Image Processing Reference
In-Depth Information
such that
∗
(
ξ
)(
x, y
)=(
D
x
ξ
iD
y
ξ
)
2
=
z
n
,
−
with
n
=0
,
±
1
,
±
2
,
···
.
(11.77)
∗
(
ξ
) is needed
6
and it makes sense to know what kind of curve families the ILST fields are projected
on,
This study is motivated because, to estimate
I
20
via theorem 11.1,
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
)
(11.78)
z
n
|
First, we establish a relationship between the operator
7
D
x
−
iD
y
and complex
derivatives as follows:
(
D
x
−
iD
y
)
[
g
(
z
)] =
D
x
[
g
(
z
)]
−
iD
y
[
g
(
z
)]
(11.79)
=
[
D
x
g
(
z
)]
−
i
[
D
y
g
(
z
)]
(11.80)
dg
dz
dg
dz
dz
dx
dz
dy
=
−
i
(11.81)
dg
dz
dg
dz
i
=
−
i
(11.82)
dg
dz
i
dg
dz
dg
dz
=
−
i
−
(11.83)
dg
dz
+
i
dg
dz
=
dg
dz
=
(11.84)
Thus, we obtain:
∗
(
ξ
(
x, y
)) = (
dg
(
z
)
dz
)
2
=
z
n
,
with
n
=0
,
±
1
,
±
2
,
···
.
(11.85)
and establish
g
(
z
)=
z
b
2
,
with
n
=0
,
±
1
,
±
2
···
(11.86)
dg
dz
=
z
n
as a solution. We integrate
2
to obtain the real and imaginary parts of
g
,
g
(
z
)=
1
z
n
2
+1
,
if
n
=
−
2;
n
2
+1
(11.87)
log(
z
)
,
if
n
=
−
2.
The scheme discussed in Sects. 11.3 and 11.4 detects the patterns that are generated
by real and imaginary parts of
g
(
z
). Such patterns are shown in Fig. 11.6 by gray
modulation:
[
g
(
z
)]) (11.88)
The 1D function
s
(
t
) = cos(
t
) is chosen for illustration purposes. The filters that are
tuned to detect the isocurves
aξ
+
bη
are not sensitive to
s
, but to the angle
ϕ
=tan
−
1
(
a, b
)
.
(11.89)
6
Since
ξ
is harmonic, to study
η
does not represent freedom of choice but is determined as
soon as
ξ
is given.
7
The properties of such linear operators will be discussed in further detail in Sect. 11.9
s
(
aξ
+
bη
)=cos(
a
[
g
(
z
)] +
b
