A null-scattering path integral formulation of light transport

1Dartmouth College 2Autodesk

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

Unbiased rendering of spectrally and spatially varying participating media could previously be accomplished using delta tracking separately for each color component (leftmost inset), but this leads to strong color noise. Spectral tracking [Kutz et al. 2017] can reduce this noise by rendering all color components together (middle left inset), but at the cost of sampling distances based on the densest color component of the medium. Our theoretical framework allows us to leverage different sampling techniques across color components (middle right), or exploit next-event estimation (NEE) (far right), and combine these into a more robust, lower-variance estimator via multiple importance sampling (MIS).


Unbiased rendering of general, heterogeneous participating media currently requires using null-collision approaches for estimating transmittance and generating free-flight distances. A long-standing limitation of these approaches, however, is that the corresponding path pdfs cannot be computed due to the black-box nature of the null-collision rejection sampling process. These techniques therefore cannot be combined with other sampling tech- niques via multiple importance sampling (MIS), which significantly limits their robustness and generality. Recently, Galtier et al. [2013] showed how to derive these algorithms directly from the radiative transfer equation (RTE). We build off this generalized RTE to derive a path integral formulation of null scattering, which reveals the sampling pdfs and allows us to devise new, express existing, and combine complementary unbiased techniques via MIS. We demonstrate the practicality of our theory by combining, for the first time, several path sampling techniques in spatially and spectrally varying media, generalizing and outperforming the prior state of the art.



Presentation slides video


All images have been rendered using PBRT. The media in Figures 1, 6, and 9 are from the OpenVDB repository and the cloud model in Figure 7 is from Disney Research. Figures 1 and 9 are illuminated by an environment map from HDRI Haven. Figure 10 is from the PBRT scene repository, and contains a dragon model from XYZ RGB Inc. and an environment map from Bernhard Vogl. The scene in Figure 11 is from Benedikt Bitterli and contains a smoke plume from the PBRT scene repository.

This work was supported by a gift from Autodesk and NSF Grants IIS-1812796 and CNS-1205521. The authors also thank John Miller at University of California, Berkeley for providing access to computing resources.


Bailey Miller, Iliyan Georgiev, Wojciech Jarosz. A null-scattering path integral formulation of light transport. ACM Transactions on Graphics (Proceedings of SIGGRAPH), 38(4), July 2019.
    author   = {Miller, Bailey and Georgiev, Iliyan and Jarosz, Wojciech},
    title    = {A null-scattering path integral formulation of light transport},
    journal  = {ACM Transactions on Graphics (Proceedings of SIGGRAPH)},
    volume   = {38},
    number   = {4},
    year     = {2019},
    month    = jul,
    issn     = {0730-0301},
    doi      = {10/gf6rzb},
    keywords = {transmittance, fractional visibility, opacity, ray marching, Woodcock tracking, delta tracking,
                pseudo-scattering, null-collision, multiple importance sampling, MIS}
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