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Graph."],"published-print":{"date-parts":[[2025,8,1]]},"abstract":"<jats:p>\n                    Low-discrepancy sequences, and more particularly Sobol' sequences are gold standard for drawing highly uniform samples for quasi-Monte Carlo applications. They produce so-called (\n                    <jats:italic toggle=\"yes\">t,s<\/jats:italic>\n                    )-sequences, that is, sequences of\n                    <jats:italic toggle=\"yes\">s<\/jats:italic>\n                    -dimensional samples whose uniformity is controlled by a non-negative integer quality factor\n                    <jats:italic toggle=\"yes\">t.<\/jats:italic>\n                    The Monte Carlo integral estimator has a convergence rate that improves as\n                    <jats:italic toggle=\"yes\">t<\/jats:italic>\n                    decreases. Sobol' construction in base 2 also provides extremely fast sampling point generation using efficient xor-based arithmetic. Computer graphics applications, such as rendering, often require high uniformity in consecutive 2D projections and in higher-dimensional projections at the same time. However, it can be shown that, in the classical Sobol' construction, only a single 2D sequence of points (up to scrambling), constructed using irreducible polynomials\n                    <jats:italic toggle=\"yes\">x<\/jats:italic>\n                    and\n                    <jats:italic toggle=\"yes\">x<\/jats:italic>\n                    + 1, achieves the ideal\n                    <jats:italic toggle=\"yes\">t<\/jats:italic>\n                    = 0 property. Reusing this sequence in projections necessarily loses high dimensional uniformity. We prove the existence and construct many 2D Sobol' sequences having\n                    <jats:italic toggle=\"yes\">t<\/jats:italic>\n                    = 1 using irreducible polynomials\n                    <jats:italic toggle=\"yes\">p<\/jats:italic>\n                    and\n                    <jats:italic toggle=\"yes\">p<\/jats:italic>\n                    <jats:sup>2<\/jats:sup>\n                    +\n                    <jats:italic toggle=\"yes\">p +<\/jats:italic>\n                    1. They can be readily combined to produce higher-dimensional low discrepancy sequences with a high-quality\n                    <jats:italic toggle=\"yes\">t<\/jats:italic>\n                    = 1, guaranteed in consecutive pairs of dimensions. We provide the initialization table that can be directly used with any existing Sobol' implementation, along with the corresponding generator matrices, for an optimized 692-dimensional Sobol' construction. In addition to guaranteeing the (1, 2)-sequence property for all consecutive pairs, we ensure that\n                    <jats:italic toggle=\"yes\">t<\/jats:italic>\n                    \u2264 4 for consecutive 4D projections up to 2\n                    <jats:sup>15<\/jats:sup>\n                    points.\n                  <\/jats:p>","DOI":"10.1145\/3730821","type":"journal-article","created":{"date-parts":[[2025,7,27]],"date-time":"2025-07-27T04:02:41Z","timestamp":1753588961000},"page":"1-16","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Sobol' Sequences with Guaranteed-Quality 2D Projections"],"prefix":"10.1145","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5243-4810","authenticated-orcid":false,"given":"Nicolas","family":"Bonneel","sequence":"first","affiliation":[{"name":"CNRS, Lyon, France"},{"name":"Universit\u00e9 Claude Bernard Lyon 1, Lyon, France"},{"name":"INSA Lyon, Lyon, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3164-8697","authenticated-orcid":false,"given":"David","family":"Coeurjolly","sequence":"additional","affiliation":[{"name":"CNRS, Lyon, France"},{"name":"Universit\u00e9 Claude Bernard Lyon 1, Lyon, France"},{"name":"INSA Lyon, Lyon, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6877-2398","authenticated-orcid":false,"given":"Jean-Claude","family":"Iehl","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Claude Bernard Lyon 1, Lyon, France"},{"name":"CNRS, Lyon, France"},{"name":"INSA Lyon, Lyon, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-3123-9388","authenticated-orcid":false,"given":"Victor","family":"Ostromoukhov","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Claude Bernard Lyon 1, Lyon, France"},{"name":"CNRS, Lyon, France"},{"name":"INSA Lyon, Lyon, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2025,7,27]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/3618308"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/3414685.3417881"},{"key":"e_1_2_2_3_1","first-page":"29","article-title":"Practical hash-based Owen scrambling","volume":"10","author":"Burley Brent","year":"2020","unstructured":"Brent Burley. 2020. 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