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8.6. Irradiance Environment Mapping 325
where θ is the angle between the surface normal and the sky hemisphere
axis. Baker and Boyd propose a cheaper approximation (described by
Taylor [1254]):
E = π
1+cosθ
2
L
sky
+
1 − cos θ
2
L
ground
, (8.43)
which is a linear interpolation between sky and ground using (cos θ +1)/2
as the interpolation factor. The approximation is reasonably close and
significantly cheaper, so it is preferable to the full expression for most
applications.
Valve uses an ambient cube representation for irradiance [848, 881]. This
is essentially a weighted blending between six irradiance values specified at
the faces of a cube:
E = E
x
+ E
y
+ E
z
,
E
x
=(n · x)
2
E
x−
, n ·x < 0,
E
x+
, n · x ≥ 0,
E
y
=(n · y)
2
E
y−
, n · y < 0,
E
y+
, n · y ≥ 0,
E
z
=(n · z)
2
E
z−
, n · z < 0,
E
z+
, n · z ≥ 0,
(8.44)
where x, y,andz are unit-length vectors aligned with the cube axes.
Forsyth [358] presents an inexpensive and flexible lighting model called
the trilight, which includes directional, bidirectional, hemispherical, and
wrap lighting as special cases.
Further Reading and Resources
A valuable reference for information on the radiometry and mathematics
used in this chapter (and much else) is Dutr´e’s free online Global Illumina-
tion Compendium [287].
The work pioneered by Paul Debevec in the area of image-based light-
ing is of great interest to anyone who needs to capture environment maps
from actual scenes. Much of this work is covered in a SIGGRAPH 2003
course [233], as well as in the book High Dynamic Range Imaging: Acqui-
sition, Display, and Image-Based Lighting by Reinhard et al. [1059].
Green’s white paper about spherical harmonic lighting [444] gives a
very accessible and thorough overview of spherical harmonic theory and
practice.
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