Graphics Reference
In-Depth Information
Principal component analysis (PCA) was described in Chapter 9 for the pur-
pose of selecting representative texture samples in bidirectional texture function
representations. A similar approach can be used for exit transfer matrices. PCA
is applied to a matrix, the columns of which are the “signals” in the language
of Section 9.3.6. The matrix for PCA is therefore constructed by flattening the
transfer matrix of each sample point into a column vector
(
Figure 10.10
)
, and
the column vectors are then arranged into the columns of the data matrix. If there
are
N
sample points, and each transfer matrix has dimension 25
×
25, the resulting
data matrix has 625 rows and
N
columns (
Figure 10.11
)
.
Basis vectors are extracted by applying PCA to the data matrix. The PCA
is actually applied to the covariance matrix of this data matrix: the data matrix
has each element replaced by its “deviation from the mean” (i.e., the average
value of the matrix is subtracted from each element) and then is multiplied by
its transpose. This results in a 625
625 symmetric matrix, which typically has
significantly fewer columns than the original 625
×
N
data matrix. PCA is applied
to the symmetric matrix, and
M
basis vectors are extracted. These correspond to
(flattened) exit transfer matrices at surface points that best represent the radiance
transfer and reflection. The remaining vectors (matrices) are constructed from
a linear combination of these basis vectors; the coefficients (weights) are found
×
625
N
(Sample Points)
625
625
625
625
Signal
625
625
625
625
Figure 10.11
Data matrix construction.
