Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 137

Conversely, a possibility measure leads to a possibility distribution:

∀ x ∈S ,π ( x )=Π( { x } ) . [8.25]

By duality, a necessity measure is defined from a possibility distribution as:

sup π ( x ) ,x/ A =inf 1 A C .

∀ A ∈C ,N ( A )=1 − ∈ − π ( x ) ,x ∈ [8.26]

In the non-normalized case, we no longer have Π( S )=1. Likewise, the properties

N ( A ) > 0 ⇒ Π( A )=1and Π( A ) < 1 ⇒ N ( A )=0are no longer true.

These definitions have a simple interpretation if we consider the problem of how
to represent the value of a variable, in which S represents the variation range of this

variable. A possibility distribution on describes the degrees to which the variable

can have each possible value. It is actually the fuzzy set of the possible values for this
variable. The degree of membership of each value to this set corresponds to the degree
of possibility for the variable to have this value. Therefore, a possibility distribution
can represent the imprecision related to the variable's exact value. Typically, a fuzzy
number is a possibility distribution that describes the possible values that this number
can have.

S

Let us consider, for example, a classification problem in image processing. Here is
a list of examples (not a comprehensive one) of possibility distributions:
-let

, defined for each object
to classify (point, area, etc.), can represent the degrees to which each object can belong
to each of these classes;
-let

S be the set of classes. A possibility distribution on S

be a characteristic space (for example, a scale of gray levels). A possibility
distribution on

S

can be defined for each class and represent, for each gray level, the
possibility for that class to appear in the image with that gray level;
-let

S

can be defined for each
class and give for each point of the image its degree of possibility of belonging to that
class.

S be the image space. A possibility distribution on S

In the definition given here, we have always considered the possibility and the
necessity of a crisp subset of S . Now, consider a fuzzy set μ of S ( μ ∈F ). The

concept of possibility must then be extended [ZAD 78]:

min μ ( x ) ,π ( x ) . Π( μ )=sup

x ∈S

[8.27]

This corresponds to the following interpretation: given a possibility distribution
π on , we can assess to which

extent “ X is μ ”. In this way, the possibility of μ combines the degree to which the
variable X has the value x and the degree of membership of x to the fuzzy set.

S , associated with a variable X taking its values in S