Graphics Reference
In-Depth Information
2
σ
A
=
1.0
−
0.5
2
+
0.33
,
(27.41)
σ
2
σ
B
=
0.45
2
+
0.09
,
(27.42)
σ
and
α
=max(
θ
i
,
θ
o
)
and
β
=min(
θ
i
,
θ
o
)
.
Until now, all of our models have used geometric optics, ignoring the effects of the
wave nature of light (such as interference and diffraction), except for the Fresnel
term. One reason for this is that working directly with Maxwell's equations proves
to be extremely difficult and computationally expensive. On the other hand, it does
predict some effects that geometric optics models miss. The work of He [He93]
makes the most compelling case for this, but the underlying physics is beyond the
scope of this topic; we refer the interested reader to the paper itself.
How important
are
wave effects? They certainly can matter, but as Pharr and
Humphreys [PH10], p. 454, note:
Nayar, Ikeuchi, and Kanade [NKK91] have shown that some
reflection models based on physical (wave) optics have substan-
tially similar characteristics to those based on geometric optics.
The geometric optics approximations don't seem to cause too
much error in practice, except on very smooth surfaces. This is
a helpful result, giving experimental basis to the general belief
that wave optics models aren't usually worth their computational
expense for computer graphics applications.
A BSDF can be represented in various ways—as a table of values to be interpo-
lated, as we saw for measured models, or as a sum of “lobes,” as in the Lafortune
model, or even in a kind of “Fourier decomposition,” using spherical harmonics,
which are the analog, on the 2-sphere, of the powers of sine and cosine on the
circle. It's also possible to represent BSDFs using sums of Gaussians, in wavelet
bases, or many other possible forms. Each choice has its advantages and disad-
vantages, and graphics has not yet arrived at a definitive ideal model.
We've discussed BSDFs and how to represent them with a general bias toward
finding models that match measured data well, which certainly seems like a good
thing. But we haven't discussed the precise criteria for “matching well.” One obvi-
ous choice is the
L
2
error: If
f
is an approximation to
f
s
, we can integrate
(
f
−
f
s
)
2
over all of
S
2
×
S
2
to determine the goodness of fit. In the sense that the difference
between
f
and
f
s
corresponds to the difference in what we see when we look at a
directly illuminated piece of the material, this seems to make intuitive sense. But
our perception is nonlinear as a function of radiance. A small error in the approx-
imated reflectance at a
(
v
o
)
pair where
f
s
is small is far more perceptually
important than the same difference at a place where
f
s
is large, but they're counted
equally in measuring the goodness of fit. This argument suggests that we should
v
i
,
