Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 167

8.8. Fuzzy modeling in fusion

Among the non-probabilistic techniques that have appeared over the past 10 years
in fusion, fuzzy set theory provides a very efficient tool for explicitly representing
imprecise information, in the form of membership functions [BAN 78, KAU 75,
ZAD 65], as we saw above. As a result, the measure M i ( x ) introduced in Chapter 1
is written in the form:

M i ( x )= μ i ( x ) , [8.100]

where μ i ( x ) refers, for example, to the degree of membership of x to the class C i
according to the source I j , or the translation of a symbolic element of information
expressed by a linguistic variable (see, for example, [DEL 92]).

In other works there are two main methods for using fuzzy sets in image processing
[BLO 96a]: the first is more symbolic in nature and expresses in the form of fuzzy rules
the membership of certain structures to a class, depending on measurements obtained
by image processing; the second uses fuzzy sets to directly represent the classes or
structures in the image, spatially covering the objects with a membership function. Let
us consider the example of the “road” class in a satellite image. In the first approach,
we would describe the road in a linguistic form such as “a road is a rather elongated
structure”. The membership of an object to the road class will then be represented by
a function associating its length with a degree in [0 , 1]. Any parallel contour detection
algorithm can then be used to assign to the objects it detects a degree of membership to
the road class, depending on their length. In the second approach, the road is directly
represented in the image by a fuzzy set, with membership degrees that are strong in
the middle of the road and close to 0 in fields or forest.

These functions are not subjected to the axiomatic constraints imposed by proba-
bilities and hence offer a greater flexibility in modeling them. This flexibility can be
seen as a disadvantage since it can easily leave the user helpless to define these func-
tions. The disadvantage of fuzzy sets is that they represent essentially the imprecise
nature of information, whereas uncertainty is represented implicitly and can only be
obtained by inference from the different membership functions.

Possibility theory [DUB 88, ZAD 78], which is derived from fuzzy sets, allows
us to represent both the imprecision and the uncertainty, by using possibility distribu-
tions π on a set S and two functions characterizing events: the possibility Π and the
necessity N .

A possibility distribution is interpreted as a function that gives the degree of pos-
sibility for a variable to have the value s , with S being the domain of the variable's
values. The distribution π is then interpreted as the membership function to the fuzzy
subset S for the possible values of this variable. In the framework of numerical fusion,