Image Processing Reference
In-Depth Information
[
Γ
{p
2
,σ
2
}
]=(
i
)
p
1
+
p
2
(
ω
x
+
iω
y
)
p
1
+
p
2
exp
ω
x
+
ω
y
2
[
Γ
{p
1
,σ
1
}
]
F
·F
−
1
σ
1
+
σ
2
(
ω
x
+
ω
y
2
2
π
(
σ
1
+
σ
2
)
σ
1
−
σ
2
)
p
1
+
p
2
−
2
π
(
σ
1
+
σ
2
)
(
i
)
p
1
+
p
2
(
ω
x
+
iω
y
)
p
1
+
p
2
exp
=
−
1
σ
1
+
σ
2
σ
1
−
σ
2
)
p
1
+
p
2
(
−
=2
π
(
σ
1
+
σ
2
)
p
1
+
p
2
Γ
{p
1
+
p
2
,
i
1
σ
1
+
σ
2
}
(
ω
x
,ω
y
)
(11.118)
σ
1
−
σ
2
−
Remembering, Eq. (11.115) we now inverse Fourier transform Eq. (11.118) by using
Eq. (11.101) and obtain:
Γ
{p
1
,σ
1
}
∗
Γ
{p
2
,σ
2
}
=
Γ
{p
1
+
p
2
,σ
1
+
σ
2
}
(11.119)
Proof of lemma 11.6
iD
y
)
ξ
is needed.
10
We write
the coordinates as a complex variable
z
=
x
+
iy
and remember their relationship be-
tween symmetry derivatives and complex derivatives that was derived in Eq. (11.84).
We use theorem 11.1 to estimate
I
20
, for which (
D
x
−
dg
dz
(
D
x
−
iD
y
)
[
g
(
z
)] =
(11.120)
dg
dz
=
z
n
But
2
, so that we can obtain the complex exponential as:
z
n
|
e
i
arg
{
[(
D
x
−iD
y
)
ξ
]
2
}
= e
i
arg([
d
dz
]
2
)
= e
i
arg(
z
n
)
=
= e
in
arg(
x
+
iy
)
z
n
|
Consequently, the expression
I
20
in equation (11.50) of theorem 11.1 reduces to
I
20
=
((
D
ξ
+
iD
η
)
f
)
2
dξdη
=
e
in
arg(
x
+
iy
)
[(
D
x
+
iD
y
)
f
]
2
dxdy
We assume that [(
D
x
+
iD
y
)
f
]
2
is discretized on a Cartesian grid and use a Gaussian
as interpolator
11
to reconstruct it from its samples:
h
(
x, y
)=2
πσ
1
k
h
k
Γ
{
0
,σ
1
}
(
x
−
x
k
,y
−
y
k
)
(11.121)
Here
h
k
represents the samples of
h
(
x, y
), and the constant 2
πσ
1
normalizes the
maximum of
Γ
{
0
,σ
1
}
to 1. We include the window function
K
n
|
n
Γ
{
0
,σ
2
}
(
x, y
),
where
K
n
is
the constant
12
that normalizes the maximum of the window function to
10
Because
ξ
is harmonic,
η
does not represent freedom of choice, but is determined as soon
as
ξ
is given.
11
This is possible by using a variety of interpolation functions, not only Gaussians. For exam-
ple, the theory of band-limited functions allows such a reconstruction via the Sinc functions
but also other functions have been discussed along with Gaussians, see interpolation and
scale space reports of [141, 152, 221].
12
At
|x
+
iy|
=(
nσ
2
)
1
/
2
, the window
K
n
|x
+
iy|
n
Γ
{
0
,σ
2
}
(
x, y
) attains the value 1 when
K
n
=2
πσ
2
(
x
+
iy
|
nσ
2
)
n/
2
.
e
