,\sigma _{2}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a51ec67ad38211d4b93434cd617afef53cafe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:34.736ex; height:3.176ex;" alt="{\displaystyle p\ \sim \ a{\mathcal {N}}(0,\,\sigma _{1}^{2})+(1-a){\mathcal {N}}(0,\,\sigma _{2}^{2})}" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc9bcce2153e3d171c8c1d3c09ba860513e8dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{1}^{2}}" /></span> is the variance of "significant" coefficients and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c44f77a3e598e15f435073db206422e8cd33a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{2}^{2}}" /></span> is the variance of "insignificant" coefficients. </p><p>Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {p}}=E(p/y)=\tau (y)y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {p}}=E(p/y)=\tau (y)y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f0cd968e770137b0a4e0f138a5f7d19fc665f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.04ex; height:2.843ex;" alt="{\displaystyle {\tilde {p}}=E(p/y)=\tau (y)y}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27990713741936f1ba1dc5e7b7ee768a856471a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.167ex; height:2.843ex;" alt="{\displaystyle \tau (y)}" /></span> is called the shrinkage factor, which depends on the prior variances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc9bcce2153e3d171c8c1d3c09ba860513e8dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{1}^{2}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c44f77a3e598e15f435073db206422e8cd33a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{2}^{2}}" /></span>. By setting coefficients that fall below a shrinkage threshold to zero, once the inverse transform is applied, an expectedly small amount of signal is lost due to the sparsity assumption. The larger coefficients are expected to primarily represent signal due to sparsity, and statistically very little of the signal, albeit the majority of the noise, is expected to be represented in such lower magnitude coefficients... therefore the zeroing-out operation is expected to remove most of the noise and not much signal. Typically, the above-threshold coefficients are not modified during this process. Some algorithms for wavelet-based denoising may attenuate larger coefficients as well, based on a statistical estimate of the amount of noise expected to be removed by such an attenuation. </p><p>At last, apply the inverse wavelet transform to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {s}}=W{\tilde {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {s}}=W{\tilde {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7618b5544a5d4c2ee9099f08fee9743f7d6c954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.185ex; height:2.509ex;" alt="{\displaystyle {\tilde {s}}=W{\tilde {p}}}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Multiscale_climate_network">Multiscale climate network</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=20" title="Edit section: Multiscale climate network"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Agarwal et al. proposed wavelet based advanced linear <sup id="cite_ref-AgarwalMaheswaranMarwan2018_42-0" class="reference"><a href="#cite_note-AgarwalMaheswaranMarwan2018-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> and nonlinear <sup id="cite_ref-AgarwalMarwanRathinasamy2017_43-0" class="reference"><a href="#cite_note-AgarwalMarwanRathinasamy2017-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> methods to construct and investigate <a href="/wiki/Climate_as_complex_networks" title="Climate as complex networks">Climate as complex networks</a> at different timescales. Climate networks constructed using <a href="/wiki/Sea_surface_temperature" title="Sea surface temperature">SST</a> datasets at different timescale averred that wavelet based multi-scale analysis of climatic processes holds the promise of better understanding the system dynamics that may be missed when processes are analyzed at one timescale only <sup id="cite_ref-AgarwalCaesarMarwan2019_44-0" class="reference"><a href="#cite_note-AgarwalCaesarMarwan2019-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="List_of_wavelets">List of wavelets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=21" title="Edit section: List of wavelets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Discrete_wavelets">Discrete wavelets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=22" title="Edit section: Discrete wavelets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Beylkin&action=edit&redlink=1" class="new" title="Beylkin (page does not exist)">Beylkin</a> (18)</li> <li>Moore Wavelet <a href="/wiki/Morlet_wavelet" title="Morlet wavelet">Morlet wavelet</a></li> <li><a href="/wiki/Biorthogonal_nearly_coiflet_basis" title="Biorthogonal nearly coiflet basis">Biorthogonal nearly coiflet (BNC) wavelets</a></li> <li><a href="/wiki/Coiflet" title="Coiflet">Coiflet</a> (6, 12, 18, 24, 30)</li> <li><a href="/wiki/Cohen-Daubechies-Feauveau_wavelet" class="mw-redirect" title="Cohen-Daubechies-Feauveau wavelet">Cohen-Daubechies-Feauveau wavelet</a> (Sometimes referred to as CDF N/P or Daubechies biorthogonal wavelets)</li> <li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies wavelet</a> (2, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.)</li> <li><a href="/wiki/Binomial_QMF" title="Binomial QMF">Binomial QMF</a> (Also referred to as Daubechies wavelet)</li> <li><a href="/wiki/Haar_wavelet" title="Haar wavelet">Haar wavelet</a></li> <li><a href="/wiki/Mathieu_wavelet" title="Mathieu wavelet">Mathieu wavelet</a></li> <li><a href="/wiki/Legendre_wavelet" title="Legendre wavelet">Legendre wavelet</a></li> <li><a href="/w/index.php?title=Villasenor_wavelet&action=edit&redlink=1" class="new" title="Villasenor wavelet (page does not exist)">Villasenor wavelet</a></li> <li><a href="/wiki/Symlet" title="Symlet">Symlet</a><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Continuous_wavelets">Continuous wavelets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=23" title="Edit section: Continuous wavelets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Real-valued">Real-valued</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=24" title="Edit section: Real-valued"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Beta_wavelet" title="Beta wavelet">Beta wavelet</a></li> <li><a href="/wiki/Hermitian_wavelet" title="Hermitian wavelet">Hermitian wavelet</a></li> <li><a href="/wiki/Meyer_wavelet" title="Meyer wavelet">Meyer wavelet</a></li> <li><a href="/wiki/Mexican_hat_wavelet" class="mw-redirect" title="Mexican hat wavelet">Mexican hat wavelet</a></li> <li><a href="/wiki/Poisson_wavelet" title="Poisson wavelet">Poisson wavelet</a></li> <li><a href="/wiki/Shannon_wavelet" title="Shannon wavelet">Shannon wavelet</a></li> <li><a href="/wiki/Spline_wavelet" title="Spline wavelet">Spline wavelet</a></li> <li><a href="/wiki/Str%C3%B6mberg_wavelet" title="Strömberg wavelet">Strömberg wavelet</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Complex-valued">Complex-valued</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=25" title="Edit section: Complex-valued"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Complex_Mexican_hat_wavelet" title="Complex Mexican hat wavelet">Complex Mexican hat wavelet</a></li> <li><a href="/wiki/Fbsp_wavelet" title="Fbsp wavelet">fbsp wavelet</a></li> <li><a href="/wiki/Morlet_wavelet" title="Morlet wavelet">Morlet wavelet</a></li> <li><a href="/wiki/Shannon_wavelet" title="Shannon wavelet">Shannon wavelet</a></li> <li><a href="/wiki/Modified_Morlet_wavelet" title="Modified Morlet wavelet">Modified Morlet wavelet</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=26" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Chirplet_transform" title="Chirplet transform">Chirplet transform</a></li> <li><a href="/wiki/Curvelet" title="Curvelet">Curvelet</a></li> <li><a href="/wiki/Digital_cinema" title="Digital cinema">Digital cinema</a></li> <li><a href="/wiki/Dimension_reduction" class="mw-redirect" title="Dimension reduction">Dimension reduction</a></li> <li><a href="/wiki/Filter_bank" title="Filter bank">Filter banks</a></li> <li><a href="/wiki/Fourier-related_transforms" class="mw-redirect" title="Fourier-related transforms">Fourier-related transforms</a></li> <li><a href="/wiki/Fractal_compression" title="Fractal compression">Fractal compression</a></li> <li><a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">Fractional Fourier transform</a></li> <li><a href="/wiki/Gabor_wavelet#Wavelet_space" title="Gabor wavelet">Gabor wavelet § Wavelet space</a><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Huygens%E2%80%93Fresnel_principle" title="Huygens–Fresnel principle">Huygens–Fresnel principle</a> (physical wavelets)</li> <li><a href="/wiki/JPEG_2000" title="JPEG 2000">JPEG 2000</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a> for computing periodicity in any including unevenly spaced data</li> <li><a href="/wiki/Morlet_wavelet" title="Morlet wavelet">Morlet wavelet</a></li> <li><a href="/wiki/Multiresolution_analysis" title="Multiresolution analysis">Multiresolution analysis</a></li> <li><a href="/wiki/Noiselet" title="Noiselet">Noiselet</a></li> <li><a href="/wiki/Non-separable_wavelet" title="Non-separable wavelet">Non-separable wavelet</a></li> <li><a href="/wiki/Scale_space" title="Scale space">Scale space</a></li> <li><a href="/wiki/Scaled_correlation" title="Scaled correlation">Scaled correlation</a></li> <li><a href="/wiki/Shearlet" title="Shearlet">Shearlet</a></li> <li><a href="/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">Short-time Fourier transform</a></li> <li><a href="/wiki/Spectrogram" title="Spectrogram">Spectrogram</a></li> <li><a href="/wiki/Ultra_wideband" class="mw-redirect" title="Ultra wideband">Ultra wideband</a> radio – transmits wavelets</li> <li><a href="/wiki/Wavelet_for_multidimensional_signals_analysis" title="Wavelet for multidimensional signals analysis">Wavelet for multidimensional signals analysis</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=27" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Wireless Communications: Principles and Practice, Prentice Hall communications engineering and emerging technologies series, T. S. Rappaport, Prentice Hall, 2002, p. 126.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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="CITEREFRicker1953" class="citation journal cs1">Ricker, Norman (1953). "Wavelet Contraction, Wavelet Expansion, and the Control of Seismic Resolution". <i>Geophysics</i>. <b>18</b> (4): <span class="nowrap">769–</span>792. <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/1953Geop...18..769R">1953Geop...18..769R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1190%2F1.1437927">10.1190/1.1437927</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geophysics&rft.atitle=Wavelet+Contraction%2C+Wavelet+Expansion%2C+and+the+Control+of+Seismic+Resolution&rft.volume=18&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E769-%3C%2Fspan%3E792&rft.date=1953&rft_id=info%3Adoi%2F10.1190%2F1.1437927&rft_id=info%3Abibcode%2F1953Geop...18..769R&rft.aulast=Ricker&rft.aufirst=Norman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://taylorandfrancis.com/knowledge/Engineering_and_technology/Engineering_support_and_special_topics/Continuous_wavelet_transform/#:~:text=The%20wavelet%20theory%20applies%20to,banks%20of%20filters%20in%20time.">"Continuous wavelet transform - Knowledge and References"</a>. <i>Taylor & Francis</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Taylor+%26+Francis&rft.atitle=Continuous+wavelet+transform+-+Knowledge+and+References&rft_id=https%3A%2F%2Ftaylorandfrancis.com%2Fknowledge%2FEngineering_and_technology%2FEngineering_support_and_special_topics%2FContinuous_wavelet_transform%2F%23%3A~%3Atext%3DThe%2520wavelet%2520theory%2520applies%2520to%2Cbanks%2520of%2520filters%2520in%2520time.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://taylorandfrancis.com/knowledge/Engineering_and_technology/Engineering_support_and_special_topics/Continuous_wavelet_transform/">"Continuous wavelet transform - Knowledge and References"</a>. <i>Taylor & Francis</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Taylor+%26+Francis&rft.atitle=Continuous+wavelet+transform+-+Knowledge+and+References&rft_id=https%3A%2F%2Ftaylorandfrancis.com%2Fknowledge%2FEngineering_and_technology%2FEngineering_support_and_special_topics%2FContinuous_wavelet_transform%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Meyer, Yves (1992), Wavelets and Operators, Cambridge, UK: Cambridge University Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-42000-8" title="Special:BookSources/0-521-42000-8">0-521-42000-8</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Chui, Charles K. (1992), An Introduction to Wavelets, San Diego, CA: Academic Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-174584-8" title="Special:BookSources/0-12-174584-8">0-12-174584-8</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Daubechies, Ingrid. (1992), Ten Lectures on Wavelets, SIAM, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-274-2" title="Special:BookSources/978-0-89871-274-2">978-0-89871-274-2</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Akansu, Ali N.; Haddad, Richard A. (1992), Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets, Boston, MA: Academic Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-047141-6" title="Special:BookSources/978-0-12-047141-6">978-0-12-047141-6</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLarson2007" class="citation cs2">Larson, David R. (2007), <i>Wavelet Analysis and Applications (See: Unitary systems and wavelet sets)</i>, Appl. Numer. Harmon. Anal., Birkhäuser, pp. <span class="nowrap">143–</span>171</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wavelet+Analysis+and+Applications+%28See%3A+Unitary+systems+and+wavelet+sets%29&rft.series=Appl.+Numer.+Harmon.+Anal.&rft.pages=%3Cspan+class%3D%22nowrap%22%3E143-%3C%2Fspan%3E171&rft.pub=Birkh%C3%A4user&rft.date=2007&rft.aulast=Larson&rft.aufirst=David+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSzuTelferLohmann1992" class="citation journal cs1">Szu, Harold H.; Telfer, Brian A.; Lohmann, Adolf W. (1992). "Causal analytical wavelet transform". <i>Optical Engineering</i>. <b>31</b> (9): 1825. <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/1992OptEn..31.1825S">1992OptEn..31.1825S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1117%2F12.59911">10.1117/12.59911</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Optical+Engineering&rft.atitle=Causal+analytical+wavelet+transform&rft.volume=31&rft.issue=9&rft.pages=1825&rft.date=1992&rft_id=info%3Adoi%2F10.1117%2F12.59911&rft_id=info%3Abibcode%2F1992OptEn..31.1825S&rft.aulast=Szu&rft.aufirst=Harold+H.&rft.au=Telfer%2C+Brian+A.&rft.au=Lohmann%2C+Adolf+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLindeberg2023" class="citation journal cs1">Lindeberg, T. (23 January 2023). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10160219">"A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time"</a>. <i>Biological Cybernetics</i>. <b>117</b> (<span class="nowrap">1–</span>2): <span class="nowrap">21–</span>59. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00422-022-00953-6">10.1007/s00422-022-00953-6</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10160219">10160219</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/36689001">36689001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biological+Cybernetics&rft.atitle=A+time-causal+and+time-recursive+scale-covariant+scale-space+representation+of+temporal+signals+and+past+time&rft.volume=117&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E21-%3C%2Fspan%3E59&rft.date=2023-01-23&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10160219%23id-name%3DPMC&rft_id=info%3Apmid%2F36689001&rft_id=info%3Adoi%2F10.1007%2Fs00422-022-00953-6&rft.aulast=Lindeberg&rft.aufirst=T.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10160219&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEMallat2009Chpt._7-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMallat2009Chpt._7_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMallat2009">Mallat 2009</a>, Chpt. 7.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">The Scientist and Engineer's Guide to Digital Signal Processing By Steven W. Smith, Ph.D. chapter 8 equation 8-1: <a rel="nofollow" class="external free" href="http://www.dspguide.com/ch8/4.htm">http://www.dspguide.com/ch8/4.htm</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHainesJones1988" class="citation journal cs1">Haines, VG. V.; Jones, Alan G. (1988). <a rel="nofollow" class="external text" href="http://homepages.dias.ie/~ajones/publications/28.pdf">"Logarithmic Fourier Transform"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Geophysical_Journal" class="mw-redirect" title="Geophysical Journal">Geophysical Journal</a></i> (92): <span class="nowrap">171–</span>178. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-246X.1988.tb01131.x">10.1111/j.1365-246X.1988.tb01131.x</a></span>. <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:9720759">9720759</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geophysical+Journal&rft.atitle=Logarithmic+Fourier+Transform&rft.issue=92&rft.pages=%3Cspan+class%3D%22nowrap%22%3E171-%3C%2Fspan%3E178&rft.date=1988&rft_id=info%3Adoi%2F10.1111%2Fj.1365-246X.1988.tb01131.x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9720759%23id-name%3DS2CID&rft.aulast=Haines&rft.aufirst=VG.+V.&rft.au=Jones%2C+Alan+G.&rft_id=http%3A%2F%2Fhomepages.dias.ie%2F~ajones%2Fpublications%2F28.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.polyvalens.com/blog/?page_id=15#7.+The+scaling+function+%5B7%5D">"A Really Friendly Guide To Wavelets – PolyValens"</a>. <i>www.polyvalens.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.polyvalens.com&rft.atitle=A+Really+Friendly+Guide+To+Wavelets+%E2%80%93+PolyValens&rft_id=http%3A%2F%2Fwww.polyvalens.com%2Fblog%2F%3Fpage_id%3D15%237.%2BThe%2Bscaling%2Bfunction%2B%255B7%255D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://scienceworld.wolfram.com/biography/Zweig.html">"Zweig, George -- from Eric Weisstein's World of Scientific Biography"</a>. <i>scienceworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-10-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=scienceworld.wolfram.com&rft.atitle=Zweig%2C+George+--+from+Eric+Weisstein%27s+World+of+Scientific+Biography&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fscienceworld.wolfram.com%2Fbiography%2FZweig.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSullivan2003" class="citation web cs1">Sullivan, Gary (8–12 December 2003). <a rel="nofollow" class="external text" href="https://www.itu.int/wftp3/av-arch/video-site/0312_Wai/VCEG-U06.doc">"General characteristics and design considerations for temporal subband video coding"</a>. <i><a href="/wiki/ITU-T" title="ITU-T">ITU-T</a></i>. <a href="/wiki/Video_Coding_Experts_Group" title="Video Coding Experts Group">Video Coding Experts Group</a><span class="reference-accessdate">. Retrieved <span class="nowrap">13 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ITU-T&rft.atitle=General+characteristics+and+design+considerations+for+temporal+subband+video+coding&rft.date=2003-12-08%2F2003-12-12&rft.aulast=Sullivan&rft.aufirst=Gary&rft_id=https%3A%2F%2Fwww.itu.int%2Fwftp3%2Fav-arch%2Fvideo-site%2F0312_Wai%2FVCEG-U06.doc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBovik2009" class="citation book cs1">Bovik, Alan C. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wXmSPPB_c_0C&pg=PA355"><i>The Essential Guide to Video Processing</i></a>. <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. p. 355. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780080922508" title="Special:BookSources/9780080922508"><bdi>9780080922508</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Essential+Guide+to+Video+Processing&rft.pages=355&rft.pub=Academic+Press&rft.date=2009&rft.isbn=9780080922508&rft.aulast=Bovik&rft.aufirst=Alan+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwXmSPPB_c_0C%26pg%3DPA355&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGallTabatabai1988" class="citation book cs1">Gall, Didier Le; Tabatabai, Ali J. (1988). "Sub-band coding of digital images using symmetric short kernel filters and arithmetic coding techniques". <i>ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing</i>. pp. 761–764 vol.2. <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%2FICASSP.1988.196696">10.1109/ICASSP.1988.196696</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:109186495">109186495</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Sub-band+coding+of+digital+images+using+symmetric+short+kernel+filters+and+arithmetic+coding+techniques&rft.btitle=ICASSP-88.%2C+International+Conference+on+Acoustics%2C+Speech%2C+and+Signal+Processing&rft.pages=761-764+vol.2&rft.date=1988&rft_id=info%3Adoi%2F10.1109%2FICASSP.1988.196696&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A109186495%23id-name%3DS2CID&rft.aulast=Gall&rft.aufirst=Didier+Le&rft.au=Tabatabai%2C+Ali+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-Said-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Said_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSaidPearlman1996" class="citation journal cs1">Said, Amir; Pearlman, William A. (June 1996). "A new fast and efficient image codec based on set partitioning in hierarchical trees". <i>IEEE Transactions on Circuits and Systems for Video Technology</i>. <b>6</b> (3): <span class="nowrap">243–</span>250. <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%2F76.499834">10.1109/76.499834</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1051-8215">1051-8215</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+Circuits+and+Systems+for+Video+Technology&rft.atitle=A+new+fast+and+efficient+image+codec+based+on+set+partitioning+in+hierarchical+trees&rft.volume=6&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E243-%3C%2Fspan%3E250&rft.date=1996-06&rft_id=info%3Adoi%2F10.1109%2F76.499834&rft.issn=1051-8215&rft.aulast=Said&rft.aufirst=Amir&rft.au=Pearlman%2C+William+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTaubmanMarcellin2012" class="citation book cs1">Taubman, David; Marcellin, Michael (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=y7HeBwAAQBAJ&pg=PA402"><i>JPEG2000 Image Compression Fundamentals, Standards and Practice: Image Compression Fundamentals, Standards and Practice</i></a>. <a href="/wiki/Springer_Science_%26_Business_Media" class="mw-redirect" title="Springer Science & Business Media">Springer Science & Business Media</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781461507994" title="Special:BookSources/9781461507994"><bdi>9781461507994</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=JPEG2000+Image+Compression+Fundamentals%2C+Standards+and+Practice%3A+Image+Compression+Fundamentals%2C+Standards+and+Practice&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=9781461507994&rft.aulast=Taubman&rft.aufirst=David&rft.au=Marcellin%2C+Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dy7HeBwAAQBAJ%26pg%3DPA402&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-Unser-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Unser_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFUnserBlu2003" class="citation journal cs1">Unser, M.; Blu, T. (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191013222932/https://pdfs.semanticscholar.org/6ed4/dece8b364416d9c390ba53df913bca7fb9a6.pdf">"Mathematical properties of the JPEG2000 wavelet filters"</a> <span class="cs1-format">(PDF)</span>. <i>IEEE Transactions on Image Processing</i>. <b>12</b> (9): <span class="nowrap">1080–</span>1090. <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/2003ITIP...12.1080U">2003ITIP...12.1080U</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%2FTIP.2003.812329">10.1109/TIP.2003.812329</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18237979">18237979</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:2765169">2765169</a>. Archived from <a rel="nofollow" class="external text" href="https://pdfs.semanticscholar.org/6ed4/dece8b364416d9c390ba53df913bca7fb9a6.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2019-10-13.</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+Image+Processing&rft.atitle=Mathematical+properties+of+the+JPEG2000+wavelet+filters&rft.volume=12&rft.issue=9&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1080-%3C%2Fspan%3E1090&rft.date=2003&rft_id=info%3Adoi%2F10.1109%2FTIP.2003.812329&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2765169%23id-name%3DS2CID&rft_id=info%3Apmid%2F18237979&rft_id=info%3Abibcode%2F2003ITIP...12.1080U&rft.aulast=Unser&rft.aufirst=M.&rft.au=Blu%2C+T.&rft_id=https%3A%2F%2Fpdfs.semanticscholar.org%2F6ed4%2Fdece8b364416d9c390ba53df913bca7fb9a6.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSwartz2005" class="citation book cs1">Swartz, Charles S. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tYw3ehoBnjkC&pg=PA147"><i>Understanding Digital Cinema: A Professional Handbook</i></a>. <a href="/wiki/Taylor_%26_Francis" title="Taylor & Francis">Taylor & Francis</a>. p. 147. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780240806174" title="Special:BookSources/9780240806174"><bdi>9780240806174</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Digital+Cinema%3A+A+Professional+Handbook&rft.pages=147&rft.pub=Taylor+%26+Francis&rft.date=2005&rft.isbn=9780240806174&rft.aulast=Swartz&rft.aufirst=Charles+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtYw3ehoBnjkC%26pg%3DPA147&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">P. Hirsch, A. Howie, R. Nicholson, D. W. Pashley and M. J. Whelan (1965/1977) <i>Electron microscopy of thin crystals</i> (Butterworths, London/Krieger, Malabar FLA) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88275-376-2" title="Special:BookSources/0-88275-376-2">0-88275-376-2</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">P. Fraundorf, J. Wang, E. Mandell and M. Rose (2006) Digital darkfield tableaus, <i>Microscopy and Microanalysis</i> <b>12</b>:S2, 1010–1011 (cf. <a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0403017">arXiv:cond-mat/0403017</a>)</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHÿtchSnoeckKilaas1998" class="citation journal cs1">Hÿtch, M. J.; Snoeck, E.; Kilaas, R. (1998). "Quantitative measurement of displacement and strain fields from HRTEM micrographs". <i>Ultramicroscopy</i>. <b>74</b> (3): <span class="nowrap">131–</span>146. <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%2Fs0304-3991%2898%2900035-7">10.1016/s0304-3991(98)00035-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ultramicroscopy&rft.atitle=Quantitative+measurement+of+displacement+and+strain+fields+from+HRTEM+micrographs&rft.volume=74&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E131-%3C%2Fspan%3E146&rft.date=1998&rft_id=info%3Adoi%2F10.1016%2Fs0304-3991%2898%2900035-7&rft.aulast=H%C3%BFtch&rft.aufirst=M.+J.&rft.au=Snoeck%2C+E.&rft.au=Kilaas%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Martin Rose (2006) <i>Spacing measurements of lattice fringes in HRTEM image using digital darkfield decomposition</i> (M.S. Thesis in Physics, U. Missouri – St. Louis)</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">F. G. Meyer and R. R. Coifman (1997) <i><a href="/wiki/Applied_and_Computational_Harmonic_Analysis" title="Applied and Computational Harmonic Analysis">Applied and Computational Harmonic Analysis</a></i> <b>4</b>:147.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">A. G. Flesia, <a href="/wiki/Hagit_Hel-Or" title="Hagit Hel-Or">H. Hel-Or</a>, A. Averbuch, <a href="/wiki/Emmanuel_Cand%C3%A8s" title="Emmanuel Candès">E. J. Candes</a>, R. R. Coifman and <a href="/wiki/David_Donoho" title="David Donoho">D. L. Donoho</a> (2001) <i>Digital implementation of ridgelet packets</i> (Academic Press, New York).</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShiZhangLiu2011" class="citation journal cs1">Shi, J.; Zhang, N.-T.; Liu, X.-P. (2011). "A novel fractional wavelet transform and its applications". <i>Sci. China Inf. Sci</i>. <b>55</b> (6): <span class="nowrap">1270–</span>1279. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11432-011-4320-x">10.1007/s11432-011-4320-x</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:255201598">255201598</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sci.+China+Inf.+Sci.&rft.atitle=A+novel+fractional+wavelet+transform+and+its+applications&rft.volume=55&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1270-%3C%2Fspan%3E1279&rft.date=2011&rft_id=info%3Adoi%2F10.1007%2Fs11432-011-4320-x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A255201598%23id-name%3DS2CID&rft.aulast=Shi&rft.aufirst=J.&rft.au=Zhang%2C+N.-T.&rft.au=Liu%2C+X.-P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">A.N. Akansu, W.A. Serdijn and I.W. Selesnick, <a rel="nofollow" class="external text" href="http://web.njit.edu/~akansu/PAPERS/ANA-IWS-WAS-ELSEVIER%20PHYSCOM%202010.pdf">Emerging applications of wavelets: A review</a>, Physical Communication, Elsevier, vol. 3, issue 1, pp. 1-18, March 2010.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Tomás, R., Li, Z., Lopez-Sanchez, J.M., Liu, P. & Singleton, A. 2016. Using wavelet tools to analyse seasonal variations from InSAR time-series data: a case study of the Huangtupo landslide. Landslides, 13, 437-450, doi: 10.1007/s10346-015-0589-y.</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLyakhovSemyonovaNagornovBergerman2023" class="citation journal cs1">Lyakhov, Pavel; Semyonova, Nataliya; Nagornov, Nikolay; Bergerman, Maxim; Abdulsalyamova, Albina (2023-11-14). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fmath11224644">"High-Speed Wavelet Image Processing Using the Winograd Method with Downsampling"</a>. <i>Mathematics</i>. <b>11</b> (22): 4644. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fmath11224644">10.3390/math11224644</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2227-7390">2227-7390</a>. <q>Wavelets are actively used to solve a wide range of image processing problems in various fields of science and technology, e.g., image denoising, reconstruction, analysis, and video analysis and processing. Wavelet processing methods are based on the discrete wavelet transform using 1D digital filtering.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics&rft.atitle=High-Speed+Wavelet+Image+Processing+Using+the+Winograd+Method+with+Downsampling&rft.volume=11&rft.issue=22&rft.pages=4644&rft.date=2023-11-14&rft_id=info%3Adoi%2F10.3390%2Fmath11224644&rft.issn=2227-7390&rft.aulast=Lyakhov&rft.aufirst=Pavel&rft.au=Semyonova%2C+Nataliya&rft.au=Nagornov%2C+Nikolay&rft.au=Bergerman%2C+Maxim&rft.au=Abdulsalyamova%2C+Albina&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fmath11224644&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDongZhangGanQiu2023" class="citation journal cs1">Dong, Liang; Zhang, Shaohua; Gan, Tiansiyu; Qiu, Yan; Song, Qinfeng; Zhao, Yongtao (2023-12-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0950061823031707">"Frequency characteristics analysis of pipe-to-soil potential under metro stray current interference using continuous wavelet transform method"</a>. <i>Construction and Building Materials</i>. <b>407</b>: 133453. <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.conbuildmat.2023.133453">10.1016/j.conbuildmat.2023.133453</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0950-0618">0950-0618</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:263317973">263317973</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Construction+and+Building+Materials&rft.atitle=Frequency+characteristics+analysis+of+pipe-to-soil+potential+under+metro+stray+current+interference+using+continuous+wavelet+transform+method&rft.volume=407&rft.pages=133453&rft.date=2023-12-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A263317973%23id-name%3DS2CID&rft.issn=0950-0618&rft_id=info%3Adoi%2F10.1016%2Fj.conbuildmat.2023.133453&rft.aulast=Dong&rft.aufirst=Liang&rft.au=Zhang%2C+Shaohua&rft.au=Gan%2C+Tiansiyu&rft.au=Qiu%2C+Yan&rft.au=Song%2C+Qinfeng&rft.au=Zhao%2C+Yongtao&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0950061823031707&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-galli-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-galli_35-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStefano_GalliO._Logvinov2008" class="citation journal cs1">Stefano Galli; O. Logvinov (July 2008). "Recent Developments in the Standardization of Power Line Communications within the IEEE". <i>IEEE Communications Magazine</i>. <b>46</b> (7): <span class="nowrap">64–</span>71. <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%2FMCOM.2008.4557044">10.1109/MCOM.2008.4557044</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:2650873">2650873</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Communications+Magazine&rft.atitle=Recent+Developments+in+the+Standardization+of+Power+Line+Communications+within+the+IEEE&rft.volume=46&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E64-%3C%2Fspan%3E71&rft.date=2008-07&rft_id=info%3Adoi%2F10.1109%2FMCOM.2008.4557044&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2650873%23id-name%3DS2CID&rft.au=Stefano+Galli&rft.au=O.+Logvinov&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span> An overview of P1901 PHY/MAC proposal.</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWotherspoonet.2009" class="citation journal cs1">Wotherspoon, T.; et., al. (2009). "Adaptation to the edge of chaos with random-wavelet feedback". <i>J. Phys. Chem</i>. <b>113</b> (1): <span class="nowrap">19–</span>22. <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/2009JPCA..113...19W">2009JPCA..113...19W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fjp804420g">10.1021/jp804420g</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19072712">19072712</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Phys.+Chem.&rft.atitle=Adaptation+to+the+edge+of+chaos+with+random-wavelet+feedback.&rft.volume=113&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E19-%3C%2Fspan%3E22&rft.date=2009&rft_id=info%3Apmid%2F19072712&rft_id=info%3Adoi%2F10.1021%2Fjp804420g&rft_id=info%3Abibcode%2F2009JPCA..113...19W&rft.aulast=Wotherspoon&rft.aufirst=T.&rft.au=et.%2C+al.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAbbott2016" class="citation journal cs1">Abbott, Benjamin P.; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Observing gravitational-wave transient GW150914 with minimal assumptions". <i><a href="/wiki/Phys._Rev._D" class="mw-redirect" title="Phys. Rev. D">Phys. Rev. D</a></i>. <b>93</b> (12): 122004. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1602.03843">1602.03843</a></span>. <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/2016PhRvD..93l2004A">2016PhRvD..93l2004A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.93.122004">10.1103/PhysRevD.93.122004</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:119313566">119313566</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+D&rft.atitle=Observing+gravitational-wave+transient+GW150914+with+minimal+assumptions&rft.volume=93&rft.issue=12&rft.pages=122004&rft.date=2016&rft_id=info%3Aarxiv%2F1602.03843&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119313566%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.93.122004&rft_id=info%3Abibcode%2F2016PhRvD..93l2004A&rft.aulast=Abbott&rft.aufirst=Benjamin+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFV_Necula,_S_Klimenko_and_G_Mitselmakher2012" class="citation journal cs1">V Necula, S Klimenko and G Mitselmakher (2012). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-6596%2F363%2F1%2F012032">"Transient analysis with fast Wilson-Daubechies time-frequency transform"</a>. <i>Journal of Physics: Conference Series</i>. <b>363</b> (1): 012032. <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/2012JPhCS.363a2032N">2012JPhCS.363a2032N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-6596%2F363%2F1%2F012032">10.1088/1742-6596/363/1/012032</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics%3A+Conference+Series&rft.atitle=Transient+analysis+with+fast+Wilson-Daubechies+time-frequency+transform&rft.volume=363&rft.issue=1&rft.pages=012032&rft.date=2012&rft_id=info%3Adoi%2F10.1088%2F1742-6596%2F363%2F1%2F012032&rft_id=info%3Abibcode%2F2012JPhCS.363a2032N&rft.au=V+Necula%2C+S+Klimenko+and+G+Mitselmakher&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F1742-6596%252F363%252F1%252F012032&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">J. Rafiee et al. Feature extraction of forearm EMG signals for prosthetics, Expert Systems with Applications 38 (2011) 4058–67.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">J. Rafiee et al. Female sexual responses using signal processing techniques, The Journal of Sexual Medicine 6 (2009) 3086–96. <a rel="nofollow" class="external text" href="http://rafiee.us/files/JSM_2009.pdf">(pdf)</a></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRafieeTse2009" class="citation journal cs1">Rafiee, J.; Tse, Peter W. (2009). "Use of autocorrelation in wavelet coefficients for fault diagnosis". <i>Mechanical Systems and Signal Processing</i>. <b>23</b> (5): <span class="nowrap">1554–</span>72. <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/2009MSSP...23.1554R">2009MSSP...23.1554R</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.ymssp.2009.02.008">10.1016/j.ymssp.2009.02.008</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mechanical+Systems+and+Signal+Processing&rft.atitle=Use+of+autocorrelation+in+wavelet+coefficients+for+fault+diagnosis&rft.volume=23&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1554-%3C%2Fspan%3E72&rft.date=2009&rft_id=info%3Adoi%2F10.1016%2Fj.ymssp.2009.02.008&rft_id=info%3Abibcode%2F2009MSSP...23.1554R&rft.aulast=Rafiee&rft.aufirst=J.&rft.au=Tse%2C+Peter+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-AgarwalMaheswaranMarwan2018-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-AgarwalMaheswaranMarwan2018_42-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAgarwalMaheswaranMarwanCaesar2018" class="citation journal cs1">Agarwal, Ankit; Maheswaran, Rathinasamy; Marwan, Norbert; Caesar, Levke; Kurths, Jürgen (November 2018). <a rel="nofollow" class="external text" href="http://mural.maynoothuniversity.ie/13175/1/CL_geography_wavelet-based.pdf">"Wavelet-based multiscale similarity measure for complex networks"</a> <span class="cs1-format">(PDF)</span>. <i>The European Physical Journal B</i>. <b>91</b> (11): 296. <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/2018EPJB...91..296A">2018EPJB...91..296A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1140%2Fepjb%2Fe2018-90460-6">10.1140/epjb/e2018-90460-6</a>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1434-6036">1434-6036</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1434-6028">1434-6028</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:125557123">125557123</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+European+Physical+Journal+B&rft.atitle=Wavelet-based+multiscale+similarity+measure+for+complex+networks&rft.volume=91&rft.issue=11&rft.pages=296&rft.date=2018-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125557123%23id-name%3DS2CID&rft.eissn=1434-6036&rft_id=info%3Abibcode%2F2018EPJB...91..296A&rft.issn=1434-6028&rft_id=info%3Adoi%2F10.1140%2Fepjb%2Fe2018-90460-6&rft.aulast=Agarwal&rft.aufirst=Ankit&rft.au=Maheswaran%2C+Rathinasamy&rft.au=Marwan%2C+Norbert&rft.au=Caesar%2C+Levke&rft.au=Kurths%2C+J%C3%BCrgen&rft_id=http%3A%2F%2Fmural.maynoothuniversity.ie%2F13175%2F1%2FCL_geography_wavelet-based.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-AgarwalMarwanRathinasamy2017-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-AgarwalMarwanRathinasamy2017_43-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAgarwalMarwanRathinasamyMerz2017" class="citation journal cs1">Agarwal, Ankit; Marwan, Norbert; Rathinasamy, Maheswaran; Merz, Bruno; Kurths, Jürgen (13 October 2017). <a rel="nofollow" class="external text" href="https://doi.org/10.5194%2Fnpg-24-599-2017">"Multi-scale event synchronization analysis for unravelling climate processes: a wavelet-based approach"</a>. <i>Nonlinear Processes in Geophysics</i>. <b>24</b> (4): <span class="nowrap">599–</span>611. <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/2017NPGeo..24..599A">2017NPGeo..24..599A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.5194%2Fnpg-24-599-2017">10.5194/npg-24-599-2017</a></span>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1607-7946">1607-7946</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:28114574">28114574</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nonlinear+Processes+in+Geophysics&rft.atitle=Multi-scale+event+synchronization+analysis+for+unravelling+climate+processes%3A+a+wavelet-based+approach&rft.volume=24&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E599-%3C%2Fspan%3E611&rft.date=2017-10-13&rft_id=info%3Adoi%2F10.5194%2Fnpg-24-599-2017&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A28114574%23id-name%3DS2CID&rft.eissn=1607-7946&rft_id=info%3Abibcode%2F2017NPGeo..24..599A&rft.aulast=Agarwal&rft.aufirst=Ankit&rft.au=Marwan%2C+Norbert&rft.au=Rathinasamy%2C+Maheswaran&rft.au=Merz%2C+Bruno&rft.au=Kurths%2C+J%C3%BCrgen&rft_id=https%3A%2F%2Fdoi.org%2F10.5194%252Fnpg-24-599-2017&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-AgarwalCaesarMarwan2019-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-AgarwalCaesarMarwan2019_44-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAgarwalCaesarMarwanMaheswaran2019" class="citation journal cs1">Agarwal, Ankit; Caesar, Levke; Marwan, Norbert; Maheswaran, Rathinasamy; Merz, Bruno; Kurths, Jürgen (19 June 2019). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6584743">"Network-based identification and characterization of teleconnections on different scales"</a>. <i>Scientific Reports</i>. <b>9</b> (1): 8808. <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/2019NatSR...9.8808A">2019NatSR...9.8808A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41598-019-45423-5">10.1038/s41598-019-45423-5</a>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2045-2322">2045-2322</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6584743">6584743</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31217490">31217490</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=Network-based+identification+and+characterization+of+teleconnections+on+different+scales&rft.volume=9&rft.issue=1&rft.pages=8808&rft.date=2019-06-19&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6584743%23id-name%3DPMC&rft.eissn=2045-2322&rft_id=info%3Abibcode%2F2019NatSR...9.8808A&rft_id=info%3Apmid%2F31217490&rft_id=info%3Adoi%2F10.1038%2Fs41598-019-45423-5&rft.aulast=Agarwal&rft.aufirst=Ankit&rft.au=Caesar%2C+Levke&rft.au=Marwan%2C+Norbert&rft.au=Maheswaran%2C+Rathinasamy&rft.au=Merz%2C+Bruno&rft.au=Kurths%2C+J%C3%BCrgen&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6584743&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><i>Matlab Toolbox</i> – URL: <a rel="nofollow" class="external free" href="http://matlab.izmiran.ru/help/toolbox/wavelet/ch06_a32.html">http://matlab.izmiran.ru/help/toolbox/wavelet/ch06_a32.html</a></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">Erik Hjelmås (1999-01-21) <i>Gabor Wavelets</i> URL: <a rel="nofollow" class="external free" href="http://www.ansatt.hig.no/erikh/papers/scia99/node6.html">http://www.ansatt.hig.no/erikh/papers/scia99/node6.html</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=28" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAddison2002" class="citation book cs1">Addison, Paul S. (2002). <i>The illustrated wavelet transform handbook: introductory theory and applications in science, engineering, medicine and finance</i>. Bristol Philadelphia: Institute of physics publ. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7503-0692-0" title="Special:BookSources/0-7503-0692-0"><bdi>0-7503-0692-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+illustrated+wavelet+transform+handbook%3A+introductory+theory+and+applications+in+science%2C+engineering%2C+medicine+and+finance&rft.place=Bristol+Philadelphia&rft.pub=Institute+of+physics+publ&rft.date=2002&rft.isbn=0-7503-0692-0&rft.aulast=Addison&rft.aufirst=Paul+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAkansuHaddad1992" class="citation book cs1"><a href="/wiki/Ali_Akansu" title="Ali Akansu">Akansu, Ali N.</a>; Haddad, Richard A. (1992). <i>Multiresolution signal decomposition: transforms, subbands, and wavelets</i>. Boston: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-047140-X" title="Special:BookSources/0-12-047140-X"><bdi>0-12-047140-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multiresolution+signal+decomposition%3A+transforms%2C+subbands%2C+and+wavelets&rft.place=Boston&rft.pub=Academic+Press&rft.date=1992&rft.isbn=0-12-047140-X&rft.aulast=Akansu&rft.aufirst=Ali+N.&rft.au=Haddad%2C+Richard+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChanShen2005" class="citation book cs1"><a href="/wiki/Tony_F._Chan" title="Tony F. Chan">Chan, Tony F.</a>; Shen, Jianhong (2005). <i>Image processing and analysis: variational, PDE, wavelet, and stochastic methods</i>. Philadelphia: Society for Industrial and Applied Mathematics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-589-X" title="Special:BookSources/0-89871-589-X"><bdi>0-89871-589-X</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/60321765">60321765</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Image+processing+and+analysis%3A+variational%2C+PDE%2C+wavelet%2C+and+stochastic+methods&rft.place=Philadelphia&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=2005&rft_id=info%3Aoclcnum%2F60321765&rft.isbn=0-89871-589-X&rft.aulast=Chan&rft.aufirst=Tony+F.&rft.au=Shen%2C+Jianhong&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDaubechies1992" class="citation book cs1"><a href="/wiki/Ingrid_Daubechies" title="Ingrid Daubechies">Daubechies, Ingrid</a> (1992). <i>Ten lectures on wavelets</i>. Philadelphia, Pa: Society for Industrial and Applied Mathematics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-274-2" title="Special:BookSources/0-89871-274-2"><bdi>0-89871-274-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ten+lectures+on+wavelets&rft.place=Philadelphia%2C+Pa&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1992&rft.isbn=0-89871-274-2&rft.aulast=Daubechies&rft.aufirst=Ingrid&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGençaySelçukWhitcher2002" class="citation book cs1">Gençay, Ramazan; Selçuk, Faruk; Whitcher, Brandon (2002). <i>An introduction to wavelets and other filtering methods in finance and economics</i>. San Diego, Calif: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-279670-5" title="Special:BookSources/0-12-279670-5"><bdi>0-12-279670-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+wavelets+and+other+filtering+methods+in+finance+and+economics&rft.place=San+Diego%2C+Calif&rft.pub=Academic+Press&rft.date=2002&rft.isbn=0-12-279670-5&rft.aulast=Gen%C3%A7ay&rft.aufirst=Ramazan&rft.au=Sel%C3%A7uk%2C+Faruk&rft.au=Whitcher%2C+Brandon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHaar1910" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Alfr%C3%A9d_Haar" title="Alfréd Haar">Haar, Alfred</a> (1910). <a rel="nofollow" class="external text" href="https://eudml.org/doc/158469">"Zur Theorie der orthogonalen Funktionensysteme: Erste Mitteilung"</a>. <i>Mathematische Annalen</i> (in German). <b>69</b> (3): <span class="nowrap">331–</span>371. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01456326">10.1007/BF01456326</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-12-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Zur+Theorie+der+orthogonalen+Funktionensysteme%3A+Erste+Mitteilung&rft.volume=69&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E331-%3C%2Fspan%3E371&rft.date=1910&rft_id=info%3Adoi%2F10.1007%2FBF01456326&rft.issn=0025-5831&rft.aulast=Haar&rft.aufirst=Alfred&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F158469&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKaiser1994" class="citation book cs1">Kaiser, Gerald (1994). <i>A friendly guide to wavelets</i>. Basel Boston: Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3711-7" title="Special:BookSources/0-8176-3711-7"><bdi>0-8176-3711-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+friendly+guide+to+wavelets&rft.place=Basel+Boston&rft.pub=Birkh%C3%A4user&rft.date=1994&rft.isbn=0-8176-3711-7&rft.aulast=Kaiser&rft.aufirst=Gerald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMallat2009" class="citation book cs1"><a href="/wiki/St%C3%A9phane_Mallat" title="Stéphane Mallat">Mallat, S. G.</a> (2009). <i>A wavelet tour of signal processing: the sparse way</i>. Amsterdam ; Boston: Elsevier/Academic Press. <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%2FB978-0-12-374370-1.X0001-8">10.1016/B978-0-12-374370-1.X0001-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-374370-1" title="Special:BookSources/978-0-12-374370-1"><bdi>978-0-12-374370-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+wavelet+tour+of+signal+processing%3A+the+sparse+way&rft.place=Amsterdam+%3B+Boston&rft.pub=Elsevier%2FAcademic+Press&rft.date=2009&rft_id=info%3Adoi%2F10.1016%2FB978-0-12-374370-1.X0001-8&rft.isbn=978-0-12-374370-1&rft.aulast=Mallat&rft.aufirst=S.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPercivalWalden2007" class="citation book cs1">Percival, Donald B.; Walden, Andrew T. (2007). <i>Wavelet methods for time series analysis</i>. Cambridge: Cambridge Univ. Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-68508-5" title="Special:BookSources/978-0-521-68508-5"><bdi>978-0-521-68508-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wavelet+methods+for+time+series+analysis&rft.place=Cambridge&rft.pub=Cambridge+Univ.+Press&rft.date=2007&rft.isbn=978-0-521-68508-5&rft.aulast=Percival&rft.aufirst=Donald+B.&rft.au=Walden%2C+Andrew+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPress2007" class="citation book cs1">Press, William H. (2007). <i>Numerical recipes : the art of scientific computing</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88068-8" title="Special:BookSources/978-0-521-88068-8"><bdi>978-0-521-88068-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1025448470">1025448470</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+recipes+%3A+the+art+of+scientific+computing&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2007&rft_id=info%3Aoclcnum%2F1025448470&rft.isbn=978-0-521-88068-8&rft.aulast=Press&rft.aufirst=William+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVaidyanathan1993" class="citation book cs1"><a href="/wiki/P._P._Vaidyanathan" title="P. P. Vaidyanathan">Vaidyanathan, P. P.</a> (1993). <i>Multirate systems and filter banks</i>. Englewood Cliffs, NJ: Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-605718-7" title="Special:BookSources/0-13-605718-7"><bdi>0-13-605718-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multirate+systems+and+filter+banks&rft.place=Englewood+Cliffs%2C+NJ&rft.pub=Prentice+Hall&rft.date=1993&rft.isbn=0-13-605718-7&rft.aulast=Vaidyanathan&rft.aufirst=P.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVetterliKovačević1995" class="citation book cs1">Vetterli, Martin; Kovačević, Jelena (1995). <i>Wavelets and subband coding</i>. Englewood Cliffs, N.J: Prentice Hall PTR. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-097080-8" title="Special:BookSources/0-13-097080-8"><bdi>0-13-097080-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wavelets+and+subband+coding&rft.place=Englewood+Cliffs%2C+N.J&rft.pub=Prentice+Hall+PTR&rft.date=1995&rft.isbn=0-13-097080-8&rft.aulast=Vetterli&rft.aufirst=Martin&rft.au=Kova%C4%8Devi%C4%87%2C+Jelena&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWickerhauser1994" class="citation book cs1">Wickerhauser, Mladen Victor (1994). <i>Adapted wavelet analysis from theory to software</i>. Wellesley, MA: A.K. Peters. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-041-5" title="Special:BookSources/1-56881-041-5"><bdi>1-56881-041-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Adapted+wavelet+analysis+from+theory+to+software&rft.place=Wellesley%2C+MA&rft.pub=A.K.+Peters&rft.date=1994&rft.isbn=1-56881-041-5&rft.aulast=Wickerhauser&rft.aufirst=Mladen+Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavelet&action=edit&section=29" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" 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href="https://www.encyclopediaofmath.org/index.php?title=Wavelet_analysis">"Wavelet analysis"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Wavelet+analysis&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DWavelet_analysis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavelet" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://web.njit.edu/~ali/s1.htm">1st NJIT Symposium on Wavelets (April 30, 1990) (First Wavelets Conference in USA)</a></li> <li><a rel="nofollow" class="external text" href="http://web.njit.edu/~ali/NJITSYMP1990/AkansuNJIT1STWAVELETSSYMPAPRIL301990.pdf">Binomial-QMF Daubechies Wavelets</a></li> <li><a rel="nofollow" class="external text" href="http://www-math.mit.edu/~gs/papers/amsci.pdf">Wavelets</a> by Gilbert Strang, American Scientist 82 (1994) 250–255. (A very short and excellent introduction)</li> <li><a rel="nofollow" class="external text" href="http://wavelets.ens.fr/ENSEIGNEMENT/COURS/UCSB/index.html">Course on Wavelets given at UC Santa Barbara, 2004</a></li> <li><a rel="nofollow" class="external text" href="http://www.isye.gatech.edu/~brani/wp/kidsA.pdf">Wavelets for Kids (PDF file)</a> (Introductory (for very smart kids!))</li> <li><a rel="nofollow" class="external text" href="http://www.laurent-duval.eu/siva-wits-where-is-the-starlet.html">WITS: Where Is The Starlet?</a> A dictionary of tens of wavelets and wavelet-related terms ending in -let, from activelets to x-lets through bandlets, contourlets, curvelets, noiselets, wedgelets.</li> <li><a rel="nofollow" class="external text" href="http://bigwww.epfl.ch/publications/blu0001.pdf">The Fractional Spline Wavelet Transform</a> describes a <a href="/wiki/Fractional_wavelet_transform" title="Fractional wavelet transform">fractional wavelet transform</a> based on fractional b-Splines.</li> <li><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/j.sigpro.2011.04.025">A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity</a> provides a tutorial on two-dimensional oriented wavelets and related geometric multiscale transforms.</li> <li><a rel="nofollow" class="external text" href="https://www.scribd.com/document/436856865/Concise-Introduction-to-Wavelets">Concise Introduction to Wavelets</a> by René Puschinger</li> <li><a rel="nofollow" class="external text" href="http://www.polyvalens.com/blog/wavelets/theory/">A Really Friendly Guide To Wavelets</a> by Clemens Valens</li> <li><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.quantamagazine.org/how-wavelets-allow-researchers-to-transform-and-understand-data-20211013/">"How Wavelets Allow Researchers to Transform — and Understand — Data"</a>. <i>Quanta Magazine</i>. 2021-10-13<span class="reference-accessdate">. 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id="Descriptive_statistics636" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="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:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="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:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</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/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</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/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</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/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</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/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</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/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</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/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</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/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection636" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="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:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</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/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</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/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</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/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</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_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</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/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference636" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="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:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</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/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="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:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</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/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</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/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</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/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</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/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</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/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</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/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</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/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis636" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="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:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</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/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</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/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</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/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</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/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</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/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</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/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis636" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="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:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</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/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</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_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="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:1%;font-weight:normal;">General</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/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</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/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</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/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</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/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a class="mw-selflink selflink">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="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:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</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/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</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/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</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/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" 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Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-04-30T13:48:05Z","dateModified":"2025-02-25T02:17:05Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/0\/0d\/Seismic_Wavelet.svg","headline":"function for integral Fourier-like transform"}</script> </body> </html>