Rendering equation

The rendering equation describes the total amount of light emitted from a point x along a particular viewing direction, given a function for incoming light and a BRDF.

In computer graphics, the rendering equation is an integral equation in which the equilibrium radiance leaving a point is given as the sum of emitted plus reflected radiance under a geometric optics approximation. It was simultaneously introduced into computer graphics by David Immel et al.[1] and James Kajiya[2] in 1986. The various realistic rendering techniques in computer graphics attempt to solve this equation.

The physical basis for the rendering equation is the law of conservation of energy. Assuming that L denotes radiance, we have that at each particular position and direction, the outgoing light (Lo) is the sum of the emitted light (Le) and the reflected light. The reflected light itself is the sum from all directions of the incoming light (Li) multiplied by the surface reflection and cosine of the incident angle.

Equation form

The rendering equation may be written in the form

L_{\text{o}}(\mathbf x,\, \omega_{\text{o}},\, \lambda,\, t) \,=\, L_e(\mathbf x,\, \omega_{\text{o}},\, \lambda,\, t) \ +\, \int_\Omega f_r(\mathbf x,\, \omega_{\text{i}},\, \omega_{\text{o}},\, \lambda,\, t)\, L_{\text{i}}(\mathbf x,\, \omega_{\text{i}},\, \lambda,\, t)\, (\omega_{\text{i}}\,\cdot\,\mathbf n)\, \operatorname d \omega_{\text{i}}

where

Two noteworthy features are: its linearityit is composed only of multiplications and additions, and its spatial homogeneityit is the same in all positions and orientations. These 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.[3]

Note this equation's spectral and time dependence L_{\text{o}}\,\! 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 t\,\!; motion blur can be produced by averaging L_{\text{o}}\,\! over some given time interval (by integrating over the time interval and dividing by the length of the interval).[4]

Note that a solution to the rendering equation is the function L_o. The function L_i is related to L_o 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 the following:

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[5] suitable for volume rendering and a transient rendering equation[6] for use with data from a time-of-flight camera.

References

  1. Immel, David S.; Cohen, Michael F.; Greenberg, Donald P. (1986), "A radiosity method for non-diffuse environments" (PDF), Siggraph 1986: 133, doi:10.1145/15922.15901, ISBN 0-89791-196-2
  2. Kajiya, James T. (1986), "The rendering equation" (PDF), Siggraph 1986: 143, doi:10.1145/15922.15902, ISBN 0-89791-196-2
  3. Watt, Alan H.; Watt, Mark (1992). Advanced Animation and Rendering Techniques: Theory and Practice. Addison-Wesley Professional. ISBN 978-0-201-54412-1.
  4. Owen, Scott (September 5, 1999). "Reflection: Theory and Mathematical Formulation". Retrieved 2008-06-22.
  5. Kajiya, James T.; Von Herzen, Brian P. (1984), "Ray tracing volume densities", Siggraph 1984 18 (3): 165, doi:10.1145/964965.808594
  6. Smith, Adam M.; Skorupski, James; Davis, James (2008). Transient Rendering (PDF) (Technical report). UC Santa Cruz. UCSC-SOE-08-26.

External links

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