Graphics Reference
In-Depth Information
The condition that no more energy leave the surface than arrives there is called
energy conservation
(with the assumption that unscattered energy is “conserved”
by appearing as heat). Not every scattering model is energy-conservative. Phong's
original model had no physical units attached, so it was impossible to tell whether
it was conservative or not! In general, conservation can be expressed as a con-
straint on the integral of the BSDF; we'll see this worked out in detail for Lam-
bertian scattering.
The other commonly used constraint on scattering is
reciprocity:
If
(
v
i
)
.
Veach [Vea97] generalizes this to include transmission: For light arriving from
direction
v
i
,
v
o
)
→
f
r
(
v
i
,
v
o
)
is the BRDF for some material, then
f
r
(
v
i
,
v
o
)=
f
r
(
v
o
,
v
i
in a medium of refractive index
n
i
, and scattering in direction
v
o
in a
medium of index
n
o
,
f
s
(
v
i
,
v
o
)
=
f
s
(
v
o
,
v
i
)
.
(27.1)
n
o
n
i
Note that when this is applied to reflection, the two refractive indices are identical,
and the equation simplifies to the usual symmetry law.
It's well known in graphics that the BRDF is symmetric, that is,
f
s
(
v
i
,
v
o
)=
f
s
(
v
i
)
, and this equality is usually attributed to Helmholtz. Veach describes
a proof of symmetry, and explains why Helmholtz's remarks, which involve
mirror reflectors and lenses, are insufficient to imply reciprocity and why sev-
eral other purported proofs have flaws in them. Despite this, the reciprocity
property is still widely known as “Helmholtz reciprocity.”
Despite Veach's proof of reciprocity, there are materials such as pearlescent
paints for which reciprocity apparently does not hold [CPMV
+
09]. Such mate-
rials do not contradict the proof, which assumes that the materials involved in
the scattering are homogeneous.
So is the BRDF symmetric or not? For a very wide range of measured
materials, the answer appears to be “It is, for almost all practical purposes.”
Snyder [SWL98] explains this in some detail.
v
o
,
n
S
2
1
Looking at the differences among solid materials, one of the first things that
attracts our attention is the range of shininess, from the matte appearance of chalk
to the very shiny appearance of a polished metal surface. Another is that some
surfaces are transparent while others are reflective. As we already saw in Chap-
ter 26, these differences are in part due to fundamental physical processes and
structures: Conductive materials, with lots of free electrons, tend to be reflective;
those whose electron orbital energy levels lack “gaps” corresponding to the ener-
gies of quanta of visible light tend to be transparent, etc. But at a higher level, it's
useful to have a language for describing kinds of scattering: reflective, transmis-
sive, mirror, impulse, glossy, diffuse, Lambertian, retroreflective, refractive. To
give these meaning, we'll consider a flat surface (see Figure 27.1) with outward
normal vector
n
(i.e.,
n
points from the material toward empty space), and the
hemispheres
S
+
(
n
)
and
S
-
(
n
)
, where
S
+
(
n
)
is the set of all unit vectors
i
o
P
S
2
2
)
?
Figure 27.1: We'll consider a sur-
face (shown shaded) with normal
vector
n
;
S
+
consists of all vec-
tors pointing
away
from the sur-
face, while
S
-
consists of vectors
pointing
into
the surface. Light
arrives from a source that's in
direction
with
v
∈
S
+
, and scatters
v
i
0, and
S
-
(
n
)
is the set of all those for which the dot product is non-
positive. Since we'll mostly be considering a single surface with a fixed normal
v
·
n
≥
in directions
v
o
,whichmaybein
either
S
-
or
S
+
.
