{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T08:15:22Z","timestamp":1774685722519,"version":"3.50.1"},"reference-count":36,"publisher":"Association for Computing Machinery (ACM)","issue":"6","license":[{"start":{"date-parts":[[2011,12,1]],"date-time":"2011-12-01T00:00:00Z","timestamp":1322697600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000143","name":"Division of Computing and Communication Foundations","doi-asserted-by":"publisher","award":["IIS 0953096CCF 0936830CMMI 0757123CCF 0811313"],"award-info":[{"award-number":["IIS 0953096CCF 0936830CMMI 0757123CCF 0811313"]}],"id":[{"id":"10.13039\/100000143","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000147","name":"Division of Civil, Mechanical and Manufacturing Innovation","doi-asserted-by":"publisher","award":["IIS 0953096CCF 0936830CMMI 0757123CCF 0811313"],"award-info":[{"award-number":["IIS 0953096CCF 0936830CMMI 0757123CCF 0811313"]}],"id":[{"id":"10.13039\/100000147","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000145","name":"Division of Information and Intelligent Systems","doi-asserted-by":"publisher","award":["IIS 0953096CCF 0936830CMMI 0757123CCF 0811313"],"award-info":[{"award-number":["IIS 0953096CCF 0936830CMMI 0757123CCF 0811313"]}],"id":[{"id":"10.13039\/100000145","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002855","name":"Ministry of Science and Technology of the People's Republic of China","doi-asserted-by":"publisher","award":["2009CB320801"],"award-info":[{"award-number":["2009CB320801"]}],"id":[{"id":"10.13039\/501100002855","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["6.09E+15"],"award-info":[{"award-number":["6.09E+15"]}],"id":[{"id":"10.13039\/501100001809","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":[[2011,12]]},"abstract":"<jats:p>\n            In this paper, we present a method for constructing a 3D\n            <jats:italic>cross-frame field<\/jats:italic>\n            , a 3D extension of the 2D cross-frame field as applied to surfaces in applications such as quadrangulation and texture synthesis. In contrast to the surface cross-frame field (equivalent to a 4-Way Rotational-Symmetry vector field), symmetry for 3D cross-frame fields cannot be formulated by simple one-parameter 2D rotations in the tangent planes. To address this critical issue, we represent the 3D frames by spherical harmonics, in a manner invariant to combinations of rotations around any axis by multiples of \u03c0\/2. With such a representation, we can formulate an efficient smoothness measure of the cross-frame field. Through minimization of this measure under certain boundary conditions, we can construct a smooth 3D cross-frame field that is aligned with the surface normal at the boundary. We visualize the resulting cross-frame field through restrictions to the boundary surface, streamline tracing in the volume, and singularities. We also demonstrate the application of the 3D cross-frame field to producing hexahedron-dominant meshes for given volumes, and discuss its potential in high-quality hexahedralization, much as its 2D counterpart has shown in quadrangulation.\n          <\/jats:p>","DOI":"10.1145\/2070781.2024177","type":"journal-article","created":{"date-parts":[[2011,11,30]],"date-time":"2011-11-30T13:58:46Z","timestamp":1322661526000},"page":"1-8","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":78,"title":["Boundary aligned smooth 3D cross-frame field"],"prefix":"10.1145","volume":"30","author":[{"given":"Jin","family":"Huang","sequence":"first","affiliation":[{"name":"Zhejiang University"}]},{"given":"Yiying","family":"Tong","sequence":"additional","affiliation":[{"name":"Michigan State University"}]},{"given":"Hongyu","family":"Wei","sequence":"additional","affiliation":[{"name":"Zhejiang University"}]},{"given":"Hujun","family":"Bao","sequence":"additional","affiliation":[{"name":"Zhejiang University"}]}],"member":"320","published-online":{"date-parts":[[2011,12,12]]},"reference":[{"key":"e_1_2_1_1_1","unstructured":"Bochkanov S. n.d. Alglib. http:\/\/www.alglib.net\/.  Bochkanov S. n.d. Alglib. http:\/\/www.alglib.net\/."},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531383"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-009-0168-8"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/34.400568"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1109\/34.1000236"},{"key":"e_1_2_1_6_1","first-page":"67","volume-title":"Computers In Mechanical Engineering","author":"Cook W."},{"key":"e_1_2_1_7_1","unstructured":"dlib n.d. dlib c++ library. http:\/\/dlib.net\/.  dlib n.d. dlib c++ library. http:\/\/dlib.net\/."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141993"},{"key":"e_1_2_1_9_1","first-page":"1","article-title":"Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds","volume":"30","author":"Edelsbrunner H.","year":"2001","journal-title":"Discrete and Computational Geometry (SoCG)"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/237218.237237"},{"key":"e_1_2_1_11_1","volume-title":"Spherical Harmonic Lighting: The Gritty Details. Archives of the Game Developers Conference (Mar.).","author":"Green R.","year":"2003"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2011.02015.x"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2007.01060.x"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2007.1032"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2009.59"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/1778765.1778856"},{"key":"e_1_2_1_17_1","volume-title":"IEEE Conference on Computer Vision and Pattern Recognition, 217--226","author":"Makadia A."},{"key":"e_1_2_1_18_1","volume-title":"Proceedings of the Conference on Visualization '02, IEEE Computer Society, Washington, DC, USA, VIS '02, 283--290","author":"Mann S."},{"key":"e_1_2_1_19_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_1_20_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2011.02014.x"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276446"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2010.121"},{"key":"e_1_2_1_23_1","unstructured":"Pointwise 2009. Gridgen - reliable cfd meshing. http:\/\/www.pointwise.com\/gridgen\/.  Pointwise 2009. Gridgen - reliable cfd meshing. http:\/\/www.pointwise.com\/gridgen\/."},{"key":"e_1_2_1_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/1183287.1183297"},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/1356682.1356683"},{"key":"e_1_2_1_26_1","unstructured":"Roca X. 2009. Paving the path towards automatic hexahedral mesh generation. PhD thesis Applied Mathematics UPC Barcelona.  Roca X. 2009. Paving the path towards automatic hexahedral mesh generation . PhD thesis Applied Mathematics UPC Barcelona."},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.finel.2010.04.005"},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1007\/s007910050004"},{"key":"e_1_2_1_29_1","volume-title":"Proceedings, 14th International Meshing Roundtable.","author":"Scott M. A."},{"key":"e_1_2_1_30_1","unstructured":"Shepherd J. 2007. Topologic and geometric constraint-based hexahedral mesh generation. PhD thesis University of Utah.   Shepherd J. 2007. Topologic and geometric constraint-based hexahedral mesh generation . PhD thesis University of Utah."},{"key":"e_1_2_1_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/1360612.1360652"},{"key":"e_1_2_1_32_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.139"},{"key":"e_1_2_1_33_1","volume-title":"Proceedings of the 18th International Meshing Roundtable, B. W. Clark, Ed. Springer Berlin Heidelberg, 377--396","author":"Vyas V."},{"key":"e_1_2_1_34_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.754"},{"key":"e_1_2_1_35_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.gmod.2010.10.006"},{"key":"e_1_2_1_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/1778765.1778855"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2070781.2024177","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2070781.2024177","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2070781.2024177","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T10:06:03Z","timestamp":1750241163000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2070781.2024177"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,12]]},"references-count":36,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2011,12]]}},"alternative-id":["10.1145\/2070781.2024177"],"URL":"https:\/\/doi.org\/10.1145\/2070781.2024177","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,12]]},"assertion":[{"value":"2011-12-12","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}