Rendering equation

The rendering equation may be written in the form

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

${\displaystyle L_{\text{r}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)=\int _{\Omega }f_{\text{r}}(\mathbf {x} ,\omega _{\text{i}},\omega _{\text{o}},\lambda ,t)L_{\text{i}}(\mathbf {x} ,\omega _{\text{i}},\lambda ,t)(\omega _{\text{i}}\cdot \mathbf {n} )\operatorname {d} \omega _{\text{i}}}$

where

• ${\displaystyle L_{\text{o}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)}$ is the total spectral radiance of wavelength ${\displaystyle \lambda }$ directed outward along direction ${\displaystyle \omega _{\text{o}}}$ at time ${\displaystyle t}$, from a particular position ${\displaystyle \mathbf {x} }$
• ${\displaystyle \mathbf {x} }$ is the location in space
• ${\displaystyle \omega _{\text{o}}}$ is the direction of the outgoing light
• ${\displaystyle \lambda }$ is a particular wavelength of light
• ${\displaystyle t}$ is time
• ${\displaystyle L_{\text{e}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)}$ is emitted spectral radiance
• ${\displaystyle L_{\text{r}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)}$ is reflected spectral radiance
• ${\displaystyle \int _{\Omega }\dots \operatorname {d} \omega _{\text{i}}}$ is an integral over ${\displaystyle \Omega }$
• ${\displaystyle \Omega }$ is the unit hemisphere centered around ${\displaystyle \mathbf {n} }$ containing all possible values for ${\displaystyle \omega _{\text{i}}}$ where ${\displaystyle \omega _{\text{i}}\cdot \mathbf {n} >0}$
• ${\displaystyle f_{\text{r}}(\mathbf {x} ,\omega _{\text{i}},\omega _{\text{o}},\lambda ,t)}$ is the bidirectional reflectance distribution function, the proportion of light reflected from ${\displaystyle \omega _{\text{i}}}$ to ${\displaystyle \omega _{\text{o}}}$ at position ${\displaystyle \mathbf {x} }$, time ${\displaystyle t}$, and at wavelength ${\displaystyle \lambda }$
• ${\displaystyle \omega _{\text{i}}}$ is the negative direction of the incoming light
• ${\displaystyle L_{\text{i}}(\mathbf {x} ,\omega _{\text{i}},\lambda ,t)}$ is spectral radiance of wavelength ${\displaystyle \lambda }$ coming inward toward ${\displaystyle \mathbf {x} }$ from direction ${\displaystyle \omega _{\text{i}}}$ at time ${\displaystyle t}$
• ${\displaystyle \mathbf {n} }$ is the surface normal at ${\displaystyle \mathbf {x} }$
• ${\displaystyle \omega _{\text{i}}\cdot \mathbf {n} }$ is the weakening factor of outward irradiance due to incident angle, 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 ${\displaystyle \cos \theta _{i}}$.

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 Fredholm integral equation of the second kind, similar to those that arise in quantum field theory.^[5]

Note this equation's spectral and time dependence — ${\displaystyle L_{\text{o}}}$ 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 ${\displaystyle t;}$ motion blur can be produced by averaging ${\displaystyle L_{\text{o}}}$ over some given time interval (by integrating over the time interval and dividing by the length of the interval).^[6]

Note that a solution to the rendering equation is the function ${\displaystyle L_{\text{o}}}$. The function ${\displaystyle L_{\text{i}}}$ is related to ${\displaystyle L_{\text{o}}}$ 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.

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.