83.

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Page 172

Table 4.1 The parameters used for generating fuzzy rules. See text for explanation

Fuzzy subspace ij	Membership grade (μ1, μ2)	β1 and β2 by operator	Class cij	Rule strength wij
i=1, j=1	Class 1=(0.3, 0.2)	β1=0.2	c11=2	w11=(0.5 − 0.2)/(0.5 +0.2)=0.43
	Class 2=(0.5, 0.6)	β2=0.5
i=1, j=2	Class 1=(0.3, 0.8)	β1=0.3	c12=2	w12=(0.4–0.3)/(0.4 +0.3) =0.14
	Class 2=(0.5, 0.4)	β2=0.4
i=2, j=l	Class 1=(0.7, 0.2)	β1=0.2	c21=2	w2l=(0.5–0.2)/(0.5 +0.2)=0.43
	Class 2=(0.5, 0.6)	β2=0.5
i=2, j=2	Class 1=(0.7, 0.8)	β1=0.7	c22=1	w22=(0.7−0.4)/(0.7 +0.4)=0.27
	Class 2=(0.5, 0.4)	β2=0.4

pixel is input to the fuzzy rule base, we first calculate the membership grades in all fuzzy subspaces for the pixel. The resulting membership grades are then combined with the rule strength as shown in step (1) of Figure 4.13b and the membership grade of an unknown pixel with respect to class x is represented by α_x (refer to step (1)). After the parameter α_x has been determined for each class x, we then allocate the pixel to the class c if α_c is the maximal, as shown in Figure 4.13b, step (2). For instance, if the unknown pixel takes values (0.6, 0.9) on the two input features, then according to Figure 4.13b, step (1) (using the minimum operator ), we obtain

4.40

where μ_i, i {1, 2} is computed using Equation (4.38). This unknown pixel is placed in class 1 because α₁>α₂ (Figure 4.13b, step (2)).

The procedures shown in Figure 4.13 can be extended to deal with higher input dimensions. However, two points should be noted. In Figure 4.13a, step (2), if several classes take the maximum value or β_x=0, for class x, the rule for the corresponding fuzzy subspace cannot be generated (one may adopt different training pixels to resolve such a problem). Similarly, in Figure 4.13b, step (2), if two or more classes take the maximum value or all α_x are zero, the input pattern will be treated as unclassified.

The strength w_ij specified in Figure 4.13a is based on the following intuitive property. If all of the training patterns within a fuzzy subspace belong to the same class, the w_ij will then reach the maximum permitted value,

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