Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 175

The generalization of all of the above to the combination of two elements of infor-
mation poses no particular difficulty (in particular, the same types of behavior are
found in CIVB operators with rules that are a little more complicated), except for
non-associative operators. The main question surrounding these operators is to know
in what order the elements of information should be combined. Several situations can
occur:
- in certain applications, each element of information has to be combined with the
others as soon as it becomes available (for example, in order to make partial decisions
based on the data available at every instant): the order is then set by the order in which
the elements of information arrive;
- the order can be imposed by priorities on the information to take into account,
and operators have been designed to respond to these needs (for example, in order to
combine database requests);
- in other situations, criteria have to be determined for finding an order adapted
to the application, particularly when the elements of information are in conflict, since
the results can be very different depending on whether the consonant or the conflicting
elements of information are combined first.

Finally, the study of the behaviors of operators in terms of the quality of the deci-
sion they lead to and of their reactions when faced with conflicting situations leads to
a final criterion for choice. An important point, however, involves the discriminating
power of the operators. Highly conjunctive or disjunctive operators (for example, the
Lukasiewicz t-norm and t-conorm) quickly saturate at 0 or 1 and therefore are often
poorly discriminatory. For example, with the t-conorm F ( a, b )=min( a + b, 1),we
have F (0 . 5 , 0 . 5) = 1, F (0 . 1 , 0 . 9) = 1,oralso F (0 . 8 , 0 . 8) = 1, whereas these three
situations have quite different interpretations.

The ability of operators to combine information that is quantitative (numerical)
or qualitative (for which only the order is known) can also be a criterion for choice.
For example, the min, the max and any rank filter are useful in this regard since they
can combine both types of information. This is because the calculation of min( x, y ),
for example, only requires knowing an order between x and y , but does not require
their numerical values to be known. Additionally, ordinal operations are imposed if
we want them to remain invariant by an increasing transformation of the membership
degrees [DUB 99].

8.11. Decision

The major rule used in fuzzy fusion is the maximum degree of membership:

C i if μ i ( x )=max μ k ( x ) , 1 n ,

x ∈ ≤ k ≤ [8.106]