Graphics Reference
In-Depth Information
N
basis vectors
+
p
w
w
625
…
w
625
…
w
625=
625 +
1
p
625 +
2
p
•
+
k
p
•
625
•
˜
w
w
w
w
N
p
1
p
M
1
2
p
M
2
p
M
k
M
N
M
p
M
0
M
0
M
1
,
M
2
,
. . . , M
N
w
1
,
w
2
,
. . . ,
w
N
: Center vector
: Basis vector
: Coefficient
Figure 10.12
Approximation using basis vectors.
by projecting the corresponding vectors onto the basis vectors
(
Figure 10.12
)
.
This reduces each transfer matrix, which ordinarily has 625 elements, to just
M
coefficients.
The data compression using PCA can be significant, especially if the surface
is densely sampled. But it actually increases the computational load, because
the exit transfer matrix at each sample point has to be reconstructed from the
basis elements. Fortunately, there is a better way of doing the interpolation. Be-
cause all the operations are linear, the basis function coefficients serve as the
coefficients to interpolate the
computed
outgoing radiance values. In other words,
the outgoing radiance can be computed at each representative sample point, and
the interpolated outgoing radiance at all the other sample points comes from a
linear combination of these values.
Data compression with local PCA.
The vectors extracted from principal
component analysis span a linear subspace; the idea is that this subspace suffi-
ciently approximates the span of the columns (the column space) of the original
matrix. As in the other applications of PCA, the column space contains elements
that are not physically meaningful. The actual set of valid exit transfer matrices
is a nonlinear subspace that may not be well approximated by a linear subspace.
The PCA approach is suboptimal in this case, as it creates a single linear approx-
imation to the true nonlinear subspace. The method of local PCA (also called
clustered PCA) described in Chapter 9 works by partitioning the data into clus-
ters small enough to be regarded as locally linear subspaces. PCA is then applied
to each cluster. If the clusters are properly chosen, a very few basis elements
are needed to adequately approximate all the elements in each cluster. In 2003,
a year after the original PRT paper was published, Peter-Pike Sloan, Jesse Hall,
