Lommel (1889) was apparently the first to derive the equation of transfer. Not only did he derive the equation of transfer, but he also solved it in some simplified cases in order to estimate reflection functions from real-world surfaces (including marble and paper) and compared his solutions to measured reflectance data from these surfaces.
Seemingly unaware of Lommel’s work, Schuster (1905) was the next researcher in radiative transfer to consider the effect of multiple scattering. He used the term self-illumination to describe the fact that each part of the medium is illuminated by every other part of the medium, and he derived differential equations that described reflection from a slab along the normal direction assuming the presence of isotropic scattering. The conceptual framework that he developed remains essentially unchanged in the field of radiative transfer.
Soon thereafter, Schwarzschild (1906) introduced the concept of radiative equilibrium, and Jackson (1910) expressed Schuster’s equation in integral form, also noting that “the obvious physical mode of solution is Liouville’s method of successive substitutions” (i.e., a Neumann series solution). Finally, King (1913) completed the rediscovery of the equation of transfer by expressing it in the general integral form. Yanovitskij (1997) traced the origin of the integral equation of transfer to Chvolson (1890), but we have been unable to find a copy of this paper.
Books by Chandrasekhar (1960), Preisendorfer (1965, 1976), and van de Hulst (1980) cover volume light transport in depth.
Blinn (1982b) first used basic volume scattering algorithms for computer graphics. The equation of transfer was first introduced to graphics by Kajiya and Von Herzen (1984). Rushmeier (1988) was the first to compute solutions of it in a general setting. Arvo (1993) first made the essential connections between previous formalizations of light transport in graphics and the equation of transfer and radiative transfer in general. Pauly, Kollig, and Keller (2000) derived the generalization of the path integral form of the light transport equation for the volume scattering case.
See also the “Further Reading” section of Chapter 11 for additional references to previous work on light scattering in participating media.
The paper by Raab et al. (2006) introduced many important sampling building-blocks for rendering participating media to graphics, including the delta-tracking algorithm for inhomogeneous media by Woodcock et al. (1965). (pbrt uses this algorithm in GridDensityMedium::Sample().) More recently, Novák et al. (2014) derived ratio tracking and residual ratio tracking, which provide unbiased estimates of the transmittance function in inhomogeneous media with considerably lower variance than delta tracking; the simpler ratio tracking algorithm is used in GridDensityMedium::Tr().
For media with substantial variation in density, delta tracking can be quite inefficient—many small steps must be taken to get through the optically thin sections. Danskin and Hanrahan (1992) presented a technique for efficient volume ray marching using a hierarchical data structure. Another way of addressing this issue was presented by Szirmay-Kalos et al. (2011), who used a grid to partition scattering volumes in cells and applied delta tracking using the maximum density cells as the ray passed through them. Yue et al. (2010) applied a similar approach but used a kd-tree, which was better able to adapt to spatially varying densities. In follow-on work, they derive an approach to estimate the efficiency of spatial partitionings and use it to construct them more effectively (Yue et al. 2011).
Kulla and Fajardo (2012) noted that techniques based on sampling beam transmittance ignore another important factor: spatial variation in the scattering coefficient. They developed a method based on computing a tabularized 1D sampling distribution for each ray passing through participating media based on the product of beam transmittance and scattering coefficient at a number of points along it. They then draw samples from this distribution, showing good results.
Accounting for illumination from emissive media is important for many effects, including fire and explosions. See Villemin and Hery (2013) for algorithms for sampling illumination from these sorts of emitters.
Researchers have recently had success in deriving closed-form expressions that describe scattering along unoccluded ray segments in participating media; these approaches can be substantially more efficient than integrating over a series of point samples. See Sun et al. (2005), Pegoraro and Parker (2009), and Pegoraro et al. (2009, 2010, 2011) for examples of such methods. (Remarkably, Pegoraro and collaborators’ work provides a closed-form expression for scattering from a point light source along a ray passing through homogeneous participating media with anisotropic phase functions.)
Light Transport Algorithms
Rushmeier and Torrance (1987) used finite-element methods for rendering participating media. Other early work in volume scattering for computer graphics includes work by Max (1986), Nishita, Miyawaki, and Nakamae (1987), Bhate and Tokuta’s approach based on spherical harmonics (Bhate and Tokuta 1992), and Blasi et al.’s two-pass Monte Carlo algorithm, where the first pass shoots energy from the lights and stores it in a grid and the second pass does final rendering using the grid to estimate illumination at points in the scene (Blasi et al. 1993). Glassner (1995) provided a thorough overview of this topic and previous applications of it in graphics, and Max’s survey article (Max 1995) also covers early work well. See Cerezo et al. (2005) for an extensive survey of approaches to rendering participating media up through 2005.
More recently, Szirmay-Kalos et al. (2005) precomputed interactions between sample points in the medium in order to more quickly compute multiple scattering. Pegoraro et al. (2008b) developed an interesting approach for improving Monte Carlo rendering of participating media by using information from previous samples to guide future sampling.
Georgiev et al. (2013) made the observation that incremental path sampling can generate particularly bad paths in participating media. They proposed new multi-vertex sampling methods that better account for all of the relevant terms in the equation of transfer.
Sampling direct illumination from lights at points inside media surrounded by a primitive is challenging; traditional direct lighting algorithms aren’t applicable at points inside the medium, as refraction through the primitive’s boundary will divert the shadow ray’s path. Walter et al. (2009) considered this problem and developed algorithms to efficiently find paths to lights accounting for this refraction.
The visual appearance of high albedo objects like clouds is striking, but many bounces may be necessary for good results. Wrenninge et al. (2013) described an approximation where after the first few bounces, the scattering coefficient, attenuation coefficient for shadow rays, and the eccentricity of the phase function are all progressively reduced. (This approach draws from ideas behind the principle of similarity from Section 15.5.1.)
All of the bidirectional light transport algorithms that will be introduced in Chapter 16 can be extended to handle participating media; most of our implementations include these extensions. See the “Further Reading” section in that chapter for references to previous work on these topics. See also Jarosz’s thesis (2008), which has extensive background on this topic (and includes a number of important contributions).
Subsurface scattering was first introduced to graphics by Hanrahan and Krueger (1993), although their approach did not attempt to simulate light that entered the object at points other than at the point being shaded. Dorsey et al. (1999) applied photon maps to simulating subsurface scattering that did include this effect, and Pharr and Hanrahan (2000) introduced an approach based on computing BSSRDFs for arbitrary scattering media with an integral over the medium’s depth.
Kajiya and Von Herzen (1984) first introduced the diffusion approximation to graphics, though Stam (1995) was the first to clearly identify many of its advantages for rendering. See Ishimaru’s book (1978) or Donner’s thesis (2006) for the derivation of the diffusion approximation and Wyman et al. (1989) for the introduction of the principle of similarity. More recently, Zhao et al. (2014) further investigated the similarity relations, derived higher order relations, and showed their application to rendering.
The dipole approximation for subsurface scattering was developed by Farrell et al. (1992). It was introduced to computer graphics by Jensen et al. (2001b). Jensen and Buhler (2002) developed an efficient hierarchical integration approach based on precomputing incident irradiance at a set of points on the primitive’s surface. The dipole approximation saw early application to production rendering via a scan-line implementation (Hery 2003).
Contini et al. (1997) generalized the dipole approach to multipoles to more accurately model finite scattering slabs. This approach was applied to subsurface scattering by Donner and Jensen (2005). However, even the multipole approach doesn’t handle all types of scattering media well; the assumptions of homogeneous media and relatively high albedos are too restrictive for many interesting objects. Li et al. (2005) developed a hybrid approach that handles the first few bounces of light with Monte Carlo path tracing but then switches to a dipole approximation. Tong et al. (2005) developed a method to capture and render materials with small deviations from an overall homogeneous appearance. Haber et al. (2005), and Wang et al. (2008b) further generalized the media supported, solving the diffusion equation on a grid of sample points. Fattal (2009) applied the discrete ordinates method, addressing a number of shortcomings of the direct application of that technique. Arbree et al. (2011) developed a finite element method to solve the diffusion equation on a tetrahedral mesh in a way that is more numerically robust than prior grid-based methods.
The photon beam diffusion approach implemented in Section 15.5 is based on the approach developed by Habel et al. (2013). It builds on the quantized diffusion model developed by d’Eon and Irving (2011), who introduced the Grosjean monopole (Grosjean 1956), the approach of Pomraning and Ganapol (1995) for computing the dipole depth in Equation (15.28), and the computation of the radiant exitance using the approach proposed by Kienle and Patterson (1997).
Frisvad et al. (2014) developed an alternative diffusion technique to model subsurface scattering due to an incident beam of light; in contrast to photon beam diffusion, which integrates isotropic sources along continuous beams, their method builds on a discrete arrangement of anisotropic (i.e., ) monopole solutions.
While much effort has gone into more accurate diffusion profiles, Christensen and Burley (2015) showed that a simple exponential approximation to these profiles fits them extremely well and is quite efficient to evaluate.
Donner et al. (2009) computed BSSRDFs with Monte Carlo simulation for a variety of scattering properties (phase function, scattering coefficients, etc.) and fit the resulting data to a low-dimensional model. This model accurately accounts for the directional variation of scattered light and the properties of medium-albedo media.
Rendering realistic human skin is a challenging problem; this problem has driven the development of a number of new methods for rendering subsurface scattering after the initial dipole work as issues of modeling the layers of skin and computing more accurate simulations of scattering between layers have been addressed. For a good overview of these issues, see Igarashi et al.’s (2007) survey on the scattering mechanisms inside skin and approaches for measuring and rendering skin. Notable research in this area includes papers by Donner and Jensen (2006), d’Eon et al. (2007), Ghosh et al. (2008), and Donner et al. (2008). Donner’s thesis includes a discussion of the importance of accurate spectral representations for high-quality skin rendering (Donner 2006, Section 8.5).
The algorithm implemented in Section 15.4.1 to find sample points for incident illumination for BSSRDFs was developed by King et al. (2013).
One key application of volume scattering algorithms in computer graphics has been simulating atmospheric scattering. Work in this area includes early papers by Klassen (1987) and Preetham et al. (1999), who introduced a physically rigorous and computationally efficient atmospheric and sky-lighting model. Haber et al. (2005a) described a model for twilight, and Hošek and Wilkie (2012, 2013) developed a comprehensive model for sky- and sun-light.
There are a number of applications of visualizing volumetric data sets for medical and engineering applications. This area is called volume rendering. In many of these applications, radiometric accuracy is substantially less important than developing techniques that help make structure in the data apparent (e.g., where the bones are in CT scan data). Early papers in this area include those by Levoy (1988, 1990a, 1990b) and Drebin, Carpenter, and Hanrahan (1988).
Moon et al. (2007) made the important observation that some of the assumptions underlying the use of the equation of transfer—that the scattering particles in the medium aren’t too close together so that scattering events can be considered to be statistically independent—aren’t in fact true for interesting scenes that include small crystals, ice, or piles of many small glass objects. They developed a new light transport algorithm for these types of discrete random media based on composing precomputed scattering solutions.
Jakob et al. (2010) derived a generalized transfer equation that describes scattering by distributions of oriented particles. They proposed a microflake scattering model as a specific example of a particle distribution (where a microflake is the volumetric analog of a microfacet on a surface) and showed a number of ways of solving this equation based on Monte Carlo, finite elements, and a dipole model. More recently, Heitz et al. (2015) derived a generalized microflake distribution, which is considerably more efficient to sample and evaluate. Their model quantifies the local scattering properties using projected areas observed from different directions, which adds a well-defined notion of volumetric level of detail.
The equation of transfer assumes that the index of refraction of a medium will only change at discrete boundaries, though many actual media have continuously varying indices of refraction. Ament et al. (2014) derived a variant of the equation of transfer that allows for this case and applied photon mapping to render images with it.
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