Rendering equation

<h2 id="equation-form">Equation form</h2>
<p>The rendering equation may be written in the form
</p>

L
          
            o
          
        
        (
        
          x
        
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          ω
          
            o
          
        
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        λ
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        t
        )
        =
        
          L
          
            e
          
        
        (
        
          x
        
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          ω
          
            o
          
        
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        λ
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        t
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        +
        
          L
          
            r
          
        
        (
        
          x
        
        ,
        
          ω
          
            o
          
        
        ,
        λ
        ,
        t
        )
      
    
    {\displaystyle L_{\text{o}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)=L_{\text{e}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)+L_{\text{r}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)}

L
          
            r
          
        
        (
        
          x
        
        ,
        
          ω
          
            o
          
        
        ,
        λ
        ,
        t
        )
        =
        
          ∫
          
            Ω
          
        
        
          f
          
            r
          
        
        (
        
          x
        
        ,
        
          ω
          
            i
          
        
        ,
        
          ω
          
            o
          
        
        ,
        λ
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        t
        )
        
          L
          
            i
          
        
        (
        
          x
        
        ,
        
          ω
          
            i
          
        
        ,
        λ
        ,
        t
        )
        (
        
          ω
          
            i
          
        
        ⋅
        
          n
        
        )
        d
        ⁡
        
          ω
          
            i
          
        
      
    
    {\displaystyle L_{\text{r}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)=\int _{\Omega }f_{\text{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}}}

<p>where
</p>
<ul><li>
  
    
      
        
          L
          
            o
          
        
        (
        
          x
        
        ,
        
          ω
          
            o
          
        
        ,
        λ
        ,
        t
        )
      
    
    {\displaystyle L_{\text{o}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)}
  
 is the total <a href="/facts/Spectral_radiance/5kS1FhpD">spectral radiance</a> of wavelength 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 directed outward along direction 
  
    
      
        
          ω
          
            o
          
        
      
    
    {\displaystyle \omega _{\text{o}}}
  
 at time 
  
    
      
        t
      
    
    {\displaystyle t}
  
, from a particular position 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
</li>
<li>
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
 is the location in space</li>
<li>
  
    
      
        
          ω
          
            o
          
        
      
    
    {\displaystyle \omega _{\text{o}}}
  
 is the direction of the outgoing light</li>
<li>
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 is a particular wavelength of light</li>
<li>
  
    
      
        t
      
    
    {\displaystyle t}
  
 is time</li>
<li>
  
    
      
        
          L
          
            e
          
        
        (
        
          x
        
        ,
        
          ω
          
            o
          
        
        ,
        λ
        ,
        t
        )
      
    
    {\displaystyle L_{\text{e}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)}
  
 is <a href="/facts/Emissivity/LUxfMRRI">emitted</a> spectral radiance</li>
<li>
  
    
      
        
          L
          
            r
          
        
        (
        
          x
        
        ,
        
          ω
          
            o
          
        
        ,
        λ
        ,
        t
        )
      
    
    {\displaystyle L_{\text{r}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)}
  
 is <a href="/facts/Reflectance/TT4sBwIw">reflected</a> spectral radiance</li>
<li>
  
    
      
        
          ∫
          
            Ω
          
        
        …
        d
        ⁡
        
          ω
          
            i
          
        
      
    
    {\displaystyle \int _{\Omega }\dots \operatorname {d} \omega _{\text{i}}}
  
 is an <a href="/facts/Integral/V7lyV997">integral</a> over 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
</li>
<li>
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 is the unit <a href="/facts/Sphere/jywYLNM2">hemisphere</a> centered around 
  
    
      
        
          n
        
      
    
    {\displaystyle \mathbf {n} }
  
 containing all possible values for 
  
    
      
        
          ω
          
            i
          
        
      
    
    {\displaystyle \omega _{\text{i}}}
  
 where 
  
    
      
        
          ω
          
            i
          
        
        ⋅
        
          n
        
        >
        0
      
    
    {\displaystyle \omega _{\text{i}}\cdot \mathbf {n} >0}
  
</li>
<li>
  
    
      
        
          f
          
            r
          
        
        (
        
          x
        
        ,
        
          ω
          
            i
          
        
        ,
        
          ω
          
            o
          
        
        ,
        λ
        ,
        t
        )
      
    
    {\displaystyle f_{\text{r}}(\mathbf {x} ,\omega _{\text{i}},\omega _{\text{o}},\lambda ,t)}
  
 is the <a href="/facts/Bidirectional_reflectance_distribution_function/9LtejD2e">bidirectional reflectance distribution function</a>, the proportion of light reflected from 
  
    
      
        
          ω
          
            i
          
        
      
    
    {\displaystyle \omega _{\text{i}}}
  
 to 
  
    
      
        
          ω
          
            o
          
        
      
    
    {\displaystyle \omega _{\text{o}}}
  
 at position 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
, time 
  
    
      
        t
      
    
    {\displaystyle t}
  
, and at wavelength 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
</li>
<li>
  
    
      
        
          ω
          
            i
          
        
      
    
    {\displaystyle \omega _{\text{i}}}
  
 is the negative direction of the incoming light</li>
<li>
  
    
      
        
          L
          
            i
          
        
        (
        
          x
        
        ,
        
          ω
          
            i
          
        
        ,
        λ
        ,
        t
        )
      
    
    {\displaystyle L_{\text{i}}(\mathbf {x} ,\omega _{\text{i}},\lambda ,t)}
  
 is spectral radiance of wavelength 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 coming inward toward 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
 from direction 
  
    
      
        
          ω
          
            i
          
        
      
    
    {\displaystyle \omega _{\text{i}}}
  
 at time 
  
    
      
        t
      
    
    {\displaystyle t}
  
</li>
<li>
  
    
      
        
          n
        
      
    
    {\displaystyle \mathbf {n} }
  
 is the <a href="/facts/Normal_(geometry)/jhtLIhcu">surface normal</a> at 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
</li>
<li>
  
    
      
        
          ω
          
            i
          
        
        ⋅
        
          n
        
      
    
    {\displaystyle \omega _{\text{i}}\cdot \mathbf {n} }
  
 is the weakening factor of outward <a href="/facts/Irradiance/fgHwu7uh">irradiance</a> due to <a href="/facts/Angle_of_incidence_(optics)/DTaCjhTU">incident angle</a>, as the light flux is smeared across a surface whose area is larger than the projected area perpendicular to the ray. This is often written as 
  
    
      
        cos
        ⁡
        
          θ
          
            i
          
        
      
    
    {\displaystyle \cos \theta _{i}}
  
.</li></ul>
<p>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. It is a <a href="/facts/Fredholm_integral_equation/euHGJbQo">Fredholm integral equation</a> of the second kind, similar to those that arise in <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>Note this equation's <a href="/facts/Spectrum/wPUjkpqI">spectral</a> and <a href="/facts/Time/HceJKUGB">time</a> dependence — 
  
    
      
        
          L
          
            o
          
        
      
    
    {\displaystyle L_{\text{o}}}
  
 may be sampled at or integrated over sections of the <a href="/facts/Visible_spectrum/frPBh8o5">visible spectrum</a> to obtain, for example, a <a href="/facts/Trichromatic/C7I3Ywds">trichromatic</a> color sample. A pixel value for a single frame in an animation may be obtained by fixing 
  
    
      
        t
        ;
      
    
    {\displaystyle t;}
  
 <a href="/facts/Motion_blur/y7jR2shk">motion blur</a> can be produced by <a href="/facts/Averaging/etawWjw4">averaging</a> 
  
    
      
        
          L
          
            o
          
        
      
    
    {\displaystyle L_{\text{o}}}
  
 over some given time interval (by integrating over the time interval and dividing by the length of the interval).<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p><p>Note that a solution to the rendering equation is the function 
  
    
      
        
          L
          
            o
          
        
      
    
    {\displaystyle L_{\text{o}}}
  
. The function 
  
    
      
        
          L
          
            i
          
        
      
    
    {\displaystyle L_{\text{i}}}
  
 is related to 
  
    
      
        
          L
          
            o
          
        
      
    
    {\displaystyle L_{\text{o}}}
  
 via a ray-tracing operation: The incoming radiance from some direction at one point is the outgoing radiance at some other point in the opposite direction.
</p>
<h2 id="applications">Applications</h2>
<p>Solving the rendering equation for any given scene is the primary challenge in <a href="/facts/Realistic_rendering/CbQX6CWM">realistic rendering</a>. One approach to solving the equation is based on <a href="/facts/Finite_element_analysis/eUcfP1vn">finite element</a> methods, leading to the <a href="/facts/Radiosity_(3D_computer_graphics)/qN75qQgV">radiosity</a> algorithm. Another approach using <a href="/facts/Monte_Carlo_method/AkHjY7jc">Monte Carlo methods</a> has led to many different algorithms including <a href="/facts/Path_tracing/fxhaM84b">path tracing</a>, <a href="/facts/Photon_mapping/t58p6V7w">photon mapping</a>, and <a href="/facts/Metropolis_light_transport/wRoPqI2f">Metropolis light transport</a>, among others.
</p>
<h2 id="limitations">Limitations</h2>
<p>Although the equation is very general, it does not capture every aspect of light reflection. Some missing aspects include the following:
</p>
<ul><li><a href="/facts/Transmission_(wave_propagation)/893q1h2K">Transmission</a>, which occurs when light is transmitted through the surface, such as when it hits a <a href="/facts/Glass/hezRsAGf">glass</a> object or a <a href="/facts/Water/0lkXnJPK">water</a> surface,</li>
<li><a href="/facts/Subsurface_scattering/rJqYNFXe">Subsurface scattering</a>, 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,</li>
<li><a href="/facts/Polarization_(waves)/fP33m0eD">Polarization</a>, where different light polarizations will sometimes have different reflection distributions, for example when light bounces at a water surface,</li>
<li><a href="/facts/Phosphorescence/WshJFsKV">Phosphorescence</a>, which occurs when light or other <a href="/facts/Electromagnetic_radiation/Ynx7ujCW">electromagnetic radiation</a> is <a href="/facts/Absorption_(electromagnetic_radiation)/xBYKt7yS">absorbed</a> at one moment and emitted at a later moment, usually with a longer <a href="/facts/Wavelength/kcJU0ymz">wavelength</a> (unless the absorbed electromagnetic radiation is very intense),</li>
<li><a href="/facts/Interference_(wave_propagation)/lI5f78I4">Interference</a>, where the wave properties of light are exhibited,</li>
<li><a href="/facts/Fluorescence/qsJVT2qP">Fluorescence</a>, where the absorbed and emitted light have different <a href="/facts/Wavelength/kcJU0ymz">wavelengths</a>,</li>
<li><a href="/facts/Nonlinear_optics/maerngXd">Non-linear</a> effects, where very intense light can increase the <a href="/facts/Energy_level/66x51Khc">energy level</a> of an <a href="/facts/Electron/L0KBo5cW">electron</a> with more energy than that of a single <a href="/facts/Photon/mFNhAEzA">photon</a> (this can occur if the electron is hit by two photons at the same time), and <a href="/facts/Emission_(electromagnetic_radiation)/fxumR717">emission</a> of light with higher frequency than the frequency of the light that hit the surface suddenly becomes possible, and</li>
<li><a href="/facts/Doppler_effect/bzrK1BKf">Doppler effect</a>, where light that bounces off an object moving at a very high speed will get its wavelength changed: if the light bounces off an object that is moving towards it, the light will be <a href="/facts/Blueshift/v8XBtGKH">blueshifted</a> and the <a href="/facts/Photon/mFNhAEzA">photons</a> will be packed more closely so the photon flux will be increased; if it bounces off an object moving away from it, it will be <a href="/facts/Redshift/Qfl4VjDF">redshifted</a> and the photon flux will be decreased. This effect becomes apparent only at speeds comparable to the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>, which is not the case for most rendering applications.</li></ul>
<p>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 <i>volume rendering equation</i><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> suitable for <a href="/facts/Volume_rendering/cpfXEhYo">volume rendering</a> and a <i>transient rendering equation</i><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> for use with data from a <a href="/facts/Time-of-flight_camera/jwaYG1VG">time-of-flight camera</a>.
</p>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://graphics.stanford.edu/courses/cs348b-00/lectures/lecture12/">Lecture notes</a> from Stanford University course CS 348B, <i>Computer Graphics: Image Synthesis Techniques</i></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Immel, David S.; Cohen, Michael F.; Greenberg, Donald P. (1986). "A radiosity method for non-diffuse environments" (PDF). In David C. Evans; RussellJ. Athay (eds.). SIGGRAPH '86. Proceedings of the 13th annual conference on Computer graphics and interactive techniques. pp. 133–142. doi:10.1145/15922.15901. ISBN 978-0-89791-196-2. S2CID 7384510. <a href="978-0-89791-196-2" target="_blank">978-0-89791-196-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Kajiya, James T. (1986). "The rendering equation" (PDF). In David C. Evans; RussellJ. Athay (eds.). SIGGRAPH '86. Proceedings of the 13th annual conference on Computer graphics and interactive techniques. pp. 143–150. doi:10.1145/15922.15902. ISBN 978-0-89791-196-2. S2CID 9226468. <a href="978-0-89791-196-2" target="_blank">978-0-89791-196-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Watt, Alan; Watt, Mark (1992). "12.2.1 The path tracing solution to the rendering equation". Advanced Animation and Rendering Techniques: Theory and Practice. Addison-Wesley Professional. p. 293. ISBN 978-0-201-54412-1. <a href="978-0-201-54412-1" target="_blank">978-0-201-54412-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Owen, Scott (September 5, 1999). "Reflection: Theory and Mathematical Formulation". Retrieved 2008-06-22. <a href="http://www.siggraph.org/education/materials/HyperGraph/illumin/reflect2.htm" target="_blank">http://www.siggraph.org/education/materials/HyperGraph/illumin/reflect2.htm</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Kajiya, James T.; Von Herzen, Brian P. (1984), "Ray tracing volume densities", ACM SIGGRAPH Computer Graphics, 18 (3): 165–174, CiteSeerX 10.1.1.128.3394, doi:10.1145/964965.808594 <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Smith, Adam M.; Skorupski, James; Davis, James (2008). Transient Rendering (PDF) (Technical report). UC Santa Cruz. UCSC-SOE-08-26. <a href="http://classes.soe.ucsc.edu/cmps290b/Fall07/TransientRendering/ucsc-soe-08-26.pdf" target="_blank">http://classes.soe.ucsc.edu/cmps290b/Fall07/TransientRendering/ucsc-soe-08-26.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Rendering equation open-in-new

Rendering equation