Rendering equation

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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.^[1] and Jim Kajiya^[2] 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 (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

$L_o(x, \mathbf w) = L_e(x, \mathbf w) + \int_\Omega f_r(x, \mathbf w', \mathbf w) L_i(x, \mathbf w') (\mathbf w' \cdot \mathbf n) d\mathbf w'$

where

$L_o(x, \mathbf w)$ is light outward at a particular position

x

and in direction $\mathbf w$ ,

$L_e(x, \mathbf w)$ is light emitted from the same position and direction,

$\int_\Omega \cdots d\mathbf w'$ is an integral over a hemisphere of inward directions,

$f_r(x, \mathbf w', \mathbf w)$ is the BRDF, the proportion of light reflected at the position (from inward direction to outward direction),

$L_i(x, \mathbf w')$ is light inward from the position and direction $\mathbf w'$ , and

$\mathbf w' \cdot \mathbf n$ is the attenuation of inward light due to incident angle.

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.

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.

[edit] References

^ Immel, David S.; Cohen, Michael F. & Greenberg, Donald P., “A radiosity method for non-diffuse environments”, SIGGRAPH 1986, DOI 10.1145/15922.15901
^ Kajiya, James T., “The rendering equation”, SIGGRAPH 1986, DOI 10.1145/15922.15902