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Unsupervised Classification of Aerial Images Based on the Otsu’s Method 177
The unknowns in FCM clustering are:
1. A fuzzy c-partition of the data, which is a c x n membership matrix U ik Vcnwith
c rows and n columns. The values in row i give the membership of all n input data in
cluster i for k=1 to n ; the k-th column of U gives the membership p of vector k (which
represents some object k) in all c clusters for i=1 to c. Each of the entries in U lies in [0,1];
each row sum is greater than zero; and each column sum equals 1.
2. The other set of unknowns in the FCM model is a set of c cluster centers or prototypes,
arrayed as the c columns of a p x c matrix V. These prototypes are vectors (points) in the
input space of p-tuples. Pairs (U,V) of coupled estimates are found by alternating
optimization through the first order necessary conditions for U and V. The objective
function minimized in the original version measured distances between data points and
prototypes in any inner product norm, and memberships were weighted with an
exponent m>1
That is:
As X x1, x2,..., xnand
the set all Vcn real matrices of dimension c x n, with 2 c n.
Can
be obtained a matrix representing the partition follow U ik Vcn.
The basic definition
FCM for m > 1 is to minimize the following objective function:
n c
m 2
m ik
k i G
k1i1 G is a matrix of dimension pxp symmetric positive definite
Where
min z ( U; v) x v
(15)
2 t
k i G k i k i
x v x v G x v
(16)
i
n
1
m
n ik m k1
ik
k1
k
v x i 1,...,
c
1
2 m1
2
xk v
i G
ik
c
2 m1
i 1,..., c; k 1,...,
n
1
2
j1
xk vj
G
The exponent m is known as exponential weight and reduces the influence of noise when
getting the centers of the clusters. The higher the m > 1, the greater this influence. More
details on fuzzy c-means clustering [11, 12].
2.4 Methods for cluster validation
Evaluation of clustering results (or cluster validation) is an important and necessary step in
cluster analysis, but it is often time-consuming and complicated work [16].
(17)
(18)