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Graph."],"published-print":{"date-parts":[[2020,4,30]]},"abstract":"<jats:p>\n            Field-guided parameterization methods have proven effective for quad meshing of surfaces; these methods compute smooth\n            <jats:italic>cross fields<\/jats:italic>\n            to guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in extending these methods to three dimensions, however, is\n            <jats:italic>representation<\/jats:italic>\n            of field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog\u2014an octahedral frame field\u2014takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These new\n            <jats:italic>odeco frames<\/jats:italic>\n            , so called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program\u2013based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.\n          <\/jats:p>","DOI":"10.1145\/3366786","type":"journal-article","created":{"date-parts":[[2020,4,5]],"date-time":"2020-04-05T12:47:58Z","timestamp":1586090878000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":35,"title":["Algebraic Representations for Volumetric Frame Fields"],"prefix":"10.1145","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1931-5673","authenticated-orcid":false,"given":"David","family":"Palmer","sequence":"first","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]},{"given":"David","family":"Bommes","sequence":"additional","affiliation":[{"name":"University of Bern, Bern, Switzerland"}]},{"given":"Justin","family":"Solomon","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]}],"member":"320","published-online":{"date-parts":[[2020,4,5]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-005-0179-9"},{"key":"e_1_2_2_2_1","volume-title":"Proceedings of the 9th IEEE International Conference on Computer Vision (ICCV\u201903)","author":"Agrawal Motilal","unstructured":"Motilal Agrawal and Larry S. Davis . 2003. Camera calibration using spheres: A semi-definite programming approach . In Proceedings of the 9th IEEE International Conference on Computer Vision (ICCV\u201903) . IEEE, Los Alamitos, CA, 782. Motilal Agrawal and Larry S. Davis. 2003. Camera calibration using spheres: A semi-definite programming approach. In Proceedings of the 9th IEEE International Conference on Computer Vision (ICCV\u201903). IEEE, Los Alamitos, CA, 782."},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.proeng.2015.10.123"},{"key":"e_1_2_2_4_1","unstructured":"Pierre-Alexandre Beaufort Jonathan Lambrechts Christophe Geuzaine and Jean-Francois Remacle. 2019. Quaternionic octahedral fields: SU (2) parameterization of 3D frames. arXiv:1910.06240.  Pierre-Alexandre Beaufort Jonathan Lambrechts Christophe Geuzaine and Jean-Francois Remacle. 2019. Quaternionic octahedral fields: SU (2) parameterization of 3D frames. arXiv:1910.06240."},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.proeng.2017.09.799"},{"key":"e_1_2_2_6_1","volume-title":"Thomas","author":"Blekherman Grigoriy","year":"2012","unstructured":"Grigoriy Blekherman , Pablo A. Parrilo , and Rekha R . Thomas . 2012 . Semidefinite Optimization and Convex Algebraic Geometry. SIAM. Grigoriy Blekherman, Pablo A. Parrilo, and Rekha R. Thomas. 2012. Semidefinite Optimization and Convex Algebraic Geometry. SIAM."},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-017-1588-6"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.5555\/2627435.2638581"},{"key":"e_1_2_2_9_1","series-title":"Lecture Notes in Computational Science and Engineering","volume-title":"International Meshing Roundtable","author":"Chemin A.","unstructured":"A. Chemin , F. Henrotte , J.-F. Remacle , and J. Van Schaftingen . 2018. Representing three-dimensional cross fields using 4th order tensors . In International Meshing Roundtable . Lecture Notes in Computational Science and Engineering , Vol. 127 . Springer , 89--108. A. Chemin, F. Henrotte, J.-F. Remacle, and J. Van Schaftingen. 2018. Representing three-dimensional cross fields using 4th order tensors. In International Meshing Roundtable. Lecture Notes in Computational Science and Engineering, Vol. 127. Springer, 89--108."},{"key":"e_1_2_2_10_1","volume-title":"Thomas","author":"Cifuentes Diego","year":"2017","unstructured":"Diego Cifuentes , Sameer Agarwal , Pablo A. Parrilo , and Rekha R . Thomas . 2017 . On the local stability of semidefinite relaxations. arXiv:1710.04287. Diego Cifuentes, Sameer Agarwal, Pablo A. Parrilo, and Rekha R. Thomas. 2017. On the local stability of semidefinite relaxations. arXiv:1710.04287."},{"key":"e_1_2_2_11_1","unstructured":"Diego Cifuentes Corey Harris and Bernd Sturmfels. 2018. 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A variational method for generating n-cross fields using higher-order Q-tensors. arXiv:1909.00922."},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/2070781.2024189"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12184"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2013.250"},{"key":"e_1_2_2_20_1","first-page":"291","article-title":"Fast arbitrary BRDF shading for low-frequency lighting using spherical harmonics","volume":"2","author":"Kautz Jan","year":"2002","unstructured":"Jan Kautz , John Snyder , and Peter-Pike J. Sloan . 2002 . Fast arbitrary BRDF shading for low-frequency lighting using spherical harmonics . Rendering Techniques 2 , 1 (2002), 291 -- 296 . Jan Kautz, John Snyder, and Peter-Pike J. Sloan. 2002. Fast arbitrary BRDF shading for low-frequency lighting using spherical harmonics. 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In Proceedings of the 2015 10th International Conference on Computer Science Education (ICCSE\u201915). 697--702."}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3366786","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3366786","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3366786","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T23:44:39Z","timestamp":1750203879000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3366786"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,5]]},"references-count":47,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,4,30]]}},"alternative-id":["10.1145\/3366786"],"URL":"https:\/\/doi.org\/10.1145\/3366786","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,5]]},"assertion":[{"value":"2019-08-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-12-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-04-05","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}