Image Processing Reference
In-Depth Information
value of the image in the window outlined by the structuring element
b
when the origin of
b
is (
x
,
y
):
{
}
[
abxy
⊕
](,)max
= −−
ax my n
(
,
)
(,)
mn b
∈
Greyscale erosion is defined as the minimum value of the image in the win-
dow outlined by the structuring element
b
when the origin of
b
is (
x
,
y
):
{
}
[
abmn
Θ
](,)min
= ++
ax my n
(
,
)
(,)
mn b
∈
The dilated image with a flat structuring element computes the maximum
value, so dilation brightens the image. Erosion is opposite to that of dilation.
Dilation and erosion using a non-flat structuring element where the grey
values vary over the domain of definition are calculated as follows:
{
}
Dilation:[ ]( ,) max( ,
abxy
⊕
= −−+
ax my nbxy
) (,)
(9.5)
(,)
mn b
∈
{
}
Erosion:
[
abmn
Θ
](,)min
= ++−
ax my nbxy
(
, )(,)
(9.6)
(,)
mn b
∈
Greyscale opening and closing are defined as
fb f bb
=
Θ
⊕
(
)
(9.7)
fb f bb
⋅= ⊕
Θ
(
)
9.3 Fuzzy Mathematical Morphology
In digital image processing, fuzzy set theory has found a promising field
of application. Fuzzy mathematical morphology is developed to soften the
binary morphology to make the operators less sensitive to image imprecision.
It is an alternative to greyscale morphology. It is studied in terms of fuzzy it-
ting [3,5,8,9,10,12,13,14,19]. The fuzziness is introduced with the degree to which
the structuring element fits into the image. The morphological operations
are modelled on a fuzzy notion. Fuzziness is introduced only in modelling
greyscale images and not in the operations. Fuzzy mathematical morpho-
logical operations are obtained by replacing ordinary set theoretic operations
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