Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 140

The major classes of operators are described in [BLO 96b, DUB 85, DUB 88,
DUB 99, YAG 91]. Among the main operators, we can mention in particular T-norms,
T-conorms [MEN 42, SCH 83], means [GRA 95, YAG 88], symmetric sums and
operators that take into account conflict measures or also the reliability of sources
[DEV 93, DUB 92a]. This sections contains the major definitions. The interpretations
in terms of set operations and information fusion will be discussed further in section
8.10.

Since most operators work point by point (i.e. by combining the membership or
plausibility degrees in the same point of ), it is sufficient to define them for the

possible values of the membership functions or possibility distributions. Therefore,
the operators are defined as functions of [0 , 1] or [0 , 1]

S

[0 , 1] in [0 , 1]. In what follows,
the letters x , y , etc. will refer to the values we wish to combine, i.e. the values in [0 , 1]
representing the degrees of membership or possibility.

×

8.5.1. Fuzzy complementation

A fuzzy complementation is a function c of [0 , 1] in [0 , 1] such that:
- c (0) = 1;
- c (1) = 0;
- c is involutive: [0 , 1] ,c ( c ( x )) = x ;

- c is strictly decreasing.

∀ x ∈

The simplest example is that given in section 8.2:

∀ x ∈ [0 , 1] ,c ( x )=1 − x. [8.36]

Since it is difficult to directly construct involutive functions, it is useful to charac-
terize them using a simpler, more general form. Thus, continuous complementations
have the following general form:

[0 , 1] ,c ( x )= ϕ − 1 1 ϕ ( x ) , ∀ x ∈ − [8.37]

with ϕ :[0 , 1] → [0 , 1], such that:

- ϕ (0) = 0;
- ϕ (1) = 1;
- ϕ is strictly increasing.

There are several functions ϕ that verify these properties and it is easy to come up
with one. The simplest example is:

[0 , 1] ,ϕ ( x )= x n ,

∀ x ∈ [8.38]