{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T21:21:27Z","timestamp":1764969687119,"version":"3.46.0"},"reference-count":30,"publisher":"Association for Computing Machinery (ACM)","issue":"6","funder":[{"name":"NWO","award":["KICH1.ST03.21.016"],"award-info":[{"award-number":["KICH1.ST03.21.016"]}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:p>\n                    A core operation in Monte Carlo volume rendering is transmittance estimation: Given a segment along a ray, the goal is to estimate the fraction of light that will pass through this segment without encountering absorption or out-scattering. A naive approach is to estimate optical depth \u03c4 using unbiased ray marching and to then use exp(-\u03c4) as transmittance estimate. However, this strategy systematically overestimates transmittance due to Jensen's inequality. On the other hand, existing unbiased transmittance estimators either suffer from high variance or have a cost governed by random decisions, which makes them less suitable for SIMD architectures. We propose a biased transmittance estimator with significantly reduced bias compared to the naive approach and a deterministic and low cost. We observe that ray marching with stratified jittered sampling results in estimates of optical depth that are nearly normal-distributed. We then apply the unique minimum variance unbiased (UMVU) estimator of exp(-\n                    <jats:italic toggle=\"yes\">\u03c4<\/jats:italic>\n                    ) based on two such estimates (using two different sets of random numbers). Bias only arises from violations of the assumption of normal-distributed inputs. We further reduce bias and variance using a variance-aware importance sampling scheme. The underlying theory can be used to estimate any analytic function of optical depth. We use this generalization to estimate multiple importance sampling (MIS) weights and introduce two integrators: Unbiased MIS with biased MIS weights and a more efficient but biased combination of MIS and transmittance estimation.\n                  <\/jats:p>","DOI":"10.1145\/3763273","type":"journal-article","created":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T17:15:39Z","timestamp":1764868539000},"page":"1-16","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Jackknife Transmittance and MIS Weight Estimation"],"prefix":"10.1145","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3939-6992","authenticated-orcid":false,"given":"Christoph","family":"Peters","sequence":"first","affiliation":[{"name":"Delft University of Technology, Delft, Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2025,12,4]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-35089-5"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.2312\/egtp.19871000"},{"volume-title":"Radiative Transfer","author":"Chandrasekhar Subrahmanyan","key":"e_1_2_2_3_1","unstructured":"Subrahmanyan Chandrasekhar. 1950. Radiative Transfer. Oxford University Press."},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.13182\/NSE68-1"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/3355089.3356559"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","unstructured":"H. L. Gray T. A. Watkins and W. R. Schucany. 1973. On the jackknife statistic and its relation to UMVU estimators in the normal case. Communications in Statistics 2 4 (1973). 10.1080\/03610927308827077","DOI":"10.1080\/03610927308827077"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4842-7185-8"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/3451256"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/3585497"},{"key":"e_1_2_2_10_1","article-title":"Hash Functions for GPU Rendering","volume":"9","author":"Jarzynski Mark","year":"2020","unstructured":"Mark Jarzynski and Marc Olano. 2020. Hash Functions for GPU Rendering. Journal of Computer Graphics Techniques 9, 3 (2020). https:\/\/www.jcgt.org\/published\/0009\/03\/02\/","journal-title":"Journal of Computer Graphics Techniques"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/3587421.3595409"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459937"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1201\/9780367805319"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2012.03148.x"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/3306346.3323025"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530160"},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/2487228.2487235"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450623.3464653"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13383"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/2661229.2661292"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","unstructured":"Jacopo Pantaleoni and Eric Heitz. 2017. Notes on optimal approximations for importance sampling. (2017). 10.48550\/arxiv.1707.08358","DOI":"10.48550\/arxiv.1707.08358"},{"key":"e_1_2_2_22_1","volume-title":"Proceedings of the Eurographics Workshop on Rendering Techniques","author":"Pauly Mark","year":"2000","unstructured":"Mark Pauly, Thomas Kollig, and Alexander Keller. 2000. Metropolis Light Transport for Participating Media. In Proceedings of the Eurographics Workshop on Rendering Techniques 2000. Springer, 11\u201322."},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/3651293"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(91)90174-P"},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/3587423.3607877"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","unstructured":"L\u00e1szl\u00f3 Szirmay-Kalos Bal\u00e1zs T\u00f3th Mil\u00e1n Magdics and Bal\u00e1zs Cs\u00e9bfalvi. 2010. Efficient Free Path Sampling in Inhomogeneous Media. 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