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