10.6.2. Other Representations

Although cube maps and spherical harmonics are the most popular representations for irradiance environment maps, other representations are possible. See Figure 10.45. Many irradiance environment maps have two dominant colors: a sky color on the top and a ground color on the bottom. Motivated by this observation, Parker et al. [1356] present a hemisphere lighting model that uses just two colors. The upper hemisphere is assumed to emit a uniform radiance L sky, and the lower hemisphere is assumed to emit a uniform radiance L ground. The irradiance integral for this case is

where θ is the angle between the surface normal and the sky hemisphere axis. Baker and Boyd propose a faster approximation (described by Taylor [1752]):

which is a linear interpolation between sky and ground, using ( cos θ + 1 ) / 2 as the interpolation factor. The term cos θ is generally fast to compute as a dot product, and in the common case where the sky hemisphere axis is one of the cardinal axes (e.g., the y- or z-axis), it does not need to be computed at all, since it is equal to one of the world-space coordinates of n. The approximation is reasonably close and significantly faster, so it is preferable to the full expression for most applications.

Forsyth [487] presents an inexpensive and flexible lighting model called the trilight, which includes directional, bidirectional, hemispherical, and wrap lighting as special cases.

Valve originally introduced the ambient cube representation (Section 10.3.1) for irradiance. In general, all the spherical function representations we have seen in Section 10.3 can be employed for precomputed irradiance. For the low-frequency signals that irradiance functions represent, we know SH is a good approximation. We tend to create special methods to simplify or use less storage than spherical harmonics.

More complex representations for high frequencies are needed if we want to evaluate occlusions and other global illumination effects, or if we want to incorporate glossy reflections (Section 10.1.1). The general idea of precomputing lighting to account for all interactions is called precomputed radiance transport (PRT) and will be discussed in Section 11.5.3. Capturing high frequencies for glossy lighting is also referred to as all-frequency lighting. Wavelet representations are often used in this context [1059] as means of compressing environment maps and to devise efficient operators in a similar fashion to the ones we have seen for spherical harmonics. Ng et al. [1269,1270] demonstrate the use of Haar wavelets to generalize irradiance environment mapping to model self-shadowing. They store both the environment map and the shadowing function, which varies over the object surface, in the wavelet basis. This representation is of note because it amounts to a transformation of an environment cube map, performing a two-dimensional wavelet projection of each of the cube faces. Thus, it can be seen as a compression technique for cube maps.

10.7 Sources of Error

To correctly perform shading, we have to evaluate integrals over non-punctual light sources. In practice, this requirement means there are many different techniques we can employ, based on the properties of the lights under consideration. Often real-time engines model a few important lights analytically, approximating integrals over the light area and computing occlusion via shadow maps. All the other light sources—distant lighting, sky, fill lights, and light bouncing over surfaces—are often represented by environment cube maps for the specular component, and spherical bases for diffuse irradiance.

Employing a mix of techniques for lighting means that we are never working directly with a given BRDF model, but with approximations that have varying degrees of error. Sometimes the BRDF approximation is explicit, as we fit intermediate models in order to compute lighting integrals—LTCs are an example. Other times we build approximations that are exact for a given BRDF under certain (often rare) conditions, but are subject to errors in general—prefiltered cube maps fall into this category.

An important aspect to take into consideration when developing real-time shading models is to make sure that the discrepancies between different forms of lighting are not evident. Having coherent light results from different representations might even be more important, visually, than the absolute approximation error committed by each.

Occlusions are also of key importance for realistic rendering, as light “leaking” where there should be none is often more noticeable than not having light where it should be. Most area light representations are not trivial to shadow. Today none of the existing real-time shadowing techniques, even when accounting for “softening” effects (Section 7.6), can accurately consider the light shape. We compute a scalar factor that we multiply to reduce the contribution of a given light when an object casts a shadow, which is not correct; we should take this occlusion into account while performing the integral with the BRDF. The case of environment lighting is particularly hard, as we do not have a defined, dominant light direction, so shadowing techniques for punctual light sources cannot be used.

Even if we have seen some fairly advanced lighting models, it is important to remember that these are not exact representations of real-world sources of illumination. For example, in the case of environment lighting, we assume infinitely distant radiance sources, ones that are never possible.

All the analytic lights we have seen work on an even stronger assumption, that the lights emit radiance uniformly over the outgoing hemisphere for each point on their surface. In practice, this assumption can be a source of error, as often real lights are strongly directional. In photographic and cinematic lighting, specially crafted masks and filters called gobos, cuculoris, or cookies are often employed for artistic effect. See for example the sophisticated cinematographic lighting in Figure 10.46, by photographer Gregory Crewdson. To restrict lighting angles while keeping a large area of emission, grids of shielding black material called honeycombs can be added in front of large light-emitting panels (so-called softboxes). Complex configurations of mirrors and reflectors can also be used in the light’s housing, such as in interior lighting, automotive headlights, and flashlights. See Figure 10.47. These optical systems create one or more virtual emitters far from the physical center radiating light, and this offset should be considered when performing falloff computations.

Note that these errors should always be evaluated in a perceptual, result-oriented framework (unless our aim is to do predictive rendering, i.e., to reliably simulate the real-world appearance of surfaces). In the hands of the artists, certain simplifications, even if not realistic, can still result in useful and expressive primitives. Physical models are useful when they make it simpler for artists to create visually plausible images, but they are not a goal in their own.

Further Reading and Resources

The book Light Science and Magic: An Introduction to Photographic Lighting by Hunter [791] is a great reference for understanding real-world photographic lights. For movie lighting Set Lighting Technician’s Handbook: Film Lighting Equipment, Practice, and Electrical Distribution [188] is a great introduction.

The work pioneered by Debevec in the area of image-based lighting is of great interest to anyone who needs to capture environment maps from real-world scenes. Much of this work is covered in a SIGGRAPH 2003 course [333], as well as in the book High Dynamic Range Imaging: Acquisition, Display, and Image-Based Lighting by Reinhard et al. [1479].

One resource that can help simulation are light profiles. The Illuminating Engineering Society (IES) publishes a handbook and file format standard for lighting measurements [960,961]. Data in this format is commonly available from many manufacturers. The IES standard is limited to describing lights by only their angular emission profile. It does not fully model the effect on the falloff due to optical systems, nor the emission over the light surface area.

Szirmay-Kalos’s [1732] state-of-the-art report on specular effects includes many references to environment mapping techniques.