{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:09:53Z","timestamp":1750306193038,"version":"3.41.0"},"reference-count":15,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2017,6,1]],"date-time":"2017-06-01T00:00:00Z","timestamp":1496275200000},"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":[[2017,6,30]]},"abstract":"<jats:p>\n            Deep compositing is an important practical tool in creating digital imagery, but there has been little theoretical analysis of the underlying mathematical operators. Motivated by finding a simple formulation of the merging operation on\n            <jats:italic>OpenEXR<\/jats:italic>\n            -style deep images, we show that the Porter-Duff\n            <jats:bold>over<\/jats:bold>\n            function is the operator of a Lie group. In its corresponding Lie algebra, the splitting and mixing functions that\n            <jats:italic>OpenEXR<\/jats:italic>\n            deep merging requires have a particularly simple form. Working in the Lie algebra, we present a novel, simple proof of the uniqueness of the mixing function.\n          <\/jats:p>\n          <jats:p>\n            The Lie group structure has many more applications, including new, correct resampling algorithms for volumetric images with alpha channels, and a deep image compression technique that outperforms that of\n            <jats:italic>OpenEXR<\/jats:italic>\n            .\n          <\/jats:p>","DOI":"10.1145\/3023386","type":"journal-article","created":{"date-parts":[[2017,6,5]],"date-time":"2017-06-05T12:50:00Z","timestamp":1496667000000},"page":"1-12","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":7,"title":["Deep Compositing Using Lie Algebras"],"prefix":"10.1145","volume":"36","author":[{"given":"Tom","family":"Duff","sequence":"first","affiliation":[{"name":"Pixar Animation Studios, Park Avenue, Emeryville, CA"}]}],"member":"320","published-online":{"date-parts":[[2017,6]]},"reference":[{"key":"e_1_2_2_1_1","volume-title":"Harris","author":"Arfken George B.","year":"2013","unstructured":"George B. Arfken , Hans-J\u00fcrgen Weber , and Frank E . Harris . 2013 . Mathematical Methods for Physicists: A Comprehensive Guide (7th ed.). Academic Press , New York, NY, USA. xiii + 1205 pages. George B. Arfken, Hans-J\u00fcrgen Weber, and Frank E. Harris. 2013. Mathematical Methods for Physicists: A Comprehensive Guide (7th ed.). Academic Press, New York, NY, USA. xiii + 1205 pages."},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/38.310740"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-079050-0.50020-5"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/37401.37414"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/2791261.2791266"},{"key":"e_1_2_2_7_1","first-page":"2","article-title":"Interpreting alpha","volume":"4","author":"Glassner Andrew","year":"2015","unstructured":"Andrew Glassner . 2015 . Interpreting alpha . Journal of Computer Graphics Techniques (JCGT) 4 , 2 (May 2015), 30--44. Andrew Glassner. 2015. Interpreting alpha. Journal of Computer Graphics Techniques (JCGT) 4, 2 (May 2015), 30--44.","journal-title":"Journal of Computer Graphics Techniques (JCGT)"},{"key":"e_1_2_2_8_1","unstructured":"Peter Hillman. 2012. The Theory of OpenEXR Deep Samples. Retrieved from http:\/\/www.openexr.com\/TheoryDeepPixels.pdf.  Peter Hillman. 2012. The Theory of OpenEXR Deep Samples. Retrieved from http:\/\/www.openexr.com\/TheoryDeepPixels.pdf."},{"key":"e_1_2_2_9_1","unstructured":"Florian Kainz. 2013. Interpreting OpenEXR Deep Pixels. Retrieved from http:\/\/www.openexr.com\/InterpretingDeepPixels.pdf.  Florian Kainz. 2013. Interpreting OpenEXR Deep Pixels. Retrieved from http:\/\/www.openexr.com\/InterpretingDeepPixels.pdf."},{"volume-title":"Springer New York","author":"Lang S.","key":"e_1_2_2_10_1","unstructured":"S. Lang . 2002. Algebra. Springer New York , New York, NY . S. Lang. 2002. Algebra. Springer New York, New York, NY."},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/344779.344958"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1137\/S00361445024180"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/964965.808606"},{"key":"e_1_2_2_15_1","unstructured":"SymPy Development Team. 2015. SymPy Documentation. http:\/\/docs.sympy.org\/latest\/index.html.  SymPy Development Team. 2015. SymPy Documentation. http:\/\/docs.sympy.org\/latest\/index.html."},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1090\/stml\/079"},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1959-0108732-6"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3023386","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3023386","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T03:50:18Z","timestamp":1750218618000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3023386"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,6,30]]}},"alternative-id":["10.1145\/3023386"],"URL":"https:\/\/doi.org\/10.1145\/3023386","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"type":"print","value":"0730-0301"},{"type":"electronic","value":"1557-7368"}],"subject":[],"published":{"date-parts":[[2017,6]]},"assertion":[{"value":"2016-09-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2016-12-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-06-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}