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Response matrix
Variation in incoming and
outgoing directions
10 3
90×90×90× 3
φ
d
Color
θ
h
θ
d
PCA
Basis vector
w
1
w
2
w
45
10 3
+
+
10 3
10 3
Basis vectors
Figure 8.18
An approximate basis of the BRDF space is constructed using principal component analy-
sis (PCA) to reduce the basis size from 103 to 45.
eigenvectors correspond to the largest eigenvalues. The authors found that 45
basis vectors (i.e., representative directions) were needed to accurately represent
all their measurements (
Figure 8.18
).
The PCA approximation resulting in 45 basis functions is certainly useful,
as it reduces the millions of samples to only 45 coefficients. However, that is
a lot of parameters for a BRDF model, and each evaluation involves computing
a linear combination of 45 vectors, each of which has 103 elements. Doing this
for each BRDF evaluation gets expensive computationally. The authors point
out a more serious theoretical concern: the subspace spanned by these basis vec-
tors includes all the measured BRDFs, it also includes some BRDFs that are not
physically plausible. Consequently, the actual useful BRDF subspace is a non-
linear subspace. Geometrically this means the subspace is a high-dimensional
curved surface (properly, a
manifold
) in the 45-dimensional BRDF space. Con-
structing and representing this is no easy task. The technique described in the
paper uses Gaussian functions to approximate the subspace; i.e., it represents a
general BRDF as a collection of Gaussians. The method employed by the au-
