Image Processing Reference
In-Depth Information
z
−
1
z
−
0
.
5
z
0
.
5
z
1
z
1
.
5
z
2
z
2
.
5
log(
z
)
Γ
{−
4
,σ
2
}
Γ
{−
3
,σ
2
}
Γ
{−
2
,σ
2
}
Γ
{−
1
,σ
2
}
Γ
{
0
,σ
2
}
Γ
{
1
,σ
2
}
Γ
{
2
,σ
2
}
Γ
{
3
,σ
2
}
n
=
−
4
n
=
−
3
n
=
−
2
n
=
−
1
n
=0
n
=1
n
=2
n
=3
Fig. 11.6. The
top row
shows the harmonic functions, Eqs. (11.87), that generate the patterns
in the
second row
. The isocurves of the images are given by a linear combination of the real
and the imaginary parts of the harmonic functions on the
top
according to Eq. (11.88) with a
constant parameter ratio, i.e.,
ϕ
=tan
−
1
(
a, b
)=
4
. The
third row
shows the filters that are
tuned to detect these curves for any
ϕ
, while the
last row
shows the symmetry order of the
filters
pattern recognition purposes, however, this will not be necessary given the control
possibility the
I
11
estimation offers. If, for an image for which 0
|
I
20
|≈
I
11
,
ϕ
0
=0is obtained, the nonprototype is known in reality too (as the
gradients come from a real image). Thus, in practice only when
=arg
I
20
|I
20
|I
11
may
pose interpretation difficulties of arg
I
20
, in which case no member of this class is a
good fit to the data anyway.
Both the Hough accumulator value
A
and the value of
I
20
will be maximal and
identical to each other when there is maximal match between the prototype and the
image edges (i.e in GST terms when arg(
I
20
)=0and
=
I
11
). However,
because of the complex votes, the two GST measurements,
I
20
and
I
11
, additionally
offer detection of other prototypes not detected by the Hough transform, e.g., the
antiprototype when arg(
I
20
)=
π
and
|
I
20
|
|
I
20
|
=
I
11
. We summarize our findings in the
following lemma.
Lemma 11.4.
The GST is GHT with complex votes. Except for a possible vote re-
duction due to edge direction mismatch, the GHT accumulator value
A
is equivalent
to
I
20
in the 0 radian direction, i.e., GST will only sharpen the accumulator peaks of
the GHT. The GST parameter
I
20
in other directions than 0 radian, along with
I
11
,
can detect and identify other prototypes not detected by GHT.
11.7 Harmonic Monomials
Here we will discuss a specific harmonic function class with member functions hav-
ing direction fields that are monomials of
z
. As will be shown, this class of harmonic
function families is easily found analytically while they constitute computationally
powerful models to process symmetric patterns in images.
Assuming
z
=
x
+
iy
, we will study those
g
(
z
)=
ξ
(
x, y
)+
iη
(
x, y
)
(11.76)
