Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 172

Fuzzy fusion operators fall into three categories. T-norms, which generalize set
intersection to fuzzy sets, are conjunctive CICB operators, since for any t-norm t ,we
have:

[0 , 1] 2 ,t ( x, y )

∀ ( x, y ) ∈ ≤ min( x, y ) .

On the other hand, t-conorms which generalize union are disjunctive CICB opera-
tors, since for any t-conorms T ,wehave:

[0 , 1] 2 ,T ( x, y ) ∀ ( x, y ) ∈ ≥ max( x, y ) .

Mean operators are also CICBs and have a compromise behavior, since they verify:

[0 , 1] 2 , min( x, y )

∀ ( x, y ) ∈ ≤ m ( x, y ) ≤ max( x, y ) .

Let us note that Bayesian fusion, in which the operator involved is a product, and
fusion of belief functions using Dempster's orthogonal sum are also conjunctive.

In the CIVB operator class we have, for example, certain symmetric sums. Gener-
ally speaking, any associative symmetric sum σ (except for medians) has the following
behavior [DUB 88]:
- conjunctive if max( x, y ) < 1 / 2: σ ( x, y ) ≤ min( x, y );

- disjunctive if min( x, y ) > 1 / 2: σ ( x, y ) ≥ max( x, y );

- compromise if x ≤ 1 / 2 ≤ y : x ≤ σ ( x, y ) ≤ y (and the opposite inequality if

y ≤ 1 / 2 ≤ x ).

Non-associative symmetric sums also have a variable behavior, but according to
less simple rules [BLO 96b].

In the CIVB operator class, we also have the operators suggested in the MYCIN
system for combining certainty factors [SHO 75].

Examples of CD operators are found in possibility theory. Earlier, we presented
operators that depended on an overall measure of the conflict between two sources of
information [DUB 92a], which are applicable to cases where one of the two elements
of information is reliable, but where we do not know which one, so that:
- they are conjunctive if the sources are consonant (low conflict): in this case, the
two sources are necessarily reliable and therefore the operator can be strict;
- they are disjunctive if the sources are dissonant (high conflict): a disjunction then
favors all of the possibilities provided by both sources;