Graphics Reference
In-Depth Information
The critical problem, when we want to include transparency into the rendering
equation, is that a pair ( P ,
S 2
v
)
M ×
is no longer associated to a unique
radiance value.
Because we are only doing surface rendering rather than volume rendering
(i.e., we're ignoring participating media), we only care about L ( P ,
) for points
P that are on some surface. That surface must have a surface normal n at P .We
can therefore define L ( P ,
v
, n ) by the rule that if
and n have a positive dot
v
v
product, then L ( P ,
, n ) represents the radiance leaving the surface in direction
v
; if their dot product is negative, then it represents the radiance arriving at the
surface in direction
v
. By using either the unit outward surface normal or its
opposite as n , we can handle both reflection and transmission. In practice, this
turns into an additional if statement at every ray-surface interaction: A surface
element has two sides (one with a normal pointing each way), and we treat the
vector
v
differently depending on whether it points in the same or the opposite
half-space as the normal vector to the “side.” In mathematical terms, we can
say that L is defined on the orientation double cover [Lee09] of the set of all scene
surfaces.
The three-argument version of L is awkward to write. As an alternative, we can
replace L altogether with two new functions: ( P ,
v
L in ( P ,
)
) and ( P ,
)
v
v
v
L out ( P ,
) , which represent light arriving at P traveling in direction
and light
v
v
leaving P in direction
, respectively; these are Arvo's field and surface radiance
functions. The reflectance equation then becomes a scattering equation:
v
v o )=
L r, out ( P ,
L in ( P ,
v i ) f s ( P ,
v i ,
v o )
| v i ·
n P |
d
v i ,
(29.10)
v i S 2 ( p )
where five things have changed.
• The integral is now over all directions of incoming light.
• The result is now L r
rather than L ref
(recall that L r denotes light either
reflected or transmitted).
• The BRDF f r has been replaced with the BSDF f s .
• The dot product now has an absolute value.
• The annotations “in” and “out” have been added to the radiance and scat-
tered radiance.
The transport equation now links incoming and outgoing light. We write the
equation in two forms, one suitable for use in ray tracing, the other for use in
photon mapping. The distinction is simply one of tracing rays in the direction of
photon propagation or in the opposite direction. The version used in raytracing is
this:
L in ( P ,
v i )= L out ( R ( P ,
v i ) ,
v i )
(29.11)
while the one that's useful in photon mapping is this:
L out ( Q ,
v o )= L in ( R ( Q ,
v o ) ,
v o )
(29.12)
Dividing the radiance field into two parts has further advantages. When we
write the radiance field this way, it naturally extends to all points P rather than
 
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