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<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="No"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>Ncorr: Open-Source 2D Digital Image Correlation Matlab Software | Experimental Mechanics</title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="Ncorr: Open-Source 2D Digital Image Correlation Matlab Software"/> <meta name="twitter:description" content="Experimental Mechanics - Digital Image Correlation (DIC) is an important and widely used non-contact technique for measuring material deformation. Considerable progress has been made in recent..."/> <meta name="twitter:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs11340-015-0009-1/MediaObjects/11340_2015_9_Fig1_HTML.gif"/> <meta name="journal_id" content="11340"/> <meta name="dc.title" content="Ncorr: Open-Source 2D Digital Image Correlation Matlab Software"/> <meta name="dc.source" content="Experimental Mechanics 2015 55:6"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2015-03-31"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2015 Society for Experimental Mechanics"/> <meta name="dc.rights" content="2015 Society for Experimental Mechanics"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="Digital Image Correlation (DIC) is an important and widely used non-contact technique for measuring material deformation. Considerable progress has been made in recent decades in both developing new experimental DIC techniques and in enhancing the performance of the relevant computational algorithms. Despite this progress, there is a distinct lack of a freely available, high-quality, flexible DIC software. This paper documents a new DIC software package Ncorr that is meant to fill that crucial gap. Ncorr is an open-source subset-based 2D DIC package that amalgamates modern DIC algorithms proposed in the literature with additional enhancements. 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Considerable progress has been made in recent decades in both developing new experimental DIC techniques and in enhancing the performance of the relevant computational algorithms. Despite this progress, there is a distinct lack of a freely available, high-quality, flexible DIC software. This paper documents a new DIC software package Ncorr that is meant to fill that crucial gap. Ncorr is an open-source subset-based 2D DIC package that amalgamates modern DIC algorithms proposed in the literature with additional enhancements. 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Considerable progress has been made in recent decades in both developing new experimental DIC techniques and in enhancing the performance of the relevant computational algorithms. Despite this progress, there is a distinct lack of a freely available, high-quality, flexible DIC software. This paper documents a new DIC software package Ncorr that is meant to fill that crucial gap. Ncorr is an open-source subset-based 2D DIC package that amalgamates modern DIC algorithms proposed in the literature with additional enhancements. 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Blaber","affiliation":[{"name":"The Woodruff School of Mechanical Engineering","address":{"name":"The Woodruff School of Mechanical Engineering, Atlanta, USA","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"B. Adair","affiliation":[{"name":"The Woodruff School of Mechanical Engineering","address":{"name":"The Woodruff School of Mechanical Engineering, Atlanta, USA","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"A. Antoniou","affiliation":[{"name":"The Woodruff School of Mechanical Engineering","address":{"name":"The Woodruff School of Mechanical Engineering, Atlanta, USA","@type":"PostalAddress"},"@type":"Organization"}],"email":"antonia.antoniou@me.gatech.edu","@type":"Person"}],"isAccessibleForFree":false,"hasPart":{"isAccessibleForFree":false,"cssSelector":".main-content","@type":"WebPageElement"},"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" >  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>   <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-visually-hidden" aria-hidden="true" data-test="darwin-icons"> <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE svg PUBLIC 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Blaber</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-B_-Adair-Aff1" data-author-popup="auth-B_-Adair-Aff1" data-author-search="Adair, B.">B. Adair</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup> & </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-A_-Antoniou-Aff1" data-author-popup="auth-A_-Antoniou-Aff1" data-author-search="Antoniou, A." data-corresp-id="c1">A. Antoniou<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff1">1</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>19k <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-citations-medium"></use> </svg>1359 <span class="app-article-metrics-bar__label">Citations</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-altmetric-medium"></use> </svg>3 <span class="app-article-metrics-bar__label">Altmetric</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1007/s11340-015-0009-1/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>Digital Image Correlation (DIC) is an important and widely used non-contact technique for measuring material deformation. Considerable progress has been made in recent decades in both developing new experimental DIC techniques and in enhancing the performance of the relevant computational algorithms. Despite this progress, there is a distinct lack of a freely available, high-quality, flexible DIC software. This paper documents a new DIC software package Ncorr that is meant to fill that crucial gap. Ncorr is an open-source subset-based 2D DIC package that amalgamates modern DIC algorithms proposed in the literature with additional enhancements. 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The fracture toughness test described in this paper was performed at the Mechanical Properties Research Lab at Georgia Tech.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">The Woodruff School of Mechanical Engineering, 801 Ferst Drive, Atlanta, GA, 30332, USA</p><p class="c-article-author-affiliation__authors-list">J. Blaber, B. Adair & A. Antoniou</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-J_-Blaber-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">J. 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Adair</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?dc.creator=B.%20Adair" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=B.%20Adair" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22B.%20Adair%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-A_-Antoniou-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">A. Antoniou</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?dc.creator=A.%20Antoniou" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=A.%20Antoniou" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22A.%20Antoniou%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:antonia.antoniou@me.gatech.edu">A. Antoniou</a>.</p></div></div></section><section data-title="Electronic supplementary material"><div class="c-article-section" id="Sec19-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec19">Electronic supplementary material</h2><div class="c-article-section__content" id="Sec19-content"><div data-test="supplementary-info"><div id="figshareContainer" class="c-article-figshare-container" data-test="figshare-container"></div><p>Below is the link to the electronic supplementary material. </p><div class="c-article-supplementary__item" data-test="supp-item" id="MOESM1"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="esm 1" href="https://static-content.springer.com/esm/art%3A10.1007%2Fs11340-015-0009-1/MediaObjects/11340_2015_9_MOESM1_ESM.pdf" data-supp-info-image="">ESM 1</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p>(PDF 5055 kb)</p></div></div></div></div></div></section><section aria-labelledby="appendices"><div class="c-article-section" id="appendices-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="appendices">Appendices</h2><div class="c-article-section__content" id="appendices-content"><h3 class="c-article__sub-heading" id="App2">Appendix A1. The Gradient and Hessian Quantities</h3><p>In order to simplify the calculations, the following assumptions are used</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{d}{d\boldsymbol{p}}\left({f}_m\right)\approx 0 $$</span></div><div class="c-article-equation__number"> (19) </div></div> <div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{d}{d\boldsymbol{p}}\left(\sqrt{{{\displaystyle \sum \left[f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{ref};\mathbf{0}\right)\right)-{f}_m\right]}}^2}\right)\approx 0 $$</span></div><div class="c-article-equation__number"> (20) </div></div> <p>The gradient for the IC-GN method based on equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ9">10</a>) is</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{l}\nabla {C}_{LS}\left(\mathbf{0}\right)=\frac{d{C}_{LS}\left(\mathbf{0}\right)}{d\boldsymbol{p}}\hfill \\ {}\approx \frac{2}{\sqrt{{{\displaystyle \sum \left[f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m\right]}}^2}}{\displaystyle \sum \left[\left[\frac{f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m}{\sqrt{{\displaystyle \sum {\left[f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m\right]}^2}}}\right.\right.}\hfill \\ {}\left.\left.-\frac{g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right)-{g}_m}{\sqrt{{{\displaystyle \sum \left[g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right)-{g}_m\right]}}^2}}\right]\left[\frac{d}{d\boldsymbol{p}}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)\right]\right]\hfill \end{array} $$</span></div><div class="c-article-equation__number"> (21) </div></div> <p>The hessian is</p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{l}\nabla \nabla {C}_{LS}\left(\mathbf{0}\right)=\frac{d^2{C}_{LS}\left(\mathbf{0}\right)}{d{\boldsymbol{p}}^2}\hfill \\ {}\approx \frac{2}{\sqrt{{\displaystyle \sum {\left[f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m\right]}^2}}}\left\{{\displaystyle \sum \left[\frac{\frac{d}{d\boldsymbol{p}}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)}{\sqrt{{\displaystyle \sum {\left[f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m\right]}^2}}}\right]}\left[\frac{d}{d\boldsymbol{p}}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+\right.\right.\right.\hfill \\ {}{\left.\left.w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)\right]}^T+{\displaystyle \sum \left[\frac{f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};0\right)\right)-{f}_m}{\sqrt{{\displaystyle \sum \left[f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m\right]}}}-\frac{g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right)-{g}_m}{\sqrt{{{\displaystyle \sum \left[g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right)-{g}_m\right]}}^2}}\right]}\hfill \\ {}\left.\left[\frac{d^2}{d\boldsymbol{p}}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)\right]\right\}\hfill \end{array} $$</span></div><div class="c-article-equation__number"> (22) </div></div> <p>Using the Gauss-Newton assumption</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \sum \left[\frac{f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m}{{\displaystyle \sum {\left[f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m\right]}^2}}-\frac{g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right)-{g}_m}{\sqrt{{\displaystyle \sum {\left[g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right)-{g}_m\right]}^2}}}\right]\left[\frac{d^2}{d{\boldsymbol{p}}^2}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)\right]\approx 0} $$</span></div><div class="c-article-equation__number"> (23) </div></div><p>yields the hessian in the final form</p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{c}\hfill \nabla \nabla {C}_{LS}\left(\mathbf{0}\right)\approx \frac{d{C}_{LS}\left(\mathbf{0}\right)}{d{\boldsymbol{p}}^2}\hfill \\ {}\hfill \approx \frac{2}{{\displaystyle \sum {\left[f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)-{f}_m\right]}^2}}{\displaystyle \sum \left[\frac{d}{d\boldsymbol{p}}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}\right.\right.}\hfill \\ {}\hfill \left.\left.+\kern0.5em w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)\right]{\left[\frac{d}{d\boldsymbol{p}}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right)\right]}^T\hfill \end{array} $$</span></div><div class="c-article-equation__number"> (24) </div></div> <h3 class="c-article__sub-heading" id="App2">Appendix A2. Biquintic B-Spline Interpolation</h3><p>The quantities <span class="mathjax-tex">$ \frac{d}{d\boldsymbol{p}}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right) $</span> and <span class="mathjax-tex">$ g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right) $</span> require some form of estimation through interpolation. Using the chain rule on <span class="mathjax-tex">$ \frac{d}{d\boldsymbol{p}}f\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};\mathbf{0}\right)\right) $</span> and equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ4">4</a>), we obtain:</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{d}{d\boldsymbol{p}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right)=\frac{\partial }{\partial {\tilde{x}}_{re{f}_i}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right)\ast \frac{d{\tilde{x}}_{re{f}_i}}{d\boldsymbol{p}}+\frac{\partial }{\partial {\tilde{y}}_{re{f}_j}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right)\ast \frac{d{\tilde{y}}_{re{f}_j}}{d\boldsymbol{p}} $$</span></div><div class="c-article-equation__number"> (25) </div></div> <p>The only two quantities we need to specifically compute for equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ24">25</a>) are <span class="mathjax-tex">$ \frac{\partial }{\partial {\tilde{x}}_{re{f}_i}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right) $</span> and <span class="mathjax-tex">$ \frac{\partial }{\partial {\tilde{y}}_{re{f}_j}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right) $</span>. These can be computed in various ways (sobel filter, finite difference, etc.), but in Ncorr, biquintic B-spline interpolation is used.</p><p>The quantity <span class="mathjax-tex">$ g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right) $</span> also requires interpolation. Once <span class="mathjax-tex">$ \frac{\partial }{\partial {\tilde{x}}_{re{f}_i}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right) $</span> and <span class="mathjax-tex">$ \frac{\partial }{\partial {\tilde{y}}_{re{f}_j}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right) $</span> are precomputed for the entire reference image, and <span class="mathjax-tex">$ g\left({\boldsymbol{\xi}}_{{\mathrm{ref}}_{\mathrm{c}}}+w\left(\varDelta {\boldsymbol{\xi}}_{\boldsymbol{ref}};{\boldsymbol{p}}_{\boldsymbol{old}}\right)\right) $</span> is computable, equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ18">19</a>) and equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ21">22</a>) can be computed and iterated with equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ9">10</a>) to find a closer approximation to <b>p</b> <span class="c-stack"> <sup>∗</sup><sub>rc</sub> </span>.</p><p>The main idea behind B-spline interpolation is to approximate the image grayscale surface with a linear combination of B-spline basis “splines.” These splines are scaled via the B-spline coefficients and then the linear combination of these scaled splines forms an approximation of the surface. Once this approximation is complete, points can be interpolated through 1-D convolutions (since biquintic B-spline interpolation is separable [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 49" title="Tevenaz P (2000) Interpolation revisited. IEEE Trans Med Imaging 19(7):739–758" href="/article/10.1007/s11340-015-0009-1#ref-CR49" id="ref-link-section-d155616590e13709">49</a>]), which reduces to a series of simple dot products. The equation for interpolation for the 1D case is</p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ g(x)={\displaystyle \sum_{k\in Z}c(k){\beta}^n\left(x-k\right)} $$</span></div><div class="c-article-equation__number"> (26) </div></div><p>where c(k),β<sup>n</sup>(x − k), and g(x) are the B-spline coefficient value at integer k, the B-spline kernel value at x − k, and the interpolated signal value at x, respectively. Here n is the B-spline kernel order, which is set to 5 (the quintic kernel) and Z is the set of integers. Note that the B-spline coefficients are not equivalent to the data samples (unlike in other forms of interpolation—i.e. bicubic keys [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Keys RG (1981) Cubic convolution interpolation for digital image processing. IEEE Trans Acoust Speech Signal Process 29(6):1153–1160" href="/article/10.1007/s11340-015-0009-1#ref-CR50" id="ref-link-section-d155616590e13789">50</a>]), and thus must be solved for directly. The equation for the B-spline kernel is</p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\beta}^n(x)=\frac{1}{n!}{\displaystyle \sum_{k=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill k\hfill \end{array}\right){\left(-1\right)}^k{\left(x-k+\frac{n+1}{2}\right)}_{+}^n} $$</span></div><div class="c-article-equation__number"> (27) </div></div> <p>When solved for the quintic case, this equation yields:</p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\beta}^5(x)=\left\{\begin{array}{ll}\frac{1}{120}{x}^5+\frac{1}{8}{x}^4+\frac{3}{4}{x}^3+\frac{9}{4}{x}^2+\frac{27}{8}x+\frac{81}{40}\hfill & -2\ge x\ge -3\hfill \\ {}-\frac{1}{24}{x}^5-\frac{3}{8}{x}^4-\frac{5}{4}{x}^3-\frac{7}{4}{x}^2-\frac{5}{8}{x}^2+\frac{17}{40}\hfill & -1\ge x\ge -2\hfill \\ {}\frac{1}{12}{x}^5+\frac{1}{4}{x}^4-\frac{1}{2}{x}^2+\frac{11}{20}\hfill & 0\kern0.5em x\ge -1\hfill \\ {}-\frac{1}{12}{x}^5-\frac{3}{8}{x}^4+\frac{5}{4}{x}^3-\frac{7}{4}{x}^2+\frac{5}{8}x+\frac{17}{40}\hfill & 1\kern0.5em x\ge 1\hfill \\ {}\frac{1}{24}{x}^5-\frac{3}{8}{x}^4+\frac{5}{4}{x}^3-\frac{7}{4}{x}^2+\frac{5}{8}x+\frac{17}{40}\hfill & 2\kern0.5em x\ge 1\hfill \\ {}-\frac{1}{120}{x}^5+\frac{1}{8}{x}^4-\frac{3}{4}{x}^3+\frac{9}{4}{x}^2-\frac{27}{8}x+\frac{81}{40}\hfill & 30x\ge 2\hfill \end{array}\right. $$</span></div><div class="c-article-equation__number"> (28) </div></div> <p>The first step of the interpolation process is to determine the B-spline coefficients. They can be found by using deconvolution. Applying Discrete Fourier Transform (DFT) to equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ25">26</a>) yields:</p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F\left\{g\right\}=F\left\{c\right\}\ast F\left\{{\beta}^{\mathrm{n}}\right\} $$</span></div><div class="c-article-equation__number"> (29) </div></div><p>where <i>F</i>{…} is the DFT. The goal is to solve for c, the B-spline coefficients. This can be done by dividing the Fourier coefficients of the B-spline kernel element-wise with the Fourier coefficients of the signal as shown:</p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F\left\{c\right\}=\frac{F\left\{{\beta}^{\mathrm{n}}\right\}}{F\left\{g\right\}} $$</span></div><div class="c-article-equation__number"> (30) </div></div> <p>Taking the inverse DFT of equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ29">30</a>) will then yield the B-spline coefficients, although caution should be exercised when using this method due to the circular nature of the DFT. To mitigate wrap-around errors, padding should be used.</p><p>After obtaining the B-spline coefficients, the image array can be interpolated point-wise by using equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ25">26</a>). This is carried out by taking a series of dot products with the columns of the B-spline coefficient array and B-spline kernel, and then taking a single dot product across the resulting row of interpolated B-spine coefficient values (note that the order of this operation doesn’t matter). The first step of the aforementioned process can be thought of as interpolating the 2D B-spline grid to obtain 1D B-spline coefficient values, and then the second step as interpolating the grayscale value from these 1D B-spline coefficient values.</p><p>The steps for obtaining the B-spline coefficients are outlined below:</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>Make a copy of the grayscale array and pad it (any method can be used; Ncorr uses the border values to expand the data as shown in the top right of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s11340-015-0009-1#Fig14">14(a)</a>). Then, sample the B-spline kernel at −2,−1,0,1, 2, and 3. This will form the quintic B-spline vector</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-14" data-title="Fig. 14"><figure><figcaption><b id="Fig14" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 14</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s11340-015-0009-1/figures/14" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs11340-015-0009-1/MediaObjects/11340_2015_9_Fig14_HTML.gif?as=webp"><img aria-describedby="Fig14" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs11340-015-0009-1/MediaObjects/11340_2015_9_Fig14_HTML.gif" alt="figure 14" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-14-desc"><p>(<b>a</b>) Schematic of the B-spline coefficient calculation. <i>Top-left</i>: original grayscale values array. <i>Top-right</i>: Copy and pad data; padding parameter here is set to 2. <i>Bottom-left</i>: Deconvolution via the DFT for each row and then each column. <i>Bottom-right</i>: The associated B-spline coefficients for the top-left image. (<b>b</b>) Extension of (<b>a</b>) with the same gray scale and B-spline coefficients. <i>Black crosses</i> represent integer pixel locations and the <i>black circle</i> (<i>top-left</i>) is the subpixel point being interpolated</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s11340-015-0009-1/figures/14" data-track-dest="link:Figure14 Full size image" aria-label="Full size image figure 14" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\mathrm{b}}_{\mathrm{o}}={\left\{\begin{array}{llllll}1/120\hfill & 13/60\hfill & 11/20\hfill & 13/60\hfill & 1/120\hfill & 0\hfill \end{array}\right\}}^{\mathrm{T}} $$</span></div><div class="c-article-equation__number"> (31) </div></div> </li> <li> <span class="u-custom-list-number">2.</span> <p>Pad the kernel with zeros to the same size as the number of columns (the width) of the image grayscale array. Take the FFT of the padded kernel, and then store it in place.</p> </li> <li> <span class="u-custom-list-number">3.</span> <p>Take the FFT of an image row, then <i>divide</i> the Fourier coefficients element-wise of the padded B-spline with the Fourier coefficients from the image row. Afterward, take the inverse FFT of the results and store them in place (in the padded grayscale array). This is done for all the image rows as shown on the bottom right of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s11340-015-0009-1#Fig14">14(a)</a>.</p> </li> <li> <span class="u-custom-list-number">4.</span> <p>Repeat steps 2–3, except column-wise, with the array obtained at the end of step 3. The result will be the B-spline coefficients of original image array as shown on the bottom right of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s11340-015-0009-1#Fig14">14(a)</a>.</p> </li> </ol> <p>Now that the B-spline coefficients have been obtained, we can interpolate values at sub pixel locations. The steps are outlined below:</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>Pick a subpixel point, <span class="mathjax-tex">$ \left({\tilde{x}}_{cur},{\tilde{y}}_{cur}\right) $</span>, within the image array to interpolate.</p> </li> <li> <span class="u-custom-list-number">2.</span> <p>Calculate Δx and Δy, where:</p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{c}\hfill \varDelta x={\tilde{x}}_{cur}-{x}_f\hfill \\ {}\hfill \varDelta y={\tilde{y}}_{cur}-{y}_f\hfill \end{array} $$</span></div><div class="c-article-equation__number"> (32) </div></div><p>where x<sub>f</sub> = floor(<span class="mathjax-tex">$ {\tilde{x}}_{cur} $</span>) and y<sub>f</sub> = floor(<i>ỹ</i> <sub> <i>cur</i> </sub>).</p> </li> <li> <span class="u-custom-list-number">3.</span> <p>Perform the operation in equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ31">32</a>) to obtain the interpolated grayscale value.</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ g\left({\tilde{x}}_{cur},{\tilde{y}}_{cur}\right)=\left[\begin{array}{llllll}1\hfill & \varDelta y\hfill & \varDelta {y}^2\hfill & \varDelta {y}^3\hfill & \varDelta {y}^4\hfill & \varDelta {y}^5\hfill \end{array}\right]\left[QK\right]\left[c\right]{}_{\left({x}_f-2:{x}_f+3,{y}_f-2:{y}_f+3\right)}{\left[QK\right]}^T\left[\begin{array}{c}\hfill 1\hfill \\ {}\hfill \varDelta x\hfill \\ {}\hfill \varDelta {x}^2\hfill \\ {}\hfill \varDelta {x}^3\hfill \\ {}\hfill \varDelta {x}^4\hfill \\ {}\hfill \varDelta {x}^5\hfill \end{array}\right] $$</span></div><div class="c-article-equation__number"> (33) </div></div><p>where [QK] is the array defined:</p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \left[QK\right]=\left[\begin{array}{cccccc}\hfill \frac{1}{120}\hfill & \hfill \frac{13}{60}\hfill & \hfill \frac{11}{20}\hfill & \hfill \frac{13}{60}\hfill & \hfill \frac{1}{120}\hfill & \hfill 0\hfill \\ {}\hfill -\frac{1}{24}\hfill & \hfill -\frac{5}{12}\hfill & \hfill 0\hfill & \hfill \frac{5}{12}\hfill & \hfill \frac{1}{24}\hfill & \hfill 0\hfill \\ {}\hfill \frac{1}{12}\hfill & \hfill \frac{1}{6}\hfill & \hfill -\frac{1}{2}\hfill & \hfill \frac{1}{6}\hfill & \hfill \frac{1}{12}\hfill & \hfill 0\hfill \\ {}\hfill -\frac{1}{12}\hfill & \hfill \frac{1}{6}\hfill & \hfill 0\hfill & \hfill -\frac{1}{6}\hfill & \hfill \frac{1}{12}\hfill & \hfill 0\hfill \\ {}\hfill \frac{1}{24}\hfill & \hfill -\frac{1}{6}\hfill & \hfill \frac{1}{4}\hfill & \hfill -\frac{1}{6}\hfill & \hfill \frac{1}{24}\hfill & \hfill 0\hfill \\ {}\hfill -\frac{1}{120}\hfill & \hfill \frac{1}{24}\hfill & \hfill -\frac{1}{12}\hfill & \hfill \frac{1}{12}\hfill & \hfill -\frac{1}{24}\hfill & \hfill \frac{1}{120}\hfill \end{array}\right] $$</span></div><div class="c-article-equation__number"> (34) </div></div><p>and<span class="mathjax-tex">$ {\left[c\right]}_{\left({x}_f-2:{x}_f+3,{y}_f-2:{y}_f+3\right)} $</span> are the B-spline coefficients as shown:</p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\left[c\right]}_{\left({x}_f-2:{x}_f+3,{y}_f-2:{y}_f+3\right)}=\left[\begin{array}{cccccc}\hfill {c}_{\left({x}_f-2,{y}_f-2\right)}\hfill & \hfill {c}_{\left({x}_f-1,{y}_f-2\right)}\hfill & \hfill {c}_{\left({x}_f,{y}_f-2\right)}\hfill & \hfill {c}_{\left({x}_f+1,{y}_f-22\right)}\hfill & \hfill {c}_{\left({x}_f+2,{y}_f-2\right)}\hfill & \hfill {c}_{\left({x}_f+3,{y}_f-2\right)}\hfill \\ {}\hfill {c}_{\left({x}_f-2,{y}_f-1\right)}\hfill & \hfill {c}_{\left({x}_f-1,{y}_f-1\right)}\hfill & \hfill {c}_{\left({x}_f+1,{y}_f-1\right)}\hfill & \hfill {c}_{\left({x}_f+1,{y}_f-1\right)}\hfill & \hfill {c}_{\left({x}_f+2,{y}_f-1\right)}\hfill & \hfill {c}_{\left({x}_f+3,{y}_f-1\right)}\hfill \\ {}\hfill {c}_{\left({x}_f-2,{y}_f\right)}\hfill & \hfill {c}_{\left({x}_f-1,{y}_f\right)}\hfill & \hfill {c}_{\left({x}_f,{y}_f\right)}\hfill & \hfill {c}_{\left({x}_f+1,{y}_f\right)}\hfill & \hfill {c}_{\left({x}_f+2,{y}_f\right)}\hfill & \hfill {c}_{\left({x}_f+3,{y}_f\right)}\hfill \\ {}\hfill {c}_{\left({x}_f-2,{y}_f+1\right)}\hfill & \hfill {c}_{\left({x}_f-1,{y}_f+1\right)}\hfill & \hfill {c}_{\left({x}_f+1,{y}_f+1\right)}\hfill & \hfill {c}_{\left({x}_f+1,{y}_f+1\right)}\hfill & \hfill {c}_{\left({x}_f+2,{y}_f+1\right)}\hfill & \hfill {c}_{\left({x}_f+3,{y}_f+1\right)}\hfill \\ {}\hfill {c}_{\left({x}_f-2,{y}_f+2\right)}\hfill & \hfill {c}_{\left({x}_f-1,{y}_f+2\right)}\hfill & \hfill {c}_{\left({x}_f,{y}_f+2\right)}\hfill & \hfill {c}_{\left({x}_f+1,{y}_f+2\right)}\hfill & \hfill {c}_{\left({x}_f+2,{y}_f+2\right)}\hfill & \hfill {c}_{\left({x}_f+3,{y}_f+2\right)}\hfill \\ {}\hfill {c}_{\left({x}_f-2,{y}_f+3\right)}\hfill & \hfill {c}_{\left({x}_f-1,{y}_f+3\right)}\hfill & \hfill {c}_{\left({x}_f,{y}_f+3\right)}\hfill & \hfill {c}_{\left({x}_f+1,{y}_f+3\right)}\hfill & \hfill {c}_{\left({x}_f+2,{y}_f+3\right)}\hfill & \hfill {c}_{\left({x}_f+3,{y}_f+3\right)}\hfill \end{array}\right] $$</span></div><div class="c-article-equation__number"> (35) </div></div> </li> </ol> <p>The position of the required B-spline coefficients within the B-spline array ultimately depends on the amount of padding used. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s11340-015-0009-1#Fig14">14(b)</a> gives an example of the location of the coefficients within the B-spline array for a given x<sub>f</sub> and y<sub>f</sub> and a padding of 2.</p><p>The left portion containing the ∆y vector and the [QK] matrix is the matrix form of resampling the quintic B-spline kernel with a shift of ∆y. Right multiplying this quantity by <span class="mathjax-tex">$ {\left[c\right]}_{\left({x}_f=2:{x}_f+3,{y}_f-2:{y}_f+3\right)} $</span> yields the interpolated B-spline coefficients which form a row of values, as shown in the top right of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s11340-015-0009-1#Fig14">14(b)</a>. When this quantity is right multiplied by [QK] and the ∆x vector, it interpolates the gray-scale value we need from the interpolated row of B-spline coefficients as shown on the bottom left of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s11340-015-0009-1#Fig14">14(b)</a>.</p><p>Lastly, examining the portion central portion containing:</p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \left[QK\right]{\left[c\right]}_{\left({x}_f-2:{x}_f+3,{y}_f-2:{y}_f+3\right)}{\left[QK\right]}^T $$</span></div><div class="c-article-equation__number"> (36) </div></div><p>this term can be precomputed to increase the speed of the program [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 37" title="Pan B, Li K (2011) A fast digital image correlation method for deformation measurement. Opt Lasers Eng 49:841–847" href="/article/10.1007/s11340-015-0009-1#ref-CR37" id="ref-link-section-d155616590e17689">37</a>]. This precomputation for biquintic B-spline interpolation requires a very large amount of storage (36 times the size of the padded B-spline coefficient array). But, the space required may be worth the trade off for the speed improvement. The largest computational bottleneck in the DIC analysis is the interpolation step when calculating the components of the hessian, so the reduction in computational time is worth the expensive memory requirement.</p><p>At this point, the <span class="mathjax-tex">$ g\left({\tilde{x}}_{cu{r}_i},{\tilde{y}}_{cu{r}_j}\right) $</span> quantity is calculable. The last quantities to address are <span class="mathjax-tex">$ \frac{\partial }{\partial {\tilde{x}}_{ref}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right) $</span> and <span class="mathjax-tex">$ \frac{\partial }{\partial {\tilde{y}}_{ref}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right) $</span>. These quantities can be computed by taking the partial derivatives of an equation of the same form as equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s11340-015-0009-1#Equ32">33</a>) and setting Δx and Δy to zero (because these are integer pixel locations) to obtain</p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{\partial }{\partial {\tilde{x}}_{ref}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right)=\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\ast \left[QK\right]\ast {\left[c\right]}_{\left({x}_f-2:{x}_f+3,{y}_f-2:{y}_f+3\right)}\ast {\left[QK\right]}^T\ast \left[\begin{array}{c}\hfill 0\hfill \\ {}\hfill 1\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \end{array}\right] $$</span></div><div class="c-article-equation__number"> (37) </div></div> <div id="Equ37" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{\partial }{\partial {\tilde{y}}_{ref}}f\left({\tilde{x}}_{re{f}_i},{\tilde{y}}_{re{f}_j}\right)=\left[\begin{array}{cccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\ast \left[QK\right]\ast {\left[c\right]}_{\left({x}_f-2:{x}_f+3,{y}_f-2:{y}_f+3\right)}\ast {\left[QK\right]}^T\ast \left[\begin{array}{c}\hfill 1\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \end{array}\right] $$</span></div><div class="c-article-equation__number"> (38) </div></div> <p>These quantities are precomputed for the entire reference image before beginning the IC-GN method.</p></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"><p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=Ncorr%3A%20Open-Source%202D%20Digital%20Image%20Correlation%20Matlab%20Software&author=J.%20Blaber%20et%20al&contentID=10.1007%2Fs11340-015-0009-1&copyright=Society%20for%20Experimental%20Mechanics&publication=0014-4851&publicationDate=2015-03-31&publisherName=SpringerNature&orderBeanReset=true">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="u-hide-print c-bibliographic-information__column c-bibliographic-information__column--border"><a data-crossmark="10.1007/s11340-015-0009-1" target="_blank" rel="noopener" href="https://crossmark.crossref.org/dialog/?doi=10.1007/s11340-015-0009-1" data-track="click" data-track-action="Click Crossmark" data-track-label="link" data-test="crossmark"><img loading="lazy" width="57" height="81" alt="Check for updates. 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