Image Processing Reference
In-Depth Information
S
=
=
S
(1
,
1)
S
(1
,
2)
S
(2
,
1)
S
(2
,
2)
,
S
(
i, j
)=
T
|
2
d
|
2
d
ωω
|
F
ω
with
ω
i
ω
j
|
F
(
ω
ω
(10.25)
)
Definition 10.2.
The matrix
S
in Eq. (10.25), which consists of the second-order
moments of the power spectrum,
|
2
, is called the
structure tensor
of the image
f
.
|
F
The matrix
J
is the
inertia tensor
of the power spectrum using a term of mechan-
ics. The matrix
S
is also called the scatter tensor of the power spectrum in statistics.
The structure tensor can be readily obtained from
J
, and vice versa via Eq. (10.24).
There is another related tensor called the
covariance tensor
or the covariance matrix
in statistics,
C
=
S
m
T
, where
m
=
ω
|
|
2
d
−
m
·
F
ω
. However, for real images
m
=0, since
is even when
f
is real. Because of the tight relationship between the
notions inertia, scatter, and covariance, they are used in an interchangeable manner
in many contexts. Since different notions of the structure tensor coexist, the follow-
ing lemma, which establishes the equivalence of
J
and
S
(and of
C
), is useful to
remember.
|
F
|
Lemma 10.2.
With eigenvalue, eigenvector pairs of
J
being
{λ
1
,
u
1
}
and
{λ
2
,
u
2
}
,
λ
1
,
u
1
}
λ
2
,
u
2
}
and those of
S
being
{
and
{
, we have
λ
1
,
u
1
}
λ
2
,
u
2
}
{
=
{
λ
1
,
u
2
}
,
and
{
=
{
λ
2
,
u
1
}
.
(10.26)
The eigenvector with a certain eigenvalue in the first matrix is an eigenvector with
the
other
eigenvalue in the second matrix. The lemma can be proven by utilizing Eq.
(10.24) and operating with
J
on
u
i
, which is assumed to be an eigenvector of
S
:
Ju
i
=(
I
S
)
u
i
=(
λ
1
+
λ
2
)
u
i
−
λ
i
u
i
·
Trace(
S
)
−
i
=1
,
2
(10.27)
The error minimization problem formulated in Eq. (10.17) is reduced to a min-
imization of a quadratic form,
k
T
Jk
with the matrix
J
given by Eq. (10.24). This
is in turn minimized by choosing
k
as the least eigenvector of the inertia matrix,
J
[231]. All eigenvalues of
J
are real and nonnegative because the error expres-
sion Eq. (10.17) is real and nonnegative. Calling the eigenvalue and eigenvector
pairs of
J
, the minimum of
e
(
k
) will occur at
e
(
k
min
)=
λ
min
. In other words, the matrix
J
, or equivalently
S
, contains suffi-
cient information to allow the computation of the optimal
k
in the TLS error sense.
We will discuss the motivation behind this choice of error in some detail in Sect.
10.10.
The matrix
S
is defined in the frequency domain, which is inconvenient, partic-
ularly if
S
must be estimated numerous times. For example, when computing the
direction for all local patches of an image, we would need to perform numerous
Fourier transformations if we attempt to directly estimate the structure tensor from
its definition. We can, however, eliminate the need for a Fourier transformation by
utilizing (Parseval-Plancherel) theorem 7.2 which states that the scalar products are
{
λ
min
,
k
min
}
and
{
λ
max
,
k
max
}
