Progressive Expectation–Maximization for hierarchical volumetric photon mapping

1Disney Research Zürich 2Cornell University 3ETH Zürich

In Computer Graphics Forum (Proceedings of EGSR), 2011

The beam radiance estimate (left) finds all photons along camera rays, which is a performance bottleneck for this CARS scene due to high volumetric depth complexity and many photons (917K). Our method (right) fits a hierarchical, anisotropic Gaussian mixture model to the photons and can render this scene faster, and with higher quality using only (4K) Gaussian components. The listed times denote the costs of the preprocessing stage (including photon tracing and hierarchy construction), as well as the final rendering stage, respectively.


State-of-the-art density estimation methods for rendering participating media rely on a dense photon representation of the radiance distribution within a scene. A critical bottleneck of such kernel-based approaches is the excessive number of photons that are required in practice to resolve fine illumination details, while controlling the amount of noise. In this paper, we propose a parametric density estimation technique that represents radiance using a hierarchical Gaussian mixture. We efficiently obtain the coefficients of this mixture using a progressive and accelerated form of the Expectation–Maximization algorithm. After this step, we are able to create noise-free renderings of high-frequency illumination using only a few thousand Gaussian terms, where millions of photons are traditionally required. Temporal coherence is trivially supported within this framework, and the compact footprint is also useful in the context of real-time visualization. We demonstrate a hierarchical ray tracing-based implementation, as well as a fast splatting approach that can interactively render animated volume caustics.



Video (full-quality)

Video (full-quality)

Video (iPhone)


Wenzel Jakob, Christian Regg, Wojciech Jarosz. Progressive Expectation–Maximization for hierarchical volumetric photon mapping. Computer Graphics Forum (Proceedings of EGSR), 30(4), June 2011.
    author  = {Jakob, Wenzel and Regg, Christian and Jarosz, Wojciech},
    title   = {Progressive {{Expectation}}--{{Maximization}} for Hierarchical Volumetric Photon Mapping},
    journal = {Computer Graphics Forum (Proceedings of EGSR)},
    volume  = {30},
    number  = {4},
    month   = jun,
    year    = {2011},
    doi     = {10/dtwcjj}
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