Image Processing Reference
InDepth Information
8.2.3.
α
cuts
The
α
cut of a fuzzy set
μ
is the binary set defined by:
μ
α
=
x
α
.
∈S
,μ
(
x
)
≥
[8.12]
Strict (or strong)
α
cuts are defined by:
μ
α
S
=
x
,μ
(
x
)
>α
.
∈S
A fuzzy set can be interpreted as its
α
cuts stacked on top of each other. It can be
reconstructed from them using several formulae, the most common of which are:
μ
(
x
)=
1
0
μ
α
(
x
)
dα,
min
α, μ
α
(
x
)
,
μ
(
x
)= sup
α
∈
]0
,
1]
αμ
α
(
x
)
.
μ
(
x
)= sup
α
∈
]0
,
1]
Most of the operations we have defined so far commute with
α
cuts. More pre
cisely, we have the following relations:
∀
2
,μ
=
ν
(
μ, ν
)
∈F
⇐⇒ ∀
α
∈
]0
,
1]
,μ
α
=
ν
α
,
2
,μ
∀
(
μ, ν
)
∈F
⊆
ν
⇐⇒ ∀
α
∈
]0
,
1]
,μ
α
⊆
ν
α
,
2
,
∀
(
μ, ν
)
∈F
∀
α
∈
[0
,
1]
,
(
μ
∩
ν
)
α
=
μ
α
∩
ν
α
,
2
,
∀
(
μ, ν
)
∈F
∀
α
∈
[0
,
1]
,
(
μ
∪
ν
)
α
=
μ
α
∪
ν
α
,
[0
,
1]
,
μ
C
α
=
μ
1
−
α
S
C
.
∀
μ
∈F
,
∀
α
∈
Choosing an
α
cut in a fuzzy set is equivalent to thresholding the membership
function in order to select the points with a level of membership of at least
α
. This
operation can be interpreted as a “defuzzification” and is used in the decision phases
after the fusion.
8.2.4.
Cardinality
In this section, we will restrict ourselves to fuzzy sets defined over a finite domain,
or that have a finite support (this will always be the case in image applications).
The cardinality of a fuzzy set
μ
is defined by:
=
x
∈S

μ

μ
(
x
)
,
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