Image Processing Reference
In-Depth Information
plenty of them by which the image can be described as
g
(
k
T
r
). Representing an im-
portant “degenerate” case, the energy in the Fourier domain will then be distributed
in such a way that there is a 2D plane (instead of an axis) containing an infinite
number of axes that give the (same) least square error. This plane is defined by the
subspace generated by linear combinations of the eigenvectors belonging to the least
eigenvalue, which has the multiplicity 2. The dimension of eigenvector subspaces is
always equal to the multiplicity of the eigenvalue to which the eigenvectors belong,
whereas the eigenvectors belonging to different eigenvalues are always orthogonal.
This is due to the fact that
J
is positive semidefinite and symmetric by definition,
Eqs. (12.5) and (12.7). Accordingly, such a 3
3 matrix always has three orthogo-
nal eigenvectors, where those possibly belonging to the same eigenvalue will not be
unique. This conclusion can be naturally extended to
N
dimensions.
×
Lemma 12.2.
An
inertia tensor
J
in
N
-D has
N
orthogonal eigenvectors with the
reservation that those belonging to the same eigenvalue, with a multiplicity 2 or
larger, are not unique.
12.2 The Direction of Lines and the Structure Tensor
Here, we will discuss what the degenerate solutions (the nonunique eigenvectors) of
the spectral line-fitting problem corresponds to in the spatial domain. We investigate
the issue first for 3D spectra. Assuming that the multiplicity of the least eigenvalue
is 2, the energy will be concentrated to a 2D plane, spanned by the eigenvectors of
the least eigenvalue. What sense will this plane make when the intention was to fit a
line through the origin and what we found is a plane (instead of a line)?
Suppose that we were intending to fit the (complex) spectral function
F
a plane
under the condition that it had to pass through the origin. Representing the normal of
the plane by
k
, we would then minimize
e
P
(
k
)=
E
3
(
d
P
(
,
k
))
2
2
dE
3
min
k
=1
ω
|
F
(
ω
)
|
(12.10)
where the
P
reminds us that we are attempting to fit a plane and
d
P
is the Euclidean
distance between the point
ω
, and the plane with the normal
k
in the spectrum,
(
d
P
(
,
k
))
2
=(
T
k
)
2
=
k
T
T
k
ω
ω
ωω
(12.11)
Compared to line-fitting, the main difference is in the distance function
d
P
(
ω
,
k
),
on the normal vector
k
, the shortest distance
to the plane. Thus Eq. (12.10) reduces to minimizing
which is now a projection of the point
ω
e
P
(
k
)=
k
T
Sk
(12.12)
where
