{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T13:48:13Z","timestamp":1785937693687,"version":"3.56.0"},"reference-count":42,"publisher":"Wiley","license":[{"start":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T00:00:00Z","timestamp":1785888000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T00:00:00Z","timestamp":1785888000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["462649663"],"award-info":[{"award-number":["462649663"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Computer Graphics Forum"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Path tracing uses Monte Carlo integration to solve the rendering equation by evaluating the integrand at random sampling points. The convergence rate of the error can be significantly improved by using correlated instead of random sampling, especially on smooth integrands. However, on integrands with discontinuities due to, e.g., occlusion, the improvement is less pronounced. Prior work has shown that the variance of the estimator is equal to the product of the power spectrum of the integrand and the expected power spectrum of the sampling pattern. Discontinuous integrands have anisotropic power spectra that exhibit high energies along the directions of the discontinuities, which need to match the low\u2010energy directions of the sampling pattern to reduce variance. However, existing anisotropic sampling patterns have at most two low\u2010energy directions. Therefore, we propose an optimization\u2010based algorithm to synthesize two\u2010dimensional correlated sampling patterns with spectra that have more than two low\u2010energy directions, leading to improved convergence behavior. Further, we propose a practical and sample\u2010efficient algorithm that estimates the directions of discontinuities in the power spectra of two\u2010dimensional integrands. We show that our algorithm can reliably estimate these directions, allowing us to align the low\u2010energy directions of anisotropic correlated sampling patterns with the predicted directions. We demonstrate in an offline path tracer with light source sampling that our aligned sampling patterns improve the convergence rate on two\u2010dimensional integrands with multiple discontinuities compared to existing anisotropic sampling patterns and thus reduce the error more quickly.<\/jats:p>","DOI":"10.1111\/cgf.70534","type":"journal-article","created":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T13:19:18Z","timestamp":1785935958000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimized and Aligned Anisotropic Monte Carlo Sampling Patterns"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-5096-9997","authenticated-orcid":false,"given":"Mirco","family":"Werner","sequence":"first","affiliation":[{"name":"Karlsruhe Institute of Technology  Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7648-1782","authenticated-orcid":false,"given":"Johannes","family":"Hanika","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology  Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4690-3574","authenticated-orcid":false,"given":"Carsten","family":"Dachsbacher","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology  Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,8,5]]},"reference":[{"key":"e_1_2_10_2_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073588"},{"key":"e_1_2_10_3_2","doi-asserted-by":"publisher","DOI":"10.1145\/2980179.2980218"},{"key":"e_1_2_10_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-45528-0"},{"key":"e_1_2_10_5_2","doi-asserted-by":"crossref","unstructured":"BridsonR.: Fast poisson disk sampling in arbitrary dimensions. InSIGGRAPH Sketches(2007) p.22. doi:10.1145\/1278780.1278807. 3","DOI":"10.1145\/1278780.1278807"},{"key":"e_1_2_10_6_2","first-page":"1","article-title":"Practical hash\u2010based owen scrambling","volume":"4","author":"Burley B.","year":"2020","journal-title":"Journal of Computer Graphics Techniques 10"},{"key":"e_1_2_10_7_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13472"},{"key":"e_1_2_10_8_2","doi-asserted-by":"publisher","DOI":"10.1145\/7529.8927"},{"key":"e_1_2_10_9_2","doi-asserted-by":"publisher","DOI":"10.1137\/0713071"},{"key":"e_1_2_10_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-336156-1.50045-8"},{"key":"e_1_2_10_11_2","doi-asserted-by":"publisher","DOI":"10.1145\/2366145.2366190"},{"key":"e_1_2_10_12_2","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073320"},{"key":"e_1_2_10_13_2","volume-title":"A frequency analysis of monte\u2010carlo and other numerical integration schemes","author":"Durand F.","year":"2011"},{"key":"e_1_2_10_14_2","unstructured":"EngelhardtT. DachsbacherC.: Octahedron environment maps. InVision Modeling and Visualization(2008) pp.383\u2013388. 11"},{"key":"e_1_2_10_15_2","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964943"},{"key":"e_1_2_10_16_2","doi-asserted-by":"publisher","DOI":"10.1145\/355588.365104"},{"key":"e_1_2_10_17_2","doi-asserted-by":"publisher","DOI":"10.1145\/2487228.2487233"},{"key":"e_1_2_10_18_2","doi-asserted-by":"crossref","unstructured":"HeitzE. VanhoeyK. ChambonT. BelcourL.: A sliced wasserstein loss for neural texture synthesis. InComputer Vision and Pattern Recognition(June2021) pp.9407\u20139415. doi:10.1109\/CVPR46437.2021.00929. 5","DOI":"10.1109\/CVPR46437.2021.00929"},{"key":"e_1_2_10_19_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13777"},{"key":"e_1_2_10_20_2","unstructured":"KingmaD. P. BaJ.: Adam: A method for stochastic optimization. InInternational Conference on Learning Representations(2017). URL:https:\/\/arxiv.org\/abs\/1412.6980. 7"},{"key":"e_1_2_10_21_2","doi-asserted-by":"publisher","DOI":"10.1145\/3180495"},{"key":"e_1_2_10_22_2","volume-title":"Correlated multi\u2010jittered sampling","author":"Kensler A.","year":"2013"},{"key":"e_1_2_10_23_2","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729694"},{"key":"e_1_2_10_24_2","doi-asserted-by":"publisher","DOI":"10.1111\/1467\u20108659.t01\u20101\u201000703"},{"key":"e_1_2_10_25_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2007.01100.x"},{"key":"e_1_2_10_26_2","doi-asserted-by":"publisher","DOI":"10.1145\/3355089.3356562"},{"key":"e_1_2_10_27_2","doi-asserted-by":"publisher","DOI":"10.1145\/122718.122736"},{"key":"e_1_2_10_28_2","doi-asserted-by":"crossref","unstructured":"OwenA. B.: Randomly permuted (t m s)\u2010nets and (t s)\u2010sequences. InMonte Carlo and Quasi\u2010Monte Carlo Methods in Scientific Computing(1995) pp.299\u2013317. doi:10.1007\/978\u20101\u20104612\u20102552\u20102_19. 3 7","DOI":"10.1007\/978-1-4612-2552-2_19"},{"key":"e_1_2_10_29_2","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392395"},{"key":"e_1_2_10_30_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13366"},{"key":"e_1_2_10_31_2","volume":"32","author":"Paszke A.","year":"2019","journal-title":"Neural Information Processing Systems"},{"key":"e_1_2_10_32_2","doi-asserted-by":"publisher","DOI":"10.1145\/2766930"},{"key":"e_1_2_10_33_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12725"},{"key":"e_1_2_10_34_2","doi-asserted-by":"publisher","DOI":"10.1145\/3550454.3555484"},{"key":"e_1_2_10_35_2","first-page":"183","volume":"91","author":"Shirley P.","year":"1991","journal-title":"Eurographics"},{"key":"e_1_2_10_36_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073656"},{"key":"e_1_2_10_37_2","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2462013"},{"key":"e_1_2_10_38_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13653"},{"key":"e_1_2_10_39_2","doi-asserted-by":"publisher","DOI":"10.1016\/0041\u20105553(67)90144\u20109"},{"key":"e_1_2_10_40_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13613"},{"key":"e_1_2_10_41_2","unstructured":"WolfeA.:Deriving the inverse CDF of a rotated square projected onto a line 2023. URL:https:\/\/blog.demofox.org\/2023\/12\/24\/deriving-the-inverse-cdf-of-a-rotated-square-projected-onto-a-line\/. 5"},{"key":"e_1_2_10_42_2","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964945"},{"key":"e_1_2_10_43_2","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185572"}],"container-title":["Computer Graphics Forum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70534","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/cgf.70534","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70534","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T13:19:20Z","timestamp":1785935960000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1111\/cgf.70534"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,5]]},"references-count":42,"alternative-id":["10.1111\/cgf.70534"],"URL":"https:\/\/doi.org\/10.1111\/cgf.70534","archive":["Portico"],"relation":{},"ISSN":["0167-7055","1467-8659"],"issn-type":[{"value":"0167-7055","type":"print"},{"value":"1467-8659","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,5]]},"assertion":[{"value":"2026-08-05","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70534"}}