## Further Reading

The general idea of tracing light-carrying paths from light sources was first investigated by Arvo (1986), who stored light in texture maps on surfaces and rendered caustics. Heckbert (1990b) built on this approach to develop a general ray-tracing-based global illumination algorithm, and Pattanaik and Mudur (1995) developed an early particle-tracing technique. Christensen (2003) surveyed applications of adjoint functions and importance to solving the LTE and related problems.

Sources of non-symmetric scattering and their impact on bidirectional light transport algorithms were first identified by Veach (1996).

Pharr and Humphreys (2004) proposed the method to sample emitted rays from environment map light sources that is used in this chapter. Dammertz and Hanika (2009) described a variation on this approach that sampled points on the visible faces of the scene bounding box rather than an oriented disk; this can lead to fewer wasted samples.

#### Photon Mapping

Approaches like Arvo’s caustic rendering algorithm (Arvo 1986) formed the basis for an improved technique that stored illumination in texture maps on surfaces developed by Collins (1994). Density estimation techniques for global illumination were first introduced by Shirley, Walter, and collaborators (Shirley et al. 1995; Walter et al. 1997).

Jensen (1995, 1996) developed the photon mapping algorithm, which introduced the key innovation of storing the light contributions in a general 3D data structure rather than in texture maps. Important improvements to the photon mapping method are described in follow-up papers and a book by Jensen (1996, 1997, 2001).

Final gathering for finite-element radiosity algorithms was first described in Reichert’s thesis (Reichert 1992). If the full photon map is stored in memory, the directional distribution of photons can be used to construct optimized final gathering techniques that importance sample directions that are likely to have large contributions (Jensen 1995). More recently, Spencer and Jones (2009a) described how to build a hierarchical kd-tree of photons such that traversal could be stopped at higher levels of the tree and showed that using the footprints of final gather rays computed using ray differentials can lead to better results than the usual approach. In another paper, Spencer and Jones (2009b) showed that a simple iterative relaxation scheme to reduce clumping in photon maps can lead to dramatic improvements in the quality of density estimates.

Havran et al. (2005) developed a final gathering photon mapping algorithm based on storing final gather intersection points in a kd-tree in the scene and then shooting photons from the lights; when a photon intersects a surface, the nearby final gather intersection records are found and the photon’s energy can be distributed to the origins of the corresponding final gather rays. Herzog et al. (2007) described an approach based on storing all of the visible points as seen from the camera and splatting photon contributions to them. Hachisuka et al. (2008b) developed the progressive photon mapping algorithm; stochastic progressive photon mapping was developed by Hachisuka and Jensen (2009).

The advantages of SPPM over traditional photon mapping are significant, and the approach was quickly adopted after its introduction. Hachisuka et al. (2010) showed how to use arbitrary density estimation kernels and how to compute error estimates during rendering to automatically determine when to stop further iterations. Knaus and Zwicker (2011) re-derived SPPM following a different approach and showed that it was possible to only maintain global statistics for values like the current search radius rather than having a separate value for each pixel. See Kaplanyan and Dachsbacher (2013a) for an extensive study of SPPM’s convergence rates and an improved (but more complex) method for updating SPPM estimates after each iteration.

The question of how to find the most effective set of photons for photon mapping is an important one: light-driven particle-tracing algorithms don’t work well for all scenes (consider, for example, a complex building model with lights in every room but where the camera sees only a single room). The earliest applications of Metropolis sampling to photon mapping was proposed in Wald’s Diploma thesis (1999). Fan et al. (2005) showed that the application of Veach’s particle-tracing theory to photon mapping provides a mechanism for generating photon paths starting from the camera. They were able to use this approach in conjunction with a Metropolis sampling algorithm to generate photon distributions. Hachisuka and Jensen (2011) used Metropolis sampling to find photon paths that were visible to the camera; their algorithm is notable for both its effectiveness and its ease of implementation. Chen et al. (2011) use a similar approach but sample additional terms of the path contribution function and distribute additional photons to parts of the image with higher error.

Jensen and Christensen (1998) were the first to generalize the photon mapping algorithm to participating media. Knaus and Zwicker (2011) showed how to render participating media using SPPM. Jarosz et al. (2008a) had the important insight that expressing the scattering integral over a beam through the medium as the measurement to be evaluated could make photon mapping’s rate of convergence much higher than if a series of point photon estimates was instead taken along each ray. Section 5.6 of Hachisuka’s thesis (2011) and Jarosz et al. (2011a, 2011b) showed how to apply this approach progressively. For another representation, see Jakob et al. (2011), who fit a sum of anisotropic Gaussians to the equilibrium radiance distribution in participating media.

#### Bidirectional Path Tracing

Bidirectional path tracing was independently developed by Lafortune and Willems (1993) and Veach and Guibas (1994). The development of multiple importance sampling was integral to the effectiveness of bidirectional path tracing (Veach and Guibas 1995). Lafortune and Willems (1996) showed how to apply bidirectional path tracing to rendering participating media, and Kollig and Keller (2000) showed how bidirectional path tracing can be modified to work with quasi-random sample patterns.

An exciting recent development has been simultaneous work by Hachisuka et al. (2012) and Georgiev et al. (2012), who developed a unified framework for both photon mapping and bidirectional path tracing. Their approaches allowed photon mapping to be included in the path space formulation of the light transport equation, which in turn made it possible to derive light transport algorithms that use both approaches to generate paths and combine them using multiple importance sampling.

Kaplanyan and Dachsbacher (2013b) noted that photon mapping algorithms use illumination from nearby points even in cases where unbiased approaches are effective. They developed a technique for regularization of light-carrying paths, where an unbiased path tracer or bidirectional path tracer is modified to treat delta distributions that cause impossible-to-sample configurations instead as having non-zero value over a small cone of directions. Thus, bias is introduced only in the challenging settings.

Vorba et al. (2014) developed an approach to compute effective sampling distributions for difficult lighting configurations over the course of rendering rather than in a preprocess and showed its applicability to bidirectional path tracing.

#### Metropolis Light Transport

Veach and Guibas (1997) first applied the Metropolis sampling algorithm to solving the light transport equation. They demonstrated how this method could be applied to image synthesis and showed that the result was a light transport algorithm that was robust to traditionally difficult lighting configurations (e.g., light shining through a slightly ajar door). Pauly, Kollig, and Keller (2000) generalized the MLT algorithm to include volume scattering. Pauly’s thesis (Pauly 1999) described the theory and implementation of bidirectional and Metropolis-based algorithms for volume light transport.

Fan et al. (2005) developed a method that let the user explicitly specify a number of important paths (e.g., through a tricky geometric configuration) that could then be used as a target state in Metropolis mutations. The energy redistribution path tracing algorithm by Cline et al. (2005) starts one or more Markov chains at every pixel of the image and runs them for a small number of iterations; the method is notable for being unbiased despite its use of non-ergodic Markov chains that can only explore a subset of path space.

Hoberock’s Ph.D. dissertation discusses a number of alternatives for the scalar contribution function, including those that adapt the sampling density to pay more attention to particular modes of light transport and those that focus on reducing noise in the final image (Hoberock 2008).

Kelemen et al. (2002) developed the “primary sample
space MLT” formulation of Metropolis light transport. They also suggested
the approach implemented in the `MLTSampler` for lazily updating sample
vector components when performing mutations. Hachisuka et al. (2014)
developed the MMLT approach that is
implemented in the `MLTIntegrator` in this chapter.

The optimal choice of the large step probability is scene dependent: for scenes with difficult-to-sample transport paths, it’s better for it to be lower, so that more successful mutations are performed with small steps once a good path is found. For scenes with simpler light transport, it’s better for the probability to be higher, so that the overall path space is explored more thoroughly. Zsolnai and Szirmay-Kalos (2013) developed a technique that gathered statistics about paths during the bootstrap phase that made it possible to automatically set this parameter to a near-optimal value.

#### Other Rendering Approaches

A number of algorithms have been developed based on a first phase of
computation that traces paths from the light sources to create “virtual
lights,” where these lights are then used to approximate indirect
illumination during a second phase. The principles behind this approach
were first introduced by Keller’s work on *instant radiosity* (1997).
The more general instant global illumination algorithm was developed by
Wald, Benthin, and collaborators (Wald et al. 2002, 2003; Benthin
et al. 2003). See Dachsbacher et al.’s recent survey article (2014) for a
summary of recent work in this area.

Building on the virtual point lights concept, Walter and collaborators (2005, 2006) developed *lightcuts*, which are based on creating
thousands of virtual point lights and then building a hierarchy by
progressively clustering nearby ones together. When a point is being
shaded, traversal of the light hierarchy is performed by computing bounds
on the error that would result from using clustered values to illuminate
the point versus continuing down the hierarchy, leading to an approach with
both guaranteed error bounds and good efficiency.

Bidirectional lightcuts (Walter et al. 2012) trace longer subpaths from the camera to obtain a family of light connection strategies; combining the strategies using multiple importance sampling eliminates bias artifacts that are commonly produced by virtual point light methods.

Jakob and Marschner (2012) expressed light transport involving specular materials as an integral over a high-dimensional manifold embedded in path space. A single light path corresponds to a point on the manifold, and nearby paths are found using a local parameterization that resembles Newton’s method; they applied a Metropolis-type method through this parameterization to explore the neighborhood of challenging specular and near-specular configurations.

Hanika et al. (2015a) apply an improved version of the local path parameterization in a pure Monte Carlo context to estimate the direct illumination through one or more dielectric boundaries; this leads to significantly better convergence when rendering glass-enclosed objects or surfaces covered with water droplets.

Kaplanyan et al. (2014) observed that the path contribution function is close to being separable when paths are parameterized using the endpoints and the half-direction vectors at intermediate vertices, which are equal to the microfacet normals in the context of microfacet reflectance models. Performing Metropolis sampling in this half-vector domain leads to a method that is particularly good at rendering glossy interreflection. An extension by Hanika et al. (2015b) improves the robustness of this approach and proposes an optimized scheme to select mutation sizes to reduce sample clumping in image space.

Another interesting approach was developed by Lehtinen and collaborators (Lehtinen et al. 2013, Manzi et al. 2014). Building on the observation that ideally, most samples from the path space should be taken around discontinuities (and not in smooth regions of the image), they developed a measurement contribution function for Metropolis sampling that focused samples on gradients in the image. They then reconstructed high-quality final images from horizontal and vertical gradient images and a coarse, noisy image. More recently, Kettunen et al. (2015) showed how this approach could be applied to regular path tracing, without Metropolis sampling. Manzi et al. (2015) showed its application to bidirectional path tracing.

Hair is particularly challenging to render; not only is it extremely geometrically complex but multiple scattering among hair also makes a significant contribution to its final appearance. Traditional light transport algorithms often have difficulty handling this case well. See the papers by Moon and Marschner (2006), Moon et al. (2008), and Zinke et al. (2008) for recent work in specialized rendering algorithms for hair.

While the rendering problem as discussed so far has been challenging enough, Jarabo et al. (2014a) showed the extension of the path integral to not include the steady-state assumption—i.e., accounting for the non-infinite speed of light. Time ends up being extremely high frequency, which makes rendering challenging; they showed successful application of density estimation to this problem.

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