Graphics Reference
In-Depth Information
we can see by considering a perfect Lambertian reflector, for which
f
s
(
v
i
,
v
o
)=
Q
2
S
+
. (We'll soon see why 1
1
/π
is the right constant.)
Suppose for a moment that for a photon arriving from a source in direction
for all
v
i
,
v
o
∈
/π
v
i
,
V
2
the probability density of scattering in direction
v
o
were the same in all directions
v
o
. To estimate the radiance at a point
Q
1
on the unit hemisphere around some
surface point
P
(see Figure 27.8), we'd take a solid angle
Ω
1
at
Q
1
and measure
the density of the light energy arriving in that solid angle. For the radiance to
be constant, as we know it is for a Lambertian surface, we'd have to get the same
value when we estimated it at
Q
2
with a solid angle
Ω
2
of the same measure.
But the amount of reflecting surface subtended by the two solid angles varies
like the inverse of the “outgoing cosine,” leading to an extra 1
i
Q
1
V
1
P
Figure 27.8: Computing the radi-
ance at points near a hypothet-
ical surface from which photons
scatter equally in all directions.
n
)
factor
in the outgoing radiance. The probability density of an incoming photon being
scattered in direction
/
(
v
o
·
v
o
from a Lambertian surface must therefore be proportional
to
f
s
(
v
i
,
v
o
)(
v
o
·
n
)
.
Figure 27.9 shows several overlapping classes of scattering that we've dis-
cussed, with the BSDF drawn in black and the scattering probability density drawn
in blue.
Models for Scattering
We now introduce a few basic scattering models that will serve several functions.
The mirror and Lambertian models are the basis for several microfacet-based
models that we'll discuss when we examine physically based models. And the
Blinn-Phong model, although not strictly physically based, is very widely used in
practice.
An ideal mirror-reflecting plane (for which all light is reflected rather than
absorbed), shown in Figure 27.10, reflects light from a source in such a way that
the emitted light distribution is exactly the same as would be produced by an iden-
tical source at some position behind the mirror's location (assuming the mirror was
removed). Because the outgoing radiance along each ray in these two situations
is the same, the mirror-reflection process evidently results in no change in the net
light energy in the scene.
Perfect mirrors are rare. More commonplace is that a mirrored surface in fact
absorbs some amount of light, and reflects the remainder. The outgoing radiance
along the mirror direction
v
r
=
v
i
−
2
(
n
·
v
i
)
n
is therefore a constant multiple of
the incoming radiance:
L
(
P
,
v
o
)=
ρ
L
(
P
,
−
v
i
)
,
(27.2)
ρ
where the
reflectivity
is a number between 0 and 1. All that's needed to specify
the mirror reflectance from a surface is the normal vector
n
and the reflectance
constant
, which is unitless. This constant may, however, have some spectral
dependence (i.e., light at different wavelengths may be reflected differently).
The spectral dependence of reflectance depends on the underlying material.
In broad terms, for insulators like plastic, the mirror-reflected light has the same
ρ
