{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T06:10:42Z","timestamp":1784268642161,"version":"3.55.0"},"reference-count":42,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2013,7,21]],"date-time":"2013-07-21T00:00:00Z","timestamp":1374364800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","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":[[2013,7,21]]},"abstract":"<jats:p>We introduce an efficient algorithm for producing provably injective mappings of tetrahedral meshes with strict bounds on their tetrahedra aspect-ratio distortion.<\/jats:p><jats:p>The algorithm takes as input a simplicial map (e.g., produced by some common deformation or volumetric parameterization technique) and<jats:italic>projects<\/jats:italic>it on the space of injective and bounded-distortion simplicial maps. Namely, finds a similar map that is both bijective and bounded-distortion. As far as we are aware, this is the first algorithm to produce injective or bounded-distortion simplicial maps of tetrahedral meshes. The construction of the algorithm was made possible due to a novel closed-form solution to the problem of finding the closest orientation-preserving bounded-distortion matrix to an arbitrary matrix in three (and higher) dimensions.<\/jats:p><jats:p>The algorithm is shown to have quadratic convergence, usually not requiring more than a handful of iterations to converge. Furthermore, it is readily generalized to simplicial maps of any dimension, including mixed dimensions. Finally, it can deal with different distortion spaces, such as bounded isometric distortion. During experiments we found the algorithm useful for producing bijective and bounded-distortion volume parameterizations and deformations of tetrahedral meshes, and improving tetrahedral meshes, increasing the tetrahedra quality produced by state-of-the-art techniques.<\/jats:p>","DOI":"10.1145\/2461912.2461931","type":"journal-article","created":{"date-parts":[[2013,7,16]],"date-time":"2013-07-16T18:06:45Z","timestamp":1373998005000},"page":"1-14","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":94,"title":["Injective and bounded distortion mappings in 3D"],"prefix":"10.1145","volume":"32","author":[{"given":"Noam","family":"Aigerman","sequence":"first","affiliation":[{"name":"Weizmann Institute of Science"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yaron","family":"Lipman","sequence":"additional","affiliation":[{"name":"Weizmann Institute of Science"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2013,7,21]]},"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":"10.1145\/1531326.1531340"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2007.1054"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","unstructured":"Bronstein A. Bronstein M. and Kimmel R. 2008. Numerical Geometry of Non-Rigid Shapes 1 ed. Springer Publishing Company Incorporated. Bronstein A. Bronstein M. and Kimmel R. 2008. Numerical Geometry of Non-Rigid Shapes 1 ed. Springer Publishing Company Incorporated.","DOI":"10.1007\/978-0-387-73301-2_1"},{"key":"e_1_2_2_5_1","volume-title":"Proceedings of the 18th International Meshing Roundtable. Springer, 463--480","author":"Casc\u00f3n J."},{"key":"e_1_2_2_6_1","unstructured":"Cgal. Cgal computational geometry algorithms library. http:\/\/www.cgal.org. Cgal. Cgal computational geometry algorithms library. http:\/\/www.cgal.org."},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/1778765.1778775"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/97880.97900"},{"key":"e_1_2_2_9_1","volume-title":"ANNALES-FACULTE DES SCIENCES TOULOUSE MATHEMATIQUES","volume":"16","author":"Dacorogna B."},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/218380.218440"},{"key":"e_1_2_2_11_1","unstructured":"Eppstein D. 2001. Global optimization of mesh quality. Tutorial at the 10th Int. Eppstein D. 2001. Global optimization of mesh quality. Tutorial at the 10th Int ."},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-004-7620-5"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2005.06.004"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-02-01466-7"},{"key":"e_1_2_2_15_1","doi-asserted-by":"crossref","unstructured":"Freitag L. A. and Knupp P. M. 2002. Tetrahedral mesh improvement via optimization of the element condition number. Freitag L. A. and Knupp P. M. 2002. Tetrahedral mesh improvement via optimization of the element condition number.","DOI":"10.1002\/nme.341"},{"key":"e_1_2_2_16_1","volume-title":"Computer Graphics Forum (Special Issue of Symposium on Geometry Processing 2011)","author":"Gregson J."},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/1028523.1028541"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964973"},{"key":"e_1_2_2_19_1","doi-asserted-by":"crossref","unstructured":"Johnen A. Remacle J.-F. and Geuzaine C. 2012. Geometrical validity of high-order triangular finite elements. Engineering with Computers 1--8. Johnen A. Remacle J.-F. and Geuzaine C. 2012. Geometrical validity of high-order triangular finite elements. Engineering with Computers 1--8.","DOI":"10.1007\/s00366-012-0305-7"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276466"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073229"},{"key":"e_1_2_2_22_1","volume-title":"Proceedings of the 16th International Meshing Roundtable, 3--23","author":"Klingner B. M."},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276448"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566590"},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/1236246.1236263"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1145\/1360612.1360677"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185604"},{"key":"e_1_2_2_28_1","volume-title":"Computer Graphics Forum","volume":"27","author":"Liu L."},{"key":"e_1_2_2_29_1","doi-asserted-by":"crossref","unstructured":"Meyer M. Desbrun M. Schr\u00f6der P. and Barr A. 2002. Discrete differential-geometry operators for triangulated 2-manifolds. Visualization and mathematics 3 7 34--57. Meyer M. Desbrun M. Schr\u00f6der P. and Barr A. 2002. Discrete differential-geometry operators for triangulated 2-manifolds. Visualization and mathematics 3 7 34--57.","DOI":"10.1007\/978-3-662-05105-4_2"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/545261.545269"},{"key":"e_1_2_2_31_1","volume-title":"7th International Meshing Roundtable","volume":"3","author":"Owen S.","year":"1998"},{"key":"e_1_2_2_32_1","doi-asserted-by":"publisher","DOI":"10.1145\/15886.15903"},{"key":"e_1_2_2_33_1","doi-asserted-by":"crossref","unstructured":"Shewchuk J. R. 2012. Combinatorial Scientific Computing. CRC Press January ch. Unstructured Mesh Generation chapter 10 257--297. Shewchuk J. R. 2012. Combinatorial Scientific Computing . CRC Press January ch. Unstructured Mesh Generation chapter 10 257--297.","DOI":"10.1201\/b11644-11"},{"key":"e_1_2_2_34_1","volume-title":"Proceedings of the fifth Eurographics symposium on Geometry processing, Eurographics Association, Aire-la-Ville","author":"Sorkine O."},{"key":"e_1_2_2_35_1","volume-title":"IN SVD AND SIGNAL PROCESSING, II: ALGORITHMS, ANALYSIS AND APPLICATIONS","author":"Stewart G. W."},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195900000152"},{"key":"e_1_2_2_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531381"},{"key":"e_1_2_2_38_1","volume-title":"Proc. Medical Imaging Computing and Computer Assisted Intervention (MICCAI), St","author":"Wang Y."},{"key":"e_1_2_2_39_1","volume-title":"Proceedings of the 2004 IEEE International Symposium on Biomedical Imaging: From Nano to Macro","author":"Wang Y.","year":"2004"},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2012.03173.x"},{"key":"e_1_2_2_41_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-13411-1_15"},{"key":"e_1_2_2_42_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2012.173"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2461912.2461931","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2461912.2461931","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T08:35:48Z","timestamp":1750235748000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2461912.2461931"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,7,21]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2013,7,21]]}},"alternative-id":["10.1145\/2461912.2461931"],"URL":"https:\/\/doi.org\/10.1145\/2461912.2461931","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,7,21]]},"assertion":[{"value":"2013-07-21","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}