On the mathematical properties of the structural similarity index

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usa-input-error-message selection-validation-message"> Name must be less than 100 characters </div> </div> <div class="action-panel-control-wrap existing-collections-controls"> <label for="collections-action-panel-add-to-existing" class="action-panel-label"> Choose a collection: </label> <select id="collections-action-panel-add-to-existing" class="action-panel-selector collections-action-add-to-existing" data-ga-category="save_share" data-ga-action="select_collection" data-ga-label="($('#collections-action-add-to-existing').val() === 'Favorites') ? 'Favorites' : 'non_favorites_collection'"> </select> <div class="collections-retry-load-on-error usa-input-error-message selection-validation-message"> Unable to load your collection due to an error<br> <a href="#">Please try again</a> </div> </div> </div> <div class="action-panel-actions"> <button class="action-panel-submit" type="submit" data-loading-label="Adding..." data-pinger-ignore data-ga-category="save_share" data-ga-action="collections" data-ga-label="add"> Add </button> <button class="action-panel-cancel" aria-label="Close 'Add to Collections' panel" ref="linksrc=close_collections_panel" aria-controls="collections-action-panel" aria-expanded="false" data-ga-category="save_share" data-ga-action="collections" data-ga-label="cancel"> Cancel </button> </div> </form> </div> </div> <div id="bibliography-action-panel" class="bibliography-action-panel action-panel in-progress-dots-panel" aria-hidden="true" data-bibliography-open-panel-enabled="false" data-bibliography-open-panel-url-hash="#open-bibliography-panel"> <div class="inner-wrap"> <h3 class="action-panel-heading"> Add to My Bibliography </h3> <form id="bibliography-action-panel-form" class="bibliography-action-panel-form action-panel-content action-form action-panel-smaller-selectors" data-add-to-bibliography-max-amount="100" data-add-to-bibliography-batch-size="10" data-bibliography-delegates-url="/list-bibliography-delegates/" data-add-to-bibliography-url="/add-to-bibliography/" data-get-article-ids-by-search-url="/get-article-ids-by-search/" data-mybib-root-url="https://www.ncbi.nlm.nih.gov/myncbi/collections/mybibliography/"> <input type="hidden" name="csrfmiddlewaretoken" value="GixcXuIgsDHNtJcr4i2w4UXjp27fHuBABMrgHWAnhXJP0mv5erxHUO5cnpchI3g7"> <div class="action-panel-control-wrap bibliographies-controls"> <div class="choice-group"> <ul class="bibliographies-action-add radio-group-items"> <li> <input name="bibliography" id="my-bibliography" class="my-bibliography" type="radio" checked/> <label for="my-bibliography">My Bibliography</label> </li> </ul> </div> </div> <div class="bibliographies-retry-load-on-error usa-input-error-message selection-validation-message"> Unable to load your delegates due to an error<br> <a href="#">Please try again</a> </div> <div class="action-panel-actions"> <button class="action-panel-submit" type="submit" data-loading-label="Adding..." data-pinger-ignore> Add </button> <button class="action-panel-cancel" aria-label="Close 'Add to bibliography' panel" ref="linksrc=close_bibliography_panel" aria-controls="bibliography-action-panel" aria-expanded="false" data-ga-category="save_share" data-ga-action="mybib" data-ga-label="cancel"> Cancel </button> </div> </form> </div> </div> <div id="saved-search-action-panel" class="saved-search-action-panel action-panel " aria-hidden="true" data-saved-search-open-panel-enabled="false" data-saved-search-open-panel-url-hash="#open-saved-search-panel"> <div class="inner-wrap"> <h2 class="action-panel-heading"> Your saved search </h2> <form id="saved-search-action-panel-form" class="saved-search-action-panel-form action-panel-content action-form" data-create-saved-search-url="/create-saved-search/" data-try-search-terms-url="/try-search-term/" data-saved-search-root-url="https://www.ncbi.nlm.nih.gov/myncbi/searches/"> <input type="hidden" name="csrfmiddlewaretoken" value="GixcXuIgsDHNtJcr4i2w4UXjp27fHuBABMrgHWAnhXJP0mv5erxHUO5cnpchI3g7"> <div class="action-panel-control-wrap"> <label for="saved-search-name" class="action-panel-label saved-search-name-label required-field-asterisk"> Name of saved search: </label> <input maxlength="200" type="text" name="saved-search-name" id="saved-search-name" class="saved-search-name" value="" required pattern="[^"&=<>\/]*" title="The following characters are not allowed in the Name field: "&=<>/"> </div> <div class="action-panel-control-wrap"> <label for="saved-search-term" class="action-panel-label required-field-asterisk"> Search terms: </label> <textarea name="saved-search-term" id="saved-search-term" class="saved-search-term" required></textarea> </div> <div class="test-search-term-wrap"> <a href="#" class="try-search-term">Test search terms</a> </div> <div class="choice-group action-panel-extra-margin-top"> <span class="action-panel-label" id="fieldset-label"> Would you like email updates of new search results? </span> <fieldset id="saved-search-alert" aria-describedby="fieldset-label"> <legend class="usa-sr-only">Saved Search Alert Radio Buttons</legend> <ul class="radio-group-items"> <li> <input type="radio" id="saved-search-alert-yes" class="saved-search-alert-yes" name="saved-search-alert" value="yes" checked> <label for="saved-search-alert-yes" class="action-panel-label">Yes</label> </li> <li> <input aria-label="No radio input" type="radio" id="saved-search-alert-no" class="saved-search-alert-no" name="saved-search-alert" value="no"> <label for="saved-search-alert-no" class="action-panel-label">No</label> </li> </ul> </fieldset> </div> <div class="alert-schedule-wrap"> <div class="action-panel-control-wrap"> <label class="action-panel-label"> Email: </label> <span aria-label="Email address" id="saved-search-email" class="action-panel-label"><span class="action-panel-label-bold"></span> (<a class="myncbi-account-settings" href="https://www.ncbi.nlm.nih.gov/account/settings/">change</a>)</span> </div> <div class="action-panel-control-wrap action-panel-extra-margin-top"> <label for="saved-search-frequency" class="action-panel-label"> Frequency: </label> <select id="saved-search-frequency" class="no-border-panel-selector saved-search-frequency"> <option value="monthly">Monthly</option> <option value="weekly">Weekly</option> <option value="daily">Daily</option> </select> </div> <div class="action-panel-control-wrap saved-search-monthly-additional"> <label for="saved-search-monthly-on-day" class="action-panel-label"> Which day? </label> <select id="saved-search-monthly-on-day" class="no-border-panel-selector"> <option value="Sunday">The first Sunday</option> <option value="Monday">The first Monday</option> <option value="Tuesday">The first Tuesday</option> <option value="Wednesday">The first Wednesday</option> <option value="Thursday">The first Thursday</option> <option value="Friday">The first Friday</option> <option value="Saturday">The first Saturday</option> <option value="day">The first day</option> <option value="weekday">The first weekday</option> </select> </div> <div class="action-panel-control-wrap saved-search-weekly-additional"> <label for="saved-search-weekly-on-day" class="action-panel-label"> Which day? </label> <select id="saved-search-weekly-on-day" class="no-border-panel-selector saved-search-weekly-on-day"> <option value="Sunday">Sunday</option> <option value="Monday">Monday</option> <option value="Tuesday">Tuesday</option> <option value="Wednesday">Wednesday</option> <option value="Thursday">Thursday</option> <option value="Friday">Friday</option> <option value="Saturday">Saturday</option> </select> </div> <div class="action-panel-control-wrap"> <label for="saved-search-report" class="action-panel-label"> Report format: </label> <select id="saved-search-report" class="no-border-panel-selector saved-search-report"> <option value="DocSum">Summary</option> <option value="DocSumText">Summary (text)</option> <option value="Abstract">Abstract</option> <option value="AbstractText">Abstract (text)</option> <option value="MEDLINE">PubMed</option> </select> </div> <div class="action-panel-control-wrap"> <label for="saved-search-amount" class="action-panel-label"> Send at most: </label> <select id="saved-search-amount" class="no-border-panel-selector saved-search-amount"> <option value="1">1 item</option> <option value="5" selected>5 items</option> <option value="10">10 items</option> <option value="20">20 items</option> <option value="50">50 items</option> <option value="100">100 items</option> <option value="200">200 items</option> </select> </div> <div> <input type="checkbox" id="saved-search-send-if-no-result" class="saved-search-send-if-no-result" name="saved-search-send-if-no-result"> <label for="saved-search-send-if-no-result" class="action-panel-label smaller-checkbox"> Send even when there aren't any new results </label> </div> <div class="action-panel-control-wrap option-text-in-email-wrap"> <label for="saved-search-email-text" class="action-panel-label"> Optional text in email: </label> <textarea name="saved-search-email-text" id="saved-search-email-text" class="saved-search-email-text"></textarea> </div> </div> <div class="action-panel-actions"> <button class="action-panel-submit" type="submit" data-loading-label="Saving..." data-ga-category="save_share" data-ga-action="alert" data-ga-label="save"> Save </button> <button class="action-panel-cancel" aria-label="Close 'Your saved search' panel" ref="linksrc=close_saved_search_panel" aria-controls="saved-search-action-panel" aria-expanded="false" data-ga-category="save_share" data-ga-action="alert" data-ga-label="cancel"> Cancel </button> </div> </form> </div> </div> <div id="citation-manager-action-panel" class="citation-manager-action-panel action-panel" aria-hidden="true"> <div class="inner-wrap"> <h2 class="action-panel-heading"> Create a file for external citation management software </h2> <form id="citation-manager-action-panel-form" class="action-panel-content action-form" action="/results-export-ids/" data-by-search-action="/results-export-search-data/" data-by-ids-action="/results-export-ids/" method="post" data-by-search-method="post" data-by-ids-method="post"> <input type="hidden" name="csrfmiddlewaretoken" value="GixcXuIgsDHNtJcr4i2w4UXjp27fHuBABMrgHWAnhXJP0mv5erxHUO5cnpchI3g7"> <input name="results-format" type="hidden" value="pubmed"/> <div class="action-panel-actions"> <button class="action-panel-submit" type="submit" data-loading-label="Sending..." data-ga-category="save_share" data-ga-action="citation_manager" data-ga-label="save"> Create file </button> <button class="action-panel-cancel" aria-label="Close 'Send citations to citation manager' panel" ref="linksrc=close_citation_manager_panel" aria-controls="citation-manager-action-panel" aria-expanded="false" data-ga-category="save_share" data-ga-action="citation_manager" data-ga-label="cancel"> Cancel </button> </div> </form> </div> </div> <div id="rss-action-panel" class="rss-action-panel action-panel " aria-hidden="true"> <div class="inner-wrap"> <h2 class="action-panel-heading"> Your RSS Feed </h2> <form id="rss-action-panel-form" class="rss-action-panel-form action-panel-content action-form" data-create-rss-feed-url="/create-rss-feed-url/" data-search-form-term-value=""> <input type="hidden" name="csrfmiddlewaretoken" value="GixcXuIgsDHNtJcr4i2w4UXjp27fHuBABMrgHWAnhXJP0mv5erxHUO5cnpchI3g7"> <div class="action-panel-control-wrap"> <label for="rss-name" class="action-panel-label required-field-asterisk"> Name of RSS Feed: </label> <input maxlength="200" placeholder="Name" type="text" name="rss-name" id="rss-name" class="rss-name" value='' required pattern="[^"&=<>\/]*" title="The following characters are not allowed in the Name field: "&=<>/"> </div> <div class="rss-limit-wrap"> <div class="action-panel-control-wrap action-panel-extra-margin-top"> <label for="rss-limit" class="action-panel-label"> Number of items displayed: </label> <select id="rss-limit" class="no-border-panel-selector rss-limit"> <option value="5">5</option> <option value="10">10</option> <option value="15" selected="selected">15</option> <option value="20">20</option> <option value="50">50</option> <option value="100">100</option> </select> </div> </div> <div class="action-panel-actions"> <button class="action-panel-submit" type="submit" data-loading-label="Creating..." data-ga-category="save_share" data-ga-action="alert" data-ga-label="save"> Create RSS </button> 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</a></li><li><a class="search-in-nlm-catalog-link dropdown-block-link" ref ="linksrc=search_in_nlm_catalog_link" href="https://www.ncbi.nlm.nih.gov/nlmcatalog?term=%22IEEE+Trans+Image+Process%22%5BTitle+Abbreviation%5D" data-ga-category="search_catalog" data-ga-action="journal_link" data-ga-label="IEEE Trans Image Process"> Search in NLM Catalog </a></li><li><a class="add-to-search-link dropdown-block-link" ref ="linksrc=add_to_search_link" role="button" data-search-term='"IEEE Trans Image Process"[jour]' data-ga-category="search" data-ga-action="add_to_search" data-ga-label='"IEEE Trans Image Process"[jour]' href="#"> Add to Search </a></li></ul></div></div></div><span class="period">. </span><span class="cit">2012 Apr;21(4):1488-99.</span> </div> <span class="citation-doi"> doi: 10.1109/TIP.2011.2173206. </span> <span class="secondary-date"> Epub 2011 Oct 24. </span> </div> <h1 class="heading-title"> On the mathematical properties of the structural similarity index </h1> <div class="inline-authors"> <div class="authors"> <div class="authors-list"> <span class="authors-list-item "><a class="full-name" href="/?term=Brunet+D&cauthor_id=22042163" ref="linksrc=author_name_link" data-ga-category="search" data-ga-action="author_link" data-ga-label="Dominique Brunet">Dominique Brunet</a><sup class="affiliation-links"><span class="author-sup-separator"> </span><a class="affiliation-link" title="Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada. dbrunet@uwaterloo.ca" href="#full-view-affiliation-1" ref="linksrc=author_aff"> 1 </a></sup><span class="comma">, </span></span><span class="authors-list-item "><a class="full-name" href="/?term=Vrscay+ER&cauthor_id=22042163" ref="linksrc=author_name_link" data-ga-category="search" data-ga-action="author_link" data-ga-label="Edward R Vrscay">Edward R Vrscay</a><span class="comma">, </span></span><span class="authors-list-item "><a class="full-name" href="/?term=Wang+Z&cauthor_id=22042163" ref="linksrc=author_name_link" data-ga-category="search" data-ga-action="author_link" data-ga-label="Zhou Wang">Zhou Wang</a></span> </div> </div> </div> <div class="short-article-details"> Affiliations <button class="more-details" id="toggle-authors" data-alt-text="Collapse" aria-controls="expanded-authors" aria-expanded="false" data-pinger-ignore data-fetch-url="/22042163/long-authors/" data-id-prefix=""> <span class="title"> Expand </span> </button> </div> <div class="extended-article-details" id="expanded-authors" aria-hidden="true"> <div class="expanded-authors" id="full-view-expanded-authors"> <div class="affiliations"> <h3 class="title"> Affiliation </h3> <ul class="item-list"> <li data-affiliation-id="full-view-affiliation-1" id="full-view-affiliation-1" ><sup class="key">1</sup> Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada. dbrunet@uwaterloo.ca</li> </ul> </div> </div> </div> <ul class="identifiers" id="full-view-identifiers"> <li> <span class="identifier pubmed"> <span class="id-label"> PMID: </span> <strong class="current-id" title="PubMed ID">22042163</strong> </span> </li> <li> <span class="identifier doi"> <span class="id-label"> DOI: </span> <a class="id-link" target="_blank" rel="noopener" ref="linksrc=article_id_link&article_id=10.1109/TIP.2011.2173206&id_type=DOI" href="https://doi.org/10.1109/tip.2011.2173206" data-ga-category="full_text" data-ga-action="DOI" > 10.1109/TIP.2011.2173206 </a> </span> </li> </ul> <div class="in-clipboard-label " hidden> Item in Clipboard </div> </div> <div class="short-view" id="short-view-heading"> <h1 class="heading-title"> On the mathematical properties of the structural similarity index </h1> <div class="short-citation"> <span class="authors-list"> <span class="authors-list-item "><span class="full-name">Dominique Brunet</span><span class="citation-separator"> et al.</span></span> </span> <span class="citation-journal"> IEEE Trans Image Process<span class="citation-separator">.</span> </span> <span class="date"> <span class="citation-year">2012 Apr</span><span class="citation-separator">.</span> </span> </div> <div class="more-article-details-section with-display-settings"> <button class="more-article-details" id="toggle-article-details" data-alt-text="Hide details" ref="linksrc=toggle_article_details" aria-controls="full-article-details" aria-expanded="false" data-fetch-url="/22042163/long-authors/" data-id-prefix=""> <span class="title">Show details</span> </button> <div class="display-options" data-format-key="format" data-default-format-value="abstract" data-current-format-value="abstract" role="region"> <button class="trigger usa-button" title="Change format" data-ga-category="display_options" data-ga-action="display_options_dialog"> <span class="button-label">Display options</span> </button> <div class="dropdown" hidden> <div class="title"> Display options </div> <div class="content"> <div class="format-container"> <span 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class="journal-actions-dropdown dropdown dropdown-container" hidden aria-label="Dropdown menu for journal IEEE transactions on image processing : a publication of the IEEE Signal Processing Society"><div class="title">Actions</div><div class="content"><ul class="journal-actions-links"><li><a class="search-in-pubmed-link dropdown-block-link" href="/?term=%22IEEE+Trans+Image+Process%22%5Bjour%5D&sort=date&sort_order=desc" data-href="/?term=%22IEEE+Trans+Image+Process%22%5Bjour%5D&sort=date&sort_order=desc" ref="linksrc=search_in_pubmed_journal_name_link&journal_abbrev=IEEE Trans Image Process" data-ga-category="search" data-ga-action="journal_link" data-ga-label="IEEE Trans Image Process"> Search in PubMed </a></li><li><a class="search-in-nlm-catalog-link dropdown-block-link" ref ="linksrc=search_in_nlm_catalog_link" href="https://www.ncbi.nlm.nih.gov/nlmcatalog?term=%22IEEE+Trans+Image+Process%22%5BTitle+Abbreviation%5D" data-ga-category="search_catalog" data-ga-action="journal_link" data-ga-label="IEEE Trans Image Process"> Search in NLM Catalog </a></li><li><a class="add-to-search-link dropdown-block-link" ref ="linksrc=add_to_search_link" role="button" data-search-term='"IEEE Trans Image Process"[jour]' data-ga-category="search" data-ga-action="add_to_search" data-ga-label='"IEEE Trans Image Process"[jour]' href="#"> Add to Search </a></li></ul></div></div></div><span class="period">. </span><span class="cit">2012 Apr;21(4):1488-99.</span> </div> <span class="citation-doi"> doi: 10.1109/TIP.2011.2173206. </span> <span class="secondary-date"> Epub 2011 Oct 24. </span> </div> <div class="expanded-authors" id="short-view-expanded-authors"> <div class="authors"> <h3 class="title"> Authors </h3> <div class="authors-list"> <span class="authors-list-item "><a class="full-name" href="/?term=Brunet+D&cauthor_id=22042163" ref="linksrc=author_name_link" data-ga-category="search" data-ga-action="author_link" data-ga-label="Dominique Brunet">Dominique Brunet</a><sup class="affiliation-links"><span class="author-sup-separator"> </span><a class="affiliation-link" title="Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada. dbrunet@uwaterloo.ca" href="#short-view-affiliation-1" ref="linksrc=author_aff"> 1 </a></sup><span class="comma">, </span></span><span class="authors-list-item "><a class="full-name" href="/?term=Vrscay+ER&cauthor_id=22042163" ref="linksrc=author_name_link" data-ga-category="search" data-ga-action="author_link" data-ga-label="Edward R Vrscay">Edward R Vrscay</a><span class="comma">, </span></span><span class="authors-list-item "><a class="full-name" href="/?term=Wang+Z&cauthor_id=22042163" ref="linksrc=author_name_link" data-ga-category="search" data-ga-action="author_link" data-ga-label="Zhou Wang">Zhou Wang</a></span> </div> </div> <div class="affiliations"> <h3 class="title"> Affiliation </h3> <ul class="item-list"> <li data-affiliation-id="short-view-affiliation-1" id="short-view-affiliation-1" ><sup class="key">1</sup> Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada. dbrunet@uwaterloo.ca</li> </ul> </div> </div> <ul class="identifiers" id="short-view-identifiers"> <li> <span class="identifier pubmed"> <span class="id-label"> PMID: </span> <strong class="current-id" title="PubMed ID">22042163</strong> </span> </li> <li> <span class="identifier doi"> <span class="id-label"> DOI: </span> <a class="id-link" target="_blank" rel="noopener" ref="linksrc=article_id_link&article_id=10.1109/TIP.2011.2173206&id_type=DOI" href="https://doi.org/10.1109/tip.2011.2173206" data-ga-category="full_text" data-ga-action="DOI" > 10.1109/TIP.2011.2173206 </a> </span> </li> </ul> <div class="in-clipboard-label " hidden> Item in Clipboard </div> </div> </div> </header> <div class="actions-buttons inline"><div class="inner-wrap"><button class="full-text-links-button full-text-links-dialog-trigger" title="Open dialog with full text links" ref="linksrc=full_text_links"> Full text links </button><button class="citation-button citation-dialog-trigger" aria-label="Open dialog with citation text in different styles" data-ga-category="save_share" data-ga-action="cite" data-ga-label="open" data-all-citations-url="/22042163/citations/" data-citation-style="nlm" data-pubmed-format-link="/22042163/export/"><span class="button-label">Cite</span></button><button class="more-actions-button more-actions-dialog-trigger" title="Open dialog with more actions to take" ref="linksrc=more_actions_btn"></button><div class="display-options" data-format-key="format" data-default-format-value="abstract" data-current-format-value="abstract" role="region"><button class="trigger usa-button" title="Change format" data-ga-category="display_options" data-ga-action="display_options_dialog"><span class="button-label">Display options</span></button><div class="dropdown" hidden><div class="title"> Display options </div><div class="content"><div class="format-container"><span class="option-label"> Format </span><select name="format" data-initial-value="abstract" aria-label="Display format" id="inline-actions-display-format"><option value="abstract">Abstract</option><option value="pubmed">PubMed</option><option value="pmid">PMID</option></select></div></div></div></div></div></div> <div class="abstract" id="abstract"> <h2 class="title"> Abstract </h2> <div class="abstract-content selected" id="eng-abstract"> <p> Since its introduction in 2004, the structural similarity (SSIM) index has gained widespread popularity as a tool to assess the quality of images and to evaluate the performance of image processing algorithms and systems. There has been also a growing interest of using SSIM as an objective function in optimization problems in a variety of image processing applications. One major issue that could strongly impede the progress of such efforts is the lack of understanding of the mathematical properties of the SSIM measure. For example, some highly desirable properties such as convexity and triangular inequality that are possessed by the mean squared error may not hold. In this paper, we first construct a series of normalized and generalized (vector-valued) metrics based on the important ingredients of SSIM. We then show that such modified measures are valid distance metrics and have many useful properties, among which the most significant ones include quasi-convexity, a region of convexity around the minimizer, and distance preservation under orthogonal or unitary transformations. The groundwork laid here extends the potentials of SSIM in both theoretical development and practical applications. </p> </div> </div> <p class="disclaimer" id="disclaimer"> <a href="/disclaimer/" target="_blank" ga_category="literature_resources" ga_action="disclaimer_link" ga_label="under_abstract">PubMed Disclaimer</a> </p> <div class="similar-articles" id="similar"> <h2 class="title"> Similar articles </h2> <ul class="articles-list" id="similar-articles-list"> <li class="full-docsum"> <div class="docsum-content" style="display:inline;"> <a class="docsum-title" href="/18482882/" ref="article_id=18482882&linksrc=similar_articles_link&ordinalpos=1" data-ga-category="similar_article" data-ga-action="18482882" data-ga-label=""> Design of linear equalizers optimized for the structural similarity index. </a> <div class="docsum-citation full-citation"> <span class="docsum-authors full-authors">Channappayya SS, Bovik AC, Caramanis C, Heath RW Jr.</span> <span class="docsum-authors 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On the mathematical properties of the structural similarity index - PubMed