Image Processing Reference
In-Depth Information
in its last
M
K
diagonal elements. By contrast, if
O
were a sleeping matrix, the
quadratic matrix
OO
T
would not be singular in general since no zero values would
be automatically generated in the diagonal elements of
ΣΣ
T
. Evidently, the number
of such null eigenvalues is a consequence of whether or not the equation
O
T
ψ
=
0
is overdetermined, which is paraphrased by the term sleeping
O
. Similarly, one can
diagonalize
O
T
O
as
−
O
T
O
=
VΣ
T
UUΣV
T
=
VΣ
T
ΣV
T
=
V ΛV
T
(15.43)
to the effect that the eigenvectors of
O
T
O
are to be found in the columns of
V
,
whereas its eigenvalues are in the diagonal elements
Σ
T
Σ
. The two diagonal matri-
ces
Σ
T
Σ
and
Σ
Σ
T
contain the same nonzero diagonal elements.
There exist effective software implementations of SVD, and even some where
only the first
K
columns of
U
are computed in case one has
K
M
. This is
valuable in image analysis, where one often fits hyperplanes with low dimensions to
data in high-dimensional (Hilbert) vector spaces. The SVD is a ubiquous tool that
can be generously applied to numerous vision problems.
