{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T00:46:54Z","timestamp":1787014014731,"version":"3.56.0"},"reference-count":40,"publisher":"Wiley","license":[{"start":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T00:00:00Z","timestamp":1775606400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"},{"start":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T00:00:00Z","timestamp":1775606400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Computer Graphics Forum"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Modeling the wave nature of light and the propagation and diffraction of electromagnetic fields is crucial for the accurate simulation of many phenomena, yet wave simulations are significantly more computationally complex than classical ray\u2010based models. In this work, we start by analyzing the classical path integral formulation of light transport and rigorously study which wave\u2010optical phenomena can be reproduced by it. We then introduce a\n                    <jats:italic>bilinear path integral<\/jats:italic>\n                    generalization for wave\u2010optical light transport that models the wave interference between paths. This formulation subsumes many existing methods that rely on shooting\u2010bouncing rays or UTD\u2010based diffractions, and serves to give insight into the challenges of such approaches and the difficulty of sampling good paths in a bilinear setting.\n                  <\/jats:p>\n                  <jats:p>\n                    With this foundation, we develop a\n                    <jats:italic>weakly\u2010local path integral<\/jats:italic>\n                    based on region\u2010to\u2010region transport using elliptical cones that allows sampling individual paths that still model wave effects accurately. As with the classic path integral form of the light transport equation, our path integral makes it possible to derive a variety of practical transport algorithms. We present a complete system for wave tracing with elliptical cones, with applications in light transport for rendering and efficient simulation of long\u2010wavelength radiation propagation and diffraction in complex environments.\n                  <\/jats:p>","DOI":"10.1111\/cgf.70322","type":"journal-article","created":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T10:54:31Z","timestamp":1775645671000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Wave Tracing: Generalizing The Path Integral To Wave Optics"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2748-4036","authenticated-orcid":false,"given":"Shlomi","family":"Steinberg","sequence":"first","affiliation":[{"name":"University of Waterloo  Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0566-8291","authenticated-orcid":false,"given":"Matt","family":"Pharr","sequence":"additional","affiliation":[{"name":"NVIDIA  United States"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,4,8]]},"reference":[{"key":"e_1_2_7_2_2","unstructured":"Arvo James. \u201cAnalytic Methods for Simulated Light Transport\u201d. PhD thesis. Yale University 19952."},{"key":"e_1_2_7_3_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073620"},{"issue":"59","key":"e_1_2_7_4_2","article-title":"Bempp-cl: A fast Python based just-in-time compiling boundary element library","volume":"6","author":"Betcke Timo","year":"2021","journal-title":"J. Open Source Softw."},{"key":"e_1_2_7_5_2","first-page":"1","volume-title":"2020 14th European Conference on Antennas and Propagation (EuCAP)","author":"Bilibashi D","year":"2020"},{"key":"e_1_2_7_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/2231816.2231820"},{"key":"e_1_2_7_7_2","doi-asserted-by":"publisher","DOI":"10.1214\/lnms\/1196285403"},{"key":"e_1_2_7_8_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.70230"},{"key":"e_1_2_7_9_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14140"},{"key":"e_1_2_7_10_2","doi-asserted-by":"publisher","DOI":"10.1145\/3414685.3417782"},{"key":"e_1_2_7_11_2","first-page":"10","volume":"8","author":"Hoydis Jakob","year":"2022","journal-title":"Sionna"},{"key":"e_1_2_7_12_2","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392094"},{"key":"e_1_2_7_13_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073621"},{"key":"e_1_2_7_14_2","unstructured":"ITU.Recommendation ITU-R P.2040-3.2023. url:https:\/\/www.itu.int\/dms_pubrec\/itu-r\/rec\/p\/R-REC-P.2040-3-202308-I!!PDF-E.pdf8\u201310."},{"issue":"10","key":"e_1_2_7_15_2","doi-asserted-by":"crossref","first-page":"2337","DOI":"10.1109\/16.88522","article-title":"The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach","volume":"38","author":"Jensen KL","year":"1991","journal-title":"IEEE Transactions on Electron Devices"},{"key":"e_1_2_7_16_2","doi-asserted-by":"publisher","DOI":"10.1145\/15922.15902"},{"key":"e_1_2_7_17_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13772"},{"key":"e_1_2_7_18_2","unstructured":"Krywonos Andrey. \u201cPredicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction\u201d. PhD thesis. University of Central Florida Jan.20062."},{"issue":"4","key":"e_1_2_7_19_2","doi-asserted-by":"crossref","first-page":"4498","DOI":"10.1103\/PhysRevE.57.4498","article-title":"\u201cDiffusing-light spectroscopies beyond the diffusion limit: The role of ballistic transport and anisotropic scattering\u201d. en","volume":"57","author":"Lemieux P-A","year":"1998","journal-title":"Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics"},{"key":"e_1_2_7_20_2","first-page":"43","volume-title":"Passive and Active Millimeter-Wave Imaging XXIV","author":"Mackay Jacob","year":"2021"},{"key":"e_1_2_7_21_2","volume-title":"Antennas & Propagation Library","author":"McNamara D A","year":"1990"},{"key":"e_1_2_7_22_2","first-page":"207","volume-title":"Lie Methods in Optics","author":"Man'ko Vladimir I","year":"2008"},{"key":"e_1_2_7_23_2","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139644105","volume-title":"Optical coherence and quantum optics","author":"Mandel Leonard","year":"1995"},{"issue":"1","key":"e_1_2_7_24_2","first-page":"014106","article-title":"Ray tracing the Wigner distribution function for optical simulations","volume":"57","author":"Mout Marco","year":"2018","journal-title":"Optical Engineering"},{"issue":"8","key":"e_1_2_7_25_2","doi-asserted-by":"crossref","first-page":"1344","DOI":"10.1109\/8.791954","article-title":"A deterministic ray tube method for microcellular wave propagation prediction model","volume":"47","author":"Son Hae-Won","year":"1999","journal-title":"IEEE Transactions on Antennas and Propagation"},{"key":"e_1_2_7_26_2","doi-asserted-by":"publisher","DOI":"10.1145\/3658166"},{"key":"e_1_2_7_27_2","doi-asserted-by":"publisher","DOI":"10.1145\/3687902"},{"key":"e_1_2_7_28_2","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530119"},{"key":"e_1_2_7_29_2","doi-asserted-by":"publisher","DOI":"10.1145\/311535.311546"},{"key":"e_1_2_7_30_2","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459791"},{"key":"e_1_2_7_31_2","doi-asserted-by":"publisher","DOI":"10.1145\/3472293"},{"key":"e_1_2_7_32_2","doi-asserted-by":"publisher","DOI":"10.1145\/3012001"},{"key":"e_1_2_7_33_2","volume-title":"Phase-Space Optics: Fundamentals and Applications","author":"Testorf Markus","year":"2010"},{"key":"e_1_2_7_34_2","unstructured":"Veach Eric. \u201cRobust Monte Carlo Methods For Light Transport Simulation\u201d. PhD thesis. Stanford University 19972\u20134."},{"key":"e_1_2_7_35_2","doi-asserted-by":"crossref","unstructured":"Veach Ericet al. \u201cMetropolis light transport\u201d.Proceedings of the 24th annual conference on Computer graphics and interactive techniques.1997 65\u2013762.","DOI":"10.1145\/258734.258775"},{"key":"e_1_2_7_36_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13347"},{"key":"e_1_2_7_37_2","doi-asserted-by":"publisher","DOI":"10.1109\/TAP.2006.875910"},{"key":"e_1_2_7_38_2","doi-asserted-by":"publisher","DOI":"10.1145\/3731174"},{"key":"e_1_2_7_39_2","doi-asserted-by":"publisher","DOI":"10.1145\/3130800.3130840"},{"key":"e_1_2_7_40_2","unstructured":"Yi Haofanet al. \u201cRay tracing meets terahertz: Challenges and opportunities\u201d.IEEE Communications Magazine(2022) 2."},{"key":"e_1_2_7_41_2","doi-asserted-by":"publisher","DOI":"10.1145\/3592414"}],"container-title":["Computer Graphics Forum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70322","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/cgf.70322","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70322","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T10:54:35Z","timestamp":1775645675000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1111\/cgf.70322"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,8]]},"references-count":40,"alternative-id":["10.1111\/cgf.70322"],"URL":"https:\/\/doi.org\/10.1111\/cgf.70322","archive":["Portico"],"relation":{},"ISSN":["0167-7055","1467-8659"],"issn-type":[{"value":"0167-7055","type":"print"},{"value":"1467-8659","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,8]]},"assertion":[{"value":"2026-04-08","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70322"}}