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represent real materials decently. Matusik in fact took this approach in compar-
ing two BRDF models, the first due to Ward [War92] and the second to Lafor-
tune [LFTG97]; he found that the Lafortune model was better able to fit the data
in many cases, but even so, there were cases for which the average error was as
large as 20%, where the difference was measured as a difference of
logarithms,
to
discount somewhat the overwhelming effect of glossy peaks in the BRDF.
One drawback to using measured BRDFs in rendering is the cost of perform-
ing effective sampling. While Matusik describes techniques for this, they require
a large amount of extra precomputed data and are still slow compared to those few
models for which explicit sampling techniques are known.
There are some limits to using measured BRDFs. We can only render images
of scenes for which we know the BRDFs of all materials, and measuring the BRDF
of a material is nontrivial; gathering observations at grazing angles is particu-
larly challenging. And we can only gather the BRDF of a material that already
exists. We can't create new BRDFs by adjusting parameters, as we can do with
the various physically based models described below. Finally, there's the problem
that the gathered data may well contain errors, errors that can make the observed
BRDF turn out to be physically unrealistic. Matusik handles this in part by pro-
jecting every measured BRDF onto the reciprocal-BRDF subspace, by replacing
f
s
(
v
i
,
v
o
)
with the average of
f
s
(
v
i
,
v
o
)
and
f
s
(
v
o
,
v
i
)
, thus ensuring reciprocity,
and by discarding certain outlying measurements.
and Diffuse Reflection
We now turn to physically based models of reflective scattering. There is a choice
to be made in attempting to explain scattering phenomena: Should we use
phys-
ical optics,
based on the wave theory of light, in conjunction with a geometric
model, to determine the scattering, or
geometric optics,
in which the reflection of
light by surfaces is determined entirely by a billiard-ball-bounce model, in which
the arriving light reflects from the surface in the mirror direction? At first glance,
the geometric optics approach seems destined for failure; after all, not every sur-
face is mirrorlike. This can, however, be addressed by examining the interaction of
light with a
rough
surface, in which the reflection is mirrorlike, but the geometry
is extremely complex, consisting of many tiny reflecting facets oriented in many
possible directions. Since the roughness can be described probabilistically, this
approach is actually feasible. In contrast, the physical optics approach presents
enormous computational challenges, in the sense that to apply it, we must apply
Maxwell's equations to relatively complex situations, where any hope of an easily
expressed formula is lost; our best hope is for rapid numerical solutions of the
equations. We'll return to this after examining the geometric optics approaches.
Geometric optics is really only appropriate when the small reflecting parts of
the surface are large compared to the wavelength of the incident light. Since the
incident light that interests us is in the visible range, we can say that the wave-
length is about 0.5 to 1.0 microns; this means that the microfacets must be at the
very least 1 micron in size. When you recall that a human hair is on the order of
15 microns in diameter, and that it's easy to feel a single hair on a flat surface, you
realize that the geometric optics assumption for ordinary materials is just barely
reasonable: 15 micron sandpaper feels about like newsprint; 2 micron sandpaper
