{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T06:01:08Z","timestamp":1784268068691,"version":"3.55.0"},"reference-count":65,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2020,8,12]],"date-time":"2020-08-12T00:00:00Z","timestamp":1597190400000},"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":[[2020,8,31]]},"abstract":"<jats:p>\n            In this paper, we introduce a numerical technique to generate sample distributions in arbitrary dimension for improved accuracy of Monte Carlo integration. We point out that optimal transport offers theoretical bounds on Monte Carlo integration error, and that the recently-introduced numerical framework of sliced optimal transport (SOT) allows us to formulate a novel and efficient approach to generating well-distributed high-dimensional pointsets. The resulting sliced optimal transport sampling, solely involving repeated 1D solves, is particularly simple and efficient for the common case of a uniform density over a\n            <jats:italic toggle=\"yes\">d<\/jats:italic>\n            -dimensional ball. We also construct a volume-preserving map from a\n            <jats:italic toggle=\"yes\">d<\/jats:italic>\n            -ball to a\n            <jats:italic toggle=\"yes\">d<\/jats:italic>\n            -cube (generalizing the Shirley-Chiu mapping to arbitrary dimensions) to offer fast SOT sampling over\n            <jats:italic toggle=\"yes\">d<\/jats:italic>\n            -cubes. We provide ample numerical evidence of the improvement in Monte Carlo integration accuracy that SOT sampling brings compared to existing QMC techniques, and derive a projective variant for rendering which rivals, and at times outperforms, current sampling strategies using low-discrepancy sequences or optimized samples.\n          <\/jats:p>","DOI":"10.1145\/3386569.3392395","type":"journal-article","created":{"date-parts":[[2020,8,12]],"date-time":"2020-08-12T11:44:27Z","timestamp":1597232667000},"update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":19,"title":["Sliced optimal transport sampling"],"prefix":"10.1145","volume":"39","author":[{"given":"Lois","family":"Paulin","sequence":"first","affiliation":[{"name":"Univ Lyon, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicolas","family":"Bonneel","sequence":"additional","affiliation":[{"name":"Univ Lyon, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Coeurjolly","sequence":"additional","affiliation":[{"name":"Univ Lyon, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Claude","family":"Iehl","sequence":"additional","affiliation":[{"name":"Univ Lyon, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Antoine","family":"Webanck","sequence":"additional","affiliation":[{"name":"Univ Lyon, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mathieu","family":"Desbrun","sequence":"additional","affiliation":[{"name":"ShanghaiTech University, Shanghai, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Victor","family":"Ostromoukhov","sequence":"additional","affiliation":[{"name":"Univ Lyon, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2020,8,12]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009187"},{"key":"e_1_2_2_2_1","volume-title":"Mantiuk","author":"B\u00e1lint Martin","year":"2019","unstructured":"Martin B\u00e1lint and Rafa\u0142 K. Mantiuk. 2019. Closed Form Transmittance in Heterogeneous Media Using Cosine Noise. In Central European Seminar on Computer Graphics. https:\/\/www.cl.cam.ac.uk\/~rkm38\/pdfs\/balint2019cosine_noise.pdf"},{"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":"10.1145\/3306346.3323021"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-014-0506-3"},{"key":"e_1_2_2_6_1","unstructured":"Nicolas Bonnotte. 2013. Unidimensional and evolution methods for optimal transportation. Ph.D. Dissertation. Paris 11."},{"key":"e_1_2_2_7_1","doi-asserted-by":"crossref","unstructured":"Robert Bridson. 2007. Fast Poisson Disk Sampling in Arbitrary Dimensions. In ACM SIGGRAPH Sketches. 22--23.","DOI":"10.1145\/1278780.1278807"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/3182162"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13472"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/7529.8927"},{"key":"e_1_2_2_11_1","first-page":"171","article-title":"Blue noise through optimal transport","volume":"31","author":"Goes Fernando De","year":"2012","unstructured":"Fernando De Goes, Katherine Breeden, Victor Ostromoukhov, and Mathieu Desbrun. 2012. Blue noise through optimal transport. ACM Trans. Graph. 31, 6 (2012), 171.","journal-title":"ACM Trans. Graph."},{"key":"e_1_2_2_12_1","volume-title":"Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration","author":"Dick Josef","unstructured":"Josef Dick and Friedrich Pillichshammer. 2010. Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press."},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/325165.325182"},{"key":"e_1_2_2_14_1","volume-title":"A frequency analysis of Monte-Carlo and other numerical integration schemes. MIT CSAIL Technical report TR-2011-052","author":"Durand Fredo","year":"2011","unstructured":"Fredo Durand. 2011. A frequency analysis of Monte-Carlo and other numerical integration schemes. MIT CSAIL Technical report TR-2011-052 (2011)."},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964943"},{"key":"e_1_2_2_16_1","doi-asserted-by":"crossref","unstructured":"Iliyan Georgiev and Marcos Fajardo. 2016. Blue-Noise Dithered Sampling. 35:1--35:1. https:\/\/doi.org\/10\/gfznbx","DOI":"10.1145\/2897839.2927430"},{"key":"e_1_2_2_17_1","volume-title":"A bi-Lipschitz continuous","author":"Griepentrog Jens Andr\u00e9","year":"2008","unstructured":"Jens Andr\u00e9 Griepentrog, Wolfgang H\u00f6ppner, Hans-Christoph Kaiser, and Joachim Rehberg. 2008. A bi-Lipschitz continuous, volume preserving map from the unit ball onto a cube. Note di Matematica 28, 1 (2008), 177--193."},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/2487228.2487233"},{"key":"e_1_2_2_19_1","volume-title":"Computer Graphics Forum","author":"Heitz Eric","unstructured":"Eric Heitz and Laurent Belcour. 2019. Distributing Monte Carlo Errors as a Blue Noise in Screen Space by Permuting Pixel Seeds Between Frames. In Computer Graphics Forum, Vol. 38. Wiley Online Library, 149--158."},{"key":"e_1_2_2_20_1","doi-asserted-by":"crossref","unstructured":"Eric Heitz Laurent Belcour Victor Ostromoukhov David Coeurjolly and Jean-Claude Iehl. 2019. A Low-Discrepancy Sampler that Distributes Monte Carlo Errors as a Blue Noise in Screen Space. In ACM SIGGRAPH Talk.","DOI":"10.1145\/3306307.3328191"},{"key":"e_1_2_2_21_1","volume-title":"Funktionen von beschr\u00e4nkter Variation in der Theorie der Gleichverteilung. Annali di Matematica Pura ed Applicata 54, 1","author":"Hlavka Edmund","year":"1961","unstructured":"Edmund Hlavka. 1961. Funktionen von beschr\u00e4nkter Variation in der Theorie der Gleichverteilung. Annali di Matematica Pura ed Applicata 54, 1 (1961), 325--333."},{"key":"e_1_2_2_22_1","unstructured":"Wenzel Jakob. 2013. Mitsuba Renderer. http:\/\/www.mitsuba-renderer.org."},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13777"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1137\/070709359"},{"key":"e_1_2_2_25_1","first-page":"89","article-title":"Functional analysis and applied mathematics","volume":"3","author":"Kantorovich Leonid Vitalevich","year":"1948","unstructured":"Leonid Vitalevich Kantorovich. 1948. Functional analysis and applied mathematics. Uspekhi Matematicheskikh Nauk 3, 6 (1948), 89--185.","journal-title":"Uspekhi Matematicheskikh Nauk"},{"key":"e_1_2_2_26_1","first-page":"52","article-title":"On a space of completely additive functions","volume":"13","author":"Kantorovich Leonid V.","year":"1958","unstructured":"Leonid V. Kantorovich and Gennady S. Rubinstein. 1958. On a space of completely additive functions. Vestnik Leningrad. Univ 13, 7 (1958), 52--59.","journal-title":"Vestnik Leningrad. Univ"},{"key":"e_1_2_2_27_1","volume-title":"Monte Carlo and Quasi-Monte Carlo Methods","author":"Keller Alexander","year":"2002","unstructured":"Alexander Keller. 2004. Stratification by rank-1 lattices. In Monte Carlo and Quasi-Monte Carlo Methods 2002, Harald Niederreiter (Ed.). Springer, 299--313."},{"key":"e_1_2_2_28_1","volume-title":"Quasi-Monte Carlo Image Synthesis in a Nutshell","author":"Keller Alexander","unstructured":"Alexander Keller. 2013. Quasi-Monte Carlo Image Synthesis in a Nutshell. In MCQMC, Josef Dick, Frances Y. Kuo, Gareth W. Peters, and Ian H. Sloan (Eds.). Springer, 213--249."},{"key":"e_1_2_2_29_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141916"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/3180495"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/1183287.1183296"},{"key":"e_1_2_2_32_1","volume-title":"Lattice Builder: A General Software Tool for Constructing Rank-1 Lattice Rules. https:\/\/github.com\/umontreal-simul\/latnetbuilder.","author":"L'Ecuyer Pierre","year":"2016","unstructured":"Pierre L'Ecuyer and David Munger. 2016. Lattice Builder: A General Software Tool for Constructing Rank-1 Lattice Rules. https:\/\/github.com\/umontreal-simul\/latnetbuilder."},{"key":"e_1_2_2_33_1","volume-title":"Monte Carlo and Quasi Monte Carlo Sampling","author":"Lemieux Christiane","unstructured":"Christiane Lemieux. 2009. Monte Carlo and Quasi Monte Carlo Sampling. Springer-Verlag New York."},{"key":"e_1_2_2_34_1","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2015055"},{"key":"e_1_2_2_35_1","volume-title":"Computer Graphics Forum","author":"M\u00e9rigot Quentin","unstructured":"Quentin M\u00e9rigot. 2011. A multiscale approach to optimal transport. In Computer Graphics Forum, Vol. 30. Wiley Online Library, 1583--1592."},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/37402.37410"},{"key":"e_1_2_2_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/127719.122736"},{"key":"e_1_2_2_38_1","article-title":"HOT","volume":"30","author":"Mullen Patrick","year":"2011","unstructured":"Patrick Mullen, Pooran Memari, Fernando de Goes, and Mathieu Desbrun. 2011. HOT: Hodge-optimized Triangulations. ACM Trans. Graph. 30, 4 (2011), Art. 103.","journal-title":"Hodge-optimized Triangulations. ACM Trans. Graph."},{"key":"e_1_2_2_39_1","doi-asserted-by":"publisher","DOI":"10.1145\/3272127.3275091"},{"key":"e_1_2_2_40_1","volume-title":"Monte Carlo Methods for Volumetric Light Transport Simulation. 37, 2","author":"Nov\u00e1k Jan","year":"2018","unstructured":"Jan Nov\u00e1k, Iliyan Georgiev, Johannes Hanika, and Wojciech Jarosz. 2018. Monte Carlo Methods for Volumetric Light Transport Simulation. 37, 2 (2018)."},{"key":"e_1_2_2_41_1","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015750"},{"key":"e_1_2_2_42_1","doi-asserted-by":"publisher","DOI":"10.1006\/jcom.1998.0487"},{"key":"e_1_2_2_43_1","doi-asserted-by":"publisher","DOI":"10.1145\/2932186"},{"key":"e_1_2_2_44_1","volume-title":"Computer Graphics Forum","author":"Perrier H\u00e9l\u00e8ne","unstructured":"H\u00e9l\u00e8ne Perrier, David Coeurjolly, Feng Xie, Matt Pharr, Pat Hanrahan, and Victor Ostromoukhov. 2018. Sequences with Low-Discrepancy Blue-Noise 2-D Projections. In Computer Graphics Forum, Vol. 37. Wiley Online Library, 339--353."},{"key":"e_1_2_2_45_1","unstructured":"Matt Pharr Wenzel Jakob and Greg Humphreys. 2016. Physically Based Rendering: From Theory to Implementation (3 ed.). Morgan-Kaufmann."},{"key":"e_1_2_2_46_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766930"},{"key":"e_1_2_2_47_1","volume-title":"IEEE Int. Conf. on Computer Vision (ICCV).","author":"Piti\u00e9 Francois","year":"2005","unstructured":"Francois Piti\u00e9, Anil C. Kokaram, and Rozenn Dahyot. 2005. N-Dimensional Probablility Density Function Transfer and Its Application to Colour Transfer. In IEEE Int. Conf. on Computer Vision (ICCV)."},{"key":"e_1_2_2_48_1","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3119910"},{"key":"e_1_2_2_49_1","volume-title":"International Conference on Scale Space and Variational Methods in Computer Vision. Springer, 435--446","author":"Rabin Julien","year":"2011","unstructured":"Julien Rabin, Gabriel Peyr\u00e9, Julie Delon, and Marc Bernot. 2011. Wasserstein barycenter and its application to texture mixing. In International Conference on Scale Space and Variational Methods in Computer Vision. Springer, 435--446."},{"key":"e_1_2_2_50_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.1986.4307775"},{"key":"e_1_2_2_51_1","volume-title":"Computer Graphics Forum","author":"Reinert Bernhard","unstructured":"Bernhard Reinert, Tobias Ritschel, Hans-Peter Seidel, and Iliyan Georgiev. 2016. Projective blue-noise sampling. In Computer Graphics Forum, Vol. 35. Wiley Online Library, 285--295."},{"key":"e_1_2_2_52_1","volume-title":"Advances in Neural Information Processing Systems.","author":"Rowland Mark","unstructured":"Mark Rowland, Krzysztof M Choromanski, Fran\u00e7ois Chalus, Aldo Pacchiano, Tamas Sarlos, Richard E Turner, and Adrian Weller. 2018. Geometrically Coupled Monte Carlo Sampling. In Advances in Neural Information Processing Systems. Vol. 31. 195--206."},{"key":"e_1_2_2_53_1","doi-asserted-by":"publisher","DOI":"10.1080\/10867651.1997.10487479"},{"key":"e_1_2_2_54_1","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073656"},{"key":"e_1_2_2_55_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13653"},{"key":"e_1_2_2_56_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766963"},{"key":"e_1_2_2_57_1","doi-asserted-by":"publisher","DOI":"10.1145\/2601097.2601175"},{"key":"e_1_2_2_58_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2462013"},{"key":"e_1_2_2_59_1","unstructured":"Peter Aundal Toft and John Aasted S\u00f8rensen. 1996. The Radon transform-theory and implementation. (1996)."},{"key":"e_1_2_2_60_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.74.061308"},{"key":"e_1_2_2_61_1","volume-title":"Digital Halftoning","author":"Ulichney Robert","unstructured":"Robert Ulichney. 1987. Digital Halftoning. MIT Press, Cambridge, MA, USA."},{"key":"e_1_2_2_62_1","volume-title":"Robust Monte Carlo methods for light transport simulation","author":"Veach Eric","unstructured":"Eric Veach. 1997. Robust Monte Carlo methods for light transport simulation. Stanford University PhD thesis."},{"key":"e_1_2_2_63_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71050-9"},{"key":"e_1_2_2_64_1","doi-asserted-by":"publisher","DOI":"10.1145\/1778765.1778816"},{"key":"e_1_2_2_65_1","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185572"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3386569.3392395","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3386569.3392395","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,25]],"date-time":"2025-06-25T05:42:32Z","timestamp":1750830152000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3386569.3392395"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,12]]},"references-count":65,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,8,31]]}},"alternative-id":["10.1145\/3386569.3392395"],"URL":"https:\/\/doi.org\/10.1145\/3386569.3392395","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,12]]},"assertion":[{"value":"2020-08-12","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}