The Rendering Equation (à la Kajiya - Siggraph 86)
An attempt to unify rendering so that all rendering had a basic model as a
basis.

where:
-
I ( x , x' ) is the intensity of light passing from point x' to point
x.
-
g ( x , x' ) is the visibility between x and x'. If they are occluded,
this is zero, otherwise it varies as the inverse square of the distance
between them. The part right to that expression is the : the "unoccluted
two point transport intensity".
-
e ( x , x' ) is the transfer emittance from
x' to x. It is related to the intensity of any light self emitted by point
x' in the direction of x
-
r ( x , x' , x'' ) is the scattering term or
bidirectional reflectivity with respect to directions x' and x''. It is
the intensity of the energy scattered towards x by a surface point located
at x' arriving from point or directions x''. The "Unoccluted three-point
transport reflectance" Related to the BRDF
-
S represents all points on all surfaces in the scene
-
i.e. the transport intensity from x' to x is the sum of emitted light
from x' to x and all light from x'' that eventually gets to x through x'.
This is of course a recursive definition !
Complexity => practical solution are aproximations
View Independant statement of the problem
-
solutions can be view independant (radiosity) or not (raytracing)
we can rewrite this equation
as
where R
is the linear integral operator
rearranging terms gives:
.
Local Reflection Models
.
only first 2 terms are used
X is the eyepoint
the g(epsilon) term is non-zero only for
light sources
R1 operates on (epsilon)
rather than g, so shadows are not computed
Basic Ray Tracing
Radiosity
by performing transformations outlined on page
293 of the text, we get
- dB(x') is the radiosity of surface element
dx'
- p0 is p( x , x' ,
x'' ) and is constant
- H(x') is the energy incident on the surface
element dx'
The Extended Two-Pass Algorithm (Sillion
1989)
.
uses the rendering equation as the basis
does not place the restriction Wollace does
of making specular surfaces perfect planar mirrors
The general equation used is:
the visibility function g is incorporated into
the reflection operator R.
p(x, x', x'') = pd(x') + ps(x, x', x'')
bidirectional diffuse specular
reflectivity
function
In the first pass, extended form factors are used to compute diffuse to diffuse
interaction that has any number of specular transfers inbetween
extended form factors: Diffuse - specular* - diffuse
The 2nd pass uses standard ray tracing to compute
specular transfer