3. A Statistical Fuzzy Neural Classifier
3.1 Fuzzy Clustering
In the fuzzy c-means algorithm, the position of a class or cluster center is found to be the average of the positions of all the patterns in that cluster, based on minimizing the sum of the variances of all variables i within a domain D for each pattern in each cluster l. And membership function
, is introduced to weight the distance measure and to define the problem of finding the fuzzy c-partitions with a fuzzy index m,
. That is, to adjust the position of , the cluster center, by minimizing the fuzzy c-means functional
 |
(6) |
where
 |
(7) |
is a fuzzy c-partition of X , X=(x
1, x
2, x
N) is a set of training N vector, and
is the membership of the i
th pattern to l
th cluster;
is the cluster center of X; In [Bezdek, 1987), the distance measure is defined by
where
denotes the matrix norm. In order to incorporate the statistical information of SAR, the distance in fuzzy c-means functional is replaced by the following form
 |
(8) |
where is class center of the l-th class, C
1 is the feature covariance matrix of class l, and Tr denotes the trace of a matrix. Without a prori information, an equal probability for each class is assumed. To merge multifrequency data, a linear combination leads to a similar distance measure as
 |
(9) |
where C
i(J) and Z
i(J) are the feature matrix and the covariance matrix for j-th band, and K is the total number of frequency bands.
Differentiating
with respect to
and applying the constrain
 |
(10) |
to find the minimum of
, we obtain
 |
(11) |
and
 |
(12) |
In fuzzy c-means algorithm, the fuzzy membership is obtained by iterative procedure of Equation (11) and (12).
3.2 Neural Implementation
The classification scheme used in this study is a neural network, called dynamic learning neural network (DLNN). DLNN has been applied in many applications. Such a neural network plays the role of connecting the input feature vector of SAR and the output membership vector. The input-output relationship can be simply expressed as
y=Wx (13)
where y is the output nodes and x is the input nodes. In this study, they are the membership vector of each pixel to a specific class and the feature vector of SAR, respectively. W is the weighting matrix of neural network. And the weighting matrix W is tuned by training process. By the use of the fuzzy membership vector as the output, DLNN becomes a fuzzy neural network (FDL)[Tzeng and Chen, 1997].
Fig 3: Configure Setup of FDL
The training process is a standard procedure for a supervised neural network. The necessary training set is formed by a set of input information and desire output. For purpose of using complete polarimetric information, the input data is the covariance matrix, and the desire output is the corresponding membership vector. The training set must be sufficient and non-ambiguous. Fig. 1 shows the configuration of the neural network.
