Image Processing Reference
In-Depth Information
Lemma 11.3 (Energy conservation).
The sum of the maximum and the minimum
error is independent of the coordinate system chosen for symmetry investigation of
the image
f
:
I
11
=
e
(
θ
max
)+
e
(
θ
min
)=
|
ξ,η
(
f
)(
ξ, η
)
|
dξdη
=
dxdy
=
∂f
∂x
2
+
∂f
∂y
2
dxdy.
|
(
f
)(
x, y
)
|
(11.47)
This suggests the interesting property
5
that
I
11
can be computed even without knowl-
edge of
ξ
. This is because
I
11
, which is the upper bound of
I
11
attains the upper bound with equality if and only if the image has linear symmetry
w.r.t.
ξ, η
curve family, is independent of the
ξ, η
. Accordingly, it follows that the
upper bound of
|
I
20
|
such that
|
I
20
|≤
is the same for all harmonic CTs,
I
11
computed for
f
w.r.t.
the (ordinary Cartesian) linear symmetry. An image cannot be both linear symmetric
w.r.t. (ordinary Cartesian) lines obtained as the linear combinations of
ξ
=
x
and
η
=
y
and w.r.t. another harmonic curve family at the same time. This means that it
is possible to construct independent shape properties of
f
by measuring
I
20
s w.r.t.
different CTs,
ξ, η
.
In summary, the elements of the GST can be found by filtering
|
I
20
|
(
f
) as stated in
the following theorem [18]:
Theorem 11.1 (Generalized structure tensor).
The structure tensor theorem holds
in harmonic coordinates. In particular, the second-order complex moments determin-
ing the minimum inertia axis of the power spectrum,
|
F
(
ω
ξ
,ω
η
)
|
2
, can be obtained
in the (Cartesian) spatial domain as:
I
20
=(
λ
max
− λ
min
)
e
i
2
ϕ
min
=
(
ω
ξ
+
iω
η
)
2
|F |
2
dω
ξ
dω
η
(11.48)
=
ξ,η
(
ξ, η
)
dξdη
=
x,y
(
ξ
)
|
x,y
(
ξ
)
|
x,y
(
f
)
dxdy
(11.49)
=
x,y
(
ξ
))
x,y
(
f
)
dxdy
exp(
i
arg
(11.50)
I
11
=
λ
max
+
λ
min
=
(
ω
ξ
+
iω
η
)(
ω
ξ
−
2
dω
ξ
dω
η
iω
η
)
|
F
|
(11.51)
=
dξdη
=
|
ξ,η
(
f
)
|
|
x,y
(
f
)
|
dxdy
(11.52)
The quantities
λ
min
,
ϕ
min
, and
λ
max
are, respectively, the minimum inertia, the di-
rection of the minimum inertia axis, and the maximum inertia of the power spectrum
of the harmonic coordinates,
|
2
.
|
F
(
ω
ξ
,ω
η
)
5
Note that this conclusion is valid for the continuous representation. In the discrete case the
integral still needs a
ξ
-dependent kernel as will be discussed further below.
