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provides the basis for the development of more robust approaches to the remote sensing classification problem. An important field of remotely sensed image analysis close to fuzzy methodology is spectral unmixing (Schowengerdt, 1996). The techniques for solving this kind of problem are considered in the final section of this chapter.

4.1 Introduction to fuzzy set theory

The difference between crisp and fuzzy sets can be characterised by means of a membership function. The membership function in a crisp set can only output two choices, {yes, no} or {0, 1}. In other words, an element of a crisp set can be a member of only one group, for which it has a membership grade of 1. The concept of the fuzzy set softens this constraint and allows the concept of partial membership such that one data element may simultaneously hold several non-zero membership grades for different groups or clusters. Clearly, in comparison with crisp approach, the fuzzy concept allows greater flexibility.

4.1.1 Fuzzy sets: definition

Let S represent a universe of discourse composed of generic elements denoted by s. A fuzzy subset G of S is determined by a membership function μ_G, which assigns a membership grade within the interval [0, 1] to each element s. The membership grade can be expressed by:

(4.1)

Such a membership mapping mechanism is quite different from the traditional crisp set approach in which membership grade must be either 0 or 1. The fuzzy set approach is much more flexible for handling problems of indistinct boundaries that are common in the natural world.

Figure 4.1a shows the traditional crisp set concept in which membership grade is normally a step function which outputs values of either 0 or 1, meaning that no intersection is allowed between the clusters, a1 and a2. However, in fuzzy set terms, as shown in Figure 4.1b, different clusters, e.g. a1 and a2, may share some units, and the membership grade of a unit will generally decrease as the distance between the unit and the cluster centre increases.

Let μ_G(s) denote the grade of membership of s in a fuzzy subset G that can be expressed as:

(4.2)

In the continuous case, G becomes:

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