Image Processing Reference
In-Depth Information
12.3 The Decomposition of the Structure Tensor
To study the decomposition in terms of its subspaces, we discuss the structure tensor
in 3D, i.e.,
S
is 3
×
3. According to the
spectral theorem
of positive semidefinite
operators [136], the structure tensor in
N
D can be written as a weighted sum of its
eigenvector subspaces. In 3D this yields,
S
=
λ
1
u
1
u
T
+
λ
2
u
2
u
T
+
λ
3
u
3
u
T
(12.23)
1
2
3
which is also called the
spectral decomposition
. As theorem 12.1 suggests, the di-
mension of the null space of
S
determines if the tensor has “succeeded” to fit a plane,
a line, or none of these to the image. Accordingly, the following three cases can be
distinguished. The analogy to 2D, see Section 10.7, is discernable, but there are now
three basic cases in 3D, instead of 2.
The spectral line: This is the case when
λ
2
equals the least significant eigenvalue:
0
=
λ
3
=
λ
2
<λ
1
(12.24)
Accordingly, the null space and the line will be given in parametric form by
S
ω
=
0
,
when
ω
=
α
u
3
+
β
u
2
,
(12.25)
S
ω
=
0
,
when
ω
=
γ
u
1
,
(12.26)
where
α, β
and
γ
are arbitrary real scalars. Alternatively, the line is given by solving
for
ω
in the underdetermined system of equations:
u
T
3
ω
=0
(12.27)
u
T
2
ω
=0
(12.28)
which is the same as searching for the space perpendicular to the null space.
The spectral plane: This is the case when
0=
λ
3
<λ
2
(12.29)
The null space and the plane will be given in parametric form by
S
ω
=
0
,
when
ω
=
α
u
3
,
(12.30)
S
ω
=
0
,
when
ω
=
β
u
2
+
γ,
u
1
(12.31)
where
α, β
, and
γ
are arbitrary real scalars. The plane can also be obtained by writing
down the condition for
ω
to be perpendicular to the null space:
u
T
3
ω
=0
(12.32)
The balanced directions: This is the case when all three eigenvalues are equal:
0
<λ
3
=
λ
2
=
λ
1
(12.33)
