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href="/w/index.php?title=Centroidal_Voronoi_tessellation&direction=prev&oldid=1195811381" title="Centroidal Voronoi tessellation">← Previous revision</a> | Latest revision (diff) | Newer revision → (diff)</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Voronoi tessellation where the generating point of each Voronoi cell is also its centroid</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="theader" style="text-align:center">Three centroidal Voronoi tessellations of five points in a square</div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:CentroidalVoronoiTessellation1.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/CentroidalVoronoiTessellation1.png/200px-CentroidalVoronoiTessellation1.png" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/CentroidalVoronoiTessellation1.png/300px-CentroidalVoronoiTessellation1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/CentroidalVoronoiTessellation1.png/400px-CentroidalVoronoiTessellation1.png 2x" data-file-width="560" data-file-height="420" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:CentroidalVoronoiTessellation2.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/CentroidalVoronoiTessellation2.png/200px-CentroidalVoronoiTessellation2.png" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/CentroidalVoronoiTessellation2.png/300px-CentroidalVoronoiTessellation2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/CentroidalVoronoiTessellation2.png/400px-CentroidalVoronoiTessellation2.png 2x" data-file-width="560" data-file-height="420" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:CentroidalVoronoiTessellation3.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/CentroidalVoronoiTessellation3.png/200px-CentroidalVoronoiTessellation3.png" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/CentroidalVoronoiTessellation3.png/300px-CentroidalVoronoiTessellation3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/CentroidalVoronoiTessellation3.png/400px-CentroidalVoronoiTessellation3.png 2x" data-file-width="560" data-file-height="420" /></a></span></div></div></div></div></div> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>centroidal Voronoi tessellation</b> (<b>CVT</b>) is a special type of <a href="/wiki/Voronoi_tessellation" class="mw-redirect" title="Voronoi tessellation">Voronoi tessellation</a> in which the generating point of each Voronoi cell is also its <a href="/wiki/Centroid" title="Centroid">centroid</a> (center of mass). It can be viewed as an optimal partition corresponding to an optimal distribution of generators. A number of algorithms can be used to generate centroidal Voronoi tessellations, including <a href="/wiki/Lloyd%27s_algorithm" title="Lloyd's algorithm">Lloyd's algorithm</a> for <a href="/wiki/K-means_clustering" title="K-means clustering">K-means clustering</a> or <a href="/wiki/Quasi-Newton_method" title="Quasi-Newton method">Quasi-Newton methods</a> like <a href="/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">BFGS</a>. <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proofs">Proofs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centroidal_Voronoi_tessellation&action=edit&section=1" title="Edit section: Proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gersho's conjecture, proven for one and two dimensions, says that "asymptotically speaking, all cells of the optimal CVT, while forming a <a href="/wiki/Tessellation" title="Tessellation">tessellation</a>, are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> to a basic cell which depends on the dimension."<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In two dimensions, the basic cell for the optimal CVT is a regular <a href="/wiki/Hexagon" title="Hexagon">hexagon</a> as it is proven to be the most dense <a href="/wiki/Circle_packing" title="Circle packing">packing of circles</a> in 2D Euclidean space. Its three dimensional equivalent is the <a href="/wiki/Rhombic_dodecahedral_honeycomb" title="Rhombic dodecahedral honeycomb">rhombic dodecahedral honeycomb</a>, derived from the most dense <a href="/wiki/Close-packing_of_equal_spheres" title="Close-packing of equal spheres">packing of spheres</a> in 3D Euclidean space. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centroidal_Voronoi_tessellation&action=edit&section=2" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Centroidal Voronoi tessellations are useful in <a href="/wiki/Data_compression" title="Data compression">data compression</a>, optimal <a href="/wiki/Numerical_integration" title="Numerical integration">quadrature</a>, optimal <a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">quantization</a>, <a href="/wiki/Data_clustering" class="mw-redirect" title="Data clustering">clustering</a>, and optimal mesh generation.<sup id="cite_ref-Du_2_3-0" class="reference"><a href="#cite_note-Du_2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>A weighted centroidal Voronoi diagrams is a CVT in which each centroid is weighted according to a certain function. For example, a <a href="/wiki/Grayscale" title="Grayscale">grayscale</a> image can be used as a density function to weight the points of a CVT, as a way to create digital <a href="/wiki/Stippling" title="Stippling">stippling</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Occurrence_in_nature">Occurrence in nature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centroidal_Voronoi_tessellation&action=edit&section=3" title="Edit section: Occurrence in nature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many <a href="/wiki/Patterns_in_nature" title="Patterns in nature">patterns seen in nature</a> are closely approximated by a centroidal Voronoi tessellation. Examples of this include the <a href="/wiki/Giant%27s_Causeway" title="Giant's Causeway">Giant's Causeway</a>, the cells of the <a href="/wiki/Cornea" title="Cornea">cornea</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and the breeding pits of the male <a href="/wiki/Tilapia" title="Tilapia">tilapia</a>.<sup id="cite_ref-Du_2_3-1" class="reference"><a href="#cite_note-Du_2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centroidal_Voronoi_tessellation&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFNocedalWright2006" class="citation book cs1">Nocedal, Jorge; Wright, Stephen J. (2006). <i>Numerical Optimization</i>. Springer Series in Operations Research and Financial Engineering (second ed.). 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"Weighted voronoi stippling." Proceedings of the 2nd international symposium on Non-photorealistic animation and rendering. ACM, 2002.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPigatto2009" class="citation journal cs1">Pigatto, João Antonio Tadeu; et al. (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1590%2FS0103-84782009005000001">"Scanning electron microscopy of the corneal endothelium of ostrich"</a>. <i>Cienc. Rural</i>. <b>39</b> (3): 926–929. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1590%2FS0103-84782009005000001">10.1590/S0103-84782009005000001</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11449%2F29422">11449/29422</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Cienc.+Rural&rft.atitle=Scanning+electron+microscopy+of+the+corneal+endothelium+of+ostrich&rft.volume=39&rft.issue=3&rft.pages=926-929&rft.date=2009&rft_id=info%3Ahdl%2F11449%2F29422&rft_id=info%3Adoi%2F10.1590%2FS0103-84782009005000001&rft.aulast=Pigatto&rft.aufirst=Jo%C3%A3o+Antonio+Tadeu&rft_id=https%3A%2F%2Fdoi.org%2F10.1590%252FS0103-84782009005000001&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentroidal+Voronoi+tessellation" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Centroidal_Voronoi_tessellation&oldid=1195811381">https://en.wikipedia.org/w/index.php?title=Centroidal_Voronoi_tessellation&oldid=1195811381</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Discrete_geometry" title="Category:Discrete geometry">Discrete geometry</a></li><li><a href="/wiki/Category:Geometric_algorithms" title="Category:Geometric algorithms">Geometric algorithms</a></li><li><a href="/wiki/Category:Diagrams" title="Category:Diagrams">Diagrams</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 15 January 2024, at 10:59<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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