Image Processing Reference
In-Depth Information
written in terms of its variables without complementation. This is equivalent to the
increasing property as was seen in Chapter 5. The reason for this is straightforward.
After threshold decomposition every binary signal in the stack
X
t
is included within
the one beneath., i.e.,
x
m
1
……
x
t
+1
x
t
x
1
(7.4)
It is in the nature of all real signals that a strict ordering as specified in Eq. 7.4 can
be observed within a stack of binary signals resulting from threshold decomposi-
tion. If this were not true, the waveform would contain holes as shown in Fig. 7.4.
Therefore, it is essential that this same ordering is preserved after filtering. It must
also hold for the binary outputs
y
t
at each threshold level. Therefore,
y
m
1
……
y
t
+1
y
t
y
1
.
(7.5)
This in turn leads to a constraint on the type of filtering that may be applied to the
binary levels. Only filters for which the following ordering is preserved may form
the basis of a stack filter.
(
x
m
1
)
……
(
x
t
+1
)
(
x
t
)
(
x
1
).
(7.6)
A necessary and sufficient condition to ensure that this ordering is preserved for all
input combinations is that the filter
be an increasing filter. This can be satisfied
by ensuring that is a binary filter based on a Boolean logic function written in a
form that contains no complementation.
A stack filter may be designed using a representative training set as described
in Chapter 2. Both the noisy and ideal data in the training set are thresholded at ev-
ery level. A sliding window is passed over the noisy signal (Fig. 7.5) and the num-
ψ
Figure 7.4
Violation of the stacking property.
