66.

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Keller, 1993, 1996) is proposed as an alternative, as it provides a means of estimating the absolute membership grade of each class independently of all other classes.

4.3 Fuzzy maximum likelihood classification

The crisp maximum likelihood classification algorithm is described in Chapter 2. It uses the ‘hard’ mean and ‘hard’ covariance matrix of each cluster. In what follows, we show that fuzzy set theory can be extended to the maximum likelihood algorithm to measure the membership grade of the pixels. This extension is used by Wang (1990) and Maselli et al. (1995).

Based on probability theory, if an event A is a precisely (hard) defined set of elements in the universe of discourse ψ, the probability density function of A denoted by P(A) can be expressed by:

(4.19)

where s denotes the element in ψ, and H_A is a hard membership function, i.e. H_A(s)=0 or 1. In the case of image classification, event A is the cluster or class, and s is a vector of feature values associated with a specific pixel. The membership function H_A thus describes s either as belonging to A (i.e. membership grade is 1) or not (i.e. membership grade is 0).

If A is regarded as a fuzzy event, which means that set A is a fuzzy subset in ψ, a probability measure of A becomes:

(4.20)

The term μ_A is the fuzzy membership function as defined in Equation (4.1). Equation (4.20) is an extension and generalisation of Equation (4.19). Even partial membership value of observation s in A can provide a contribution to the total probability P(A).

The mean and variance of fuzzy set A relative to a probability measure can similarly be quantified as:

(4.21)

and

(4.22)

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