Image Processing Reference
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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
[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
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;
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