Image Processing Reference
In-Depth Information
ξ
=log(
x
2
+
y
2
)=constant (11.20)
η
=tan
−
1
(
x, y
)=constant
and
∂
∂ξ
∂
as invariants. Because
D
ξ
=
∂η
as a rotator, the di-
rection given by
θ
min
represents the amount of scaling versus rotation leaving
f
un-
changed(inpractice,leastchanged).The
e
(
θ
min
) containsinformationastowhether
θ
min
, which represents the symmetry “direction”, is or is not significant.
acts as a zoomer and
D
η
=
The basis of the recognition is that
e
(
θ
) is small for some
θ
. We define
k
(
θ
)=(cos
θ,
sin
θ
)
T
(11.21)
and we rewrite the error function in quadratic form:
e
(
θ
)=
k
(
θ
)
T
∂
∂ξ
∂f
∂η
∂f
k
(
θ
)
dξdη
=
k
(
θ
)
T
S
k
(
θ
)
∂ξ
,
∂f
(11.22)
∂η
where
S
is a matrix defined as follows:
Definition 11.2.
The matrix
S
=
∂
∂ξ
∂
∂ξ
dξdη,
∂
∂ξ
∂
∂η
dξdη
∂
∂ξ
∂
∂η
dξdη,
∂
∂η
∂
∂η
dξdη
(11.23)
is called the generalized structure tensor (GST).
Applying Parseval-Plancherel identity to the elements of matrix
S
gives
S
=
μ
20
(
|
2
)
,μ
11
(
|
2
)
|
F
(
ω
ξ
,ω
η
)
|
F
(
ω
ξ
,ω
η
)
(11.24)
|
2
)
,μ
02
(
|
2
)
μ
11
(
|
F
(
ω
ξ
,ω
η
)
|
F
(
ω
ξ
,ω
η
)
Hence, elements of
S
correspond to the second-order moments of the power spec-
trum (
|
2
), which is the Fourier transform of image
f
(
ξ, η
) taken in the
ξη
coordinates. It should be emphasized that the Fourier transform of
f
w.r.t. the (
ξ, η
)
coordinates is not the same as when taken w.r.t. the (
xy
) coordinates.
We know from Chap. 10 that the second-order moments can also be represented
by the second-order complex moments,
I
pq
with
p
+
q
=2, see Eq. 10.36, so that
we can conveniently use another matrix as defined next.
|
F
(
ω
ξ
,ω
η
)
Definition 11.3.
The complex representation of the generalized structure tensor is
given as :
I
11
(
Z
=
1
2
|
2
)
|
2
)
|
F
(
ξ, η
)
−
iI
20
(
|
F
(
ξ, η
)
(11.25)
|
2
)
∗
|
2
)
iI
20
(
|
F
(
ξ, η
)
I
11
(
|
F
(
ξ, η
)
where
I
20
=
S
(1
,
1)
S
(2
,
2) +
i
2
S
(1
,
2)
I
11
=
S
(1
,
1) +
S
(2
,
2)
−
(11.26)
with
S
(
k, l
)
being the elements of
S
according to Eq. (11.23).
