67.

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Equations (4.21) and (4.22) determine the fuzzy mean and fuzzy variance, respectively, both of which are derived for the continuous case. In practice, the discrete fuzzy mean v_i and fuzzy covariance matrix F_i for class i can be expressed as:

(4.23)

and

(4.24)

where x_j denotes the feature vector for pixel j. If the exponent m in Equation (4.18) is set to 1, Equation (4.24) becomes equivalent to Equation (4.18), which is used in the optimum clustering algorithm. It can be inferred that, when the value of μ_i(x_j) becomes either 0 or 1 (i.e. crisp membership function), Equations (4.23) and (4.24) become the conventional crisp mean and covariance matrix formulations.

A fuzzy set is characterised by its membership function. Wang (1990) defines the membership grade for each land cover class based on the maximum likelihood classification algorithm with fuzzy mean and fuzzy covariance matrix as shown in Equations (4.23) and (4.24) as follows:

(4.25)

where k is the land cover class and probability P_i(x_j) denotes the class-conditional probability for class i given the observation x_j which is described in Chapter 2, Equation (2.23), except that the crisp mean and covariance matrix in Equation (2.23) are replaced by the fuzzy mean and fuzzy covariance matrix. This is the core of the fuzzy maximum likelihood algorithm. Calculating membership grades in terms of Equation (4.25) is equivalent to normalising the probabilities of the pixel to all of the information classes. Although such a method is quite straightforward, its validity requires further investigation.

In applications of the fuzzy maximum likelihood approach, it is questionable whether or not mixed pixels (i.e. a pixel that is a mixture of two or

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