Integral formulations of volumetric transmittance

1Autodesk 2Dartmouth College 3The University of Tokyo 4McGill University 5Charles University and Chaos Czech a. s.

In ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia), 2019

Equal-lookup comparisons of various estimators in a scene with both homogeneous (chess board) and heterogeneous (chess pieces) media. Our proposed estimators (in bold) trade more extinction lookups for lower-variance estimates of transmittance. RMSE and render times are for the entire image.


Computing the light attenuation between two given points is an essential yet expensive task in volumetric light transport simulation. Existing unbiased transmittance estimators are all based on "null-scattering" random walks enabled by augmenting the media with fictitious matter. This formulation prevents the use of traditional Monte Carlo estimator variance analysis, thus the efficiency of such methods is understood from a mostly empirical perspective. In this paper, we present several novel integral formulations of volumetric transmittance in which existing estimators arise as direct Monte Carlo estimators. Breaking from physical intuition, we show that the null-scattering concept is not strictly required for unbiased transmittance estimation, but is a form of control variates for effectively reducing variance. Our formulations bring new insight into the problem and the efficiency of existing estimators. They also provide a framework for devising new types of transmittance estimators with distinct and complementary performance tradeoffs, as well as a clear recipe for applying sample stratification.



Presentation slides video


All images have been rendered using PBRT. The chessboard and pieces in Figs. 1 and 8 are from TurboSquid. The OpenVDB cloud model in Fig. 9 is from Disney Research. The Fox model in Fig. 11 is from Thingiverse.

This work was supported by a gift from Autodesk, National Science Foundation Grant IIS-1812796, Czech Science Foundation Grant 19-07626S, and Charles University Grant SVV-2017-260452.


Iliyan Georgiev, Zackary Misso, Toshiya Hachisuka, Derek Nowrouzezahrai, Jaroslav Křivánek, Wojciech Jarosz. Integral formulations of volumetric transmittance. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia), 38(6), November 2019.
    author   = {Georgiev, Iliyan and Misso, Zackary and Hachisuka, Toshiya and Nowrouzezahrai, Derek and K\v
                riv\'anek, Jaroslav and Jarosz, Wojciech},
    title    = {Integral formulations of volumetric transmittance},
    journal  = {ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia)},
    volume   = {38},
    number   = {6},
    year     = {2019},
    month    = nov,
    doi      = {10/dffn},
    keywords = {participating media, transmittance, null collision, null scattering, stochastic sampling, Monte Carlo
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