{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T21:02:02Z","timestamp":1773090122539,"version":"3.50.1"},"reference-count":43,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2010,7,26]],"date-time":"2010-07-26T00:00:00Z","timestamp":1280102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["ERC-StG-205693"],"award-info":[{"award-number":["ERC-StG-205693"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2010,7,26]]},"abstract":"<jats:p>\n            This paper introduces\n            <jats:italic>L<\/jats:italic>\n            <jats:sub>\n              <jats:italic>p<\/jats:italic>\n            <\/jats:sub>\n            -Centroidal Voronoi Tessellation (\n            <jats:italic>L<\/jats:italic>\n            <jats:sub>\n              <jats:italic>p<\/jats:italic>\n            <\/jats:sub>\n            -CVT), a generalization of CVT that minimizes a higher-order moment of the coordinates on the Voronoi cells. This generalization allows for aligning the axes of the Voronoi cells with a predefined background tensor field (anisotropy).\n            <jats:italic>L<\/jats:italic>\n            <jats:sub>\n              <jats:italic>p<\/jats:italic>\n            <\/jats:sub>\n            -CVT is computed by a quasi-Newton optimization framework, based on closed-form derivations of the objective function and its gradient. The derivations are given for both surface meshing (\u03a9 is a triangulated mesh with per-facet anisotropy) and volume meshing (\u03a9 is the interior of a closed triangulated mesh with a 3D anisotropy field). Applications to anisotropic, quad-dominant surface remeshing and to hexdominant volume meshing are presented. Unlike previous work,\n            <jats:italic>L<\/jats:italic>\n            <jats:sub>\n              <jats:italic>p<\/jats:italic>\n            <\/jats:sub>\n            -CVT captures sharp features and intersections without requiring any pre-tagging.\n          <\/jats:p>","DOI":"10.1145\/1778765.1778856","type":"journal-article","created":{"date-parts":[[2010,7,15]],"date-time":"2010-07-15T12:48:46Z","timestamp":1279198126000},"page":"1-11","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":159,"title":["<i>L<\/i>\n            <sub>\n              <i>p<\/i>\n            <\/sub>\n            Centroidal Voronoi Tessellation and its applications"],"prefix":"10.1145","volume":"29","author":[{"given":"Bruno","family":"L\u00e9vy","sequence":"first","affiliation":[{"name":"INRIA-ALICE"}]},{"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"INRIA-ALICE"}]}],"member":"320","published-online":{"date-parts":[[2010,7,26]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882296"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073238"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/1137856.1137906"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531383"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/1629255.1629259"},{"key":"e_1_2_2_6_1","first-page":"299","article-title":"Optimal Delaunay triangulations","volume":"22","author":"Chen L.","year":"2004","journal-title":"Journal of Computational Mathematics"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015817"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.5555\/1735603.1735627"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.3390\/a2041327"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141993"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.476"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.616"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503428527"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144599352836"},{"key":"e_1_2_2_15_1","doi-asserted-by":"crossref","unstructured":"Edelsbrunner H. and Shah N. R. 1997. Triangulating topological spaces. International Journal of Computational Geometry &amp; Applications 7 4 365--378.  Edelsbrunner H. and Shah N. R. 1997. Triangulating topological spaces. International Journal of Computational Geometry &amp; Applications 7 4 365--378.","DOI":"10.1142\/S0218195997000223"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/1409060.1409100"},{"key":"e_1_2_2_17_1","volume-title":"Proc. IFIP, 273--288","author":"Iri M."},{"key":"e_1_2_2_18_1","unstructured":"Itoh T. and Shimada K. 2001. Automatic conversion of triangular meshes into quadrilateral meshes with directionality. International Journal of CAD\/CAM.  Itoh T. and Shimada K. 2001. Automatic conversion of triangular meshes into quadrilateral meshes with directionality. International Journal of CAD\/CAM."},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2007.01060.x"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/777792.777822"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/1364901.1364921"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.2001.11919812"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/1559755.1559758"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1982.1056489"},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-04319-2_5"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1145\/1364901.1364938"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1002\/1097-0207(20000910\/20)49:1\/2<17::AID-NME920>3.0.CO;2-U"},{"key":"e_1_2_2_28_1","unstructured":"Minka T. 1997. Old and new matrix algebra useful for statistics. Tech. rep. MIT Media Lab. revised 12\/00.  Minka T. 1997. Old and new matrix algebra useful for statistics. Tech. rep. MIT Media Lab. revised 12\/00."},{"key":"e_1_2_2_29_1","volume-title":"Proceedings CGC Workshop on Computational Geometry, 33--40","author":"Mount D. M."},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1002\/1097-0207(20000910\/20)49:1\/2<289::AID-NME934>3.0.CO;2-L"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/1183287.1183297"},{"key":"e_1_2_2_32_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-008-0091-4"},{"key":"e_1_2_2_33_1","volume-title":"9th Intl. Meshing Roundtable conference proceedings.","author":"Shepherd J."},{"key":"e_1_2_2_34_1","volume-title":"International Meshing Roundtable conference proceedings.","author":"Staten M. L."},{"key":"e_1_2_2_35_1","first-page":"2","article-title":"Planar parameterization for closed manifold genus-g meshes using any type of positive weights","volume":"5","author":"Steiner D.","year":"2005","journal-title":"Journal of Computational Methods in Information Science and Engineering"},{"key":"e_1_2_2_36_1","volume-title":"15th International Meshing Roundtable conference proceedings.","author":"Tchon K.-F."},{"key":"e_1_2_2_37_1","volume-title":"SGP '06: Proceedings of the fourth Eurographics symposium on Geometry processing, 201--210","author":"Tong Y."},{"key":"e_1_2_2_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531381"},{"key":"e_1_2_2_39_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2007.70430"},{"key":"e_1_2_2_40_1","volume-title":"18th International Meshing Roundtable conference proceedings.","author":"Vyas V."},{"key":"e_1_2_2_41_1","doi-asserted-by":"crossref","unstructured":"Yamakawa S. and Shimada K. 2003. Anisotropic tetrahedral meshing via bubble packing and advancing front. International Journal for Numerical Methods in Engineering 57.  Yamakawa S. and Shimada K. 2003. Anisotropic tetrahedral meshing via bubble packing and advancing front. International Journal for Numerical Methods in Engineering 57.","DOI":"10.1002\/nme.750"},{"key":"e_1_2_2_42_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.754"},{"key":"e_1_2_2_43_1","doi-asserted-by":"publisher","DOI":"10.5555\/1735603.1735629"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1778765.1778856","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1778765.1778856","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T11:39:47Z","timestamp":1750246787000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1778765.1778856"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,7,26]]},"references-count":43,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,7,26]]}},"alternative-id":["10.1145\/1778765.1778856"],"URL":"https:\/\/doi.org\/10.1145\/1778765.1778856","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,7,26]]},"assertion":[{"value":"2010-07-26","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}