Image Processing Reference
In-Depth 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
( x )
[8.12]
Strict (or strong) α -cuts are defined by:
μ α S = x
( x ) .
∈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 = ν
( μ, ν )
∈F
⇐⇒ ∀
α
]0 , 1] α = ν α ,
2
( μ, ν )
∈F
ν
⇐⇒ ∀
α
]0 , 1] 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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