Image Processing Reference
In-Depth Information
14.5 Discrete Group Direction Tensor by Tensor Sampling
|
2
to
estimate the group direction. These computations can also be carried out in the spatial
domain by direct tensor sampling, in analogy with the discussion in Sect. 10.11. We
can write the integral of the complex moment
I
n,
0
as follows, if we assume that the
spectrum is represented in its Cartesian coordinates,
F
(
ω
x
,ω
y
), and
n
is nonzero
and even:
Theorem 14.1 suggests computing complex moments of the power spectrum
|
F
I
n,
0
=
(
ω
x
+
iω
y
)
n
(
ω
x
−
iω
y
)
0
2
dω
x
dω
y
|
F
(
ω
x
,ω
y
)
|
E
2
=
iω
y
)
n
2
F
]
∗
[(
ω
x
+
iω
y
)
n
[(
ω
x
−
2
F
(
ω
x
,ω
y
)]
dω
x
dω
y
(14.10)
E
2
By use of theorem 7.2, due to Parseval-Plancherel, the complex moments integral
can be computed in the spatial domain:
2
π
)
2
I
n,
0
=(
1
iD
y
)
n
2
f
(
x, y
)]
∗
[(
D
x
+
iD
y
)
n
[(
D
x
−
2
f
(
x, y
)]d
x
dy
,
E
2
2
π
)
2
=(
1
[(
D
x
+
iD
y
)
n
2
f
(
x, y
)]
2
d
x
d
y,
(14.11)
E
2
where
f
is the inverse FT of
F
. Further, we can identify (
D
x
+
iD
y
)
n
2
f
as a
symmetry
derivative
of
f
, introduced in Section 11.9. Then, assuming that
f
is band-limited,
there exists an interpolation function (
μ
1
below) by which we can reconstruct
f
via
its discrete samples
f
(
r
)=
j
f
j
μ
1
(
r
−
r
j
)
(14.12)
and
r
j
=(
x
j
,y
j
)
T
. However, the function (
D
x
+
iD
y
)
n
where
r
=(
x, y
)
T
f
is
2
band-limited too, and can be reconstructed from the samples of
f
:
2
f
(
r
)=
j
(
D
x
+
iD
y
)
n
f
j
(
D
x
+
iD
y
)
n
2
μ
1
(
r
−
r
j
)
(14.13)
by linear filtering. Evidently, this function can also be sampled without loss of infor-
mation on the same grid as the samples
f
j
are defined, so that
2
f
(
r
k
)=
j
(
D
x
+
iD
y
)
n
f
j
(
D
x
+
iD
y
)
n
2
μ
1
(
r
k
−
r
j
)
(14.14)
Assuming that
μ
1
is a Gaussian with the variance
σ
p
, the discrete symmetry deriva-
tives of
f
can be computed by an ordinary discrete linear filtering with the filter
exp
r
j
2
2
σ
p
1
2
πσ
p
r
k
−
n
2
,σ
p
}
(
r
k
−
r
j
)=(
D
x
+
iD
y
)
n
Γ
{
(14.15)
2
