Radiance Transfer - The Magic of Computer Graphics - page 360

, l is the vectorized set of
illumination samples, and A is a matrix consisting of elements a ij .
The matrix A in Equation (10.13) represents radiance transfer, but in a funda-
mentally different way than the transfer matrices employed in the PRT method as
defined in Equation (10.7). One difference is that the light vectors

where r is the vectorized set of reflectances R ( p i , v )

contain the
SH coefficients, and the transfer matrix elements t ij represent how the coefficients
change. The resulting light vector r is a set of SH coefficients for the expansion of

the outgoing radiance in all directions at the sample point. In contrast, the vector
l in Equation (10.13) represents point samples in the environment map, and the
resulting vector

r is radiance to the viewpoint at all sample points on the object
in the specific view direction. Another difference concerns what the matrices rep-
resent: while there is a separate PRT transfer matrix for each sample point on
the object, the matrix A represents the transfer over all the sample points on the
object.
The matrix product formulation of Equation (10.13) decouples the lighting
samples l from the visibility and BRDF effects, which are stored in the matrix A .
Relighting the scene therefore amounts to simply multiplying A , which is constant
for the object and viewpoint, by a different environment map encoded into the
elements of l . However, the computational cost of this operation can be extreme.
In the basic case, there is one element l j for each pixel in the environment map,
and one element R

for each pixel in the image. The matrix A is therefore a
huge matrix, having something like a trillion elements—the storage requirement
alone for the matrix A is prohibitive. The PRT transfer matrices are minute by
comparison: each has at most 625 elements (although there is one such matrix for
each surface sample point).
One way of looking at the matrix A is to consider each directional sample

( p i , v )

s j as
coming from a separate directional light source, as illustrated in Figure 10.19. If
the scene is illuminated only by the source corresponding to sample s j , then only
column j of A is relevant in the computation of the outgoing radiance. In other
words, the columns of A correspond to the effects of the individual illumination
samples. On the other hand, each row of A contributes only to the computation of
outgoing sample r i = , which corresponds to a single surface sample point.

The summation involved in a row multiplication corresponds to Equation (10.12),
which is the reflection integral approximation. The elements in a matrix row
therefore correspond to all the samples points over the sphere; each element is the
visibility of the associated directional source times the cosine-weighted BRDF.
(The matrix A can be extended to include other effects, such as interreflection
and subsurface scattering.) The rows of A therefore correspond, loosely, to the
transfer vectors in the original PRT paper. The difference is that the PRT vectors
assume diffuse reflectance so there is only one radiance value produced, which

R ( p i )