Graphics Reference
In-Depth Information
r
n
C
α
p
Figure 1.13.
Visualization of cone of specular importance sampling.
We successfully applied this method to various specular models such as Phong,
Blinn-Phong, Beckmann and GGX [Walter et al. 07]. However, it is worth noting
that a cone shape is a relaxed bound on the actual distribution 3D shape, the
complexity of which varies over different BRDF. Therefore, the cone shape is
only approximate. In the case of Phong, the cone is actually a perfect match.
However, in the case of Blinn-Phong, due to the use of the half vector, a pyramid
with an ellipse base or even more complex shape would provide tighter, more
accurate bounds.
Now we have an approximation model for the most important samples of our
specular function. In the case of a single area light source, we would be interested
in calculating the integral of our specular function over the portion of area of the
light subtending the cone—defining integration limits. We can approximate the
final solution by applying reasoning and methodology similar to Section 1.3.3.
Again, a varied database was prepared, an integral numerically calculated, and
the most important points estimated. In the end we found out that the geometric
center of the integration limits area is a good point to estimate the unbounded
specular integral. Let's call it
p
sc
. Therefore, to obtain the final result, we must
calculate
Specular
(
p,ω
o
) for a single point light at
p
sc
and normalize the result
by the integration limits area (see Figure 1.14).
To simplify our problem, we move to the light space (see Figure 1.15) by
projecting our specular cone onto the light plane. In that case we are looking at
a 2D problem, of finding the area of intersection between an ellipsoid (projected
specular cone) and a given light shape (in our case a disk or rectangle, as spheres
prove to be easily solvable working exclusively with solid angles).
