Progressive Expectation‐Maximization for Hierarchical Volumetric Photon Mapping

Abstract

Beam Radiance Estimation [Jarosz et al. 2008] Our Method 917K Photons (13 + 268 = 281 s) 917K Photons (13 + 268 = 281 s) Per-Pixel Render-time Per-Pixel Render-time 4K Gaussian fit (54 + 71 = 125 s) 4K Gaussian fit (54 + 71 = 125 s) Per-Pixel Render-time Per-Pixel Render-time Figure 1: 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. AbstractState-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.

Realistically rendering polluted water

“…The beam radiance estimate allows 2D kernels to be applied in the volumetric photon mapping. Based on the beam radiance estimate theory, Jakob et al [14] used 2D Gaussian mixture to fit the light distribution during photon tracing. They approximated the medium radiance using their Gaussian mixture model.…”

Continuous projection for fast L 1 reconstruction

“…To describe complex data distributions using a simple model, Gaussian mixture models have been used in various scientific fields, like image segmentation [Garcia et al 2010], object recognition [Vasconcelos 1998] and rendering [Walter et al 2008;Jakob et al 2011]. We employ this model to obtain a simple but expressive representation of the point distribution in an input point set.…”

“…In contrast to previous authors [Jakob et al 2011;Walter et al 2008], who choose ρ to be the n-th globally smallest occurring distance between Gaussians, we try to avoid such a global computation, but rather choose ρ to be a good compromise between clustering efficiency (large, relaxed ρ), and geometric accuracy (small, restrictive ρ). To provide an intuitive control over ρ, we suggest a free parameter α, so that ρ = α 2 /2, which has a simple interpretation: If two Gaussians have equal covariances, thus presumably representing similarly oriented geometry, Eq.…”

Real-time online learning of Gaussian mixture model for opacity mapping

“…We rely on the fact that stepwise EM is a generalization of batch EM and replace u k N (Equation 8) with u k j (Equation 12). Note that for α = 1 they are equivalent after processing the N samples or N th (i.e.…”

“…The GMM has been adopted for spatial density estimation of radiance in the context of volumetric photon mapping [12]. Given a large number of photons, they used the accelerated EM [13] to partition the space into cells, whose statistics describing enclosed photons rather than the original ones are adopted to train the model.…”