Image Processing Reference
In-Depth Information
S
=
C
l
T
f
l
)
μ
l
(
∇
f
l
∇
(10.76)
μ
l
=
C
l
(
D
x
f
l
)
2
(
D
x
f
l
)(
D
y
f
l
)
(
D
x
f
l
)(
D
y
f
l
)
D
y
f
l
)
2
(10.77)
where
μ
l
is a discrete Gaussian with
σ
=
σ
p
+
σ
w
.
In analogy with Eqs. (10.63)-(10.65), the quantity
D
x
f
(
r
l
)+
iD
y
f
(
r
l
)
(10.78)
can be obtained by two convolutions using real filters, one for
D
x
f
(
r
l
) and one for
D
y
f
(
r
l
). After that, the complex result depicted by Eq. (10.78) is squared to yield
the ILST image:
(
f
)(
r
l
)=(
D
x
f
(
r
l
)+
iD
y
f
(
r
l
))
2
(10.79)
In consequence of theorem 10.2, the following theorem then holds true:
Theorem 10.4 (Discrete structure tensor II).
Assuming a Gaussian interpolator
with
σ
p
and a Gaussian window with
σ
w
, the optimal
discrete structure tensor
com-
plex elements are given by
I
11
−
Z
=
1
2
iI
20
(10.80)
iI
20
I
11
where
I
20
=
C
l
(
D
x
f
l
+
iD
y
)
2
μ
l
(10.81)
I
11
=
C
l
D
x
f
l
+
iD
y
|
2
μ
l
|
(10.82)
with
μ
l
being a discrete Gaussian with
σ
=
σ
p
+
σ
w
.
Figure 10.15 shows a frequency-modulated test (FM-test) image. The test im-
age has axes marked by the spatial frequencies of the waves in the horizontal and
vertical directions from the image center. The absolute frequency of the waves de-
creases exponentially radially, whereas the direction of the waves changes uniformly
angularly. The exponential decrease occurs between the spatial frequencies 0
.
4
π
and
0
.
9
π
. The image on the right represents the color code of the ideal orientation in
double-angle representation, i.e., exp 2
ϕ
, where
ϕ
is the polar angle coordinate of
a point in the image. In half of the image, spatially uncorrelated Gaussian noise
X
,
with mean 0.5 and variance 1/36, has been added to the image signal,
f
, according
to
pf
+(1
−
p
)
X
, where the weight coefficient
p
=0
.
3, and
X
is the noise. On
