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3. The clusters are reconstructed according to the approximation error in the
sets of basis vectors. That is, each response vector in the entire data set is
projected onto the set of basis vectors for each cluster. The response vector
is moved, if necessary, to the cluster for which the approximation has the
smallest error.
4. The existing basis vectors are discarded, and PCA is applied to construct
a new set of basis vectors for each cluster. A representative cluster mean
is computed from the mean of the updated PCA basis vectors, and this is
subtracted from each cluster response vector before PCA is applied. This
way, the vectors contain the “deviation from the mean” values as described
earlier.
5. Steps 3 and 4 are repeated until the approximation error for every response
vector is sufficiently small.
This iterative PCA algorithm may seem computationally expensive, but the
clustering method is highly accurate. As a result, only a few basis vectors are nec-
essary for each cluster, which makes the computation of the approximation error
very fast. As a result, memory consumption and computation time in the whole
process is significantly reduced. Another advantage of this kind of clustering in
general is that once the basis vectors are constructed, the synthesis (rendering)
step is very fast, and is also independent of the other clusters.
9.3.5 Local PCA and BTF Rendering
In the image-capture method described in the eigen-textures paper, the object is
placed on a turntable and the lighting and viewing direction remain fixed. The
captured image set is much smaller than that of a BTF data set, as it does not
include relative variations in the lighting/viewing directions. However, the tech-
nique itself can be applied to more general sets of captured images. An extension
of the eigen-texture approach to full BTF data sets was presented in the paper
“Compression and Real-Time Rendering of Measured BTFs using Local PCA”
by Gero Muller, Jan Meseth, and Reinhard Klein [Muller et al. 03].
The basic idea is similar to that of the eigen-texture method. In that method,
the response vectors come from the captured image “cells.” In the technique of
Gero Muller et al., all the pixels corresponding to a particular small area on the
surface for all the images are collected from BTF data and arranged into a sin-
gle response vector. Each response vector thus represents the response to the
change in lighting and viewing directions at each sample point on the object. In
this sense, each response vector represents the total characteristics of reflection
