Image Processing Reference
In-Depth Information
The elemental erosion tests if the corresponding structuring element B i fits beneath
the input signal. While in computational morphology theory the input signal is not
thresholded, in practice this test is put into the context of a series of threshold in-
puts. This is explained by the following function
i
i
1
i
i
1 .
yx xx
=⋅
(8.15)
0
1
2
It can be seen that the output at each threshold level y i is formed by taking an AND
function of the three inputs ( x 0 i 1 , x 1 i , x 2 i 1 ). These correspond to the (temporally)
central input signal x 1 i (from the same level i ) and the two on either side x 0 i 1 and
x 2 i 1 (taken from the next level down, i- 1). Provided that all three of these signals
are 1, then the structuring element B i fits below the input signal x and hence the out-
put at that threshold level y i =1.
For consistency it is usually assumed that inputs x i = 1 for i < 0, hence the out-
puts are defined as
3
2
3
2
y xxx
=⋅
0
1
2
2
1
2
1
yxxx
=⋅
0
1
2
(8.16)
1
0
1
0
yxxx
=⋅
0
1
2
0
0
yx
=
.
1
The circuit to implement this grayscale morphological erosion is shown in Fig. 8.9.
8.5 Computational Morphology and Aperture Filters
A general computational morphology filter would be capable of implementing any
function of any complexity, linear or nonlinear. However, the function used to cre-
ate each output level would require access to all levels of the thresholded input. It
would therefore be very complex.
It would also be very difficult to design, since the unconstrained filter requires
that every possible combination of inputs is seen a sufficient number of times to es-
timate the output conditional probabilities.
Design of such filters is normally complex and consists of determining the con-
tent of the kernel of structuring elements. This translates into estimating the func-
tions
ψ
i in Eq. 8.3. It is essential that the functions are ordered in such a way that
ψ
≥ψ
i +1 for all i and for all combinations of inputs.
(8.17)
i
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