Image Processing Reference
In-Depth Information
Fig. 11.1. The HFPs used in Examples 11.1 (
top
) and 11.2 (
bottom
), respectively
Example 11.1. Let
w
(
z
) be defined as the identity coordinate transformation:
w
(
z
)=
z
=
ξ
(
x, y
)+
iη
(
x, y
)=
x
+
iy
Since
w
is an analytic function in
z
, (
x, y
) is a HFP. For illustration, we let the one-
dimensional function
g
be
g
(
τ
)=(1+cos
τ
)
/
2
(11.8)
while bearing in mind that
g
can be any 1D function. The argument
τ
is replaced by
aξ
+
bη
to generate the family of isocurves defined by this transformation. Figure
11.1 (top) illustrates the two basis patterns,
g
(
ξ
)=constant, and
g
(
η
)=constant
respectively.Thelinearcombinationsofthesetwopatternsgeneratenewpatterns,all
ofwhichbelongtothesamefamilyofcurves,thosethatarelinearlysymmetricw.r.t.
the Cartesian
x
and
y
, which we studied in Chap. 10.
Example 11.2. By using the same
g
as in Example 11.1, we can illustrate the trans-
formation defined by
w
(
z
)=log
z
=
ξ
(
x, y
)+
iη
(
x, y
)=log
x
2
+
y
2
+
i
tan
−
1
(
x, y
)
which is analytic everywhere except at the origin. We assume the principal branch
as the value set of
w
to avoid multiple-valued functions. Figure 11.1 (bottom) shows
