Rendering equation

The rendering equation , also known as the rendering equation , is used in 3D computer graphics . It was developed in 1986 by Jim Kajiya and at the same time by David Immel et al. released. It is an integral equation that describes the conservation of energy during the propagation of light rays and thus forms the mathematical basis for all algorithms for global lighting .

History and classification

In principle, everything that was necessary to calculate three-dimensional images was available long before the rendering equation: Maxwell's equations (1861–1864), special relativity (1905) and quantum mechanics (1920s) explain the interaction of light and matter so precisely that theoretically it would be possible to calculate realistic images as desired.

For 3D computer graphics, however, it turned out very early on that it was completely impractical to work with these basic models; in most cases they require a computational effort that cannot be managed even with today's computers. However, it also became apparent that it was not necessary to work with such exact models: quantum mechanics explains small-scale effects that are not perceptible in everyday human life (cf. for example double-slit experiment ), the theory of relativity explains large-scale facts, mainly in astronomical orders of magnitude develop their effect (see, for example, space-time ) and even some of the effects of Maxwell's equations (see, for example, interference ) are often meaningless for the practice of computer graphics.

The researchers therefore worked with geometric optics , which goes back to ancient Greece and describes the behavior of light on a large scale - that is, neglecting its wave properties. This resulted in approaches and techniques that instead of complex waves follow simple rays of light through the scene and lead to acceptable results in a manageable amount of time, namely ray tracing and radiosity .

In 1986 Jim Kajiya published the rendering equation in line with this development . Kajiya showed that all rendering techniques widely used to date can be derived directly from the rendering equation. For the first time there was a common mathematical foundation on which the techniques could be compared.

The rendering equation not only led to a systematization of the field of knowledge, but also inspired numerous further developments. Today it is considered to be so fundamental that many mistakenly assume that ray tracing originated from the render equation or that the previously created radiosity equation emerged from it as a transformation.

formula

Original formula

The rendering equation is:

${\displaystyle L(x,x')=g(x,x')\cdot \left(L_{e}(x,x')+\int _{S}b(x,x',x'')L(x',x'')\mathrm {d} x''\right)}$

It describes how much light reaches one surface point from another surface point . A third surface point is taken into account, the light of which first hits and is reflected from there . The individual parts have the following meaning: ${\displaystyle x}$ ${\displaystyle x'}$ ${\displaystyle x''}$ ${\displaystyle x'}$ ${\displaystyle x}$

• The energy flow indicates how much light from reaching out. It is a radiance with the unit W · m ⁻² · sr ⁻¹ . The same applies to the term . ${\displaystyle L(x,x')}$ ${\displaystyle x}$ ${\displaystyle x'}$ ${\displaystyle L(x',x'')}$
• The geometric term describes the mutual position of the points in the scene. Usually the term has the value where is the distance from and . It then indicates how much of the outgoing light actually hits. This also applies if an intermediate surface is completely transparent; in this case the surface absorbs the light on one side and emits it again on the other side. However , if there is an opaque surface between and , the term is 0, which means that no light arrives directly from. ${\displaystyle g(x,x')}$ ${\displaystyle {1}/{r^{2}}}$ ${\displaystyle r}$ ${\displaystyle x}$ ${\displaystyle x'}$ ${\displaystyle x'}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x'}$ ${\displaystyle x}$ ${\displaystyle x'}$
• The emission term indicates how much light is radiated from to (if a light source represents the scene). This is again a radiance with the unit W · m ⁻² · sr ⁻¹ . ${\displaystyle L_{e}(x,x')}$ ${\displaystyle x'}$ ${\displaystyle x}$ ${\displaystyle x'}$
• The scattering term is, which part of the light from reaching out in the direction is reflected. This is a bidirectional reflection distribution function (BRDF). ${\displaystyle b(x,x',x'')}$ ${\displaystyle x'}$ ${\displaystyle x''}$ ${\displaystyle x}$
• ${\displaystyle S}$ is the entirety of all surfaces in the scene.

Kajiya presented the rendering equation in a slightly modified form, but this representation has now proven to be more useful. In its original form, the emission term was not radiance and the scattering term was a dimensionless construct rather than a BRDF.

Equivalent representation

Images from Kajiya's publication in 1986. The image on the left was calculated with normal ray tracing, the one on the right with path tracing. By emitting rays on all surfaces, light effects like this caustic are possible.

Equivalent forms of representation of the rendering equation are chosen in order to describe other applications more clearly. The following illustration is widespread and describes how much light is emitted from the surface point in the direction of the vector : ${\displaystyle x}$ ${\displaystyle {\vec {\omega }}}$

${\displaystyle L(x,{\vec {\omega }})=L_{e}(x,{\vec {\omega }})+\int _{\Omega }{f_{r}(x,{\vec {\omega }}',{\vec {\omega }})L(x,{\vec {\omega }}')({\vec {\omega }}'\cdot {\vec {n}})\mathrm {d} {\vec {\omega }}'}}$

The individual parts have essentially the same meaning as in the other illustration, but are given depending on the direction instead of a second or third point:

• The energy flow indicates how much light is radiated from in the direction ; here, too, it is the radiance. ${\displaystyle L(x,{\vec {\omega }})}$ ${\displaystyle x}$ ${\displaystyle {\vec {\omega }}}$
• The emission term indicates how much light is emitted from the direction (if the point itself is a light source). ${\displaystyle L_{e}(x,{\vec {\omega }})}$ ${\displaystyle x}$ ${\displaystyle {\vec {\omega }}}$
• The scatter term is a BRDF with angle of incidence and angle of reflection . ${\displaystyle f_{r}(x,{\vec {\omega }}',{\vec {\omega }})}$ ${\displaystyle {\vec {\omega }}'}$ ${\displaystyle {\vec {\omega }}}$
• The term describes how much light from direction reaches the point . ${\displaystyle L(x,{\vec {\omega }}')}$ ${\displaystyle {\vec {\omega }}'}$ ${\displaystyle x}$
• ${\displaystyle {\vec {n}}}$is the normal of the surface at the point . ${\displaystyle x}$
• ${\displaystyle \Omega }$is the total of all angles of the hemisphere above the surface.

The values ​​within the integral can be e.g. B. by ray tracing , i.e. by sending a light beam in the direction . The approximation of the integral by a Monte Carlo simulation and the recursive emission of light rays leads to path tracing , which Kajiya described together with the rendering equation. ${\displaystyle {\vec {\omega }}'}$

literature

1. ^ David S. Immel, Michael F. Cohen, Donald P. Greenberg: A radiosity method for non-diffuse environments. In: Proceedings of the 13th annual conference on Computer graphics and interactive techniques ( SIGGRAPH ) 1986, ACM Press, doi : 10.1145 / 15922.15901 , ISBN 0-89791-196-2 , ( PDF )
2. ↑ James T. Kajiya: The rendering equation. In: Proceedings of the 13th annual conference on Computer graphics and interactive techniques ( SIGGRAPH ) 1986, ACM Press, pp. 143-150 ( PDF )