Graphics Reference
In-Depth Information
brighter picture sooner, [CCWG88] describes a way of introducing an initial ambient
term.
10.3.1
Form Factors: The Hemicube Method
The lengthiest computations of the radiosity method are evaluating the form factors.
These factors arise from an integral of the form
cos
f
cos
f
1
i
j
Ú
Ú
F
=
dA dA
(10.9)
ij
j
i
A
2
A
A
p
r
i
j
i
These factors are not easy to compute. Some analytical formulas are known dealing
with areas such as polygon to polygon (without occlusion), differential area to
polygon, and some special cases such as rectangle to rectangle and point to disk. They
are, however, purely geometric and do not depend on the aspects of light, such as
reflectivity or emissivity. One mostly has to use numerical approaches.
The integral in (10.9) can be approximated in different ways. One can use the
trapezoidal rule or Simpson's rule, for example. The standard approach there is to
sample at evenly spaced points. On the other hand, if one knows something about the
integrand, then one might be able to sample less by sampling at the most significant
places. One of the early and efficient methods of computing form factors is what is
called the hemicube method . This is the only method we shall describe here.
To begin with, let us make the assumption that patches are far apart relative to
their size. Two simplifications follow from this assumption. First, the inner integral
is essentially constant, so that the outer integral with the 1/A i term essentially disap-
pears and we may assume an approximation of the form
cos
f
cos
f
i
j
Ú
FF
ª
dA
(10.10)
ij
dij
j
2
A j
p
r
Second, two patches that have the same projection onto the surface of a hemi-
sphere centered on the patch A i have the same form factor. See Figure 10.9. Rather
than dealing with a hemisphere, it is more convenient to use a hemicube centered on
A i . See Figure 10.10. Subdivide this hemicube surface into small rectangles Q st . The
form factor for each Q st has an approximation
st = cos
f
cos
f
i
j
D
F
D
A
,
Q
st
2
p
r
where DA st is the area of the rectangle Q st . One precomputes these form factors and
stores them in a lookup table. To compute the form factor F ij one simply adds up the
form factors for all the rectangles Q st onto which the patch A j projects, that is,
= Â D
F
F
,
(10.11)
ij
Q
Q
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