## Further Reading

The first application of Monte Carlo to global illumination for creating synthetic images that we are aware of was described in Tregenza’s paper on lighting design (Tregenza 1983). Cook’s distribution ray-tracing algorithm computed glossy reflections, soft shadows from area lights, motion blur, and depth of field with Monte Carlo sampling (Cook et al. 1984; Cook 1986), although the general form of the light transport equation wasn’t stated until papers by Kajiya (1986) and Immel, Cohen, and Greenberg (1986).

Additional important theoretical work on light transport has been done by
Arvo (1993, 1995a), who has investigated the connection between rendering
algorithms in graphics and previous work in *transport theory*, which
applies classical physics to particles and their interactions to predict
their overall behavior. Our description of the path integral form of the
LTE follows the framework in Veach’s Ph.D. thesis, which has thorough
coverage of different forms of the LTE and its mathematical structure
(Veach 1997).

#### Monte Carlo Techniques

Russian roulette and splitting were introduced to graphics by Arvo and Kirk (1990). Hall and Greenberg (1983) had previously suggested adaptively terminating ray trees by not tracing rays with less than some minimum contribution. Arvo and Kirk’s technique is unbiased, although in some situations, bias and less noise may be the more desirable artifact.

Cook and collaborators first introduced random sampling for integration in rendering (Cook et al. 1984; Cook 1986), and Kajiya (1986) developed the general-purpose path-tracing algorithm. Other important early work on Monte Carlo in rendering includes Shirley’s Ph.D. thesis (1990) and a paper by Kirk and Arvo (1991) on sources of bias in rendering algorithms. Shirley (1992) described a number of useful recipes for warping uniform random numbers to useful distributions for rendering.

Keller and collaborators have written extensively on the application of quasi–Monte Carlo integration in graphics (Keller 1996, Keller 2001; Friedel and Keller 2000; Kollig and Keller 2000, 2002). Keller’s “Quasi-Monte Carlo image synthesis in a nutshell” (2012) is a good introduction to quasi–Monte Carlo for rendering.

Talbot et al. (2005) applied *importance resampling* to
rendering, showing that this variant of standard importance sampling is
applicable to a number of problems in graphics, and Pegoraro et al. (2008a)
implemented an approach that found sampling PDFs for global
illumination over the course of rendering the image.

Many researchers have investigated techniques for adaptive sampling, adding more samples in parts of the integrand that are complex, and adaptive reconstruction, where final results are reconstructed from noisy samples with more sophisticated techniques than the simple reconstruction filters from Chapter 7. See Zwicker et al.’s survey article (2015) for a thorough summary of recent work in this area.

#### Sampling BSDFs

The approach for directly sampling the microfacet visible normal distribution implemented in Section 14.1.1 was developed by Heitz and d’Eon (2014). See also Heitz (2014a) for an overview of traditional sampling techniques for various microfacet distribution functions that sample the regular function directly.

When dealing with refraction through rough dielectrics, a modified change of variables term is needed to account for the mapping from half vectors to outgoing direction. A model based on this approach was originally developed by Stam (2001); Walter et al. (2007) proposed improvements and provided an alternative geometric justification of the half vector mapping.

Lawrence et al. (2004) developed methods for sampling arbitrary BRDF
models, including those based on measured reflectance data. They applied
methods that factor the 4D BRDF into a product of two 2D functions, both of
which are guaranteed to always be greater than 0, thus making it
possible to use them as importance sampling distributions. Most of the
recently developed parametric models for fitting BRDFs described in
Chapter 8’s “Further Reading” section have been developed with importance
sampling in mind. The sampling technique for the `FourierBSDF` was
developed by Jakob et al. (2014a).

A number of researchers have investigated effective sampling of hair reflection models (typically Marschner et al.’s model (2003)). See, for example, papers by Ou et al. (2012), Hery and Ramamoorthi (2012), and d’Eon et al. (2013). More recently, Pekelis et al. (2015) developed a more efficient approach to sampling the Marschner model.

The idea of applying statistical hypothesis tests to verify the correctness of
graphics-related Monte Carlo sampling routines such as BSDF models was
introduced by Subr and Arvo (2007a). The test variant for validating the
BSDF model implementations in `pbrt` was originally developed as part of the
Mitsuba renderer by Jakob (2010).

#### Direct Lighting

Algorithms to render soft shadows from area lights were first developed by Amanatides (1984) and Cook, Porter, and Carpenter (1984). Shirley et al. (1996) derived methods for sampling a number of shapes for use as area light sources. Arvo showed how to sample the projection of a triangle on the sphere of directions with respect to a reference point; this approach can give better results than sampling the area of the triangle directly (Arvo 1995b). Ureña et al. (2013) and Pekelis and Hery (2014) developed analogous techniques for sampling projected quadrilateral light sources. The approach implemented in Section 14.2.2 to convert an angle in a cone to a point on a sphere was derived by Akalin (2015).

Subr and Arvo (2007) developed an efficient technique for sampling environment map light sources that not only accounts for the term from the scattering equation but also only generates samples in the hemisphere around the surface normal.

When environment maps are used for illuminating indoor scenes, many incident directions may be blocked by the building structure. Bashford-Rogers et al. (2013) developed a two-pass algorithm where a first pass from the camera finds directions that reach the environment map; this information is used to create sampling distributions that are used during a second rendering pass. Bitterli et al. (2015) developed an interesting solution to this issue: they rectify the environment map so that rectangular portals in the building map to rectangular regions of the environment map. In turn, at a given point receiving illumination, they compute the projection of portals to the outside and can efficiently sample the environment map using a summed area table.

As described in the “Further Reading” section of Chapter 12, a useful generalization of environment maps for illumination allows the emitted radiance to vary by both position and direction. Lu et al. (2015) developed techniques for efficiently importance sampling these light sources and describes previous work in this area.

The expense of tracing shadow rays to light sources can be significant; a number of interesting approaches have been developed to improve the efficiency of this part of the rendering computation. Billen et al. (2013) propose a technique where only a random subset of potential occluders are tested for intersections; a compensation term ensures that the result is unbiased. Following work shows how to use simplified geometry for some shadow tests while still computing the correct result overall (Billen et al. 2014). Another approach to reducing the cost of shadow rays is visibility caching, where the point-to-point visibility function’s value is cached for clusters of points on surfaces in the scene (Clarberg and Akenine-Möller 2008b; Popov et al. 2013).

A number of approaches have been developed to efficiently render scenes with hundreds or thousands of light sources. (For densely occluded environments, many of the lights in the scene may have little or no contribution to the part of the scene visible from the camera.) Early work on this issue was done by Ward (1991a) and Shirley et al. (1996). Wald et al. (2003) suggested rendering an image with path tracing and a very low sampling rate (e.g., one path per pixel), recording information about which of the light sources made some contribution to the image. This information is then used to set probabilities for sampling each light. Donikian et al. (2006) adaptively found PDFs for sampling lights through an iterative process of taking a number of light samples, noting which ones were effective, and reusing this information at nearby pixels. The “lightcuts” algorithm, described in the “Further Reading” section of Chapter 16, also addresses this problem.

`pbrt`’s direct lighting routines are based on using multiple importance
sampling to combine samples taken from the BSDF and the light sources’
sampling distributions; while this works well in many cases, it can be
ineffective in cases where the product of these two functions has a
significantly different distribution from either one individually. A
number of more efficient approaches have been developed to sample directly
from the product distribution (Burke et al. 2005; Cline
et al. 2006). Clarberg, Rousselle, and collaborators developed
techniques based on representing BSDFs and illumination in the wavelet
basis and efficiently sampling from their product (Clarberg et al. 2005;
Rousselle et al. 2008; Clarberg and
Akenine-Möller 2008a). Efficiency of the direct
lighting calculation can
be further improved by sampling from the *triple product* distribution
of BSDF, illumination, and visibility; this issue was investigated by Ghosh and Heidrich (2006) and Clarberg and Akenine-Möller (2008b). Finally,
Wang and Åkerlund (2009) have developed a technique
that incorporates
an approximation to the distribution of indirect illumination in the light
sampling distribution used in these approaches.

Subr et al. (2014) analyzed the combination of multiple importance sampling and jittered sampling for direct lighting calculations and propose sampling improvements to improve convergence rates.

#### Other Topics

With full spectral rendering, it’s sometimes necessary to perform a sampling operation for a ray based on a single wavelength from the SPD (e.g., with wavelength-dependent indices of refraction). For this case, Radziszewski et al. (2009) introduced an application of multiple importance sampling that reduces variance in this case.

Ward and collaborators developed the irradiance caching algorithm, which is
described in a series of papers (Ward et al. 1988; Ward 1994). The basic
idea is to cache irradiance from indirect illumination at a sparse set of
points on surfaces in the scene; because indirect lighting is generally
slowly changing, irradiance can often be safely interpolated. Tabellion and Lamorlette (2004) described a number of additional improvements to
irradiance caching that made it viable for rendering for movie
productions. Křivánek and collaborators generalized irradiance
caching to *radiance caching*, where a more complex directional
distribution of incident radiance is stored, so that more accurate shading
from glossy surfaces is possible (Křivánek et al. 2005).
Recent work by Schwarzhaupt et al. proposed a better way of assessing the
validity of a cache point using a second-order expansion of the incident
lighting (Schwarzhaupt et al. 2012).

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