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Traditional methods typically rely on basis expansions for both the input and output functions. However, using fixed output bases can restrict the precision of output reconstruction and limit the compactness of the kernel representation. In this work, we introduce a new method that approximates both the kernel and the input function using Gaussian mixtures. This formulation allows the integral operator to be evaluated analytically, leading to improved flexibility in kernel storage and output representation. Moreover, our method naturally supports the sequential application of multiple operators and enables closed-form operator composition, which is particularly beneficial in tasks involving chains of operators. We demonstrate the versatility and effectiveness of our approach across a variety of graphics problems, including environment map relighting, boundary value problems, and fluorescence rendering.<\/jats:p>","DOI":"10.1145\/3763321","type":"journal-article","created":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T17:15:39Z","timestamp":1764868539000},"page":"1-15","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Gaussian Integral Linear Operators for Precomputed Graphics"],"prefix":"10.1145","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-2595-2493","authenticated-orcid":false,"given":"Haolin","family":"Lu","sequence":"first","affiliation":[{"name":"MPI for Informatics, Saarbrucken, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-7070-2845","authenticated-orcid":false,"given":"Yash","family":"Belhe","sequence":"additional","affiliation":[{"name":"UC San Diego, La Jolla, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0970-5835","authenticated-orcid":false,"given":"Gurprit","family":"Singh","sequence":"additional","affiliation":[{"name":"MPI for Informatics, Saarbrucken, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5443-470X","authenticated-orcid":false,"given":"Tzu-Mao","family":"Li","sequence":"additional","affiliation":[{"name":"UC San Diego, La Jolla, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0284-3776","authenticated-orcid":false,"given":"Toshiya","family":"Hachisuka","sequence":"additional","affiliation":[{"name":"University of Waterloo, Waterloo, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2025,12,4]]},"reference":[{"unstructured":"Anima Anandkumar Kamyar Azizzadenesheli Kaushik Bhattacharya Nikola Kovachki Zongyi Li Burigede Liu and Andrew Stuart. 2019. 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