{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T09:29:07Z","timestamp":1780392547260,"version":"3.54.1"},"reference-count":30,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2007,7,29]],"date-time":"2007-07-29T00:00:00Z","timestamp":1185667200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2007,7,29]]},"abstract":"<jats:p>We present a symmetrization algorithm for geometric objects. Our algorithm enhances approximate symmetries of a model while minimally altering its shape. Symmetrizing deformations are formulated as an optimization process that couples the spatial domain with a transformation configuration space, where symmetries can be expressed more naturally and compactly as parametrized point-pair mappings. We derive closed-form solution for the optimal symmetry transformations, given a set of corresponding sample pairs. The resulting optimal displacement vectors are used to drive a constrained deformation model that pulls the shape towards symmetry. We show how our algorithm successfully symmetrizes both the geometry and the discretization of complex 2D and 3D shapes and discuss various applications of such symmetrizing deformations.<\/jats:p>","DOI":"10.1145\/1276377.1276456","type":"journal-article","created":{"date-parts":[[2007,9,14]],"date-time":"2007-09-14T13:44:55Z","timestamp":1189777495000},"page":"63","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":85,"title":["Symmetrization"],"prefix":"10.1145","volume":"26","author":[{"given":"Niloy J.","family":"Mitra","sequence":"first","affiliation":[{"name":"TU Vienna"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leonidas J.","family":"Guibas","sequence":"additional","affiliation":[{"name":"Stanford University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Pauly","sequence":"additional","affiliation":[{"name":"ETH Zurich"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2007,7,29]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073238"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/TC.1985.1676605"},{"key":"e_1_2_2_3_1","unstructured":"Attali D. Boissonnat J. and Edelsbrunner H. 2004. Stability and computation of the medial axis --- a state-of-the-art report. Mathematical Foundations of Scientific Visualization Computer Graphics and Massive Data Exploration.  Attali D. Boissonnat J. and Edelsbrunner H. 2004. Stability and computation of the medial axis --- a state-of-the-art report. Mathematical Foundations of Scientific Visualization Computer Graphics and Massive Data Exploration ."},{"key":"e_1_2_2_4_1","volume-title":"Models for the Perception of Speech and Visual Forms","author":"Blum H."},{"key":"e_1_2_2_5_1","volume-title":"Proc. Symposium on Geometry Processing, 11--22","author":"Botsch M."},{"key":"e_1_2_2_6_1","unstructured":"Cox T. and Cox M. 1994. Multidimensional Scaling. Chapman and Hall London.  Cox T. and Cox M. 1994. Multidimensional Scaling . Chapman and Hall London."},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/s001380050048"},{"key":"e_1_2_2_8_1","unstructured":"Faber G. 1920. Ueber potentialtheorie und konforme abbildung. Sitzungsber. Bayer. Akad. Wiss. Math.-Naturwiss. Kl. 49--64.  Faber G. 1920. Ueber potentialtheorie und konforme abbildung. Sitzungsber. Bayer. Akad. Wiss. Math.-Naturwiss. Kl. 49--64."},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/358669.358692"},{"key":"e_1_2_2_10_1","volume-title":"Symposium on Geometry Processing, 131--142","author":"Funkhouser T."},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/1122501.1122507"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/258734.258849"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/007\/0156259"},{"key":"e_1_2_2_14_1","volume-title":"Oberflaeche und Isoperimetrie","author":"Hadwiger H."},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00371-003-0221-3"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073323"},{"key":"e_1_2_2_17_1","doi-asserted-by":"crossref","unstructured":"Kazhdan M. M. Chazelle B. Dobkin D. P. Finkelstein A. and Funkhouser T. A. 2002. A reflective symmetry descriptor. In ECCV 642--656.   Kazhdan M. M. Chazelle B. Dobkin D. P. Finkelstein A. and Funkhouser T. A. 2002. A reflective symmetry descriptor. In ECCV 642--656.","DOI":"10.1007\/3-540-47967-8_43"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/1057432.1057448"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/1138450.1138462"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141924"},{"key":"e_1_2_2_21_1","volume-title":"Proceedings of Eurographics, 281--289","author":"Pauly M."},{"key":"e_1_2_2_22_1","volume-title":"Symposium on Geometry Processing, 23--32","author":"Pauly M."},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141923"},{"key":"e_1_2_2_24_1","doi-asserted-by":"crossref","volume-title":"Tensor analysis for physicists","author":"Schouten A.","DOI":"10.1063\/1.3067367"},{"key":"e_1_2_2_25_1","volume-title":"Proc. Symposium on Geometry Processing.","author":"Simari P."},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.2005.221"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1109\/99.641604"},{"key":"e_1_2_2_28_1","doi-asserted-by":"crossref","unstructured":"Wolter J. Woo T. and Volz R. 1985. Optimal algorithms for symmetry detection in two and three dimensions. The Visual Computer 37--48.  Wolter J. Woo T. and Volz R. 1985. Optimal algorithms for symmetry detection in two and three dimensions. The Visual Computer 37--48.","DOI":"10.1007\/BF01901268"},{"key":"e_1_2_2_29_1","doi-asserted-by":"publisher","DOI":"10.1006\/cviu.1996.0506"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1109\/34.476508"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1276377.1276456","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1276377.1276456","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T14:52:18Z","timestamp":1750258338000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1276377.1276456"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,7,29]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,7,29]]}},"alternative-id":["10.1145\/1276377.1276456"],"URL":"https:\/\/doi.org\/10.1145\/1276377.1276456","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,7,29]]},"assertion":[{"value":"2007-07-29","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}