Fuzzy Logic Introduction
Laboratoire Antennes Radar Telecom, F.R.E CNRS 2272, Equipe Radar Polarimetrie, Universit´ de Rennes 1, UFR S.P.M, Campus de Beaulieu - Bat. 22, e 263 Avenue General Leclerc, CS 74205, 35042 Rennes Cedex, France
1.I NTRODUCTION Fuzzy Logic was initiated in 1965 , , , by Lotﬁ A. Zadeh , professor for computer science at the University of California in Berkeley. Basically, Fuzzy Logic (FL) is a multivalued logic, that allows intermediate values to be deﬁned between conventional evaluations like true/false, yes/no, high/low, etc. Notions like rather tall or very fast can be formulated mathematically and processed by computers, in order to apply a more human-like way of thinking in the programming of computers . Fuzzy systems is an alternative to traditional notions of set membership and logic that has its origins in ancient Greek philosophy. The precision of mathematics owes its success in large part to the efforts of Aristotle and the philosophers who preceded him. In their efforts to devise a concise theory of logic, and later mathematics, the so-called ”Laws of Thought” were posited . One of these, the ”Law of the Excluded Middle,” states that every proposition must either be True or False. Even when Parminedes proposed the ﬁrst version of this law (around 400 B.C.) there were strong and immediate objections: for example, Heraclitus proposed that things could be simultaneously True and not True. It was Plato who laid the foundation for what would become fuzzy logic, indicating that there was a third region (beyond True and False) where these opposites ”tumbled about.” Other, more modern philosophers echoed his sentiments, notably Hegel, Marx, and Engels. But it was Lukasiewicz who ﬁrst proposed a systematic alternative to the bi–valued logic of Aristotle . Even in the present time some Greeks are still outstanding examples for fussiness and fuzziness, (note: the connection to logic got lost somewhere during the last 2 mileniums ). Fuzzy Logic has emerged as a a proﬁtable tool for the controlling and steering of of systems and complex industrial processes, as well as for household and entertainment electronics, as well as for other expert systems and applications like the classiﬁcation of SAR data. 2. F UZZY S ETS AND C RISP S ETS The very basic notion of fuzzy systems is a fuzzy (sub)set. In classical mathematics we are familiar with what we call crisp sets. For example, the possible interferometric coherence g values are the set X of all real numbers between 0 and 1. From this set X a subset A can be deﬁned, (e.g. all values 0 ≤ g ≤ 0.2). The characteristic function of A, (i.e. this function assigns a number 1 or 0 to each element in X, depending on whether the element is in the subset A or not) is shown in Fig.1. The elements which have been assigned the number 1 can be interpreted as the elements that are in the set A and the elements which have assigned the number 0 as the elements that are not in the set
Figure 1: Characteristic Function of a Crisp Set
A. This concept is sufﬁcient for many areas of applications, but it can easily be seen, that it lacks in ﬂexibility for some applications like classiﬁcation of remotely sensed data analysis. For example it is well known that water shows low interferometric coherence g in SAR images. Since g starts at 0, the lower range of this set ought to be clear. The upper range, on the other hand, is rather hard to deﬁne. As a ﬁrst attempt, we set the upper range to 0.2. Therefore we get B as a crisp interval B=[0,0.2]. But this means that a g value of 0.20 is low but a g value of 0.21 not. Obviously, this is a structural problem, for if we moved the upper boundary of the range from g =0.20 to an arbitrary point we can pose the same question. A more natural way to construct the set B would be to relax the strict separation between low and not low. This can be done by allowing not only the (crisp) decision...
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