Graphics Reference
In-Depth Information
Cluster
A
basis = {
a
1
,
a
2
,
…
a
3
}
→→
→
Global
Local
Cluster centers
a
0
Response vector
K
at point
P
P
Projection
Cluster
B
basis = {
b
1
,
b
2
,
…
b
3
}
→→
→
b
0
c
0
Cluster
C
basis = {
c
1
,
c
2
,
…
c
3
}
→→
→
Plane
a
0
+
w
1
a
1
+
w
2
a
2
+
…
+
w
n
a
n
→
→
→
→
→
b
0
+
w'
1
b
1
+
w'
2
b
2
+
…
+
w'
n
b
n
→
→
→
→
→
c
0
+
w''
1
c
1
+
w''
2
c
2
+
…
+
w''
n
c
n
→ → →
Assign point
P
to cluster
that gives minimum error
A small region on sphere
can be approximated by
a plane (linear)
Sphere is nonlinear
Examine
errors
(a)
(b)
Figure 9.13
Local PCA (Leen's iterative PCA).
Applying local PCA to the response matrix defined above can improve the
quality of the compression and therefore the interpolated images. Each column
contains the cell image for a particular view direction, and each row contains
the change in appearance of each point in the cell with changing view direction.
Rows with less variation among the values correspond to points that appear sim-
ilar from any viewpoint; clustering these rows together leaves rows for points
whose appearance differs substantially. These points are those that fall in and
out of shadow and exhibit specular reflection, and the corresponding rows can
be clustered together. PCA can be applied to the clusters separately, as rows of
a separate matrix, which results in a different set of basis eigenvectors for each
cluster. Fewer basis vectors are necessary for the cluster having the least variation
in pixel appearance.
The key to effective local PCA is the method of clustering. The
k
-means
PCA algorithm (Section 8.1) is one way to perform the clustering. In order to
perform clustering that most accurately approximates all the response vectors in
the region, the PCA computation is performed simultaneously with the clustering
calculation, which is more efficient than performing PCA calculation after the
clustering is done. This simultaneous approach is known as
Leen's iterative PCA
algorithm. Its process is as follows (see also
Figure 9.13
)
.
1. Cluster centers are initialized, and the entire data set is partitioned into
N
initial regions.
2. Basic PCA is employed in each cluster to compute basis vectors for the
cluster.
