Image Processing Reference
In-Depth Information
Theorem 10.2 (Structure tensor II).
The minimum and the maximum inertia as
well as the axis of minimum inertia of the power spectrum are given by
1
4
π
2
λ
min
)
e
i
2
ϕ
min
=
I
20
=(
λ
max
−
(
f
)(
x, y
)
d
x
(10.37)
1
4
π
2
I
11
=
λ
max
+
λ
min
=
|
(
f
)(
x, y
)
|
d
x
(10.38)
with the
infinitesimal linear symmetry tensor
(ILST) defined as
2
x,y
(
f
)(
x, y
)=
∂f
2
∂x
+
i
∂f
(10.39)
∂y
The quantities
λ
min
,
λ
max
, and
ϕ
min
are, respectively, the minimum inertia, the max-
imum inertia, and the axis of the minimum inertia of the power spectrum.
The eigenvalues of the tensor in theorem 10.1 and the
λ
's appearing in this theorem
are identical. Likewise, the direction of the major eigenvector
k
max
and the
ϕ
min
,
of theorem 10.2 coincide. Accordingly, the eigenvector information is encoded ex-
plicitly in an offdiagonal element of
Z
, i.e.,
I
20
whereas the sum and the difference
of the eigenvalues are encoded in the diagonal element as
I
11
and in the offdiagonal
element as
, respectively.
For completeness, we provide the eigenvectors of
Z
as well. Because the ar-
gument angle of
I
20
is twice the direction angle of
k
max
, the direction of latter is
obtained by the direction of the square root of
I
20
. The eigenvectors of
S
are related
to the eigenvectors of
Z
via Eq. (10.33), so that we obtain
k
max
=
γ
(
I
20
,i
I
2
0
)
T
|
I
20
|
(10.40)
k
min
=
γ
(
I
20
,i
I
20
)
T
−
−
(10.41)
where
γ
is a scalar that normalizes the norms of the vectors to 1.
Summary of the complex structure tensor
•
Independence under merging. Averaging the “square” of the complex field
(
D
x
f
+
iD
y
f
)
2
and its magnitude (scalar) field
|
D
x
f
+
iD
y
f
|
2
, automatically
fits an optimal axis to the spectrum,
•
Schwartz inequality. The inequality
|
I
20
|≤
I
11
holds with equality if and only
if the image is linearly symmetric,
•
Rotation-invariance and covariance. If the image is rotated, the absolute values
of the elements of
Z
, i.e.,
as well as
I
11
, will be invariant to the rotation,
while the argument of
I
20
will change linearly with the rotation.
In the next two sections we discuss two simpler tensors that will be used as basis
tensors for d
ecomposing the structure tensor.
2
The symbol is pronounced as “doleth” or “daleth”, which is intended to be a mnemonic
for the fact that it is not an ordinary gradient delivering a vector constisting of derivatives
but is a (complex)
scalar
comprised of derivatives.
|I
20
|
