Direct Transmittance Estimation in Heterogeneous Participating Media Using Approximated Taylor Expansions

Abstract

Evaluating the transmittance between two points along a ray is a key component in solving the light transport through heterogeneous participating media and entails computing an intractable exponential of the integrated medium's extinction coefficient. While algorithms for estimating this transmittance exist, there is a lack of theoretical knowledge about their behaviour, which also prevent new theoretically sound algorithms from being developed. For this purpose, we introduce a new class of unbiased transmittance estimators based on random sampling or truncation of a Taylor expansion of the exponential function. In contrast to classical tracking algorithms, these estimators are non-analogous to the physical light transport process and directly sample the underlying extinction function without performing incremental advancement. We present several versions of the new class of estimators, based on either importance sampling or Russian roulette to provide finite unbiased estimators of the infinite Taylor series expansion. We also show that the well known ratio tracking algorithm can be seen as a special case of the new class of estimators. Lastly, we conduct performance evaluations on both the central processing unit (CPU) and the graphics processing unit (GPU), and the results demonstrate that the new algorithms outperform traditional algorithms for heterogeneous mediums.

An unbiased ray-marching transmittance estimator

“…One such form, which is used to estimate the exponential of the Hamiltonian in particle physics Markov-chain Monte Carlo simulations, due to Bhanot and Kennedy [Bhanot and Kennedy 1985;Lin et al 2000;Wagner 1987Wagner , 1988, can be applied directly to transmittance estimation. Related applications of this idea to transmittance estimation were independently presented by several authors [El-Hafi et al 2018;Georgiev et al 2019;Jonsson et al 2020;Longo 2002]. Similar work has been proposed by [Lyne et al 2015] in the context of Bayesian inference.…”

“…where 𝑁 ∼ Po(τ r ) points 𝑥 𝑖 in the interval (𝑎, 𝑏) are generated by sampling a PPP with residual intensity 𝜆(𝑥) = μ(𝑥)−𝜇 c (𝑥) ≥ 0. While different articles may propose different approaches for setting the residual intensity 𝜆(𝑥) of the PPP, this estimator is conceptually equivalent to the one known as the Poisson estimator [Beskos et al 2006;Chen and Huang 2012;Fearnhead et al 2008;Jacob et al 2015;Jonsson et al 2020;Papaspiliopoulos 2011], first presented by Wagner [1987]. Concurrently, Jonsson et al [2020] have also connected the ratio tracking and Poisson estimator literature and proposed several new variants of RRT that use online estimation of a constant control.…”

“…where 𝑃(𝑁 ) is the probability mass function of 𝑁 . Using deltatracking results in a Poisson distribution, 𝑁 ∼ Po(τ r ), but other distributions can be used [Fearnhead et al 2008;Jonsson et al 2020].…”

“…To discern more about how this could be achieved, we need a detailed picture of how these parameters influence variance. While it is known that the efficiency of residual ratio tracking improves with increasing majorant [Georgiev et al 2019] and basic heuristics for setting control variates have been discussed [Jonsson et al 2020;Novák et al 2014], to the best of our knowledge, little to no detailed investigation of query size 𝑀 or expansion parameter 𝑐 has been presented for any truncated estimator.…”

Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients

“…These estimators can be used to exploit known medium properties, for example by introducing control variates [Kutz et al 2017;Novák et al 2014]. More recently, this framework has been built upon further [Jonsson et al 2020;Kettunen et al 2021] and looks to offer a solid foundation for future volumetric rendering work. By expressing our estimators in Sec.…”