{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T08:17:00Z","timestamp":1774599420252,"version":"3.50.1"},"reference-count":24,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2007,7,29]],"date-time":"2007-07-29T00:00:00Z","timestamp":1185667200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2007,7,29]]},"abstract":"<jats:p>Blockwise or Clustered Principal Component Analysis (CPCA) is commonly used to achieve real-time rendering of shadows and glossy reflections with precomputed radiance transfer (PRT). The vertices or pixels are partitioned into smaller coherent regions, and light transport in each region is approximated by a<jats:italic>locally low-dimensional subspace<\/jats:italic>using PCA. Many earlier techniques such as surface light field and reflectance field compression use a similar paradigm. However, there has been no clear theoretical understanding of how light transport dimensionality increases with local patch size, nor of the optimal block size or number of clusters.<\/jats:p><jats:p>In this paper, we develop a theory of locally low dimensional light transport, by using Szego's eigenvalue theorem to analytically derive the eigenvalues of the covariance matrix for canonical cases. We show mathematically that for symmetric patches of area<jats:italic>A<\/jats:italic>, the number of basis functions for glossy reflections increases linearly with<jats:italic>A<\/jats:italic>, while for simple cast shadows, it often increases as \u221a<jats:italic>A<\/jats:italic>. These results are confirmed numerically on a number of test scenes. Next, we carry out an analysis of the cost of rendering, trading off local dimensionality and the number of patches, deriving an optimal block size. Based on this analysis, we provide useful practical insights for setting parameters in CPCA and also derive a new adaptive subdivision algorithm. Moreover, we show that rendering time scales sub-linearly with the resolution of the image, allowing for interactive all-frequency relighting of 1024 x 1024 images.<\/jats:p>","DOI":"10.1145\/1276377.1276454","type":"journal-article","created":{"date-parts":[[2007,9,14]],"date-time":"2007-09-14T13:44:55Z","timestamp":1189777495000},"page":"62","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":23,"title":["A theory of locally low dimensional light transport"],"prefix":"10.1145","volume":"26","author":[{"given":"Dhruv","family":"Mahajan","sequence":"first","affiliation":[{"name":"Columbia University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ira Kemelmacher","family":"Shlizerman","sequence":"additional","affiliation":[{"name":"Weizmann Institute of Science"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ravi","family":"Ramamoorthi","sequence":"additional","affiliation":[{"name":"Columbia University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Belhumeur","sequence":"additional","affiliation":[{"name":"Columbia University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2007,7,29]]},"reference":[{"key":"e_1_2_2_1_1","volume-title":"International Conference on Computer Vision, 383--390","author":"Basri R.","unstructured":"Basri , R. , and Jacobs , D . 2001. Lambertian reflectance and linear subspaces . In International Conference on Computer Vision, 383--390 . Basri, R., and Jacobs, D. 2001. Lambertian reflectance and linear subspaces. In International Conference on Computer Vision, 383--390."},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566601"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/38.365003"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073320"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.2312\/EGWR\/EGSR06\/251-262"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1561\/0100000006"},{"key":"e_1_2_2_7_1","doi-asserted-by":"crossref","unstructured":"Grenander U. and Szego G. 1958. Toeplitz Forms and Their Applications. University of Calif. Press Berkeley and Los Angeles. Grenander U. and Szego G. 1958. Toeplitz Forms and Their Applications . University of Calif. Press Berkeley and Los Angeles.","DOI":"10.1525\/9780520355408"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141949"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.2312\/EGWR\/EGSR04\/337-344"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566599"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/1027411.1027414"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882280"},{"key":"e_1_2_2_13_1","volume-title":"EuroGraphics Workshop on Rendering, 359--373","author":"Nimeroff J.","unstructured":"Nimeroff , J. , Simoncelli , E. , and Dorsey , J . 1994. Efficient re-rendering of naturally illuminated environments . In EuroGraphics Workshop on Rendering, 359--373 . Nimeroff, J., Simoncelli, E., and Dorsey, J. 1994. Efficient re-rendering of naturally illuminated environments. In EuroGraphics Workshop on Rendering, 359--373."},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1109\/34.969116"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2005.193"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-96-00188-9"},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/1027411.1027416"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2005.22"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2002.1039204"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.2005.168"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566612"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882281"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/280814.280927"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/1138450.1138456"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1276377.1276454","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1276377.1276454","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T14:52:18Z","timestamp":1750258338000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1276377.1276454"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,7,29]]},"references-count":24,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,7,29]]}},"alternative-id":["10.1145\/1276377.1276454"],"URL":"https:\/\/doi.org\/10.1145\/1276377.1276454","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,7,29]]},"assertion":[{"value":"2007-07-29","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}