Image Processing Reference

In-Depth Information

The support of a fuzzy set
μ
is the set of points with strictly positive membership

to
μ
(it is a binary set):

Supp(
μ
)=
x

,μ
(
x
)
>
0
.

∈S

[8.2]

The core of a fuzzy set
μ
is the set of points that are completely included in
μ
(it

is also a binary set):

Core(
μ
)=
x

,μ
(
x
)=1
.

∈S

[8.3]

A fuzzy set
μ
is said to be normalized if at least one point is completely included

in the set
μ
(which is equivalent to Core(
μ
)

=

∅

):

∃

x

∈S

,μ
(
x
)=1
.

[8.4]

A fuzzy set
μ
is described as unimodal if there is only one point
x
such that

μ
(
x
)=1. A less restrictive definition allows the core to be a compact set, not neces-

sarily reduced to a point.

8.2.2.
Set operations: Zadeh's original definitions

Since fuzzy sets were introduced to generalize the concept of sets, the first oper-

ations defined were set operations. In this section, we will present Zadeh's original

definitions [ZAD 65]. More general classes of operations will be presented in section

8.5.

The equality of two fuzzy sets is defined by the equality of their membership func-

tions:

μ
=
ν

⇐⇒ ∀

x

∈S

,μ
(
x
)=
ν
(
x
)
.

[8.5]

The inclusion of a fuzzy set in another is defined by an inequality between the two

membership functions:

μ

⊆

ν

⇐⇒ ∀

x

∈S

,μ
(
x
)

≤

ν
(
x
)
.

[8.6]

The equality of
μ
and
ν
is of course equivalent to having the inclusion in both

directions.

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