Graphics Reference
In-Depth Information
x
d
0
tl
d
1
c
0
c
1
c
R
0
R
1
br
y
Figure 1.17.
Finding the area and center of the intersection of rectangles.
integral calculation, we not only approximate the final result, but we also limit
the model to radially symmetric PDFs. In the case of most microfacet BRDFs
based on half vectors (equation (1.9)), the initial shape of the specular cone would
be similar to an elliptical base, which would result in an ellipse shape on the light
plane—thus an integral over an ellipse.
This is a rather crude approximation; however, it proved good enough in
visual assessment of final results, when tested with Phong, Blinn-Phong, and
GGX, using radially symmetrical light shapes (see Figures 1.18 and 1.19).
1.3.5 Nonuniform Light Sources
Up to this point we were only considering light sources with constant
L
i
(
p,ω
i
).
We will now change this assumption and for simplicity assume that light inten-
sity
I
is constant over the light and that
L
i
(
p,ω
i
) returns a normalized, wave-
length dependent value. Looking at integrals from equations (1.11) and (1.12)
and following intuitions from Sections 1.3.3 and 1.3.4, we can see that in order
to acquire the correct result, we could pre-integrate equations (1.11) and (1.12)
with varying
L
i
(
p,ω
i
) over the light source and then normalize by the source
area. Assuming we can approximate diffuse and specular integrals at runtime,
due to equations (1.16) and (1.20), we would just need to multiply those results
by the pre-integrated, normalized full integral from equations (1.11) and (1.12)
at our most important points for diffuse and specular integrals, respectively.
In the case of a diffuse (equations (1.11) and (1.16)), we need to integrate
Diffuse
Lookup
(
p
d
,r
)=
A
L
i
(
p
d
+
rn
l
,ω
i
)
dω
i
,
