Image Processing Reference
In-Depth Information
3
1
2
0.8
1
0.6
0
1
0.4
2
0.2
3
0
2
3
4
5
6
7
2
3
4
5
6
7
Fig. 10.17. The graphs in the
left image
represent the estimated (arg(
I
20
),
solid
) as well as
the ideal direction angle (
dashed
) on a ring in the FM test image. The graphs on the
right
show
|I
20
|
(
solid
)and
I
11
(
dashed
) on the same ring
In the left graph of Fig. 10.17 we show in solid green the direction measurements,
arg(
I
20
), extracted along the circle passing through the points 1 to 4, see also Fig.
10.15. We show by the dotted black curve the reference direction measurements. The
direction measurements agree with the reference values nearly exactly in the noise-
free parts of the image, whereas they follow the reference quite well in the noisy parts
of the image. The estimated significance of the direction measurements is given by
the graphs of the right image. The solid curve shows
, whereas the dotted curve
shows
I
11
. As predicted by the theory, we have 0
|I
20
|
=
I
11
|I
20
|
in the noise-free
part, whereas 0
and
I
11
are invariant to
directional changes, which is manifested by the fact that they both equal a constant
(one) in the noise-free part of the test image.
Likewise, in the left graph of Fig. 10.18 we display in blue the analogous mea-
surements, but for arg(
I
20
) extracted along the horizontal line joining point 6 to the
center, as marked in Fig. 10.15. This curve is reasonably horizontal even in the noisy
part. The constant represented by the green line shows the direction measurements
extracted along the line joining point 5 to the center. Being in the noise-free part, this
line adheres nearly perfectly to the reference measurements, shown as a dotted. The
estimated significance of the direction measurements is given by the graphs of the
right image. Here, the solid, and the dashed curves at the top represent the measure-
ments of
≈|
I
20
|
<I
11
in the noisy part. Both
|
I
20
|
, and
I
11
respectively for the clean signal, (from point 5 to the center).
The corresponding measurements for the noisy part (from Point 6 to the center) are
given by the solid and the dashed curves at the bottom, respectively. The direction
estimation quality is assured by the condition that
|
I
20
|
is high and is close to its up-
per bound,
I
11
. By contrast, when the linear symmetry in the signal is disturbed poor
estimations of the direction are obtained. This is manifested by the fact that
|
I
20
|
|
I
20
|
is
close to zero while
I
11
is weak.
The structure tensor measurements are implemented by using the results of linear
operators. First, the partial derivative operator
D
x
+
iD
y
using a Gaussian derivative
