Graphics Reference
In-Depth Information
,
ω
)
is an abstract representation of the
directional distribution of light reaching a surface point—only in the simplest
cases can it be expressed directly as a mathematical expression. If a direction
ω
corresponds to a light source, the value of
L
i
is normally the radiance emitted
by the source. Otherwise it represents indirect light, which might include light
reflected from the point
x
itself. (This interdependence is part of what makes
photorealistic rendering such a difficult problem, and is discussed extensively in
coming chapters.) In the case of a point light source,
L
i
(
The incident radiance function
L
i
(
x
,
ω
)
x
has a singularity
ω
is the direction to the light source; elsewhere the value is zero. This
can be modeled in terms of a Dirac
where
, which is a function that has an infinitely
thin “spike” at a single point, and is zero elsewhere.
4
The radiance produced by a
point source can thus be regarded as being infinite.
δ
1.2.3 Surface Reflection and BRDFs
A
bidirectional reflectance distribution function
(BRDF) provides one model of
local reflection. A BRDF is a function
f
r
(
,
ω
,
ω
)
x
that determines how much
ω
is reflected out in direction
light coming in from direction
at surface point
x
.
It might seem natural to define the BRDF as the ratio of the outgoing radi-
ance
L
r
ω
,
ω
)
(
,
ω
)
(
; however, it is actually defined in
terms of the
irradiance
at the surface point caused by light in a thin cone around
the incident direction
x
to the incoming radiance
L
i
x
ω
. More formally, the BRDF is defined as the ratio of the
differential outgoing radiance
L
r
(
x
,
ω
)
to the differential surface
irradiance
at
x
,
ω
)
caused by the incoming radiance
L
i
(
x
spread through a thin cone of directions
ω
about the incident direction
ω
:
d
,
ω
)
dE
i
(
(
dL
r
(
,
ω
)
dL
r
x
x
,
ω
,
ω
)=
f
r
(
x
=
ω
.
(1.7)
,
ω
)
θ
d
x
)
dL
i
(
x
cos
While perhaps less intuitive, this definition has proven more useful.
One way of visualizing a BRDF is to fix the surface point
x
and the incoming
direction
ω
, and consider the behavior of
f
r
as a function of only the outgoing
ω
direction
(see
Figure 1.5
). The shaded area, known as a
lobe
, represents the
relative distribution of the surface reflection of light from the fixed direction ω
at
point
x
.
The BRDF of a highly reflective surface has long thin lobes centered near
the mirror direction of ω
, while dull surfaces have more uniform BRDF lobes.
It is worth mentioning that the value of a BRDF, while never negative, can get
4
In its basic form, the Dirac δ,or
impulse function
is a function of a single variable that satisfies
−
∞
δ
(
x
)
dx
=
1 and, for any function
f
,
−
∞
δ
(
x
)
f
(
x
)
dx
=
f
(
0
)
. To be precise,
δ
is a kind of
generalized function—no real function can satisfy these properties.