•Defining the rendering equation
In computer graphics, the rendering equation is an integral equation that expresses the amount of light leaving a point on a surface as the sum of emitted light and reflected light. It was independently introduced into computer graphics by David Immel et al. and James Kajiya in 1986. The equation is important in the theory of physically based rendering, describing the relationships between the bidirectional reflectance distribution function (BRDF) and the radiometric quantities used in rendering.
The rendering equation is defined at every point on every surface in the scene being rendered, including points hidden from the camera. The incoming light quantities on the right side of the equation usually come from the left (outgoing) side at other points in the scene, and ray casting can be used to find these other points. The radiosity rendering method solves a discrete approximation of this system of equations. In distributed ray tracing, the integral on the right side of the equation may be evaluated using Monte Carlo integration by randomly sampling possible incoming light directions. Path tracing improves and simplifies this method.
The equation form
The outgoing radiance at a point equals the emitted radiance plus the reflected radiance. The reflected component is computed as an integral over the unit hemisphere centered around the surface normal, containing all possible incoming directions. This integral combines the bidirectional reflectance distribution function, which describes the proportion of light reflected from an incoming direction to an outgoing direction, with the incoming spectral radiance, and includes a weakening factor for the incident angle. That factor, often written as cosine of the incident angle, accounts for the fact that light flux is smeared across a surface area larger than the projected area perpendicular to the ray.
Two noteworthy features are its linearity—it is composed only of multiplications and additions—and its spatial homogeneity—it is the same in all positions and orientations. These properties mean a wide range of factorings and rearrangements of the equation are possible. It is a Fredholm integral equation of the second kind, similar to those that arise in quantum field theory.
The equation carries spectral and time dependence. The outgoing radiance may be sampled at or integrated over sections of the visible spectrum to obtain, for example, a trichromatic color sample. A pixel value for a single frame in an animation may be obtained by fixing time, while motion blur can be produced by averaging the outgoing radiance over some given time interval.
A solution to the rendering equation is the function representing outgoing radiance. The incoming radiance is related to the outgoing radiance via a ray-tracing operation: the incoming radiance from some direction at one point is the outgoing radiance at some other point in the opposite direction.
Applications
Solving the rendering equation for any given scene is the primary challenge in realistic rendering. One approach to solving the equation is based on finite element methods, leading to the radiosity algorithm. Another approach using Monte Carlo methods has led to many different algorithms including path tracing, photon mapping, and Metropolis light transport, among others.
Limitations
Although the equation is very general, it does not capture every aspect of light reflection. Some missing aspects include transmission, which occurs when light is transmitted through the surface, such as when it hits a glass object or a water surface. It also omits subsurface scattering, where the spatial locations for incoming and departing light are different. Surfaces rendered without accounting for subsurface scattering may appear unnaturally opaque, though it is not necessary to account for this if transmission is included in the equation, since that will effectively include also light scattered under the surface.
Other aspects not captured include polarization, where different light polarizations will sometimes have different reflection distributions, such as when light bounces at a water surface. Phosphorescence, which occurs when light or other electromagnetic radiation is absorbed at one moment and emitted at a later moment, usually with a longer wavelength, is likewise excluded. The equation does not reflect interference, where the wave properties of light are exhibited, nor fluorescence, where the absorbed and emitted light have different wavelengths.
Non-linear effects are also absent, where very intense light can increase the energy level of an electron with more energy than that of a single photon, making emission of light with higher frequency suddenly possible. The Doppler effect, where light bouncing off an object moving at very high speed gets its wavelength changed, is not captured either; this effect becomes apparent only at speeds comparable to the speed of light, which is not the case for most rendering applications.
For scenes that are either not composed of simple surfaces in a vacuum or for which the travel time for light is an important factor, researchers have generalized the rendering equation to produce a volume rendering equation suitable for volume rendering and a transient rendering equation for use with data from a time-of-flight camera.
This explanation is adapted from Wikipedia and is available under CC BY-SA 4.0.
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