{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T08:17:51Z","timestamp":1783066671558,"version":"3.54.6"},"reference-count":40,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T00:00:00Z","timestamp":1783036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["62372257"],"award-info":[{"award-number":["62372257"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2026,7,3]]},"abstract":"<jats:p>\n                    Spherical Harmonics (SH) are fundamental mathematical tools for compactly representing low-frequency spherical functions in computer graphics. Evaluating products of SH-represented functions is a core operation in many rendering algorithms and often becomes a major computational bottleneck. Existing SH product methods face two key challenges: insufficient computational efficiency, and the inflexible requirement that all input functions share the same SH order. To overcome these limitations, we introduce the\n                    <jats:italic toggle=\"yes\">Generalized Spherical Harmonics Product<\/jats:italic>\n                    , a novel formulation that supports inputs with different SH orders and allows flexible truncation of the output.\n                  <\/jats:p>\n                  <jats:p>We further propose an efficient and practical computation approach based on Spherical Grids (SphGrid). We establish explicit bidirectional conversions between SphGrid representations and SH coefficients, thus enabling SphGrid to serve as an intermediate representation. Under this framework, SH product computation is reduced to simple point-wise multiplications on the grid. We then derive sufficient conditions on the grid resolution to guarantee exact computation. Using lower resolutions yields an approximate variant that provides a favorable trade-off between accuracy and efficiency.<\/jats:p>\n                  <jats:p>Our approach is not only more general but also demonstrates superior performance even for the specialized case of identical input orders. In particular, our accurate method achieves a 3.5\u20136.0\u00d7 speedup over the state-of-the-art method in a practical setting for graphics, while our approximate variant yields lower errors and a 2.5\u20136.5\u00d7 speedup over the state-of-the-art approximate method. We rigorously analyze the correctness and efficiency of our method and demonstrate its effectiveness in real-time rendering applications.<\/jats:p>","DOI":"10.1145\/3811322","type":"journal-article","created":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T07:05:51Z","timestamp":1783062351000},"page":"1-18","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Generalized Spherical Harmonics Products using Spherical Grids"],"prefix":"10.1145","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6112-2816","authenticated-orcid":false,"given":"Di","family":"An","sequence":"first","affiliation":[{"name":"Tsinghua University, Beijing, China, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-8256-0592","authenticated-orcid":false,"given":"Jiaqi","family":"Wu","sequence":"additional","affiliation":[{"name":"Tsinghua University, Beijing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-5986-6907","authenticated-orcid":false,"given":"Bowen","family":"Xu","sequence":"additional","affiliation":[{"name":"Tsinghua University, Beijing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9379-094X","authenticated-orcid":false,"given":"Lingqi","family":"Yan","sequence":"additional","affiliation":[{"name":"Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2671-4170","authenticated-orcid":false,"given":"Kun","family":"Xu","sequence":"additional","affiliation":[{"name":"Tsinghua University, Beijing, China, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2026,7,3]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-6660-4_6"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1006\/aama.1994.1008"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1086\/427976"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11200-011-0029-y"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/990002.990004"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-003-0018-9"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/3592433"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2005.83"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/1289603.1289604"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/2601097.2601149"},{"key":"e_1_2_1_11_1","volume-title":"All-Frequency Precomputed Radiance Transfer for Glossy Objects. Rendering Techniques 2004","author":"Liu Xinguo","year":"2004","unstructured":"Xinguo Liu, Peter-Pike J Sloan, Heung-Yeung Shum, and John Snyder. 2004. All-Frequency Precomputed Radiance Transfer for Glossy Objects. Rendering Techniques 2004 (2004)."},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12190"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01261607"},{"key":"e_1_2_1_14_1","volume-title":"Instant neural graphics primitives with a multiresolution hash encoding. ACM transactions on graphics (TOG) 41, 4","author":"M\u00fcller Thomas","year":"2022","unstructured":"Thomas M\u00fcller, Alex Evans, Christoph Schied, and Alexander Keller. 2022. Instant neural graphics primitives with a multiresolution hash encoding. ACM transactions on graphics (TOG) 41, 4 (2022), 1\u201315."},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015749"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/2167076.2167081"},{"key":"e_1_2_1_17_1","volume-title":"Numerical recipes","author":"Press William H","unstructured":"William H Press. 2007. Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press.","edition":"3"},{"key":"e_1_2_1_18_1","doi-asserted-by":"crossref","unstructured":"Ravi Ramamoorthi et al. 2009. Precomputation-based rendering. Foundations and Trends\u00ae in Computer Graphics and Vision 3 4 (2009) 281\u2013369.","DOI":"10.1561\/0600000021"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/383259.383317"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/566570.566611"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141982"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1137\/050623073"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1038\/ngeo643"},{"key":"e_1_2_1_24_1","doi-asserted-by":"publisher","DOI":"10.1002\/ggge.20071"},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/566570.566612"},{"key":"e_1_2_1_26_1","volume-title":"Precomputed Radiance Transfer: Theory and Practice. SIGGRAPH courses","author":"Sloan Peter-Pike","year":"2005","unstructured":"Peter-Pike Sloan, Jaakko Lehtinen, and Jan Kautz. 2005a. Precomputed Radiance Transfer: Theory and Practice. SIGGRAPH courses (2005)."},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882279"},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073335"},{"key":"e_1_2_1_29_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-01-01386-2"},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141980"},{"key":"e_1_2_1_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/3197517.3201291"},{"key":"e_1_2_1_32_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00371-006-0057-8"},{"key":"e_1_2_1_33_1","first-page":"2574","article-title":"SHTools: Tools for working with spherical harmonics. Geochemistry, Geophysics","volume":"19","author":"Wieczorek Mark A","year":"2018","unstructured":"Mark A Wieczorek and Matthias Meschede. 2018. SHTools: Tools for working with spherical harmonics. Geochemistry, Geophysics, Geosystems 19, 8 (2018), 2574\u20132592.","journal-title":"Geosystems"},{"key":"e_1_2_1_34_1","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392373"},{"key":"e_1_2_1_35_1","doi-asserted-by":"publisher","DOI":"10.1145\/3478513.3480563"},{"key":"e_1_2_1_36_1","doi-asserted-by":"publisher","DOI":"10.1007\/s41095-023-0367-z"},{"key":"e_1_2_1_37_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2007.01077.x"},{"key":"e_1_2_1_38_1","doi-asserted-by":"publisher","DOI":"10.26599\/CVM.2025.9450445"},{"key":"e_1_2_1_39_1","doi-asserted-by":"publisher","DOI":"10.1145\/3658139"},{"key":"e_1_2_1_40_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073332"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","deposited":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T07:41:02Z","timestamp":1783064462000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3811322"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,3]]},"references-count":40,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,7,3]]}},"alternative-id":["10.1145\/3811322"],"URL":"https:\/\/doi.org\/10.1145\/3811322","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,3]]},"assertion":[{"value":"2026-01-22","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-03-27","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-07-03","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}