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a possible application of this theory consists of choosing S = D (the class set) and
defining the measure M i
M i ( x )= π j C i ,
i.e. as the possibility degree for the class to which x belongs to have the value C i ,
according to the source I j . This defines a possibility distribution for each source and
each element x . The possibility and the necessity for each class are then written:
Π j C i = π j C i ,
N j C i =inf 1
π j C k ,C k
= C i .
For any subset A of D , the possibility and the necessity are calculated using for-
mulae [8.24] and [8.26].
This modeling assumes that classes are crisp, whereas the fuzzy model defined by
equation [8.100] assumes fuzzy classes.
Generally speaking, there are three interpretations of fuzziness in other works, in
terms of plausibilities, similarities, preferences [DUB 99]. The same interpretations
are used in signal and image fusion. The interpretation in terms of plausibilities is
used for the membership to a class, in the definition of a fuzzy spatial object (an
object with imprecise limits). The interpretation in terms of similarities is that used
for the definition of a fuzzy class in a characteristic space as a function of the distance
to a prototype, for example, of linguistic variables representing information or knowl-
edge about spatial objects, or also of degrees of satisfaction of a relation, a constraint.
Finally, preferences are used in the expression of choice criteria (for example, for
planning applications in robotics), which are often related to constraints or knowledge
outside the image.
8.9. Defining membership functions or possibility distributions
Constructing membership functions or possibility distributions can be done in sev-
eral ways.
In most applications, this construction is done either by taking ideas directly from
probabilistic learning methods, from heuristics, from neuromimetic methods used for
learning the parameters of particular forms of membership functions, or finally by
minimizing classification criteria [BEZ 81]. Below is a description of the major meth-
A first method consists of defining a fuzzy class membership function based on
the image's intensity function I (the gray levels):
μ i ( x )= F i I ( x ) ,
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