Graphics Reference
In-Depth Information
When do high powers of a linear operator approach the zero operator? We
can answer this by looking at eigenvalues: If all eigenvalues are strictly less
than one, then T k L e
. In rendering, this more or less corre-
sponds to there being no perfect reflectors in a scene; indeed, one can imagine
a scene consisting of two enormous planar mirrors that face each other, and a
point light source between them. Equal amounts of light moving left and right
constitute an eigenvector of the light-scattering operator T : After reflection, we
once again have equal amounts of light moving left and right. So in this situ-
ation, T has an eigenvalue of 1, and iterative computation is not guaranteed to
converge. Indeed, if the light source puts out some light, a moment later that
light will be reflected by the mirrors and will be added to new light sent out
by the source, etc., so that the transported light goes to infinity. The unrealis-
tic assumption of perfect mirrors leads to the unrealistic prediction of infinite
light transport (and the nonconvergence of the iterative method for solving the
equation).
In practice, most surfaces we encounter have relatively low reflectance, and
an iterative computation not only converges, it converges fairly quickly. Unfor-
tunately, the convergence isn't necessarily the kind we want: Our estimate of
the radiance field L , after a few iterations, may be very close to the true radiance
field L 0 , but the scene's appearance to a human observer might be very differ-
ent. For instance, if the scene consists of a room lit by a tiny pinhole, behind
which there's a light source, the true light in the room is very small . . . and
therefore very similar to no light in the room; similar, that is, when we com-
pare using the standard mathematical measure of similarity. When we compare
using a perceptual metric, the difference is clear: A tiny bit of light when you
awaken at night lets you avoid stubbing your toe, while no light at all does not!
0as k goes to
31.13 Alternative Formulations of Light
Transport
We've described light transport in term of the radiance field, L , which is defined on
R 3
S 2 or
S 2 , where
is the set of all surface points in a scene. (Since radi-
ance is constant along rays in empty space, knowing L at points of
×
M ×
M
determines
its values on all of R 3 .) And we've used the scattering operator, which transforms
an incoming radiance field to an outgoing one in writing the rendering equation.
But there are alternative formulations.
Arvo [Arv95] describes light transport in terms of two separate operators. The
first operator, G , takes the surface radiance on
M
and converts it to the field radi-
ance, essentially by ray casting: Surface radiance leaving a point P in a direction
v
M
. The second
operator, K , takes field radiance at a point P and combines it with the BRDF at
P to produce surface radiance (i.e., it describes single-bounce scattering locally).
Thus, the transport operator T can be expressed as T = K
becomes field radiance at the point Q where the ray first hits
M
G .
Kajiya [Kaj86] takes a different approach in which light directly transported
from any point P
M is represented by a value I ( P , Q ) ;if
P and Q are not mutually visible, then I ( P , Q ) is zero. Kajiya calls the quantity I
the unoccluded two-point transport intensity. (The letters I ,
M to any point Q
ρ
,
, M , and g used
 
 
 
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