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# Rendering equation

In computer graphics, the rendering equation describes the flow of light energy throughout a scene. Based on the physics of light, it provides theoretically perfect results, in contrast to the various rendering techniques, which approximate this ideal.

The physical basis for the rendering equation is the law of conservation of energy. At a particular position and direction, the outgoing light (Lo) is the sum of the emitted light (Le) and the reflected light. The reflected light itself is the sum of the incoming light (Li) from all directions, multiplied by the surface reflection and incoming angle.

Mathematically, the equation is expressed as follows:

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

Broken down:

$L_o(x, \vec w)$ is light outward at a particular position x and in direction $\vec w$.
$L_e(x, \vec w)$ is light emitted from the same position and direction.
$\int_\Omega ... d\vec w'$ is an infinitesimal sum over a hemisphere of inward directions.
$f_r(x, \vec w', \vec w)$ is the proportion of light reflected at the position (from inward direction to outward direction).
$L_i(x, \vec w')$ is light inward from the position and direction $\vec w'$.
$(\vec w' \cdot \vec n)$ 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 arrangements are (easily) possible for solutions.

The rendering equation is the key academic/theoretical concept in rendering. It serves as the most abstract formal expression of the non-perceptual aspect of rendering. By connecting outward light to inward light, via an interaction point, this equation stands for the whole 'light transport' in a scene. All more complete algorithms can be seen as solutions to particular formulations of this equation.

• Lecture notes from Stanford University course CS 348B, Computer Graphics: Image Synthesis Techniques

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