{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T01:22:48Z","timestamp":1775870568340,"version":"3.50.1"},"reference-count":43,"publisher":"Association for Computing Machinery (ACM)","issue":"6","license":[{"start":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T00:00:00Z","timestamp":1638316800000},"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":[[2021,12]]},"abstract":"<jats:p>\n            Injective parameterizations of triangulated meshes are critical across applications but remain challenging to compute. Existing algorithms to find injectivity either require initialization from an injective starting state, which is currently only possible without positional constraints, or else can only prevent triangle inversion, which is insufficient to ensure injectivity. Here we present, to our knowledge, the first algorithm for recovering a globally injective parameterization from an arbitrary non-injective initial mesh subject to stationary constraints. These initial meshes can be inverted, wound about interior vertices and\/or overlapping. Our algorithm in turn enables globally injective mapping for meshes with arbitrary positional constraints. Our key contribution is a new energy, called\n            <jats:italic>smooth excess area<\/jats:italic>\n            (SEA), that measures non-injectivity in a map. This energy is well-defined across both injective and non-injective maps and is smooth almost everywhere, making it readily minimizable using standard gradient-based solvers starting from a non-injective initial state. Importantly, we show that maps minimizing SEA are guaranteed to be locally injective and\n            <jats:italic>almost<\/jats:italic>\n            globally injective, in the sense that the overlapping area can be made arbitrarily small. Analyzing SEA's behavior over a new benchmark set designed to test injective mapping, we find that optimizing SEA successfully recovers globally injective maps for 85% of the benchmark and obtains locally injective maps for 90%. In contrast, state-of-the-art methods for removing triangle inversion obtain locally injective maps for less than 6% of the benchmark, and achieve global injectivity (largely by chance as prior methods are not designed to recover it) on less than 4%.\n          <\/jats:p>","DOI":"10.1145\/3478513.3480556","type":"journal-article","created":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T18:28:45Z","timestamp":1639160925000},"page":"1-18","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":9,"title":["Optimizing global injectivity for constrained parameterization"],"prefix":"10.1145","volume":"40","author":[{"given":"Xingyi","family":"Du","sequence":"first","affiliation":[{"name":"Washington University in St. Louis"}]},{"given":"Danny M.","family":"Kaufman","sequence":"additional","affiliation":[{"name":"Adobe Research"}]},{"given":"Qingnan","family":"Zhou","sequence":"additional","affiliation":[{"name":"Adobe Research"}]},{"given":"Shahar Z.","family":"Kovalsky","sequence":"additional","affiliation":[{"name":"Duke University"}]},{"given":"Yajie","family":"Yan","sequence":"additional","affiliation":[{"name":"Facebook"}]},{"given":"Noam","family":"Aigerman","sequence":"additional","affiliation":[{"name":"Adobe Research"}]},{"given":"Tao","family":"Ju","sequence":"additional","affiliation":[{"name":"Washington University in St. Louis"}]}],"member":"320","published-online":{"date-parts":[[2021,12,10]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/1377676.1377726"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461931"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/2816795.2818099"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566623"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1137\/080737617"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/2897824.2925890"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/147508.147511"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13243"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392484"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459757"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-02-01466-7"},{"key":"e_1_2_2_12_1","volume-title":"Advances in multiresolution for geometric modelling","author":"Floater Michael S","unstructured":"Michael S Floater and Kai Hormann . 2005. Surface parameterization: a tutorial and survey . In Advances in multiresolution for geometric modelling . Springer , 157--186. Michael S Floater and Kai Hormann. 2005. Surface parameterization: a tutorial and survey. In Advances in multiresolution for geometric modelling. Springer, 157--186."},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/2980179.2980231"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766938"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.5555\/1133946.1648437"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1531188190"},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/2070781.2024171"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/1399504.1360622"},{"key":"e_1_2_2_19_1","volume-title":"Computer Graphics Forum","author":"Hefetz Eden Fedida","unstructured":"Eden Fedida Hefetz , Edward Chien , and Ofir Weber . 2019. A Subspace Method for Fast Locally Injective Harmonic Mapping . In Computer Graphics Forum , Vol. 38 . Wiley Online Library , 105--119. Eden Fedida Hefetz, Edward Chien, and Ofir Weber. 2019. A Subspace Method for Fast Locally Injective Harmonic Mapping. In Computer Graphics Forum, Vol. 38. Wiley Online Library, 105--119."},{"key":"e_1_2_2_20_1","volume-title":"Proceedings","volume":"1","author":"Hormann Kai","year":"2000","unstructured":"Kai Hormann and Giinther Greiner . 2000 . MIPS: An Efficient Global Parametrization Method. France on 1-7 July 1999 . Proceedings , Volume 1 . Curve and Surface Design. F61775-99-WF068 (2000), 153. Kai Hormann and Giinther Greiner. 2000. MIPS: An Efficient Global Parametrization Method. France on 1-7 July 1999. Proceedings, Volume 1. Curve and Surface Design. F61775-99-WF068 (2000), 153."},{"key":"e_1_2_2_21_1","doi-asserted-by":"crossref","unstructured":"Kai Hormann Bruno L\u00e9vy and Alla Sheffer. 2007. Mesh parameterization: Theory and practice. (2007).  Kai Hormann Bruno L\u00e9vy and Alla Sheffer. 2007. Mesh parameterization: Theory and practice. (2007).","DOI":"10.1145\/1281500.1281510"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/1186822.1073323"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/3130800.3130895"},{"key":"e_1_2_2_24_1","unstructured":"Steven G. Johnson. [n.d.]. The NLopt nonlinear-optimization package. http:\/\/github.com\/stevengj\/nlopt  Steven G. Johnson. [n.d.]. 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