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.
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