Image Processing Reference
In-Depth Information
The kernel
K
[
ψ
] may be divided into L slices without loss of generalization.
K
[
ψ
]={
S
0
[
ψ
]
∪
S
1
[
ψ
]
∪
……………….
S
L
1
[
ψ
]},
(7.10)
where slice
S
k
[
] contains the input values
x
giving a filter output of
k
.
This means that the filter is partitioned into a set of slices each corresponding to
a different level
k
of the output. If the value of the output,
Y
=k
for a given input
x
k
,
then that value will be contained in the slice corresponding to the output
k
, i.e.,
ψ
x
k
∈
S
k
[
ψ
].
A kernel
K
k
[
ψ
] may also be defined for each level. The relationship between
kernel
K
k
[
] con-
tains only those inputs
x
k
for which
Y
=k
, i.e., where
k
is the highest level of the
stack for which the output is 1. On the other hand, the kernel
K
k
[
ψ
] and slice
S
k
[
ψ
] for a given level
k
is a subtle one. The slice
S
k
[
ψ
ψ
] contains all in-
puts
x
k
for which that level of the stack is 1, i.e.,
Y
k
.
In other words, the slice
S
k
[
] contains inputs for which the output exactly cor-
responds to level
k
and the kernel contains inputs for which the output is level
k
or
greater. The kernel
K
k
[
ψ
ψ
] may therefore be written as
K
k
[
ψ
]={
S
k
[
ψ
]
∪
S
k
+1
[
ψ
]
∪
……………….
S
L
1
[
ψ
]}.
(7.11)
Dougherty showed that any operation linear or nonlinear may be placed in the
context of computational morphology.
7,8
The framework is shown in Fig. 7.11. The
input signal is subjected to an elemental erosion by a set of kernels of structuring el-
ements. The output from each elemental erosion produces a binary signal for the
appropriate level of the output. All of these binary signals are stacked to produce
the grayscale output signal.
Although a 1D signal is shown here, the principle may be extended to images in
which the structuring elements correspond to windows of gray level values. Unlike
stack filters, there is no thresholding of the input signal; the full grayscale signal is
subjected to the elemental erosion for every level of the output. In the most general
case, the kernel for every output level is different although there must be an order-
ing such that
K
1
[
ψ
]
⊇
K
2
[
ψ
]
⊇
……………….
K
L
[
ψ
].
(7.12)
This is a more general condition than the equivalent stacking property in stack
filters required to preserve the stacking property of the outputs in computational
morphology.
