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Existing approaches heavily rely on Newton iterations in path space, which are limited to obtaining at most a single solution each time and easily diverge when initialized with improper seeds.<\/jats:p>\n          <jats:p>\n            We propose\n            <jats:italic>specular polynomials<\/jats:italic>\n            , a Newton iteration-free methodology for finding a complete set of admissible specular paths connecting two arbitrary endpoints in a scene. The core is a reformulation of specular constraints into polynomial systems, which makes it possible to reduce the task to a univariate root-finding problem. We first derive bivariate systems utilizing rational coordinate mapping between the coordinates of consecutive vertices. Subsequently, we adopt the hidden variable resultant method for variable elimination, converting the problem into finding zeros of the determinant of univariate matrix polynomials. This can be effectively solved through Laplacian expansion for one bounce and a bisection solver for more bounces.\n          <\/jats:p>\n          <jats:p>Our solution is generic, completely deterministic, accurate for the case of one bounce, and GPU-friendly. We develop efficient CPU and GPU implementations and apply them to challenging glints and caustic rendering. Experiments on various scenarios demonstrate the superiority of specular polynomial-based solutions compared to Newton iteration-based counterparts. Our implementation is available at https:\/\/github.com\/mollnn\/spoly.<\/jats:p>","DOI":"10.1145\/3658132","type":"journal-article","created":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T14:47:57Z","timestamp":1721400477000},"page":"1-13","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["Specular Polynomials"],"prefix":"10.1145","volume":"43","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-6620-1900","authenticated-orcid":false,"given":"Zhimin","family":"Fan","sequence":"first","affiliation":[{"name":"Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4176-7617","authenticated-orcid":false,"given":"Jie","family":"Guo","sequence":"additional","affiliation":[{"name":"Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-6212-4224","authenticated-orcid":false,"given":"Yiming","family":"Wang","sequence":"additional","affiliation":[{"name":"Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-5558-4687","authenticated-orcid":false,"given":"Tianyu","family":"Xiao","sequence":"additional","affiliation":[{"name":"Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-1145-783X","authenticated-orcid":false,"given":"Hao","family":"Zhang","sequence":"additional","affiliation":[{"name":"Southeast University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-7673-718X","authenticated-orcid":false,"given":"Chenxi","family":"Zhou","sequence":"additional","affiliation":[{"name":"Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5995-8051","authenticated-orcid":false,"given":"Zhenyu","family":"Chen","sequence":"additional","affiliation":[{"name":"Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-6345-0927","authenticated-orcid":false,"given":"Pengpei","family":"Hong","sequence":"additional","affiliation":[{"name":"University of Utah, Salt Lake City, United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7605-5206","authenticated-orcid":false,"given":"Yanwen","family":"Guo","sequence":"additional","affiliation":[{"name":"Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9379-094X","authenticated-orcid":false,"given":"Ling-Qi","family":"Yan","sequence":"additional","affiliation":[{"name":"University of California, Santa Barbara, Santa Barbara, United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2024,7,19]]},"reference":[{"key":"e_1_2_2_1_1","volume-title":"GPU, & Game Tools 2","author":"Akenine-M\u00f6ller Tomas","year":"1997","unstructured":"Tomas Akenine-M\u00f6ller and Ben Trumbore. 1997. 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