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on which principal component analysis can be applied. Here
N
is the total number
of states, and
M
is the number of points in the deformation model. Each displace-
ment vector
u
i
is a 3D column vector, so the matrix
A
u
is actually a 3
M
×
N
matrix.
Row
i
ofthedatamatrix
A
u
corresponds to the set of displacements of a par-
ticular point
i
. At nearby points, the corresponding matrix rows are likely to be
similar, and therefore redundant. PCA is applied to select a representative basis
for the matrix rows. This is done using the singular value decomposition (see
Section 9.3): the matrix
A
u
is decomposed into the product of three matrices
U
u
Σ
u
V
u
.
A
u
=
Σ
u
represent the importance of
the corresponding row in the matrix
A
u
, and each row of this matrix corresponds
to a vertex. Therefore, important vertices in terms of vertices representing the
characteristics of deformation can be chosen by setting all components to zero
except the largest values of the diagonal elements. Intuitively this SVD reduction
amounts to extracting a small number of points that represent the characteristics
of the deformation, and this suggests that the dynamic precomputation or the run-
time simulation should be performed only on these representative points.
The selection of the reduced coordinates from the SVD thus amounts to se-
lecting the
k
largest values from the diagonal matrix
The sizes of the diagonal elements in the matrix
and setting the remaining
diagonal elements to zero
(
Figure 10.23
)
. Denoting the reduced diagonal matrix
Σ
Matrix
Q
u
Matrix
V
u
T
Matrix
U
u
Matrix
S
u
σ
1
σ
2
0
σ
s
0
σ
M
u
1
u
2
u
i
u
N
σ
1
σ
2
σ
k
0
0
0
0
Figure 10.23
Dimensional reduction of state space.
