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Concept
Rendering equation
Summary
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. and James Kajiya 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 (Lo) is the sum of the emitted light (Le) and the reflected light (Lr). The reflected light itself is the sum from all directions of the incoming light (Li) multiplied by the surface reflection and cosine of the incident angle.
The rendering equation may be written in the form
where
is the total spectral radiance of wavelength directed outward along direction at time , from a particular position
is the location in space
is the direction of the outgoing light
is a particular wavelength of light
is time
is emitted spectral radiance
is reflected spectral radiance
is an integral over
is the unit hemisphere centered around containing all possible values for where
is the bidirectional reflectance distribution function, the proportion of light reflected from to at position , time , and at wavelength
is the negative direction of the incoming light
is spectral radiance of wavelength coming inward toward from direction at time
is the surface normal at
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 .
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.
Official source
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