A Null-Scattering Path Integral Formulation of Light Transport

A null-scattering path integral formulation of light transport BAILEY MILLER∗, Dartmouth College, USA ILIYAN GEORGIEV∗, Autodesk, United Kingdom WOJCIECH JAROSZ, Dartmouth College, USA RMSE: 0.045 RMSE: 0.041 RMSE: 0.037 RMSE: 0.032 LTUV: 4.33M LTUV: 3.91M LTUV: 3.02M LTUV: 2.31M Time: 252s Time: 267s Time: 273s Time: 275s Single scattering Single scattering Multiple scattering µt µs Multiple scattering RMSE: 0.088 RMSE: 0.089 RMSE: 0.074 RMSE: 0.069 LTUV: 16.0M LTUV: 16.5M LTUV: 12.0M LTUV: 10.5M Time: 237s Time: 234s Time: 275s Time: 270s Reference Independent tracking Spectral tracking Spectral MIS (ours) Spectral+NEE MIS (ours) Fig. 1. 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), andcombine these into a more robust, lower-variance estimator via multiple importance sampling (MIS). See Tables 1 and2 for descriptions of the methods and the medium. Unbiased rendering of general, heterogeneous participating media currently CCS Concepts: • Computing methodologies → Ray tracing. requires using null-collision approaches for estimating transmittance and generating free-flight distances. A long-standing limitation of these ap- Additional Key Words and Phrases: global illumination, light transport, proaches, however, is that the corresponding path pdfs cannot be computed participating media, null scattering, Monte Carlo integration due to the black-box nature of the null-collision rejection sampling process. ACM Reference Format: These techniques therefore cannot be combined with other sampling tech- Bailey Miller, Iliyan Georgiev, and Wojciech Jarosz. 2019. A null-scattering niques via multiple importance sampling (MIS), which significantly limits path integral formulation of light transport. ACM Trans. Graph. 38, 4, Arti- their robustness and generality. Recently, Galtier et al. [2013] showed how to cle 44 (July 2019), 13 pages. https://doi.org/10.1145/3306346.3323025 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, 1 INTRODUCTION express existing, and combine complementary unbiased techniques via MIS. The world around us is filled with participating media which vol- We demonstrate the practicality of our theory by combining, for the first umetrically attenuates and scatters light as it travels from light time, several path sampling techniques in spatially and spectrally varying media, generalizing and outperforming the prior state of the art. sources to our eyes. While important in many fields, simulating this transport efficiently and accurately is unfortunately a notoriously ∗Authors with equal contribution. difficult problem, since it requires solving not only the rendering equation [Kajiya 1986; Immel et al. 1986], but also its volumetric gen- Authors’ addresses: Bailey Miller, [email protected], Dartmouth Col- lege, 9 Maynard Street, Hanover, NH, 03755, USA; Iliyan Georgiev, iliyan.georgiev@ eralization, the radiative transfer equation [Chandrasekhar 1960]. autodesk.com, Autodesk, 17 Broadwick Street, London, W1F 0DE, United Kingdom; Monte Carlo path sampling methods such as (bidirectional) path Wojciech Jarosz, [email protected], Dartmouth College, 9 Maynard tracing [Kajiya 1986; Lafortune and Willems 1993; Veach and Guibas Street, Hanover, NH, 03755, USA. 1995] and its volumetric variants [Lafortune and Willems 1996; © 2019 Copyright held by the owner/author(s). Publication rights licensed to ACM. Georgiev et al. 2013] have been investigated in academia for decades This is the author’s version of the work. It is posted here for your personal use. Not for due to their elegant simplicity, generality, and accuracy. Moreover, redistribution. The definitive Version of Record was published in ACM Transactions on Graphics, https://doi.org/10.1145/3306346.3323025. revision 2 (22 May 2019) by now most major film production renderers have adopted such approaches as their dominant rendering algorithms [Georgiev et al. ACM Trans. Graph., Vol. 38, No. 4, Article 44. Publication date: July 2019. 44:2 • Miller, Georgiev, and Jarosz 2018; Burley et al. 2018; Fascione et al. 2018; Christensen et al. 2018; 2.1 Classical formulation Christensen and Jarosz 2016; Fascione et al. 2017]. In scenes containing media with particles distributed statistically These approaches operate by stochastically constructing light independently from each other, the radiance equilibrium is described transport paths between sensors and emitters. Thanks to decades of by the radiative transfer equation (RTE) [Chandrasekhar 1960] research [Novák et al. 2018], we now have an arsenal of strategies ¹ · rº ¹ º − ¹ º ¹ º ¹ º m¹ º for constructing such paths in participating media [Raab et al. 2008; ω L x; ω = µt x L x; ω + µt x L x; ω ; (1) Kulla and Fajardo 2012; Georgiev et al. 2013; Křivánek et al. 2014]. losses gains One core strength of such methods is the ability to leverage multiple which relates the differential change in the radiance L at a point importance sampling (MIS) [Veach and Guibas 1995] to combine x in direction ω to losses from absorption and out-scattering and m m complementary sampling techniques into a single, provably good gains from self-emission Le and in-scattering L : estimator with improved robustness. MIS operates by weighting ¹ m µa¹xº m µs¹xº 0 0 0 each technique proportionally to its path sampling pdf. L ¹x; ωº = Le ¹x; ωº + ρm¹ω; x; ω ºL¹x; ω º dω: (2) µt¹xº µt¹xº S2 Unfortunately, combining multiple unbiased sampling techniques 2 in spatially and spectrally varying participating media has remained Here S is the unit sphere, ρm is the medium phase function, and elusive. Such media require the use of so-called null-collision ap- µt¹xº = µa¹xº + µs¹xº is the medium extinction coefficient, where µa proaches to estimate transmittance and generate light transport and µs are the absorption and scattering coefficients, respectively. paths. These approaches, also called “delta scattering” or “Wood- Solving this ordinary linear differential equation for the radiance s cock tracking”, were initially developed and analyzed in the 1950s L¹x; ωº, and using the outgoing radiance L ¹z; ωº at the first visible and 1960s [Butcher and Messel 1958, 1960; Zerby et al. 1961; Bertini surface point z = x − zω as a boundary condition, yields the volume 1963; Woodcock et al. 1965; Miller 1967; Coleman 1968]. They were rendering equation (VRE) [Arvo 1993]: ¹ z only recently adopted [Raab et al. 2008] and extended upon in graph- m s L¹x; ωº = T ¹x; yºµt¹yºL ¹y; ωº dy + T ¹x; zºL ¹z; ωº; (3) ics [Jarosz et al. 2011b; Novák et al. 2014; Kutz et al. 2017; Szirmay- 0 Kalos et al. 2011, 2017, 2018]. A derivation of these algorithms di- where the medium radiance Lm is integrated at points y = x − yω rectly from the RTE [Galtier et al. 2013] was recently introduced along the line, and like Lm (2), Ls is a sum of self-emitted and in- to graphics [Kutz et al. 2017; Novák et al. 2018], demonstrating not scattered radiance [Kajiya 1986; Immel et al. 1986]: only their correctness, but also providing a convenient framework ¹ for postulating new variants. Unfortunately, the authors also noted s s 0 0 0 0 L ¹z; ωº = Le¹z; ωº + ρs¹ω; z; −ω ºL¹z; ω ºjn¹zº · ω j dω : (4) that “the most limiting drawback of these methods is their inability S2 to quantify the pdf of individual samples”. Since MIS requires access Here ρs is the bidirectional scattering distribution function (BSDF) to these pdfs, it has so far been limited to (piecewise) homogeneous and n¹zº is the surface normal at z. Finally, the medium extinction media [Křivánek et al. 2014; Georgiev et al. 2013; Wilkie et al. 2014]. transmittance is defined as ¯ y Heterogeneous media must either forego MIS [Kutz et al. 2017] or −τ ¹x;yº − µ ¹x−sωº ds T ¹x; yº = e = e 0 t ; (5) forego null-collision methods in favor of regular tracking [Sutton et al. 1999; Wilkie et al. 2014; Fascione et al. 2018] or tabulated where τ ¹x; yº is the medium extinction optical thickness between x sampling [Szirmay-Kalos et al. 2017; Gamito 2018] to support MIS. and y. Note that there is one VRE (3) for each color component, i.e. We solve this long-standing problem by turning the null-scattering wavelength or red, green, and blue (R, G, B) channel. RTE into a generalized null-scattering path integral formulation of To compute the radiance at a point, the VRE can be estimated volumetric light transport. Our formulation allows unbiased path recursively via Monte Carlo integration. The traditional approach, sampling and analytic pdf evaluation in spatially and spectrally vary- known as analog sampling, is to importance sample the non-recursive ing media for the first time. This enables us to generalize, improve terms under the integrals in Eqs. (2) to (4)[Novák et al . 2018]. For the performance of, or remove bias from recent techniques like spec- Eq. (3) this involves sampling a distance y with density T ¹x; yºµt¹yº. tral decomposition tracking [Kutz et al. 2017] and hero wavelength This, however, is possible only when the corresponding cumulative sampling [Wilkie et al. 2014], all while enabling new techniques distribution function is invertible, e.g. in media with (piece-wise) and allowing their combination with complementary existing tech- simple density (see Fig.

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