Image Processing Reference
In-Depth Information
λ
2
)
u
1
u
T
λ
3
)(
u
1
u
T
+
u
2
u
T
S
=(
λ
1
−
+(
λ
2
−
)+
1
1
2
+
λ
3
(
u
1
u
T
+
u
2
u
T
+
u
3
u
T
)
(12.41)
1
2
3
(12.42)
This rearrangement allows a direct interpretation of the eigenvalues in terms of our
three fundamental cases. The line case is the dominating structure when 0
λ
2
λ
1
, i.e., the first term is largest. The plane case is the dominating structure when
0
≈
λ
2
, i.e., the second term is largest. The balanced direction case is the
dominating structure when 0
≈
λ
3
λ
3
≈
λ
2
≈
λ
1
, i.e., the last term is largest. Accord-
ingly,
λ
1
=
C
L
=
λ
1
− λ
2
λ
2
=
C
P
=
λ
2
− λ
3
λ
3
=
C
B
=
λ
3
U
1
=
u
1
u
T
1
U
2
=
u
1
u
T
+
u
2
u
T
with
(12.43)
1
2
U
3
=
u
1
u
T
+
u
2
u
T
+
u
3
u
T
=
I
1
2
3
where the three coordinates,
C
L
,
C
P
,
C
B
, can be used as a certainty or saliency
for
S
representing a line, a plane, and a balanced directions structure. That
u
1
u
T
+
1
u
2
u
T
+
u
3
u
T
equals the identity matrix
I
in the last row follows from the fact that
I
can be expanded in the orthonormal basis
U
j
=
u
j
u
j
2
3
by using the scalar product,
Eq. (3.45):
=
k,l
l
)=
k
2
=1
U
j
,
I
u
j
(
k
)
u
j
(
l
)
δ
(
k
−
|
u
j
(
k
)
|
(12.44)
Following a similar procedure, these results can be generalized to the spectral
decomposition of the an
N
×
N
structure tensor as follows:
Lemma 12.5 (Structure tensor decomposition).
Assuming that
λ
j
,
u
j
constitute
the eigenvalue and eigenvector pairs of the symmetric positive definite matrix
S
,the
spectral decomposition of
S
yields
S
=
j
=
j
λ
j
u
j
u
j
λ
j
U
j
(12.45)
where
λ
1
=
λ
1
−
U
1
=
u
1
u
T
λ
2
,
λ
2
=
λ
2
− λ
3
,
···
λ
N
,
1
U
2
=
u
1
u
T
+
u
2
u
T
,
1
2
with
···
U
N−
1
=
u
1
u
T
+
u
2
u
T
···
u
N−
1
u
N−
1
,
=
λ
N−
1
−
λ
N
,
+
1
2
···
u
N
u
N
=
I
.
(12.46)
Fo r
j<N
,
λ
j
, represents the certainty for
S
to represent a
j
-dimensional hyper-
plane, whereas
λ
N
represents the certainty for
S
to represent a perfectly balanced
structure.
λ
N
U
N
=
u
1
u
T
+
u
2
u
T
=
λ
N
,
+
1
2
