A radiative transfer framework for non-exponential media

1Dartmouth College 2Disney Research 3ETH Zürich 4Pixar Animation Studios 5Cornell University

In ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia), 2018
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We show a cloud rendered with a traditional exponential transmittance (left) and with a non-exponential, long-tailed transmittance curve (right). The non-exponential transmittance leads to both deeper light penetration as well as a softer appearance near the surface, allowing for a richer appearance.


We develop a new theory of volumetric light transport for media with non-exponential free-flight distributions. Recent insights from atmospheric sciences and neutron transport demonstrate that such distributions arise in the presence of correlated scatterers, which are naturally produced by processes such as cloud condensation and fractal-pattern formation. Our theory formulates a non-exponential path integral as the result of averaging stochastic classical media, and we introduce practical models to solve the resulting averaging problem efficiently. Our theory results in a generalized path integral which allows us to handle non-exponential media using the full range of Monte Carlo rendering algorithms while enriching the range of achievable appearance. We propose parametric models for controlling the statistical correlations by leveraging work on stochastic processes, and we develop a method to combine such unresolved correlations (and the resulting non-exponential free-flight behavior) with explicitly modeled macroscopic heterogeneity. This provides a powerful authoring approach where artists can freely design the shape of the attenuation profile separately from the macroscopic heterogeneous density, while our theory provides a physically consistent interpretation in terms of a path space integral. We address important considerations for graphics including reciprocity and bidirectional rendering algorithms, all in the presence of surfaces and correlated media.



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This work is concurrent with A Radiative Transfer Framework for Spatially-Correlated Materials by Jarabo et al.



We thank Adrian Jarabo, Carlos Aliaga and Diego Gutierrez for helpful discussions, and the reviewers for their insightful feedback. We would also like to thank the participants of ICTT 2017 – particularly Edward W. Larsen, Richard Vasques, Brian C. Kiedrowski, Joel Kulesza and Norman McCormick – for fruitful cross-disciplinary discussions on transport theory. This work was partially supported by a grant from Microsoft Azure, as well as the National Science Foundation (Grant CNS-1205521).


Benedikt Bitterli, Srinath Ravichandran, Thomas Müller, Magnus Wrenninge, Jan Novák, Steve Marschner, Wojciech Jarosz. A radiative transfer framework for non-exponential media. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia), 37(6):225:1–225:17, November 2018.
    author   = {Bitterli, Benedikt and Ravichandran, Srinath and M\"uller, Thomas and Wrenninge, Magnus and Nov\'ak,
                Jan and Marschner, Steve and Jarosz, Wojciech},
    title    = {A radiative transfer framework for non-exponential media},
    journal  = {ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia)},
    volume   = {37},
    number   = {6},
    pages    = {225:1--225:17},
    year     = {2018},
    month    = nov,
    doi      = {10/gfz2cm},
    keywords = {physically based rendering, participating media, non-classical transport, artistic editing, appearance
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