Graphics Reference
In-Depth Information
in this and the following section will not be used again; we are merely explaining
the correspondence between his notation and ours.) Kajiya's version of the BRDF
is not expressed in terms of a point and two directions, but rather in terms of three
points; he writes
( P , Q , R ) for the amount of light from R to Q that's scattered
toward the point P . His “emitted light” function also has points as parameters
rather than point-direction parameters:
ρ
( P , Q ) is the amount of light emitted from
Q in the direction of P . Kajiya's quantities exclude various cosines that appear
in our formulation of the rendering equation, including them instead as part of
the integration (his integrals are over the set
of all surfaces in the scene, while
ours are usually over hemispheres around a point; the change-of-variables for-
mula introduces the necessary cosines, as described in Section 26.6.4). Kajiya's
formulation of the rendering equation is therefore
I ( P , Q )= g ( P , Q )
M
( P , Q , R ) I ( Q , R ) dR ,
( P , Q )+
R M ρ
(31.54)
where g ( P , Q ) is a “geometry” term that in part determines the mutual visibility of
P and Q : It's zero if P is occluded from Q . Expressing this in terms of operators,
he writes
I = g
+ gMI ,
(31.55)
where M is the operator that combines I with
ρ
in the integral. The series solution
then becomes
+ g ( Mg ) 3
I = g
+ gMg
+ gMgMg
+
...
(31.56)
This formulation has the advantage that the computation of visibility is
explicit: Every occurrence of g represents a visibility (or ray-casting) operation.
31.14 Approximations of the Series Solution
As we mentioned, summing an infinite series to solve the rendering equation is not
really practical. But several approximate approaches have worked well in practice.
We follow Kajiya's discussion closely.
The earliest widely used approximate solution consisted (roughly) of the
following:
• Limiting the emission function to point lights
• Computing only one-bounce scattering (i.e., paths of the form LDE )
That is to say, the approximation to Equation 31.56 used was
I = g
+ gM
0 ,
(31.57)
0 denotes the use of only point lights. (The first term has
where
, because it was
possible to render directly visible area lights.) Note that the second term should
have been gMg
0 , that is, it should have accounted for whether the illumination
could be seen by the surface (i.e., was the surface illuminated?). But such visibility
computations were too expensive for the hardware, with the result that these early
pictures lacked shadows.
 
 
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