57.

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Chapter 4
Methods based on fuzzy set theory

Everyday language and decision-making are not generally deterministic but are usually characterised by some level of fuzziness or uncertainty. Concepts such as ‘hot’, ‘cold’, ‘good’, or ‘difficult’ contain elements of subjectivity, which is another way of saying that they cannot be completely (or deterministically) specified. One person’s ‘hot’ may well overlap with another person’s ‘warm’, for example.

The same problem can also occur in the classification of remotely sensed imagery. A considerable number of identification errors are due to pixels that show an affinity with several information classes. This type of pixel is often described as ‘mixed’, and it may be more realistic to consider an approach that acknowledges this problem, although it is not clear whether the term ‘classification’ applies to these methods (Mather, 1999b). Traditional classification methods (e.g. the k-mean clustering algorithm, or the parallelepiped classifier, described in Chapter 2) do not provide a good mechanism for coping with such uncertainty and imprecision. For example, in the case of the k-mean clustering algorithm, the formation of each cluster is in terms of competitive logic so that, once a pixel has been assigned to one cluster, its effect on other clusters is nil. For those pixels located in an inter-class overlapping area of feature space, there will be a high probability that one cluster may incorrectly include some pixels properly belonging to some other cluster. These outliers will shift the mean of the cluster and result in clustering bias.

Fuzzy set theory (Zadeh, 1965), which was triggered by these considerations, provides a conceptual framework for solving knowledge representation and classification problems in an ambiguous environment. The fuzzy concept has been adopted in different fields such as fuzzy logic control (Yamakawa, 1993; Kong and Kosko, 1992), fuzzy neural networks (Pal and Mitra, 1992; Blonda and Bennardo, 1996), and fuzzy rule base (Ishibuchi et al., 1992, 1995). The fuzzy concept is also a valuable tool for dealing with classification problems. In remote sensing classification, fuzzy-based classifiers are becoming increasingly popular. This chapter describes the main fuzzy-based classifiers. The introduction to fuzzy methodology

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