79.

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In comparison with traditional crisp set classification, the valuable property of the three-step fuzzy classification method is that it recognises the fuzziness of the real world, and provides a means of accommodating such fuzziness. Reference to Sections 4.2 (fuzzy c-means) and 4.3 (fuzzy maximum likelihood) shows that the spirit of the fuzzy methodology is to try to involve any possible contribution (even if it is small) from the universe of discourse, then generate the solution. To deal with classification problems in this way is often more realistic than using crisp concepts.

4.5 Image classification using fuzzy rules

This section introduces an image classification methodology using fuzzy rules, as proposed by Ishibuchi et al. (1992, 1995). The core of the method is the same as the three steps described in Section 4.4, namely fuzzification, inference and defuzzification. One important property of the method is that the generation of fuzzy partitions and fuzzy rules is more automated. The user defines the type of membership functions and the number of fuzzy partitions, and selects the training pixels. Fuzzy rules are then extracted automatically according to the characteristics of the training data.

4.5.1 Introductory methodology

For simplicity, a two-dimensional input case is examined. The input features are normalised on to the range [0, 1]. Each dimension in the input space is then partitioned into k fuzzy subspaces denoted by {A₁, A₂,…A_k}, where A_i indicates the ith fuzzy subspace. A symmetric triangular membership function is adopted, and the fuzzy subspace for A_i is then defined by:

(4.38)

where s is the input pixel value, a_i is the triangular centre for fuzzy subspace i and λ is the membership function width. Both a_i and λ are defined as:

(4.39)

where k is the maximum number of partitions for the corresponding input dimension, and i runs from 1 to k.

If the same partition function and the same parameters are used to generate fuzzy subspaces for both dimensions of the input space, the result is k² fuzzy subspaces. An example of fuzzy partitions with the parameter k set

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