Graphics Reference
In-Depth Information
BRDF
matrix
Rotation
matrix
Transfer matrix
Source light vector
Outgoing light vector
e = BR p T p
M p
Exit transfer
matrix
Outgoing light
(rendering solution
for point p )
Incident
light
Figure 10.9 Exit transfer matrix.
particular render-time assumptions. For example, it might be more efficient to
transform the BRDF matrices to the global coordinate system if the viewpoint re-
mains fixed as described above. The exit transfer matrix also depends on which
effects the transfer matrix includes. But the basic idea is always the same: the
incident light vector is transformed to an outgoing light vector by a single ma-
trix. As the outgoing light vector contains the SH coefficients of the outgoing
radiance function, the actual outgoing radiance is therefore the value of the SH
approximation in the viewing direction
v :
n 2
i = 1 e p , i y i ( v ) .
In Equation (10.10), the SH approximation is in local surface coordinates at p ,so
the viewing vector
r p (
v
)=
v also has to be in local surface coordinates at p .
A principal benefit of the exit transfer matrix is that it represents all the render-
ing steps in a single matrix, including the effects of shadows, and interreflections
(if they are included). At render time, the rendering computation reduces to a
matrix-vector product and vector dot products. Although the sizes of the vectors
and matrices are much larger than the 3D or 4D vectors and matrices normally
used in computer graphics, the computation is still quite fast, and enables real-
time computation on contemporary hardware. The exit transfer matrix depends
only on the geometry; changing the incident light vector does not affect the ma-
trix. In fact, if the BRDF matrix is independent of the transfer matrix (which is
not the case if interreflection is included) then the surface reflectance can be re-
computed dynamically at render time as well. Another very useful property of
the approach is that the outgoing radiance is represented as a smooth function,
approximated by an expansion in spherical harmonics. The outgoing radiance
 
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