- Rendering equation
In

computer graphics , the**rendering equation**is anintegral equation in which the equilibriumradiance leaving a point is given as the sum of emitted plus reflected radiance under a geometricoptics approximation. It was simultaneously introduced into computer graphics by David Immel et al. [*Citation*] and

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 = 10.1145/15922.15901Jim Kajiya [*Citation*] in 1986. The various realistic rendering techniques in computer graphics attempt to solve this equation.

last1 = Kajiya | first1 = James T.

title = The rendering equation

journal = SIGGRAPH 1986

doi = 10.1145/15922.15902The 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 (L_{o}) is the sum of the emitted light (L_{e}) and the reflected light. The reflected light itself is the sum of the incoming light (L_{i}) from all directions, multiplied by the surface reflection and cosine of the incident angle.The rendering equation may be written in the form

: $L\_o(x,\; mathbf\; w,\; lambda,\; t)\; =\; L\_e(x,\; mathbf\; w,\; lambda,\; t)\; +\; int\_Omega\; f\_r(x,\; mathbf\; w\text{'},\; mathbf\; w,\; lambda,\; t)\; L\_i(x,\; mathbf\; w\text{'},\; lambda,\; t)\; (-mathbf\; w\text{'}\; cdot\; mathbf\; n)\; dmathbf\; w\text{'}$

where

*$lambda,!$ is a particular wavelength of light

*$t,!$ is time

*$L\_o(x,\; mathbf\; w,\; lambda,\; t)$ is the total amount of light of wavelength $lambda,!$ directed outward along direction $mathbf\; w$ at time $t,!$, from a particular position $x,!$

*$L\_e(x,\; mathbf\; w,\; lambda,\; t)$ is emitted light

*$int\_Omega\; cdots\; dmathbf\; w\text{'}$ is an integral over a hemisphere of inward directions

*$f\_r(x,\; mathbf\; w\text{'},\; mathbf\; w,\; lambda,\; t)$ is the BRDF, the proportion of light reflected from $mathbf\; w\text{'}$ to $mathbf\; w$ at position $x,!$, time $t,!$, and at wavelength $lambda,!$

*$L\_i(x,\; mathbf\; w\text{'},\; lambda,\; t)$ 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 angleTwo 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 mean a wide range of factorings and rearrangements of the equation are possible.

Note this equation's spectral and

time dependence—$L\_o,!$ may be sampled at or integrated over sections of thevisible spectrum to obtain, for example, atrichromatic 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 = 2008-06-22Solving 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 usingMonte Carlo method s has led to many different algorithms includingpath tracing ,photon mapping , andMetropolis light transport , among others.**References****External links*** [

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

*Wikimedia Foundation.
2010.*

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