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Let us discuss computations of the threshold level α for some selected classes of continuous membership functions

1.  Triangular membership functions. Assume the membership function of the form

For given α the values of a1 and a2 are equal

and

Computing the corresponding areas and solving the resulting quadratic equation with respect to α we derive

and

Obviously, only the first root becomes accepted as it satisfies the requirement formulated by V. Note that the level of α does not depend on the values of “a” and “b”. These bounds appear in the formulas describing the shadow itself, namely

2.  Consider the nonlinear membership function assuming the form

Here a1 = a + α2(b-a) and a2 = a + (1-α)2(b-a).

Proceeding in the same way as before, the problem produces a fifth-order polynomial equation - the only root satisfying the imposed requirement is α = 0.405.

For some other types of membership functions one can easily derive the pertaining threshold values either by solving some algebraic equations or finding a minimum of the introduced performance index. For instance, a Gaussian membership function comes with α =0.3950. For the Γ-membership function one gets α = 0.3850.

The use of shadowed sets or shadowed relations can also be substantial in image processing especially in the problem of determining boundaries of objects. Consider a blurred circle as in Fig. 3.12. Due to a spectrum of different levels of brightness, this object can be conveniently regarded as a two variable fuzzy relation. The minimization of V leads to α = 0.395, Fig. 3.13, and the produced residual structure (shadow of the fuzzy relation) identifying the boundary of the circle is visualized in Fig. 3.14.


Figure 3.12  An image of a blurred circle


Figure 3.13  V as a function of α


Figure 3.14  Shadow of the circle

The basic operations on shadowed sets (union, intersection, and complement) are defined pointwise (as already introduced in the case of fuzzy sets).

As before, the properties such as symmetry, distributivity and associativity are fully retained. Essentially, the above operations are isomorphic with the logic connectives encountered in three-valued logic, and Lukasiewicz logic, in particular.

The operations can also be carried out in a sort of mixed mode embracing both fuzzy sets and shadowed sets. Note that for any a in [0, 1] we have

and

hence the operations of union and intersection are straightforward. We remark that the result of mixing fuzzy sets and shadowed sets arises in the form of interval valued sets.

3.16. The frame of cognition

So far we have discussed a single fuzzy set and proposed its several conceptually different scalar characterizations. What really matters in most applications of fuzzy sets technology are the families of fuzzy sets. We usually refer to them as a frame of cognition.

3.16.1. Basic definition

The notion of a frame of cognition forms a fundamental concept when dealing with fuzzy sets. It will become evident that this notion emerges in the fuzzy modelling, fuzzy controllers, classifiers, etc. Primarily, any use of fuzzy sets call for some form of interfacing with any real world process. Generally speaking, the frame consists of several normal fuzzy sets called also linguistic labels or linguistic landmarks that are used as basic reference points for fuzzy information processing. Sometimes in order to emphasize their focal role in this processing they are referred to as linguistic landmarks. When the aspects of fuzzy information processing need to be emphasized, we may refer to these fuzzy sets as a fuzzy codebook - a concept widely exploited in information coding and its transmission. By adjusting the granularity of the labels one can easily implement the principle of incompatibility. In particular, this allows us to cover a broad spectrum of cases concerning information granularity spreading between that of a qualitative form (conveyed by symbols) and numbers. Definitely, the numerical information exhibits the highest level of information granularity.

Let us now get into a more formal definition. A frame of cognition (Pedrycz, 1990; 1992)

is a collection of the fuzzy sets above is defined in the same universe of discourse X and satisfying the following conditions:

(i)  coverage: “covers” X, that is any element of x ∈ X belongs to at least one label of . More precisely, this requirement can be written down in the form

Being more stringent, we may demand an ε-level of coverage of X, that formalizes in the following form,

where ε ∈ [0, 1] stands for the coverage level. This simply means that any element of X belongs to at least one label to a degree not less than ε. Otherwise, we can regard this label as a representative of this element to a nonzero extent. The condition of coverage assures us that each element of X is sufficiently represented by . Moreover, if the membership functions sum up to 1 over X

then the frame of cognition is referred to as a fuzzy partition.

(ii)  semantic soundness of : this condition translates into a general requirement of a linguistic “interpretability” of its elements. Especially, we may pose a few more detailed conditions characterizing this notion in more detail; see also Pedrycz and Oliveira(1993):

  Ais are unimodal and normal fuzzy sets; in this way they identify the regions of X that are semantically equivalent with the linguistic terms,
  Ai are sufficiently disjoint; this requirement assures that the terms are sufficiently distinct and therefore become linguistically meaningful.
  The number of the elements of is usually quite reduced; some psychological findings suggest 7 ± 2 linguistic constitute an upper limit for the cardinality of the frame of cognition when being perceived in the sense of a basic vocabulary of the linguistic terms.


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