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We further provide a theoretical background that explains how our spherical framework can be extended to the hemispherical domain. We use our framework to estimate the variance convergence rate of different state-of-the-art sampling patterns in both the Euclidean and spherical domains, as the number of samples increases. Furthermore, we formulate design principles for constructing sampling methods that can be tailored according to available resources. We validate our theoretical framework by performing numerical integration over several integrands sampled using different sampling patterns.<\/jats:p>","DOI":"10.1145\/2766930","type":"journal-article","created":{"date-parts":[[2015,7,28]],"date-time":"2015-07-28T12:26:38Z","timestamp":1438086398000},"page":"1-14","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":50,"title":["Variance analysis for Monte Carlo integration"],"prefix":"10.1145","volume":"34","author":[{"given":"Adrien","family":"Pilleboue","sequence":"first","affiliation":[{"name":"Universit\u00e9 Lyon"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gurprit","family":"Singh","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Lyon"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Coeurjolly","sequence":"additional","affiliation":[{"name":"CNRS\/LIRIS UMR"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Kazhdan","sequence":"additional","affiliation":[{"name":"Johns Hopkins University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Victor","family":"Ostromoukhov","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Lyon and F@CNRS\/LIRIS UMR"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2015,7,27]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/218380.218500"},{"key":"e_1_2_2_2_1","volume-title":"SIGGRAPH 2001 Course Notes 29","author":"Arvo J.","year":"2001","unstructured":"Arvo , J. 2001 . 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