Image Processing Reference
In-Depth Information
conserved under the Fourier transform. Applying it to Eq. (10.28), the computation
of the matrix elements will be lifted from the Fourier domain to the spatial domain:
S
(
i, j
)=
∂f
∂x
i
1
4
π
2
∂f
∂x
j
d
x
|
2
d
ω
i
ω
j
|
F
(
ω
)
ω
=
i, j
:1
,
2
(10.28)
where
x
1
=
x
,
x
2
=
y
,
d
x
=
dxdy
, and the integral is a double integral over the entire
2D plane. This is rephrased in matrix form,
S
=
(
1
4
π
2
T
|
2
d
T
f
)
d
x
ωω
|
F
ω
=
∇
f
)(
∇
(10.29)
where
f
=
D
x
f, D
y
f
)
T
T
=(
∂f
(
r
)
∂x
,
∂f
(
r
)
∂y
∇
.
(10.30)
We summarize our finding on 2D direction estimation via the following theorem,
where the integrals are double integrals taken over the 2D spatial domain.
Theorem 10.1 (Structure tensor I).
The extremal inertia axes of the power spec-
trum,
|
2
are determined by the eigenvectors of the
structure tensor
:
|
F
(
1
4
π
2
T
f
)
d
x
S
=
∇
f
)(
∇
(10.31)
(
D
x
f
)
2
d
x
(
D
x
f
)(
D
y
f
)
d
x
1
4
π
2
(
D
x
f
)(
D
y
f
)
d
x
(
D
y
f
)
2
d
x
=
(10.32)
The eigenvalues
λ
min
,
λ
max
and the corresponding eigenvectors
k
min
,
k
max
of the
tensor represent the minimum inertia, the maximum inertia, the axis of the maximum
inertia, and the axis of the minimum inertia of the power spectrum, respectively.
We note that
k
min
is the least eigenvector, but it represents the axis of the maximum
inertia. This is because the inertia tensor
J
is tightly related to the scatter tensor
S
according to lemma 10.2. The two tensors share eigenvalues in 2D, although the
correspondence between the eigenvalues and the eigenvectors is reversed.
While the major eigenvector of
S
fits the minimum inertia axis to the power spec-
trum, the image itself does not need to be Fourier transformed according to the the-
orem. The eigenvalue
λ
max
represents the largest inertia or error, which is achieved
with the inertia axis having the direction
k
min
. The worst error is useful too, because
it indicates the scale of the error when judging the size of the smallest error,
λ
min
(the
range problem
). By contrast, the axis of the maximum inertia provides no addi-
tional information, because it is always orthogonal to the minimum inertia axis as a
consequence of the matrix
S
being symmetric and positive semidefinite.
10.4 The Complex Representation of the Structure Tensor
Estimating the structure tensor, and thereby the direction of an image, can be sim-
plified further by utilizing the algebraic properties of the complex z-plane. Multi-
plication and division are well-defined in complex numbers, whereas conventional
