Image Processing Reference
In-Depth Information
Figure 7.10
Brief illustration of elemental erosion. Unlike standard erosion, the structuring el-
ement is anchored to the
x
axis and slides horizontally. The output is binary and requires the
entire structuring element to lie beneath the signal.
Note that Eq. 7.8 only applies to increasing filters. This is because it is based on a
maximum (union) operation. Provided that the input exceeds a given level, it will
cause the output to be 1. It is assumed that all inputs greater than this will also cause
the output to be 1. However, for a nonincreasing filter, this will not be the case and
the input must be further tested to determine if it falls within an
interval
. This con-
cept is a generalization of the hit-or-miss transform discussed in Chapter 5. Further
details are given in Dougherty.
5
Although the output from the elemental erosion is binary, it may be used to
model the behavior of a grayscale filter by representing a single output level
k
.A
number of elemental erosions, each with a different kernel, are carried out in paral-
lel—one for each level of the grayscale filter.
Consider the most general grayscale filter
ψ
. It can be represented by its kernel
K
[
]. This is very similar to a look-up table that returns an output value for any in-
put combination.
If input
x
to the filter is a vector of
n
values all between 0 and
m
- 1, and the out-
put
Y
is a single grayscale value lying between 0 and
L
- 1, this may be written as
ψ
{0, 1,…
m
-1}
n
x
∈
and
Y
∈
{0, 1,…
L
- 1}.
(7.9)
