Biomedical Engineering Reference
In-Depth Information
Figure 19.
The level set for each image after alignment. The last image is the average level
set, overlapped with the average shape after training. See attached CD for color version.
where
U
is an
N
=
N
n
matrixwhose columns represent the orthogonal modes of
variation in the shape, and Σ is an
n
×
×
n
diagonal matrix whose diagonal elements
represent the corresponding nonzero eigenvalues. The
N
elements of the
i
th
column of
U
, denoted by
U
i
, are arranged back into the structure of the
N
=
N
1
×
N
2
rectangular image grid (by undoing the earlier lexicographical concatenation of
the grid columns) to yield Φ
i
, the
i
th principal mode or eigenshape. Based on this
approach, a maximum of
n
different eigenshapes
are generated.
In most cases, the dimension of the matrix
n
SS
T
is large so that the calculation of
the eigenvectors and eigenvalues of this matrix is computationally expensive. In
practice, the eigenvectors and eigenvalues of
n
SS
T
can be efficiently computed
from a much smaller
n
{
Φ
1
,
Φ
2
,...,
Φ
n
}
n
matrix
W
given by
n
S
T
S
. It is straightforward to
show that if
d
is an eigenvector of
W
with corresponding eigenvalue
λ
, then
Sd
is
an eigenvector of
n
SS
T
with eigenvalue
λ
.
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