Image Processing Reference
In-Depth Information
11.8 “Steerability” of Harmonic Monomials
π
4
In Fig. 11.6, the angle is fixed to
ϕ
=
4 and 3. Each
n
represents a separate isocurve family. By changing
ϕ
and keeping
n
fixed, the
parameter pair (
a, b
) is rotated to (
a
,b
). Except for the patterns with
n
=
, and
n
is varied between
−
2,
which we will come back to next, this results in rotating the isocurves, since for
n
−
2 and
g
(
z
)=
z
n
=
−
2
+1
we have:
a
ξ
+
b
η
=
[(
a
−
ib
)(
ξ
+
iη
)] =
[(
a
−
ib
)
g
(
z
)]
(11.90)
2
+1
]=
[(
a − ib
)
g
(
z
e
i
1
ϕ
=
[(
a − ib
)e
iϕ
z
n
n
2
+1
)]
(11.91)
=
aξ
+
bη
(11.92)
Here
ξ
and
η
are rotated versions of the harmonic pair
ξ
and
η
,sothat
g
(
z
exp
iϕ
0
)=
ξ
+
iη
for some
ϕ
0
. The top row of Fig. 11.3 displays the curves generated by Eq.
(11.88), for increasing values of
ϕ
in (
a, b
) = (cos(
ϕ
)
,
sin(
ϕ
)), to illustrate that a
coefficient rotation results in a pattern rotation.
When
n
=
−
2, we obtain the isocurves via the function
g
(
x
+
iy
)=log(
|
x
+
iy
|
)+
i
arg(
x
+
iy
)
which is special in that it represents the only case when a change of the ratio between
a
and
b
does not result in a rotation of the image pattern. Instead, changing the angle
ϕ
bends the isocurves,
cos(
ϕ
)log(
|
x
+
iy
|
)+sin(
ϕ
)arg(
x
+
iy
)=constant
(11.93)
That is, the spirals become “tighter” or “looser” until the limit patterns, circles,
and radial patterns, corresponding to infinitely tight and infinitely loose spirals, are
reached.
The relationships (11.90)-(11.92) show that the isocurves of the patterns that are
modeled by harmonic monomials are obtained as a linear combination of the non-
rotated isocurves, except
g
(
z
)=log(
z
). Yet half of these patterns fail to fulfill the
steerability condition [75, 180]. The steerability condition foresees that the angu-
lar Fourier series expansion function,
f
, must have a limited number of elem
e
nts
to allow steering by linear weighting of basis elements. In our case,
g
(
z
)=
z
n
2
+1
does not satisfy the steerability condition on angular band-limitedness when
n
is odd
since it is not possible to expand odd powers of a square root with a limited number
of (integer) angular frequencies. The same holds evidently for the isocurve family
represented by the CT log(
z
), the members of which cannot be rotated by changing
the linear coefficients. Consequently, the patterns with odd
n
or with
n
=
2 can-
not be generated by weighted sums of a low number of
steerable functions
. In turn,
this makes it impossible to detect the mentioned patterns by correlating the
original
gray images
with steerable filters. Yet, these patterns can be accurately generated by
analytic functions.
8
−
They can even be detected by steerable filters, as will be shown
8
Analytic functions are harmonic, but they do not necessarily meet the steerablity condition
of [75, 180].
