Image Processing Reference
In-Depth Information
As such, a more plausible model of the edge is a ramp rather than a step. For a ramp
function given by:
-
ax
$
%
1 - 0.5
e
x
0
ux
() =
(4.28)
ax
0.5
e
x
< 0
where
a
is a positive constant depending on the image acquisition system's characteristics.
A suggested range for
a
is between 0.5 and 3.0. The derived filter (which is optimal for
these ramp edges) is:
#
&
&
e
ar
(
K
sin(
Ar
) +
K
cos(
Ar
)) +
e
-ar
(
K
sin(
Ar
) +
K
cos(
Ar
)) +
K
+
- 0
K e
wr
sr
1
2
3
4
5
6
(4.29)
fr
() =
- (-
f
r
)
0 <
rw
where
w
is the size of the filtering operator. Optimal values for the constants
K
1
,
K
2
,
K
3
,
K
4
,
K
5
,
K
6
,
A
and
w
were determined, leading to templates which can be used to detect ramp
edges. In application, the window size
w
is fixed first, followed by appropriate choice of
a
that leads to appropriate selection of the template coefficients. Since the process is based
on ramp edges, and because of limits imposed by its formulation, the Petrou operator uses
templates that are 12 pixels wide at minimum, in order to preserve optimal properties. As
such, the operator can impose greater computational complexity but is a natural candidate
for applications with the conditions for which its properties were formulated. The operator
has been implemented in a similar manner to the Spacek operator. An example showing
application of the Petrou operator is shown in Figure
4.30
(c). The scale of the action of the
operator is clear since many small features are omitted, leaving only large-scale image
features, as expected. Note that the (black) regions at the border of the picture are larger,
due to the larger size of windowing operator.
4.5
Comparison of edge detection operators
Naturally, the selection of an edge operator for a particular application depends on the
application itself. As has been suggested, it is not usual to require the sophistication of the
advanced operators in many applications. This is reflected in analysis of the performance
of the edge operators on the eye image. In order to provide a different basis for comparison,
we shall consider the difficulty of low-level feature extraction in ultrasound images. As has
been seen earlier (Section 3.5.4), ultrasound images are very
noisy
and require
filtering
prior to analysis. Figure
4.31
(a) is part of the ultrasound image which could have been
filtered using the truncated median operator (Section 3.5.3). The image contains a feature
called the pitus (it's the 'splodge' in the middle) and we shall see how different edge
operators can be used to detect its perimeter, though without noise filtering. Earlier, in
Section 3.5.4, we considered a comparison of statistical operators on ultrasound images.
The median is actually perhaps the most popular of these processes for general (i.e. non-
ultrasound) applications. Accordingly, it is of interest that one study (Bovik, 1987) has
suggested that the known advantages of median filtering (the removal of noise with the
preservation of edges, especially for salt and pepper noise) are shown to good effect if used
as a prefilter to first- and second-order approaches, though naturally with the cost of the
median filter. However, we will not consider median filtering here: its choice depends more
on suitability to a particular application.
