75.

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ranges [100, 115] and [135, 150] trigger the rule at lower strength, and pixel values outside the range [100, 150] have a membership grade of 0.

The example given above is an extremely simple case. In practical situations, a multidimensional input space and multiple classes are involved. In addition, the rules being triggered can be numerous, because the fuzzy membership functions normally overlap with each other. Hence, a pixel value falling within the overlap area will simultaneously trigger several rules. A final solution requires the use of inference and defuzzification.

4.4.2 Inference

The inference stage computes the strength contributed by the triggered rules, and aggregates those triggered rules. The process can be illustrated by the following example showing an automatic air-conditioner controller with two-dimensional inputs: one is a measure of the current room temperature, denoted by s1, the other is a measure of how quickly the room temperature increases, denoted by s2. Assume that the automatic airconditioner controller is handled by the following three rules:

Rule 1: IF room temperature is high and the increase in room temperature is high

THEN the control should be turned to high

Rule 2: IF room temperature is low and the increase in room temperature is middle

THEN the control should be turned to low

Rule 3: IF room temperature is middle and the increase in room temperature is low

THEN the control should be turned to medium

where if-clauses high, middle, low, and then-clauses high, low, medium are all modelled by a triangular membership function as shown in Figure 4.10. For each rule, the inference engine firstly generates the membership grades for the measurement s1 and s2, respectively. Both membership grades are then combined through an interaction () operator, which is defined in Section 4.1.2 to be equal to the minimum operator.

If both membership grades are equal to one, i.e. the rule condition is fully satisfied, then the then-clause in the rule should be fully adopted (i.e. with full strength). On the other hand, if the rule condition is only partially satisfied, the then-clause should be partially weighted. Two weighting approaches, known as multiplication and minimisation, are commonly used. In mathematical terms, the multiplication weighting method for rule 1 can be expressed as:

(4.30)

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