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| [[File:Rendering eq.png|right|thumb|300px|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 [[Bidirectional reflectance distribution function|BRDF]].]]
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| 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.<ref>
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| {{Citation
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| | last1 = Immel | first1 = David S.
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| | last2 = Cohen | first2 = Michael F.
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| | last3 = Greenberg | first3 = Donald P.
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| | title = A radiosity method for non-diffuse environments
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| | url = http://pages.cpsc.ucalgary.ca/~mario/courses/591-691-W06/project/radiosity/p133-immel.pdf
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| | journal = Siggraph 1986
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| | doi = 10.1145/15922.15901
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| | year = 1986
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| | pages = 133
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| | isbn = 0-89791-196-2}}</ref> and [[James Kajiya]]<ref>
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| {{Citation
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| | last1 = Kajiya | first1 = James T.
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| | title = The rendering equation
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| | journal = Siggraph 1986
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| | url = http://www.cse.chalmers.se/edu/year/2011/course/TDA361/2007/rend_eq.pdf
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| | doi = 10.1145/15922.15902
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| | year = 1986
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| | pages = 143
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| | isbn = 0-89791-196-2}}</ref> in 1986. The various realistic [[Rendering (computer graphics)|rendering]] techniques in computer graphics attempt to solve this equation.
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| 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 (L<sub>o</sub>) is the sum of the emitted light (L<sub>e</sub>) and the reflected light. The reflected light itself is the sum of the incoming light (L<sub>i</sub>) from all directions, multiplied by the surface reflection and cosine of the incident angle.
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| == Equation form ==
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| The rendering equation may be written in the form
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| : <math>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}}</math>
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| where
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| *<math>\lambda\,\!</math> is a particular wavelength of light
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| *<math>t\,\!</math> is time
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| *<math>\mathbf x</math> is the location in space
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| *<math>\omega_{\text{o}}</math> is the direction of the outgoing light
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| *<math>\omega_{\text{i}}</math> is the negative direction of the incoming light
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| *<math>L_{\text{o}}(\mathbf x,\, \omega_{\text{o}},\, \lambda,\, t)</math> is the total [[spectral radiance]] of wavelength <math>\lambda\,\!</math> directed outward along direction <math>\omega_{\text{o}}</math> at time <math>t\,\!</math>, from a particular position <math>\mathbf x\,\!</math>
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| *<math>L_e(\mathbf x,\, \omega_{\text{o}},\, \lambda,\, t)</math> is [[emissivity|emitted]] spectral radiance
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| *<math>\Omega</math> is the unit [[hemisphere]] containing all possible values for <math>\omega_{\text{i}}</math>
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| *<math>\int_\Omega \dots\, \operatorname d\omega_{\text{i}}</math> is an [[integral]] over <math>\Omega</math>
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| *<math>f_r(\mathbf x,\, \omega_{\text{i}},\, \omega_{\text{o}},\, \lambda,\, t)</math> is the [[bidirectional reflectance distribution function]], the proportion of light reflected from <math>\omega_{\text{i}}</math> to <math>\omega_{\text{o}}</math> at position <math>\mathbf x\,\!</math>, time <math>t\,\!</math>, and at wavelength <math>\lambda\,\!</math>
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| *<math>L_{\text{i}}(\mathbf x,\, \omega_{\text{i}},\, \lambda,\, t)</math> is spectral radiance of wavelength <math>\lambda\,\!</math> coming inward toward <math>\mathbf x\,\!</math> from direction <math>\omega_{\text{i}}</math> at time <math>t\,\!</math>
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| *<math>\omega_{\text{i}} \cdot \mathbf n</math> is the weakening factor of inward [[irradiance]] due to [[Angle of incidence#Optics|incident angle]], as the light flux is smeared across a surface whose area is larger than the projected area perpendicular to the ray
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| Two 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.
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| Note this equation's [[spectrum|spectral]] and [[time]] dependence — <math>L_{\text{o}}\,\!</math> 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 <math>t\,\!;</math> [[motion blur]] can be produced by [[averaging]] <math>L_{\text{o}}\,\!</math> over some given time interval (by integrating over the time interval and dividing by the length of the interval).<ref>{{cite web
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| | last = Owen
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| | first = Scott
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| | title = Reflection: Theory and Mathematical Formulation
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| | date = September 5, 1999
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| | url = http://www.siggraph.org/education/materials/HyperGraph/illumin/reflect2.htm
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| | accessdate = 2008-06-22}}</ref>
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| == Applications ==
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| 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 analysis|finite element]] methods, leading to the [[radiosity (3D computer graphics)|radiosity]] algorithm. Another approach using [[Monte Carlo method]]s has led to many different algorithms including [[path tracing]], [[photon mapping]], and [[Metropolis light transport]], among others.
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| == Limitations ==
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| Although the equation is very general, it does not capture every aspect of light reflection. Some missing aspects include the following:
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| * [[Transmission (wave propagation)|Transmission]], which occurs when light is transmitted through the surface, like for example when it hits a [[glass]] object or a [[water]] surface,
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| * [[Subsurface scattering]], where the spatial locations for incoming and departing light are different. Surfaces rendered without accounting for subsurface scattering may appear unnaturally opaque — however, it is not necessary to account for this if transmission is included in the equation, since that will effectively include also light scattered under the surface,
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| * [[Polarization (waves)|Polarization]], where different light polarizations will sometimes have different reflection distributions, for example when light bounces at a water surface,
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| * [[Phosphorescence]], which occurs when light or other [[electromagnetic radiation]] is [[Absorption (electromagnetic radiation)|absorbed]] at one moment in time and emitted at a later moment in time, usually with a longer [[wavelength]] (unless the absorbed electromagnetic radiation is very intense),
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| * [[Interference (wave propagation)|Interference]], where the wave properties of light are exhibited,
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| * [[Fluorescence]], where the absorbed and emitted light have different [[wavelength]]s,
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| * [[Non-linear]] effects, where very intense light can increase the [[energy level]] of an [[electron]] with more energy than that of a single [[photon]] (this can occur if the electron is hit by two photons at the same time), and [[Emission (electromagnetic radiation)|emission]] of light with higher frequency than the frequency of the light that hit the surface suddenly becomes possible, and
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| * [[Relativistic Doppler effect]], where light that bounces on an object that is moving in a very high speed will get its wavelength changed; if the light bounces at an object that is moving towards it, the impact will compress the [[photon]]s, so the wavelength will become shorter and the light will be [[blueshift]]ed and the photons will be packed more closely so the photon flux will be increased; if it bounces at an object that is moving away from it, it will be [[redshift]]ed and the photons will be packed more sparsely so the photon flux will be decreased.
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| 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''<ref>{{Citation
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| | last1 = Kajiya | first1 = James T.
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| | last2 = Von Herzen | first2 = Brian P.
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| | title = Ray tracing volume densities
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| | journal = Siggraph 1984
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| | doi = 10.1145/964965.808594
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| | year = 1984
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| | volume = 18
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| | pages = 165
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| | issue = 3}}
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| </ref> suitable for [[volume rendering]] and a ''transient rendering equation''<ref>{{Cite techreport | |
| | first=Adam M.
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| | last=Smith
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| | coauthors=Skorupski, James, Davis, James
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| | title=Transient Rendering
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| | number=UCSC-SOE-08-26
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| | institution=UC Santa Cruz
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| | year=2008
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| | url=http://classes.soe.ucsc.edu/cmps290b/Fall07/TransientRendering/ucsc-soe-08-26.pdf
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| }}</ref> for use with data from a [[time-of-flight camera]].
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| ==References== | |
| <references/>
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| ==External links==
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| * [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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| [[Category:Computer graphics]]
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