{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T19:30:01Z","timestamp":1783107001303,"version":"3.54.6"},"reference-count":41,"publisher":"Association for Computing Machinery (ACM)","issue":"5","license":[{"start":{"date-parts":[[2017,10,11]],"date-time":"2017-10-11T00:00:00Z","timestamp":1507680000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"name":"State Key Development Program of Basic Research of China","award":["2015CB352501"],"award-info":[{"award-number":["2015CB352501"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["61772097 and U1401252"],"award-info":[{"award-number":["61772097 and U1401252"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2017,10,31]]},"abstract":"<jats:p>In this article, we present a multi-class blue noise sampling algorithm by throwing samples as the constrained Wasserstein barycenter of multiple density distributions. Using an entropic regularization term, a constrained transport plan in the optimal transport problem is provided to break the partition required by the previous Capacity-Constrained Voronoi Tessellation method. The entropic regularization term cannot only control spatial regularity of blue noise sampling, but it also reduces conflicts between the desired centroids of Vornoi cells for multi-class sampling. Moreover, the adaptive blue noise property is guaranteed for each individual class, as well as their combined class. Our method can be easily extended to multi-class sampling on a point set surface. We also demonstrate applications in object distribution and color stippling.<\/jats:p>","DOI":"10.1145\/3119910","type":"journal-article","created":{"date-parts":[[2017,10,12]],"date-time":"2017-10-12T12:52:50Z","timestamp":1507812770000},"page":"1-13","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":22,"title":["Wasserstein Blue Noise Sampling"],"prefix":"10.1145","volume":"36","author":[{"given":"Hongxing","family":"Qin","sequence":"first","affiliation":[{"name":"Chongqing Key Laboratory of Computational Intelligence, Chonqging University of Posts and Telecommunications, Shandong University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Computer Science and Technology, Chonqging University of Posts and Telecommunications"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinlong","family":"He","sequence":"additional","affiliation":[{"name":"College of Computer Science and Technology, Chonqging University of Posts and Telecommunications"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Baoquan","family":"Chen","sequence":"additional","affiliation":[{"name":"Shandong University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2017,10,11]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/100805741"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/VISUAL.2001.964489"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531392"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1137\/141000439"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/2897824.2925918"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2014.2346594"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2012.94"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/7529.8927"},{"key":"e_1_2_2_9_1","unstructured":"Marco Cuturi. 2013. Sinkhorn distances: Lightspeed computation of optimal transport. In Adv. Neural Info. Process. Syst. 2292--2300.  Marco Cuturi. 2013. Sinkhorn distances: Lightspeed computation of optimal transport. In Adv. Neural Info. Process. Syst. 2292--2300."},{"key":"e_1_2_2_10_1","volume-title":"Proceedings of The 31st International Conference on Machine Learning","author":"Cuturi Marco","year":"2014","unstructured":"Marco Cuturi and Arnaud Doucet . 2014 . Fast computation of Wasserstein barycenters . Proceedings of The 31st International Conference on Machine Learning (2014), 685--693. Marco Cuturi and Arnaud Doucet. 2014. Fast computation of Wasserstein barycenters. Proceedings of The 31st International Conference on Machine Learning (2014), 685--693."},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/2366145.2366190"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/325165.325182"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141915"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964944"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2012.03059.x"},{"key":"e_1_2_2_16_1","volume-title":"Blue-noise point sampling using kernel density model. 30, 4","author":"Fattal Raanan","year":"2011","unstructured":"Raanan Fattal . 2011. Blue-noise point sampling using kernel density model. 30, 4 ( 2011 ), 48:1--48:10. Raanan Fattal. 2011. Blue-noise point sampling using kernel density model. 30, 4 (2011), 48:1--48:10."},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/1640443.1640451"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/2816795.2818102"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1080\/2151237X.2006.10129217"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1289238"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2007.01100.x"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1982.1056489"},{"key":"e_1_2_2_23_1","volume-title":"Proceedings of the Conference on Graphics Interface","volume":"92","author":"McCool Michael","year":"1992","unstructured":"Michael McCool and Eugene Fiume . 1992 . Hierarchical Poisson disk sampling distributions . In Proceedings of the Conference on Graphics Interface , Vol. 92 . 94--105. Michael McCool and Eugene Fiume. 1992. Hierarchical Poisson disk sampling distributions. In Proceedings of the Conference on Graphics Interface, Vol. 92. 94--105."},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/37401.37410"},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/1882261.1866190"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-10605-2_48"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1080\/2151237X.2011.609773"},{"key":"e_1_2_2_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/2018323.2018345"},{"key":"e_1_2_2_29_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2012.03072.x"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766963"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1109\/5.3288"},{"key":"e_1_2_2_32_1","volume-title":"Optimal Transport: Old and New.","author":"Villani C\u00e9dric","year":"2008","unstructured":"C\u00e9dric Villani . 2008 . Optimal Transport: Old and New. Vol. 338 . Springer Science 8 Business Media. C\u00e9dric Villani. 2008. Optimal Transport: Old and New. Vol. 338. Springer Science 8 Business Media."},{"key":"e_1_2_2_33_1","volume-title":"Proceedings of the International Conference on Image Processing. 328--332","author":"Wang Muge","unstructured":"Muge Wang and Kevin J. Parker . 1999. Properties of combined blue noise patterns . In Proceedings of the International Conference on Image Processing. 328--332 . Muge Wang and Kevin J. Parker. 1999. Properties of combined blue noise patterns. In Proceedings of the International Conference on Image Processing. 328--332."},{"key":"e_1_2_2_34_1","first-page":"20","article-title":"Parallel Poisson disk sampling","volume":"27","author":"Wei Li-Yi","year":"2008","unstructured":"Li-Yi Wei . 2008 . Parallel Poisson disk sampling . ACM Trans. Grap. (TOG) 27 , 3 (2008), 20 . Li-Yi Wei. 2008. Parallel Poisson disk sampling. ACM Trans. Grap. (TOG) 27, 3 (2008), 20.","journal-title":"ACM Trans. Grap. (TOG)"},{"key":"e_1_2_2_35_1","doi-asserted-by":"publisher","DOI":"10.1145\/1778765.1778816"},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964945"},{"key":"e_1_2_2_37_1","doi-asserted-by":"publisher","DOI":"10.1109\/RT.2007.4342600"},{"key":"e_1_2_2_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/2980179.2982428"},{"key":"e_1_2_2_39_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cag.2012.02.005"},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cag.2011.03.031"},{"key":"e_1_2_2_41_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12538"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3119910","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3119910","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T19:05:04Z","timestamp":1750273504000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3119910"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,11]]},"references-count":41,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2017,10,31]]}},"alternative-id":["10.1145\/3119910"],"URL":"https:\/\/doi.org\/10.1145\/3119910","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,10,11]]},"assertion":[{"value":"2016-10-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-05-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-10-11","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}