Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 127

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 ) ,