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Fuzzy Logic in Development of Fundamentals of Pattern Recognition.

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Processes of pattern recognition still remain an intriguing and challenging area of human activity. A human being can easily cope with a variety of recognition problems that are far beyond the capabilities of advanced computer programs. This paper addresses some issues of primordial interest in the understanding of principles of recognition and classification. These issues indicate clearly some properties that are essentially tied in with significant aspects of development of an appropriate cognitive perspective and linguistic problems of interpretation of classification results and their multi-membership character. How fuzzy sets, and fuzzy logic in particular, can handle numerical and symbolic computations used in classification procedures is discussed. In an application study, a role of neural networks and neurocomputations is clarified. It will be indicated how the symbolic part of computations is handled by fuzzy sets and how neural nets can contribute to specific numerical problems. A role of a suitable interface is strongly underlined ..

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PATTERN RECOGNITION: OPEN PROBLEMS AND THE ROLE OF FUZZY LOGIC.

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They are interested in two issues that play a central role in the formation of a general designing environment for classification procedures : • A cognitive perspective (and relevant computational schemes responsible for its formation and implementation ) • A membership character of classification problems and a linguistic way to evaluate grades of class membership.

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Cognitive Perspective: Formation and Development.

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there are two general approaches:. Symbolic computations - Viewed as almost a synonym of AI techniques ( Charniak and McDermott ), these generate a spectrum of very powerful representation schemes. In pattern recognition one can refer, for example , to syntactic methods being representative of them where different classes of patterns are generated by corresponding grammars. In this sense a pattern being classified is described by a string of symbols, and afterwards one checks as whether it could be generated by a certain grammar . Symbolic computations do not cope with any numerical information . Even when this type of information is available it is simply converted into plain symbolic form..

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Coping with Linguistic Description of Class Membership.

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It is also seen that this description of class membership is free of the strongly restricted assumptions that arise in probabilistic pattern classification and states that the sum of (posteriori) probabilities describing class membership equals 1. In the simplest case, for two classes w1 and w2 it reads as a constraint stating that p(w1)+ p(w1)= 1. Thus for p(w2) given the probability p(w1) is calculated as 1 - p(w2)..

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fundamental assumption that truth values of propositions are viewed as fuzzy sets defined in a unit interval. These sets represent the so-called linguistic truth values existing in commonsense reasoning. Unlike the pointwise truth values characteristic of two-valued or many-valued logics, fuzzy logic imposes a new dimension on reasoning and by means of a set-theoretic form of truth values generates a significant ability to deal with the ambiguity of reasoning processes . As said before, the linguistic truth values are represented by fuzzy sets. For instance, we can speak about truth values such as true, more or less true, false , and very true. Truth values of two terms true and false, to give an example, are specified as.

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Fundamentals of Pattern Recognition. In limits, the truth values in binary logic, false and true, are represented as single points in the unit interval, namely, 0 and 1 . Now let us address a problem of truth evaluation that immediately implies a method of matching two fuzzy statements. Consider the two statements where.

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Maximize B under constraints generated by A.. To highlight selected properties of this truth transformation, it is worthwhile to discuss some examples. We assume that both A and B have continuous membership functions..

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3. Now let B(x) be an interval-valued quantity distributed over a set X. Then T(v) is again a set of truth values defined in [0, II. It is calculated if ve VI, v2 1 0 otherwise where and v2 are determined from the relationships VI = inf A(x) v2 = sup A(x) .xex (3) (4) The second task concerns an aspect of the determination of the member- ship function of B given A and T. The truth value T acts as a functional modifier imposed on A, namely, that is, B(x) = (5) (5').

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these fuzzy truth values are converted into the corresponding truth values describing class membership of the pattern. The entire process of conversion, considered in a certain logical context, can be efficiently handled by a neural network, the main role of which is to perform the computations, translating the results in the logical space of truth valuation into the logical space of class membership. At the third step, the second problem associated with (1) refers to the determination of B when A and the truth valuation constraint are specified. Following (5), an induced fuzzy set B results as a modified fuzzy set A, where the modification is performed with the aid of r,.

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Of course, if r is the unitor function, B = A. When r is equal identically to 1, r(v)= 1, then B(x) is equal identically to 1. This indicates that the induced fuzzy set is meaningless, conveying no useful information. Now that we are equipped with the primary mechanisms of handling fuzzy logic, we can discuss the general structure of the classifier within which they will be incorporated (see Figure 4)..

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They consider a simple numerical example to illustrate how the structure of the classifier is designed and how all basic designing steps are accomplished. To avoid a complex exposition, we restrict ourselves to one of the simplest situations . Thus we discuss only one feature and distinguish two classes. For this feature, three labels forming the associated cognitive perspective are specified as trapezoidal or triangular membership functions. They are uniquely characterized by four numbers (for a trapezoidal shape) or by three numbers which holds for triangular membership functions). The labels are:.

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The learning family of patterns consists of non-fuzzy or interval-valued values of the feature and class membership. We consider the following elements of this family:.

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In this study we have summarized selected aspects of problems of pattern recognition underlying the need to cope with them in the design of classification tion structures. A key issue that should not be neglected refers to the emulation of human abilities in recognition tasks. We indicated that the notion of cognitive perspective plays a vital role. In addition to that, a relevant representation framework is necessary..

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THANK YOU!.