{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T16:07:52Z","timestamp":1773504472995,"version":"3.50.1"},"reference-count":59,"publisher":"Association for Computing Machinery (ACM)","issue":"2","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2026,4,30]]},"abstract":"<jats:p>Designing a quad mesh that meets aesthetic, anatomical, and numerical requirements often requires meticulous manual effort in conventional methods, making quadrilateral remeshing an \u201cart of design\u201d. Neural networks hold significant promise for automating this process. However, current approaches that directly predict cross fields cannot properly handle the discontinuous behavior of smooth cross fields: minor shape variations can lead to substantial changes in the cross field, even when singularities remain largely unchanged. Therefore, such methods often result in non-smooth outputs when combining multiple singularity instances. To avoid such discontinuity, we propose to learn the sparse singularities, including their locations and indices, then let the non-neural conventional method to smoothly connect them. The imbalanced ratio of singular and regular vertices poses a significant challenge for learning. We convert them into a geodesic distance field and an over-sampled index field to address it. This carefully designed two-stage strategy satisfies several key requirements, such as coordinate invariance and tessellation insensitivity, while enabling the generation of smooth cross fields with varying topologies. By shifting the focus from directly learning the cross field to learning singularities, we also simplify the dataset preparation process by requiring only sparse annotations.<\/jats:p>","DOI":"10.1145\/3787520","type":"journal-article","created":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T21:09:22Z","timestamp":1768252162000},"page":"1-15","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Learning Sparse Singularities for Cross Field Design"],"prefix":"10.1145","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-0988-3288","authenticated-orcid":false,"given":"Xiaohu","family":"Zhang","sequence":"first","affiliation":[{"name":"State Key Lab of CAD&CG, Zhejiang University","place":["Hangzhou, China"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2662-0334","authenticated-orcid":false,"given":"Hujun","family":"Bao","sequence":"additional","affiliation":[{"name":"State Key Lab of CAD&CG, Zhejiang University","place":["Hangzhou, China"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2549-6810","authenticated-orcid":false,"given":"Jin","family":"Huang","sequence":"additional","affiliation":[{"name":"State Key Lab of CAD&CG, Zhejiang University","place":["Hangzhou, China"]}]}],"member":"320","published-online":{"date-parts":[[2026,3,9]]},"reference":[{"key":"e_1_3_3_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICCVW.2011.6130444"},{"key":"e_1_3_3_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/3618379"},{"key":"e_1_3_3_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2017.591"},{"key":"e_1_3_3_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2462014"},{"key":"e_1_3_3_6_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2011.01868.x"},{"key":"e_1_3_3_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531383"},{"key":"e_1_3_3_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2018.07.011"},{"key":"e_1_3_3_9_1","volume-title":"Shigley\u2019s Mechanical Engineering Design","unstructured":"Richard Gordon Budynas and J. Keith Nisbett. 2014. Shigley\u2019s Mechanical Engineering Design. McGraw-Hill Higher Education - VST E+p. Retrieved from https:\/\/books.google.co.jp\/books?id=B7wivgAACAAJ"},{"key":"e_1_3_3_10_1","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2017.16"},{"key":"e_1_3_3_11_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13593"},{"key":"e_1_3_3_12_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2010.01761.x"},{"key":"e_1_3_3_13_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01520.x"},{"key":"e_1_3_3_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/1409060.1409101"},{"key":"e_1_3_3_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV48922.2021.01198"},{"key":"e_1_3_3_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766906"},{"key":"e_1_3_3_17_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14366"},{"key":"e_1_3_3_18_1","doi-asserted-by":"publisher","DOI":"10.2514\/6.2019-1988"},{"key":"e_1_3_3_19_1","doi-asserted-by":"publisher","unstructured":"Qiujie Dong Jiepeng Wang Rui Xu Cheng Lin Yuan Liu Shiqing Xin Zichun Zhong Xin Li Changhe Tu Taku Komura Leif Kobbelt Scott Schaefer and Wenping Wang. 2025. CrossGen: Learning and generating cross fields for quad meshing. ACM Trans. Graph. 44 6 (December 2025). DOI:10.1145\/3763299","DOI":"10.1145\/3763299"},{"key":"e_1_3_3_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/3731159"},{"key":"e_1_3_3_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/2508363.2508372"},{"key":"e_1_3_3_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/2661229.2661240"},{"key":"e_1_3_3_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/2980179.2982413"},{"key":"e_1_3_3_24_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-002-2885-2"},{"key":"e_1_3_3_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/3197517.3201354"},{"key":"e_1_3_3_26_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2021.3136199"},{"key":"e_1_3_3_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/2897824.2925957"},{"key":"e_1_3_3_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/3197517.3201375"},{"key":"e_1_3_3_29_1","doi-asserted-by":"publisher","DOI":"10.1145\/3478513.3480523"},{"key":"e_1_3_3_30_1","doi-asserted-by":"publisher","DOI":"10.5555\/3692070.3692654"},{"key":"e_1_3_3_31_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2008.110"},{"key":"e_1_3_3_32_1","doi-asserted-by":"publisher","DOI":"10.1109\/TKDE.2008.239"},{"key":"e_1_3_3_33_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2012.01.001"},{"key":"e_1_3_3_34_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cag.2012.03.025"},{"key":"e_1_3_3_35_1","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR42600.2020.00492"},{"key":"e_1_3_3_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2462005"},{"key":"e_1_3_3_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/3592457"},{"key":"e_1_3_3_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/2816795.2818013"},{"key":"e_1_3_3_39_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14365"},{"key":"e_1_3_3_40_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2024.3418892"},{"key":"e_1_3_3_41_1","doi-asserted-by":"publisher","DOI":"10.1137\/0216045"},{"key":"e_1_3_3_42_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461970"},{"key":"e_1_3_3_43_1","doi-asserted-by":"publisher","DOI":"10.1145\/2024156.2024175"},{"key":"e_1_3_3_44_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.10.045"},{"key":"e_1_3_3_45_1","doi-asserted-by":"publisher","DOI":"10.1145\/1640443.1640444"},{"key":"e_1_3_3_46_1","doi-asserted-by":"publisher","DOI":"10.1145\/1356682.1356683"},{"key":"e_1_3_3_47_1","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2016.91"},{"key":"e_1_3_3_48_1","doi-asserted-by":"publisher","DOI":"10.1145\/3507905"},{"key":"e_1_3_3_49_1","doi-asserted-by":"publisher","DOI":"10.1145\/3243651"},{"key":"e_1_3_3_50_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01515.x"},{"key":"e_1_3_3_51_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12443"},{"key":"e_1_3_3_52_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01610.x"},{"key":"e_1_3_3_53_1","doi-asserted-by":"publisher","DOI":"10.1145\/2070781.2024176"},{"key":"e_1_3_3_54_1","article-title":"The topology toolkit","author":"Tierny Julien","year":"2017","unstructured":"Julien Tierny, Guillaume Favelier, Joshua A. Levine, Charles Gueunet, and Michael Michaux. 2017. The topology toolkit. IEEE Transactions on Visualization and Computer Graphics 24, 1 (2017), 832\u2013842. Retrieved from https:\/\/topology-tool-kit.github.io\/","journal-title":"IEEE Transactions on Visualization and Computer Graphics 24, 1"},{"key":"e_1_3_3_55_1","doi-asserted-by":"publisher","unstructured":"Amir Vaxman Marcel Campen Olga Diamanti Daniele Panozzo David Bommes Klaus Hildebrandt and Mirela Ben-Chen. 2016. Directional field synthesis design and processing. Computer Graphics Forum (2016). DOI:10.1111\/cgf.12864","DOI":"10.1111\/cgf.12864"},{"key":"e_1_3_3_56_1","doi-asserted-by":"publisher","unstructured":"Chaman Singh Verma and Krishnan Suresh. 2017. A robust combinatorial approach to reduce singularities in quadrilateral meshes. Computer-Aided Design 85 (2017) 99\u2013110. DOI:10.1016\/j.cad.2016.07.008","DOI":"10.1016\/j.cad.2016.07.008"},{"key":"e_1_3_3_57_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2024.102294"},{"key":"e_1_3_3_58_1","doi-asserted-by":"publisher","DOI":"10.1145\/3410700.3425436"},{"key":"e_1_3_3_59_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2008.03.037"},{"key":"e_1_3_3_60_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2018.03.003"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3787520","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T12:46:57Z","timestamp":1773492417000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3787520"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,9]]},"references-count":59,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4,30]]}},"alternative-id":["10.1145\/3787520"],"URL":"https:\/\/doi.org\/10.1145\/3787520","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,9]]},"assertion":[{"value":"2024-11-27","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-12-18","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-03-09","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}