68.

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more information classes; see Section 4.6) should be used to construct the fuzzy mean and covariance matrix as suggested by Wang (1990). The classification results given by Wang (1990) show some level of improvement in comparison with those derived from the crisp mean and crisp covariance matrix.

4.4 Fuzzy rule base

The rule-based classification methodology described in Chapter 2 is an attempt to gather a human operator’s knowledge in order to approach the classification task. Traditional rule-based methods (such as the hierarchical decision tree method illustrated in Chapter 2) can be regarded as crisp versions. That is, each rule has equal weight (or effectiveness). The value of this kind of crisp rule-based classifier may face some problems. For instance, in the case of a hierarchical decision rule base, the classification is implemented by traversing the decision tree until an end node is reached. This traversing process is equivalent to gradually defining crisp partitions of the solution space. If the boundaries between classes are well defined, a hierarchical decision rule base should perform well. In remotely sensed imagery, information classes often overlap with each other in feature space. For those pixels lying in the overlap area of feature space, the use of the crisp decision tree methodology can result in classification error. If the crisp rule base is inferred simultaneously rather than hierarchically, the problem of rule collision (i.e. rules being triggered against each other) is also likely to happen. One solution for solving these problems is to use a fuzzy rule base, and to apply a fuzzy inference mechanism. The idea is illustrated below.

The difference between the fuzzy rule and crisp rule is that each rule in a fuzzy rule base contains a strength (also known as a weighting or a certainty) parameter. In a fuzzy rule base, the simultaneously triggering of several rules is allowed, even though these triggered rules may act against each other (for instance, rule a may classify the pixel as class 1, while rule b may classify pixel as class 2). However, since each rule is triggered by the level of strength (or certainty), a decision can be made in favour of the rule that contains the greatest strength.

A fuzzy rule base (or fuzzy system) used for classification generally comprises of three principal steps, as shown in Figure 4.5. The first step, fuzzification, involves the division of the input feature space into fuzzy subspaces, each specified by a fuzzy membership function (defined below). Fuzzy rules are then generated from each fuzzy subspace. The second step, inference, requires the calculation of the strength of each rule being triggered. The final step, defuzzification, combines all triggered rules and generates a non-fuzzy (i.e. crisp) outcome.

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