Radiance Transfer - The Magic of Computer Graphics - page 349

vector. The principal difference is that BRDF matrices include only the surface
reflection according to the BRDF, while general transfer matrices can include lo-
cal geometry effects, including shadowing, interreflection, and even subsurface
scattering.

10.2.3 Efficient PRT

Exit transfer matrices. Incorporating a BRDF matrix into the precomputed
radiance transfer method is even simpler than it is for BRDF textures, because
the BRDF matrix is itself just another transfer matrix. There is, however, the
coordinate system compatibility problem: normally the incident light vector is
expanded in a global coordinate system, while the BRDF matrix is constructed in
local surface coordinates. Whether a transfer matrix applies to a light vector in lo-
cal or global coordinates depends on the particular assumptions (or the particular
implementation). In any case, the transformation from global to local coordi-
nates is a rotation, and as described above, the corresponding transformation of
SH coefficients is itself a linear transformation represented by a high-dimensional
“rotation” matrix. If the surface is fixed and rigid, this matrix can be precomputed
for each surface point. If the surface is allowed to rotate or deform after the pre-
computation phase, then the rotation matrix must be recomputed. As described
earlier, this is not a fast computation, but it is not very complicated either. The
cost of this computation depends on the size of the light vectors, i.e., the order of
the SH approximations.
Assuming that the transfer matrix accounts for self-shadowing only (i.e., there
is no interreflection involved) then the transfer matrix is independent of the BRDF
matrix. Assuming also that the transfer matrix is constructed in a global coordi-
nate system, then the outgoing radiance light vector at a surface point p can be
computed from the source light vector in three steps. First, the transfer matrix
T p is applied to the source light vector , which produces an incident light vec-

tor accounting for self-shadowing. This light vector is then transformed to the
local surface coordinates by a rotation matrix R p , and finally the BRDF matrix
B is applied to produce the outgoing light vector. The composition of all three
transformations—the self-shadowing transfer, the rotation, and the reflection—
can be combined into a single matrix by the ordinary matrix product

BR p T p = M p . e p = (10.10)

The combined matrix M p is known as the exit transfer matrix (see also Fig-
ure 10.9 ) .
The precise formulation of an exit transfer matrix depends on the coordinate
systems of the light vectors and transfer matrices, which in turn depends on the