Image Processing Reference
In-Depth Information
In the above equation representing the most general filter, the value of the out-
put at every level
y
i
is a function of all
x
t
i
, i.e., samples of the input derived from ev-
ery level
i
and every time delay
t
. It was stated in the previous chapter that
Dougherty
17
has shown that any filter, linear or nonlinear, may be represented in
terms of computational morphology and hence may be placed in this form.
In practice, special cases of filters such as stack filters and grayscale morphol-
ogy result in restricted forms of the functions
ψ
i.
The
unstacking
operation consists of digital logic that converts the maximum
value of
i
for which
y
i
= 1 to a binary number. This is a matter of straightforward (if
tedious) logic design. The interesting part of the process lies in the functions
ψ
i
which link
y
i
and
x
t
i
.
8.3 Stack Filter
The simplest case of nonlinear filtering in this context is the
stack filter
. Figure 8.5
shows the structure of a stack filter with
T
= 3 and
L
= 4. The general model is sim-
plified to
)
(
i
i
i
i
y
=
ψ
x
,
x
,
x
(8.4)
01
T
−
1
The value of
y
i
is determined by a combinatorial binary function of the time-de-
layed versions of
x
i
. Note that
y
i
(the output at level
i
) is only dependent on the in-
puts
x
t
i
, i.e., those derived from the same threshold level
i
. Also, the binary function
ψ
. Many increasing filters may be represented via
stack filters including morphological operations with flat structuring elements and
the median and weighted median. Some linear FIR filters with positive coefficients
may also be represented as a stack filter.
A specific case of the stack filter is the morphological operation of a
three-point erosion
of the signal
X
by a flat structuring element
B
to produce an out-
put signal
Y
, i.e.,
is the same for all levels:
ψ
i
=
ψ
Y
=
X
Θ
B
(8.5)
where
is the erosion operator and
B
is a three-point flat structuring element. In
terms of logic, this reduces to a three-input Boolean AND operation applied to the
individual elements of the signals, i.e.,
Θ
i
i
i
i
yxxx
=⋅
⋅
(8.6)
01 2
The output signal therefore consists of an AND function of the time-delayed inputs
at the same threshold level. This is shown in Fig. 8.6.
