Rendering equation

The rendering equation, also called rendering equation is used in 3D computer graphics. It was published in 1986 by Jim Kajiya. Is an integral equation that describes the energy conservation for the propagation of light rays and thus forms the basis for all the mathematical algorithms for global illumination.

History and classification

In principle, was all that was necessary to calculate three-dimensional images, long before the rendering equation exists: Maxwell's equations (1861-1864), the special theory of relativity (1905 ) and quantum mechanics ( 1920 ) explain the interaction of light and matter as accurately that theoretically calculating any realistic images would be possible with them.

For 3D computer graphics, however, proved a very early stage, that it is completely impractical to work with these basic models; they require in most cases a self with today's computers can not cope with computational complexity. However, it also became clear that it was not necessary to work with such exact models: Quantum mechanics explains effects in the little ones that are in our everyday lives imperceptible ( see, for example double-slit experiment ), the theory of relativity explains issues in great, with mainly astronomical magnitudes exert their effect ( see, for example space-time ), and even some of the effects of Maxwell's equations ( cf., eg, interference ) are often irrelevant to the practice of computer graphics.

Therefore, the researchers worked with the geometrical optics, which goes back to ancient Greece and the behavior of light in the large - that is, neglecting its wave properties - describes. Thus arose approaches and techniques that track instead of complex waves simple light rays through the scene and resulted in a manageable time to passable results, namely ray tracing and radiosity.

In this development into Jim Kajiya published 1986 render equation. Kajiya showed that all can be derived directly from the rendering equation to date popular rendering techniques. Therefore, there was the first time a common mathematical foundation on which the techniques could be compared.

The rendering equation subsequently led not only to a systematization of the field of knowledge, but also inspired numerous further developments. It is now regarded as so fundamental that many mistakenly believe, ray tracing was conceived out of the rendering equation or the previously created radiosity equation had emerged as forming from her.

Formula

Original formula

The rendering equation is:

It describes how much light a surface point x is reached by another surface point x ' from. In this case, a third surface point x is taken into account '', which light is first incident on x ' and is reflected from there to x. The parts have the following meaning:

• The " energy flow " L (x, x ' ) indicates how much light x of x' reaches out. There is a radiance in the unit [ W / ( m2 sr) ]. The same applies to the term L (x ', x '').
• The " geometric term " g (x, x ' ) describes the relative position of points in the scene. Normally, the term has the value 1/r2, where r is the distance from X and X ' is. He then indicates how much of the x ' outgoing light x actually fits exactly. Between X and X ', a further surface, however, as the term is 0, that is when x is no light from x' to a direct route. This is true even if the intermediate surface is completely transparent; in this case, the surface absorbs the light on one side and radiates it again on the other side.
• The " emission term " Le (x, x ' ) indicates how much light from x' to x is emitted from (if x 'is a source of light ). This is again a beam density with the unit [ W / ( m2 sr) ].
• The " scattering term " b (x, x ', x '' ) indicates what fraction of the light that x' obtained from x'' from, is reflected in the direction x. This is a bidirectional reflectance distribution function ( BRDF ).
• S is the set of all surfaces in the scene.

Kajiya introduced the rendering equation before in a slightly different form, this representation has now, however, proved to be more appropriate. In the original form of the emission was not term radiance and dispersion was a dimensionless term construct instead of a BRDF.

Equivalent representation

Equivalent forms of representation of the rendering equation can be chosen to describe other applications clearer. It is a widespread following diagram, which describes how much light from the surface point x of radiated in the direction of the vector:

The individual parts have been the same meaning as in the other display, but are a function of the direction, instead of a second or third essentially point:

• The energy flow indicates how much light is emitted from x in direction; it is also about the beam density.
• The emission term indicates how much light is emitted from x in direction ( if the item itself is a light source).
• The scattering term is a BRDF with angle of incidence and angle of reflection.
• The term describes how much light from the direction of the point x is reached.
• Is the normal of the surface at the point x.
• Ω is the set of all angles of the hemisphere above the surface.

The values ​​inside the integral can be eg by ray tracing, so by emitting a light beam toward calculate. The convergence of the integral by a Monte Carlo simulation and the recursive emitting light beams leads to the path tracing, described the Kajiya, along with the rendering equation.