Image Processing Reference
In-Depth Information
It is to be noted that the principle of duality plays an important role in fuzzy
dilation and erosion, since one operation can be deduced from another. If
the complementation is μ
C
(
x
) = 1 − μ(
x
), then the duality with respect to the
complementation between erosion and dilation in relation to the structuring
element is expressed as [3]
D
()
μ
=− −
11
E
(
μ
)()
x
Example
:
An example of a matrix is shown for dilation and erosion (using
Sinha and Dougherty) where erosion and dilation are defined using the
membership function. Consider
A
as a fuzzy image,
B
as a fuzzy structuring
element and μ
A
and μ
B
as the membership functions of the image and the
structuring element, respectively.
Erosion and dilation are defined as
⎡
⎢
⎤
⎥
(
)
EAB
,
=
μ
( )min min[
x
=
1
+ +−
μ
(
xy y
)
μ
()]
AB
Θ
A
B
yB
∈
⎡
⎢
⎤
⎥
=
min,min
1
[
1
+
μ
(
xy y
+
)
−
μ
( )]
A
B
y
∈
B
⎡
⎢
⎤
⎥
(
)
DAB
,
=
μ
( )max max[
x
=
μ
(
xy y
−
)
+
μ
() ]
−
1
AB
⊕
A
B
yB
∈
⎡
⎢
⎤
⎥
max,max
0
[( )
xy y
( )]
1
=
μ
−+ −
μ
A
B
y
∈
B
Fuzzy erosion and dilation have membership functions within the interval
[0, 1]:
03 10 08 09 04
04 08 09 10 05
04 09 03 08 03
⎥
.
.
.
.
.
⎡
⎤
⎢
⎢
⎢
⎥
⎥
μ
A
=
.
.
.
.
.
.
.
.
.
.
⎣
⎦
μ
=
⎣
08 09
., .
,
μ
=
†
08 09
., .
⎦
⎣
⎦
B
−
B
The arrows denote the origin of the coordinate system.
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