Graphics Reference
In-Depth Information
e
1,1
e
2,1
e
k,1
X
1, 1
X
2,1
X
M,1
e
1,2
e
2, 2
e
k,2
X
1,2
X
2, 2
X
M,2
e
1,3N
e
2,3N
e
k, 3N
X
1, 3N
X
2, 3N
X
M, 3N
k
eigen-textures
M
k
A sequence of
cell images
Figure 9.11
Cell images (left) are converted to a matrix
M
where each column consists of the pixel
values in a cell image. The selection of basis eigenvectors reduces the
M
columns to
k
representative eigenvectors (middle), which correspond to cell images known as eigen-
textures (right). (After images courtesy of Ko Nishino.)
interpolating only the linear coefficients. Not only does this reduce the amount of
computation, it also provides a better interpolation, as it emphasizes the strongest
details.
To make the eigen-texture process more concrete, suppose each image has
N
pixels, and there are
M
rotation angles, or “poses.” Each cell image has 3
N
values (three color channels per pixel), which results in a 3
N
×
1 column vector
X
m
for the image cell corresponding to pose
i
,andan
M
3
N
matrix
X
for all
poses. The PCA is actually applied to the (symmetric) matrix
Q
×
XX
T
,anda
=
,
,...,
set of representative basis eigenvectors
e
k
are selected corresponding
to the
k
largest eigenvalues. The choice of
k
depends on the particular data set,
but as mentioned above, it is much less than 3
N
. Because the basis vectors are
orthogonal, the projection of each remaining eigenvector
e
1
e
2
v
m
onto the basis set is
just a dot product,
a
m
,
i
=
e
i
. Each basis vector still has 3
M
entries, but only
k
such coefficients are needed to represent each eigenvector; this is a significant
reduction from the 3
N
elements in the original (unprojected) vectors. In total, the
storage requirements are reduced from 3
MN
to
kM
v
·
3
kN
. Figure 9.11 illustrates
the basic reduction, and a particular set of eigen-textures.
As noted previously, synthesis of an image from a new rotation angle is done
directly from the basis eigenvectors and the projection coefficients. For simplic-
ity in the interpolation, the basis eigenvectors can themselves be represented by
k
trivial projection coefficients. Figure 9.12 illustrates the concept of a linear
approximation of a synthesized cell image from eigen-textures.
The eigen-texture method uses PCA effectively; it selects basis vectors from
the eigenvectors of the response matrix based on the value of the eigenvalues,
starting with the largest. The largest PCA eigenvalue usually corresponds to the
phenomenon that is most numerically significant. Unfortunately, it is not always
the most visually significant. For example, diffuse reflection usually has more
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