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overflow: hidden; text-overflow: ellipsis; -webkit-line-clamp: 3; -webkit-box-orient: vertical; }</style><div class="col-xs-12 clearfix"><div class="u-floatLeft"><h1 class="PageHeader-title u-m0x u-fs30">Fourier</h1><div class="u-tcGrayDark">26 Followers</div><div class="u-tcGrayDark u-mt2x">Recent papers in <b>Fourier</b></div></div></div></div></div></div><div class="TabbedNavigation"><div class="container"><div class="row"><div class="col-xs-12 clearfix"><ul class="nav u-m0x u-p0x list-inline u-displayFlex"><li class="active"><a href="https://www.academia.edu/Documents/in/Fourier">Top Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Fourier/MostCited">Most Cited Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Fourier/MostDownloaded">Most Downloaded Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Fourier/MostRecent">Newest Papers</a></li><li><a class="" href="https://www.academia.edu/People/Fourier">People</a></li></ul></div><style type="text/css">ul.nav{flex-direction:row}@media(max-width: 567px){ul.nav{flex-direction:column}.TabbedNavigation li{max-width:100%}.TabbedNavigation li.active{background-color:var(--background-grey, #dddde2)}.TabbedNavigation li.active:before,.TabbedNavigation li.active:after{display:none}}</style></div></div></div><div class="container"><div class="row"><div class="col-xs-12"><div class="u-displayFlex"><div class="u-flexGrow1"><div class="works"><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_62988107" data-work_id="62988107" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/62988107/Calcul_symbolique_et_calcul_int%C3%A9gral_de_Lagrange_%C3%A0_Cauchy">Calcul symbolique et calcul intégral de Lagrange à Cauchy</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Dans un memoire publie en 1774, Lagrange utilise des methodes reposant sur l&#39;analogie des puissances positives et des differences, et des puissances negatives et des sommes, qui lui permettent, notamment, d&#39;obtenir diverses... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_62988107" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Dans un memoire publie en 1774, Lagrange utilise des methodes reposant sur l&#39;analogie des puissances positives et des differences, et des puissances negatives et des sommes, qui lui permettent, notamment, d&#39;obtenir diverses formules d&#39;integration. D&#39;autres auteurs s&#39;engagent alors dans cette voie. Les problemes de calcul integral jouent un role important dans le developpement de diverses formes de calcul symbolique et celui-ci fait la preuve de son efficacite dans ce domaine: il permet de generaliser ou de retrouver rapidement des resultats anciens, introduit de la clarte dans des pratiques d&#39;integration numerique, unifie les procedures d&#39;integration des divers types d&#39;equations lineaires, et conduit a la resolution de nouvelles equations aux derivees partielles. Cependant, les notations et les fondements memes des nouveaux procedes restent longtemps l&#39;objet d&#39;interrogations. Dans les annees 1820, Cauchy apporte une reponse conforme a sa conce...</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/62988107" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a1335cb8c3018d8196589d068a400ff9" rel="nofollow" data-download="{"attachment_id":75569113,"asset_id":62988107,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/75569113/download_file?st=MTczMjc4MzYyNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="112336677" href="https://independent.academia.edu/jeanpierrelubet">jean-pierre lubet</a><script data-card-contents-for-user="112336677" type="text/json">{"id":112336677,"first_name":"jean-pierre","last_name":"lubet","domain_name":"independent","page_name":"jeanpierrelubet","display_name":"jean-pierre lubet","profile_url":"https://independent.academia.edu/jeanpierrelubet?f_ri=600430","photo":"https://0.academia-photos.com/112336677/30478175/28239859/s65_jean-pierre.lubet.jpg"}</script></span></span></li><li class="js-paper-rank-work_62988107 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="62988107"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 62988107, container: ".js-paper-rank-work_62988107", }); 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u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_1953778" data-work_id="1953778" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/1953778/Wavelet_based_spatio_temporal_fusion_of_observed_rainfall_with_NDVI_in_Sri_Lanka">Wavelet-based spatio-temporal fusion of observed rainfall with NDVI in Sri Lanka</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Availability of rainfall time-series is limited in many parts of the World, and the continuity of such records is variable. This research endeavors to extend actual daily rainfall observations to ungauged areas, taking into account events... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_1953778" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Availability of rainfall time-series is limited in many parts of the World, and the continuity of such records is variable. This research endeavors to extend actual daily rainfall observations to ungauged areas, taking into account events of rainfall as well as cumulative total daily rainfall, over a period of 11 years. Results show that rainfall events histograms can be reconstructed, and that total cumulative rainfall is estimated with 85% accuracy, using  a surrounding network of rain gauges at 30-50 Km of distance from the point of study. This research can strengthen various types of research and applications such as ungauged basins research, regional climate modeling, food security early warning systems, agricultural insurance systems, etc.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/1953778" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="9dc47a81671e128617380251fba06e7e" rel="nofollow" data-download="{"attachment_id":28248834,"asset_id":1953778,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/28248834/download_file?st=MTczMjc4MzYyNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="1223613" href="https://iwmi.academia.edu/YannChemin">Yann Chemin</a><script data-card-contents-for-user="1223613" type="text/json">{"id":1223613,"first_name":"Yann","last_name":"Chemin","domain_name":"iwmi","page_name":"YannChemin","display_name":"Yann Chemin","profile_url":"https://iwmi.academia.edu/YannChemin?f_ri=600430","photo":"https://0.academia-photos.com/1223613/442696/984097/s65_yann.chemin.jpg"}</script></span></span></li><li class="js-paper-rank-work_1953778 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="1953778"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 1953778, container: ".js-paper-rank-work_1953778", }); 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This research endeavors to extend actual daily rainfall observations to ungauged areas, taking into account events of rainfall as well as cumulative total daily rainfall, over a period of 11 years. Results show that rainfall events histograms can be reconstructed, and that total cumulative rainfall is estimated with 85% accuracy, using a surrounding network of rain gauges at 30-50 Km of distance from the point of study. This research can strengthen various types of research and applications such as ungauged basins research, regional climate modeling, food security early warning systems, agricultural insurance systems, etc.","downloadable_attachments":[{"id":28248834,"asset_id":1953778,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":1223613,"first_name":"Yann","last_name":"Chemin","domain_name":"iwmi","page_name":"YannChemin","display_name":"Yann Chemin","profile_url":"https://iwmi.academia.edu/YannChemin?f_ri=600430","photo":"https://0.academia-photos.com/1223613/442696/984097/s65_yann.chemin.jpg"}],"research_interests":[{"id":9850,"name":"Sri Lanka","url":"https://www.academia.edu/Documents/in/Sri_Lanka?f_ri=600430","nofollow":false},{"id":14852,"name":"Data Fusion (Engineering)","url":"https://www.academia.edu/Documents/in/Data_Fusion_Engineering_?f_ri=600430","nofollow":false},{"id":15355,"name":"Fourier and Wavelet Processing","url":"https://www.academia.edu/Documents/in/Fourier_and_Wavelet_Processing?f_ri=600430","nofollow":false},{"id":28120,"name":"Spatio Temporal Analysis","url":"https://www.academia.edu/Documents/in/Spatio_Temporal_Analysis?f_ri=600430","nofollow":false},{"id":43974,"name":"Wavelets","url":"https://www.academia.edu/Documents/in/Wavelets?f_ri=600430"},{"id":58906,"name":"Fourier Analysis","url":"https://www.academia.edu/Documents/in/Fourier_Analysis?f_ri=600430"},{"id":76299,"name":"MODIS","url":"https://www.academia.edu/Documents/in/MODIS?f_ri=600430"},{"id":146693,"name":"Ndvi","url":"https://www.academia.edu/Documents/in/Ndvi?f_ri=600430"},{"id":158331,"name":"Python Programming","url":"https://www.academia.edu/Documents/in/Python_Programming?f_ri=600430"},{"id":201862,"name":"Rainfall","url":"https://www.academia.edu/Documents/in/Rainfall?f_ri=600430"},{"id":600430,"name":"Fourier","url":"https://www.academia.edu/Documents/in/Fourier?f_ri=600430"},{"id":600431,"name":"Daily","url":"https://www.academia.edu/Documents/in/Daily?f_ri=600430"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_1536947" data-work_id="1536947" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/1536947/Greenhouse_Gas_GHG_The_Runaway_Effect_Earth_s_Surface_Heat_Out_of_Balance">Greenhouse Gas (GHG): The Runaway Effect-Earth’s Surface Heat Out of Balance</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">We are entering a new era of carbon and nitrogenous waste management. There is one particular biochemical phenomena of interest in this ‘tug-of-war’ for preserving life on Earth as we know it: the greenhouse effect. The greenhouse effect... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_1536947" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">We are entering a new era of carbon and nitrogenous waste management. There is one particular biochemical phenomena of interest in this ‘tug-of-war’ for preserving life on Earth as we know it: the greenhouse effect. The greenhouse effect is a process by which thermal radiation from a planetary surface is absorbed by atmospheric greenhouse gases (carbon dioxide, methane, water vapour, nitrous oxide), and is re-radiated in all directions. It is not a new phenomena.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/1536947" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="afb6dfa13737d778629afdba6dccfa78" rel="nofollow" data-download="{"attachment_id":12821852,"asset_id":1536947,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/12821852/download_file?st=MTczMjc4MzYyNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="991987" href="https://udenver.academia.edu/KeliRae">Keli Rae</a><script data-card-contents-for-user="991987" type="text/json">{"id":991987,"first_name":"Keli","last_name":"Rae","domain_name":"udenver","page_name":"KeliRae","display_name":"Keli Rae","profile_url":"https://udenver.academia.edu/KeliRae?f_ri=600430","photo":"https://0.academia-photos.com/991987/581209/722297/s65_keli.rae.jpg"}</script></span></span></li><li class="js-paper-rank-work_1536947 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="1536947"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 1536947, container: ".js-paper-rank-work_1536947", }); 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itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/74811836/Curvelet_feature_based_fingerprint_recognition_Using_fourier_enhancement">Curvelet feature based fingerprint recognition: Using fourier enhancement</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Fingerprints are the oldest and most widely used form of biometric identification. Uniqueness of one&#39;s Fingerprint remains same throughout the lifetime. Therefore, fingerprints are being used in many biometric systems such as civilian... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_74811836" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Fingerprints are the oldest and most widely used form of biometric identification. Uniqueness of one&#39;s Fingerprint remains same throughout the lifetime. Therefore, fingerprints are being used in many biometric systems such as civilian and commercial identification devices and forensic investigation systems. In our paper we have proposed a fingerprint recognition system as a combination of three significant stages: preprocessing, post-processing and matching stage that improves accuracy by overcoming the challenges previous systems presented. We have applied several enhancement methods in the pre-processing stage. For fingerprint image alignment purpose we have used core point based `Poincare Index value&#39; method, which is one of the widely used method for core point detection. Curvelet transformation, a new multi-resolution and multi-orientations tool is therefore used for the extraction of feature set from the image in different scale and orientation. Extracted features vectors from Curvelet co-efficient matrix are then classified using k nearest neighbor classifier in order to recognize the specific fingerprint.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/74811836" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="184016682" href="https://independent.academia.edu/MdMarufkhanKhan">Md Maruf khan Khan</a><script data-card-contents-for-user="184016682" type="text/json">{"id":184016682,"first_name":"Md Maruf khan","last_name":"Khan","domain_name":"independent","page_name":"MdMarufkhanKhan","display_name":"Md Maruf khan Khan","profile_url":"https://independent.academia.edu/MdMarufkhanKhan?f_ri=600430","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_74811836 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="74811836"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 74811836, container: ".js-paper-rank-work_74811836", }); 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Uniqueness of one\u0026#39;s Fingerprint remains same throughout the lifetime. Therefore, fingerprints are being used in many biometric systems such as civilian and commercial identification devices and forensic investigation systems. In our paper we have proposed a fingerprint recognition system as a combination of three significant stages: preprocessing, post-processing and matching stage that improves accuracy by overcoming the challenges previous systems presented. We have applied several enhancement methods in the pre-processing stage. For fingerprint image alignment purpose we have used core point based `Poincare Index value\u0026#39; method, which is one of the widely used method for core point detection. Curvelet transformation, a new multi-resolution and multi-orientations tool is therefore used for the extraction of feature set from the image in different scale and orientation. Extracted features vectors from Curvelet co-efficient matrix are then classified using k nearest neighbor classifier in order to recognize the specific fingerprint.","downloadable_attachments":[],"ordered_authors":[{"id":184016682,"first_name":"Md Maruf khan","last_name":"Khan","domain_name":"independent","page_name":"MdMarufkhanKhan","display_name":"Md Maruf khan Khan","profile_url":"https://independent.academia.edu/MdMarufkhanKhan?f_ri=600430","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":27360,"name":"Databases","url":"https://www.academia.edu/Documents/in/Databases?f_ri=600430","nofollow":false},{"id":56368,"name":"Image Classification","url":"https://www.academia.edu/Documents/in/Image_Classification?f_ri=600430","nofollow":false},{"id":148960,"name":"Fingerprint","url":"https://www.academia.edu/Documents/in/Fingerprint?f_ri=600430","nofollow":false},{"id":160144,"name":"Feature Extraction","url":"https://www.academia.edu/Documents/in/Feature_Extraction?f_ri=600430","nofollow":false},{"id":284729,"name":"Fingerprint Recognition","url":"https://www.academia.edu/Documents/in/Fingerprint_Recognition?f_ri=600430"},{"id":403692,"name":"Curvelet","url":"https://www.academia.edu/Documents/in/Curvelet?f_ri=600430"},{"id":600430,"name":"Fourier","url":"https://www.academia.edu/Documents/in/Fourier?f_ri=600430"},{"id":1136390,"name":"Biometric Identification","url":"https://www.academia.edu/Documents/in/Biometric_Identification?f_ri=600430"},{"id":3455089,"name":"image resolution","url":"https://www.academia.edu/Documents/in/image_resolution?f_ri=600430"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_74697079" data-work_id="74697079" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 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spectroscopy","url":"https://www.academia.edu/Documents/in/Infrared_spectroscopy?f_ri=600430"},{"id":148225,"name":"Boron","url":"https://www.academia.edu/Documents/in/Boron?f_ri=600430"},{"id":159482,"name":"Investigations","url":"https://www.academia.edu/Documents/in/Investigations?f_ri=600430"},{"id":260118,"name":"CHEMICAL SCIENCES","url":"https://www.academia.edu/Documents/in/CHEMICAL_SCIENCES?f_ri=600430"},{"id":335361,"name":"Infrared","url":"https://www.academia.edu/Documents/in/Infrared?f_ri=600430"},{"id":362531,"name":"Bismuth","url":"https://www.academia.edu/Documents/in/Bismuth?f_ri=600430"},{"id":386304,"name":"Nanoceramic","url":"https://www.academia.edu/Documents/in/Nanoceramic?f_ri=600430"},{"id":390056,"name":"Fourier transform","url":"https://www.academia.edu/Documents/in/Fourier_transform?f_ri=600430"},{"id":600430,"name":"Fourier","url":"https://www.academia.edu/Documents/in/Fourier?f_ri=600430"},{"id":963367,"name":"FT IR","url":"https://www.academia.edu/Documents/in/FT_IR?f_ri=600430"},{"id":1133030,"name":"Nanoceramics","url":"https://www.academia.edu/Documents/in/Nanoceramics?f_ri=600430"},{"id":1279532,"name":"Doped","url":"https://www.academia.edu/Documents/in/Doped?f_ri=600430"},{"id":1282185,"name":"Indium","url":"https://www.academia.edu/Documents/in/Indium?f_ri=600430"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_47685202" data-work_id="47685202" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/47685202/Information_Models_of_Acupuncture_Analgesia_and_Meridian_Channels">Information Models of Acupuncture Analgesia and Meridian Channels</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/47685202" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="852dbb1074c9fc1c38cb74c54c8d7d1e" rel="nofollow" data-download="{"attachment_id":66649705,"asset_id":47685202,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/66649705/download_file?st=MTczMjc4MzYyNSw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="120674408" href="https://independent.academia.edu/KangCheng6">Kang Cheng</a><script data-card-contents-for-user="120674408" type="text/json">{"id":120674408,"first_name":"Kang","last_name":"Cheng","domain_name":"independent","page_name":"KangCheng6","display_name":"Kang Cheng","profile_url":"https://independent.academia.edu/KangCheng6?f_ri=600430","photo":"https://0.academia-photos.com/120674408/29951816/27814126/s65_kang.cheng.jpg"}</script></span></span></li><li class="js-paper-rank-work_47685202 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" 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InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="47685202"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 47685202; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=47685202]").text(description); $(".js-view-count-work_47685202").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_47685202").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="47685202"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">10</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="11035" href="https://www.academia.edu/Documents/in/Regulation">Regulation</a>, <script data-card-contents-for-ri="11035" type="text/json">{"id":11035,"name":"Regulation","url":"https://www.academia.edu/Documents/in/Regulation?f_ri=600430","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="48999" href="https://www.academia.edu/Documents/in/Information">Information</a>, <script data-card-contents-for-ri="48999" type="text/json">{"id":48999,"name":"Information","url":"https://www.academia.edu/Documents/in/Information?f_ri=600430","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="100657" href="https://www.academia.edu/Documents/in/Therapy">Therapy</a>, <script data-card-contents-for-ri="100657" type="text/json">{"id":100657,"name":"Therapy","url":"https://www.academia.edu/Documents/in/Therapy?f_ri=600430","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="144046" href="https://www.academia.edu/Documents/in/Frequency">Frequency</a><script data-card-contents-for-ri="144046" type="text/json">{"id":144046,"name":"Frequency","url":"https://www.academia.edu/Documents/in/Frequency?f_ri=600430","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=47685202]'), work: {"id":47685202,"title":"Information Models of Acupuncture Analgesia and Meridian Channels","created_at":"2021-04-23T16:01:56.703-07:00","url":"https://www.academia.edu/47685202/Information_Models_of_Acupuncture_Analgesia_and_Meridian_Channels?f_ri=600430","dom_id":"work_47685202","summary":null,"downloadable_attachments":[{"id":66649705,"asset_id":47685202,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":120674408,"first_name":"Kang","last_name":"Cheng","domain_name":"independent","page_name":"KangCheng6","display_name":"Kang Cheng","profile_url":"https://independent.academia.edu/KangCheng6?f_ri=600430","photo":"https://0.academia-photos.com/120674408/29951816/27814126/s65_kang.cheng.jpg"}],"research_interests":[{"id":11035,"name":"Regulation","url":"https://www.academia.edu/Documents/in/Regulation?f_ri=600430","nofollow":false},{"id":48999,"name":"Information","url":"https://www.academia.edu/Documents/in/Information?f_ri=600430","nofollow":false},{"id":100657,"name":"Therapy","url":"https://www.academia.edu/Documents/in/Therapy?f_ri=600430","nofollow":false},{"id":144046,"name":"Frequency","url":"https://www.academia.edu/Documents/in/Frequency?f_ri=600430","nofollow":false},{"id":397902,"name":"Series","url":"https://www.academia.edu/Documents/in/Series?f_ri=600430"},{"id":447209,"name":"Coupling","url":"https://www.academia.edu/Documents/in/Coupling?f_ri=600430"},{"id":600430,"name":"Fourier","url":"https://www.academia.edu/Documents/in/Fourier?f_ri=600430"},{"id":963184,"name":"Beat","url":"https://www.academia.edu/Documents/in/Beat?f_ri=600430"},{"id":1554800,"name":"Amplitude","url":"https://www.academia.edu/Documents/in/Amplitude?f_ri=600430"},{"id":3069705,"name":"Information model","url":"https://www.academia.edu/Documents/in/Information_model?f_ri=600430"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_36145770 coauthored" data-work_id="36145770" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/36145770/A_Simple_Spectral_Observer">A Simple Spectral Observer</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. This paper proposes a novel spectral observer... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_36145770" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. This paper proposes a novel spectral observer with an adjustable constant gain for reconstructing a given signal by means of the recursive identification of the coefficients of a Fourier series. The reconstruction or estimation of a signal in the context of this work means to find the coefficients of a linear combination of sines a cosines that fits a signal such that it can be reproduced. The design procedure of the spectral observer is presented along with the following applications: (1) the reconstruction of a simple periodical signal, (2) the approximation of both a square and a triangular signal, (3) the edge detection in signals by using the Fourier coefficients and (4) the fitting of the historical Bitcoin market data from 2014-12-01 to 2018-01-08.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/36145770" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a67133ebc6e31eeac28aa2027cbc05d8" rel="nofollow" data-download="{"attachment_id":56038449,"asset_id":36145770,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/56038449/download_file?st=MTczMjc4MzYyNSw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="3758847" href="https://unam.academia.edu/LizethTorres">LIZETH TORRES</a><script data-card-contents-for-user="3758847" type="text/json">{"id":3758847,"first_name":"LIZETH","last_name":"TORRES","domain_name":"unam","page_name":"LizethTorres","display_name":"LIZETH TORRES","profile_url":"https://unam.academia.edu/LizethTorres?f_ri=600430","photo":"https://0.academia-photos.com/3758847/1356365/4383390/s65_lizeth.torres_ortiz.jpg"}</script></span></span><span class="u-displayInlineBlock InlineList-item-text"> and <span class="u-textDecorationUnderline u-clickable InlineList-item-text js-work-more-authors-36145770">+1</span><div class="hidden js-additional-users-36145770"><div><span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a href="https://unam.academia.edu/PabloP%C3%A9rez">Pablo Pérez</a></span></div></div></span><script>(function(){ var popoverSettings = { el: $('.js-work-more-authors-36145770'), placement: 'bottom', hide_delay: 200, html: true, content: function(){ return $('.js-additional-users-36145770').html(); 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feature based fingerprint recognition: Using fourier enhancement</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Fingerprints are the oldest and most widely used form of biometric identification. Uniqueness of one&#39;s Fingerprint remains same throughout the lifetime. Therefore, fingerprints are being used in many biometric systems such as civilian... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_66912078" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Fingerprints are the oldest and most widely used form of biometric identification. Uniqueness of one&#39;s Fingerprint remains same throughout the lifetime. Therefore, fingerprints are being used in many biometric systems such as civilian and commercial identification devices and forensic investigation systems. In our paper we have proposed a fingerprint recognition system as a combination of three significant stages: preprocessing, post-processing and matching stage that improves accuracy by overcoming the challenges previous systems presented. We have applied several enhancement methods in the pre-processing stage. For fingerprint image alignment purpose we have used core point based `Poincare Index value&#39; method, which is one of the widely used method for core point detection. Curvelet transformation, a new multi-resolution and multi-orientations tool is therefore used for the extraction of feature set from the image in different scale and orientation. Extracted features vectors from Curvelet co-efficient matrix are then classified using k nearest neighbor classifier in order to recognize the specific fingerprint.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/66912078" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="162807128" href="https://independent.academia.edu/bushranazia">nazia bushra</a><script data-card-contents-for-user="162807128" type="text/json">{"id":162807128,"first_name":"nazia","last_name":"bushra","domain_name":"independent","page_name":"bushranazia","display_name":"nazia bushra","profile_url":"https://independent.academia.edu/bushranazia?f_ri=600430","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_66912078 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="66912078"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 66912078, container: ".js-paper-rank-work_66912078", }); 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Uniqueness of one\u0026#39;s Fingerprint remains same throughout the lifetime. Therefore, fingerprints are being used in many biometric systems such as civilian and commercial identification devices and forensic investigation systems. In our paper we have proposed a fingerprint recognition system as a combination of three significant stages: preprocessing, post-processing and matching stage that improves accuracy by overcoming the challenges previous systems presented. We have applied several enhancement methods in the pre-processing stage. For fingerprint image alignment purpose we have used core point based `Poincare Index value\u0026#39; method, which is one of the widely used method for core point detection. Curvelet transformation, a new multi-resolution and multi-orientations tool is therefore used for the extraction of feature set from the image in different scale and orientation. Extracted features vectors from Curvelet co-efficient matrix are then classified using k nearest neighbor classifier in order to recognize the specific fingerprint.","downloadable_attachments":[],"ordered_authors":[{"id":162807128,"first_name":"nazia","last_name":"bushra","domain_name":"independent","page_name":"bushranazia","display_name":"nazia bushra","profile_url":"https://independent.academia.edu/bushranazia?f_ri=600430","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":27360,"name":"Databases","url":"https://www.academia.edu/Documents/in/Databases?f_ri=600430","nofollow":false},{"id":56368,"name":"Image Classification","url":"https://www.academia.edu/Documents/in/Image_Classification?f_ri=600430","nofollow":false},{"id":148960,"name":"Fingerprint","url":"https://www.academia.edu/Documents/in/Fingerprint?f_ri=600430","nofollow":false},{"id":160144,"name":"Feature 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u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/63026462/Seizure_detection_seizure_prediction_and_closed_loop_warning_systems_in_epilepsy">Seizure detection, seizure prediction, and closed-loop warning systems in epilepsy</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/63026462" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="fb33f38344c6e6fb56b937436b66db55" rel="nofollow" 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type="text/json">{"id":5069,"name":"Principal Component Analysis","url":"https://www.academia.edu/Documents/in/Principal_Component_Analysis?f_ri=600430","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="7648" href="https://www.academia.edu/Documents/in/Epilepsy">Epilepsy</a>, <script data-card-contents-for-ri="7648" type="text/json">{"id":7648,"name":"Epilepsy","url":"https://www.academia.edu/Documents/in/Epilepsy?f_ri=600430","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="9739" href="https://www.academia.edu/Documents/in/Probabilistic_Markov_Modeling">Probabilistic Markov Modeling</a><script data-card-contents-for-ri="9739" type="text/json">{"id":9739,"name":"Probabilistic Markov Modeling","url":"https://www.academia.edu/Documents/in/Probabilistic_Markov_Modeling?f_ri=600430","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: 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href="https://www.academia.edu/6888491/One_dimensional_Fourier_transform_coefficients_for_rotation_invariant_texture_classification">One-dimensional Fourier transform coefficients for rotation invariant texture classification</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper introduces a texture descriptor that is invariant to rotation. The new texture descriptor utilizes the property of the magnitudes of Fourier transform coefficients that do not change with spatial shift of input elements. Since... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_6888491" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper introduces a texture descriptor that is invariant to rotation. The new texture descriptor utilizes the property of the magnitudes of Fourier transform coefficients that do not change with spatial shift of input elements. Since rotating an image by an arbitrary angle does not change pixel intensities in an image but shift them in circular motion, the notion of producing textural features invariant to rotation using 1D Fourier transform coefficients can be realized if the relationship between circular motion and spatial shift can be established. By analyzing pixels in a circular neighborhood in an image, a number of FOurier transform coefficients can be generated to describe local properties of the neighborhood. From the magnitudes of these coefficients, several rotation invariant features are obtained to represent each texture class. Based on these features, an unknown image is assigned to one of the known classes using a nearest neighbor classifier. All of the feature samples for the classifier are extracted from unrotated texture images only. The new texture descriptor outperformed the circular simultaneous autoregressive model in classifying rotated texture images taken from 30 texture classes. ©2005 Copyright SPIE - The International Society for Optical Engineering. <br /> <br /><a href="http://spie.org/Publications/Proceedings/Paper/10.1117/12.257258" rel="nofollow">http://spie.org/Publications/Proceedings/Paper/10.1117/12.257258</a>  <br /><a href="http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1025128" rel="nofollow">http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1025128</a></div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/6888491" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="562a7faa3f9387037b6ed6f008430ddb" rel="nofollow" data-download="{"attachment_id":33571963,"asset_id":6888491,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/33571963/download_file?st=MTczMjc4MzYyNSw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="11362128" href="https://malaya.academia.edu/HamzahArof">Hamzah Arof</a><script data-card-contents-for-user="11362128" type="text/json">{"id":11362128,"first_name":"Hamzah","last_name":"Arof","domain_name":"malaya","page_name":"HamzahArof","display_name":"Hamzah Arof","profile_url":"https://malaya.academia.edu/HamzahArof?f_ri=600430","photo":"https://0.academia-photos.com/11362128/3358432/3951609/s65_hamzah.arof.jpg"}</script></span></span></li><li class="js-paper-rank-work_6888491 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="6888491"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 6888491, container: ".js-paper-rank-work_6888491", }); 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The new texture descriptor utilizes the property of the magnitudes of Fourier transform coefficients that do not change with spatial shift of input elements. Since rotating an image by an arbitrary angle does not change pixel intensities in an image but shift them in circular motion, the notion of producing textural features invariant to rotation using 1D Fourier transform coefficients can be realized if the relationship between circular motion and spatial shift can be established. By analyzing pixels in a circular neighborhood in an image, a number of FOurier transform coefficients can be generated to describe local properties of the neighborhood. From the magnitudes of these coefficients, several rotation invariant features are obtained to represent each texture class. Based on these features, an unknown image is assigned to one of the known classes using a nearest neighbor classifier. All of the feature samples for the classifier are extracted from unrotated texture images only. The new texture descriptor outperformed the circular simultaneous autoregressive model in classifying rotated texture images taken from 30 texture classes. ©2005 Copyright SPIE - The International Society for Optical Engineering.\r\n\r\nhttp://spie.org/Publications/Proceedings/Paper/10.1117/12.257258 \r\nhttp://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1025128 ","downloadable_attachments":[{"id":33571963,"asset_id":6888491,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":11362128,"first_name":"Hamzah","last_name":"Arof","domain_name":"malaya","page_name":"HamzahArof","display_name":"Hamzah Arof","profile_url":"https://malaya.academia.edu/HamzahArof?f_ri=600430","photo":"https://0.academia-photos.com/11362128/3358432/3951609/s65_hamzah.arof.jpg"}],"research_interests":[{"id":854,"name":"Computer Vision","url":"https://www.academia.edu/Documents/in/Computer_Vision?f_ri=600430","nofollow":false},{"id":1185,"name":"Image 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In this paper we ask when it is possible to find a subset $\Lambda$ in $\br^d$ such that the corresponding complex exponential functions $e_\lambda$ indexed by... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_29164674" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Let $d$ be a positive integer, and let $\mu$ be a finite measure on $\br^d$. In this paper we ask when it is possible to find a subset $\Lambda$ in $\br^d$ such that the corresponding complex exponential functions $e_\lambda$ indexed by $\Lambda$ are orthogonal and total in $L^2(\mu)$. If this happens, we say that $(\mu, \Lambda)$ is a spectral pair.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/29164674" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="5126d66d8b10a6cd0bf2240521f03196" rel="nofollow" data-download="{"attachment_id":49610463,"asset_id":29164674,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/49610463/download_file?st=MTczMjc4MzYyNSw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="14880054" href="https://uiowa.academia.edu/PalleJorgensen">Palle Jorgensen</a><script data-card-contents-for-user="14880054" type="text/json">{"id":14880054,"first_name":"Palle","last_name":"Jorgensen","domain_name":"uiowa","page_name":"PalleJorgensen","display_name":"Palle Jorgensen","profile_url":"https://uiowa.academia.edu/PalleJorgensen?f_ri=600430","photo":"https://0.academia-photos.com/14880054/11454360/12775954/s65_palle.jorgensen.jpg_oh_9d313fab31f41b8f5b3158c764bad23d_oe_572cd8ae"}</script></span></span></li><li class="js-paper-rank-work_29164674 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="29164674"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 29164674, container: ".js-paper-rank-work_29164674", }); 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