3.1 Fuzzy sets and fuzzy representation Let X be a space of pixels with a generic element of X denoted by x, thus X={x) A={x,ƒA(x)}; xεX (1) Membership in a fuzzy set A of X is characterized by a membership function ƒA, which is associated with each x, a real number in [0, 1]. ƒA(x) represents the “grade of membership” of x in A (Zadeh, 1965). The closer the value of ƒA(x) is to 1, the more x belongs to A. A fuzzy set does not have sharply defined boundaries, and a set element may have partial membership (Wang, 1990). 3.2 Fuzzy parameters and Training data The membership function ƒA(x) of fuzzy sets are chosen either on the basis of expert knowledge (e. g. boundary values of discriminating criteria chosen by custom, law or an external taxonomy), or by using methods of numerical taxonomy (Burrought and McDonnell, 1998). The first approach is known as the Semantic Import (SI) model, where classes based on expert knowledge are usually imposed or imported classes that are set up without direct reference to the local data set. The second approach is known as the Similarity Relationship (SR) model, where the value of the membership function is a function of the classifier used. One commonly used version of SR model is fuzzy k-means or c-means method. In this research, the fuzzy supervised classification proposed by Wang (1990) was adapted to classify the ASTER imagery. In the supervised mode of fuzzy classification, the fuzzy-membership functions are elicited from training data that support an approximate yet more explicit definition of land-cover memberships. Such a fuzzy classifier is built on histograms of spectral data (Zhang and Goodchild, 2001). A histogram (Figure 1) indicates the distribution of the relative frequencies of certain measurements (e.g., digital numbers of spectral reflectance). Such a histogram was created from training data consisting of a number of pixels representative of their class prototypes, which were rarely uniformly distributed. An important feature for the fuzzy definition of class membership is the maximum relative frequency of a named class, relevant to a specific data source, as denoted by ƒmax in Figure 1.  Fig.1 Histogram for defining fuzzy class memberships Suppose a pixel x is to be fuzzily classified, and has a digital number dn, which is associated in the histogram with a relative frequency ƒ(x). It seems reasonable to take the ratio ƒ(x)/ƒmax as approximating the fuzzy membership of pixel x in class UA, that is: UA (x) = ƒ(x)/ƒmax (2) Where ƒx and ƒmax stand respectively for the relative frequency of value dn and the maximum relative frequency recorded for class UA. In this way, the partition matrix of the training classes can be developed. After defining the fuzzy partition matrix, Equations (3) and (4) are applied to each row of the matrix to generate a fuzzy mean and fuzzy covariance matrix for each class.    The study area is about 400km south of Perth and covers an area of 40 square kilometers. This is a part of the Lower Great Southern region of Western Australia. Via the south coast highway, it is 20 km to the north-east of Albany, the regional center of the Great Southern. Figure 2 shows the location of the study area.  Fig. 2 Location of the study area with training sites 5. Implementation of the fuzzy supervised classification In this study, the fuzzy algorithms were tested on three visible bands of ASTER imagery. Five land cover classes were selected after intensive study of the aerial photographs and vegetation maps of the study area. The land cover classes are mainly pine forest, old eucalyptus, young eucalyptus, mixed forest and bare ground. 5.1 Selecting the training sites The training sites of these classes were digitized from the ASTER image. Location of the training sites is also shown in figure 2. These training sites were created as polygons in vector format. They were then converted into raster format to implement the fuzzy classification. |