# Rendering equation

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Rendering equation

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. [ Citation
last1 = Immel | first1 = David S.
last2 = Cohen | first2 = Michael F.
last3 = Greenberg | first3 = Donald P.
title = A radiosity method for non-diffuse environments
journal = SIGGRAPH 1986
doi = 10.1145/15922.15901
] and Jim Kajiya [ Citation
last1 = Kajiya | first1 = James T.
title = The rendering equation
journal = SIGGRAPH 1986
doi = 10.1145/15922.15902
] 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 of the incoming light (Li) from all directions, multiplied by the surface reflection and cosine of the incident angle.

The rendering equation may be written in the form

: $L_o\left(x, mathbf w, lambda, t\right) = L_e\left(x, mathbf w, lambda, t\right) + int_Omega f_r\left(x, mathbf w\text{'}, mathbf w, lambda, t\right) L_i\left(x, mathbf w\text{'}, lambda, t\right) \left(-mathbf w\text{'} cdot mathbf n\right) dmathbf w\text{'}$

where
*$lambda,!$ is a particular wavelength of light
*$t,!$ is time
*$L_o\left(x, mathbf w, lambda, t\right)$ is the total amount of light of wavelength $lambda,!$ directed outward along direction $mathbf w$ at time $t,!$, from a particular position $x,!$
*$L_e\left(x, mathbf w, lambda, t\right)$ is emitted light
*$int_Omega cdots dmathbf w\text{'}$ is an integral over a hemisphere of inward directions
*$f_r\left(x, mathbf w\text{'}, mathbf w, lambda, t\right)$ is the BRDF, the proportion of light reflected from $mathbf w\text{'}$ to $mathbf w$ at position $x,!$, time $t,!$, and at wavelength $lambda,!$
*$L_i\left(x, mathbf w\text{'}, lambda, t\right)$ is light of wavelength $lambda,!$ coming inward toward $x,!$ from direction $mathbf w\text{'}$ at time $t,!$
*$-mathbf w\text{'} cdot mathbf n$ is the attenuation of inward light due to incident angle

Two noteworthy features are: its linearity&mdash;it is composed only of multiplications and additions, and its spatial homogeneity&mdash;it is the same in all positions and orientations. These mean a wide range of factorings and rearrangements of the equation are possible.

Note this equation's spectral and time dependence&mdash;$L_o,!$ 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 integrating $L_o,!$ over $t,!$. [cite web
last = Owen
first = Scott
title = Reflection: Theory and Mathematical Formulation
date = September 5, 1999
url = http://www.siggraph.org/education/materials/HyperGraph/illumin/reflect2.htm
accessdate = 2008-06-22
]

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.

References

* [http://graphics.stanford.edu/courses/cs348b-00/lectures/lecture12/ Lecture notes] from Stanford University course CS 348B, "Computer Graphics: Image Synthesis Techniques"

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