## Further Reading

The books written by van de Hulst (1980) and Preisendorfer (1965, 1976) are excellent introductions to volume light transport. The seminal book by Chandrasekhar (1960) is another excellent resource, although it is mathematically challenging. See also the “Further Reading” section of Chapter 15 for more references on this topic.

The Henyey–Greenstein phase function was originally described by Henyey and Greenstein (1941). Detailed discussion of scattering and phase functions, along with derivations of phase functions that describe scattering from independent spheres, cylinders, and other simple shapes, can be found in van de Hulst’s book (1981). Extensive discussion of the Mie and Rayleigh scattering models is also available there. Hansen and Travis’s survey article is also a good introduction to the variety of commonly used phase functions (Hansen and Travis 1974).

While the Henyey–Greenstein model often works well, there are many materials that it can’t represent accurately. Gkioulekas et al. (2013b) showed that sums of Henyey–Greenstein and von Mises-Fisher lobes are more accurate for many materials than Henyey–Greenstein alone and derived a 2D parameter space that allows for intuitive control of translucent appearence.

Just as procedural modeling of textures is an effective technique for shading surfaces, procedural modeling of volume densities can be used to describe realistic-looking volumetric objects like clouds and smoke. Perlin and Hoffert (1989) described early work in this area, and the book by Ebert et al. (2003) has a number of sections devoted to this topic, including further references. More recently, accurate physical simulation of the dynamics of smoke and fire has led to extremely realistic volume data sets, including the ones used in this chapter; see, for example, Fedkiw, Stam, and Jensen (2001). See the book by Wrenninge (2012) for further information about modeling participating media, with particular focus on techniques used in modern feature film production.

In this chapter, we have ignored all issues related to sampling and antialiasing of volume density functions that are represented by samples in a 3D grid, although these issues should be considered, especially in the case of a volume that occupies just a few pixels on the screen. Furthermore, we have used a simple triangle filter to reconstruct densities at intermediate positions, which is suboptimal for the same reasons as the triangle filter is not a high-quality image reconstruction filter. Marschner and Lobb (1994) presented the theory and practice of sampling and reconstruction for 3D data sets, applying ideas similar to those in Chapter 7. See also the paper by Theußl, Hauser, and Gröller (2000) for a comparison of a variety of windowing functions for volume reconstruction with the sinc function and a discussion of how to derive optimal parameters for volume reconstruction filter functions.

Acquiring volumetric scattering properties of real-world objects is particularly difficult, requiring solving the inverse problem of determining the values that lead to the measured result. See Jensen et al. (2001b), Goesele et al. (2004), Narasimhan et al. (2006), and Peers et al. (2006) for recent work on acquiring scattering properties for subsurface scattering. More recently, Gkioulekas et al. (2013a) produced accurate measurements of a variety of media. Hawkins et al. (2005) have developed techniques to measure properties of media like smoke, acquiring measurements in real time. Another interesting approach to this problem was introduced by Frisvad et al. (2007), who developed methods to compute these properties from a lower-level characterization of the scattering properties of the medium.

Acquiring the volumetric density variation of participating media is also challenging. See work by Fuchs et al. (2007), Atcheson et al. (2008), and Gu et al. (2013a) for a variety of approaches to this problem, generally based on illuminating the medium in particular ways while photographing it from one or more viewpoints.

The medium representation used by `GridDensityMedium` doesn’t adapt its
spatial sampling rate as the amount of local detail in the underlying medium changes.
Furthermore, its on-disk representation is a fairly inefficient string of
floating-point values encoded as text. See Museth’s VDB format (2013), or
the Field3D system, which is described by Wrenninge (2015), for
industrial-strength volume representation formats and libraries.

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