Page 299 - Digital Analysis of Remotely Sensed Imagery

Spectral Image Analysis 261

calculated from the pixels belonging to respective clusters using the
following formula:
k
n
SSE = ∑ ∑ [ DN( , ) m j ] 2 (7.4)
−
ij
= j 1 = i 1
where n = number of pixels enclosed in a given cluster. Its
specific value varies from cluster to cluster
DN(i, j) = value of the ith pixel in the jth cluster
m = mean of the jth cluster
j
Moving clustering is carried out iteratively. At the end of each
iteration, the center of each cluster m is updated with the mean value
j
of all pixels comprising that group. The entire process of assigning
pixels to one of the updated candidate clusters is reiterated using the
newly derived cluster mean. As the clustering process continues, the
updated cluster mean gradually approaches the genuine mean. In
other words, SSE is going to become stabilized and leveled off. There
are two means by which the iteration process is terminated: either the
number of iterations reaches the specified value or the SSE conver-
gence threshold (e.g., the amount of variation in the membership of
all clusters from one iteration to the next) is reached. Obviously, the
number of iterations required to reach the SSE threshold is affected
by the initial arbitrarily selected cluster centers. The closer these cen-
ters are located to the genuine ones, the fewer number of iterations
are required to reach the final result. A sensible approach of allocating
the initial means is to determine the DN range in each band. The
mean is obtained by dividing this range by the number of clusters.
Then the means for each cluster equals the increment of the quotient
plus the minimum DN.
The process of K-means clustering analysis is best understood by
examining Fig. 7.4 in which there are eight pixels in the two spectral-
band domain. During the first iteration, two cluster centers are ran-
domly chosen by the computer. Five of the pixels fall into the first
cluster while the remaining three are grouped into the second cluster
(Fig. 7.4a). After this iteration the two clusters produce an SSE value
of 93. During the second iteration, the cluster means have been
updated using the member pixels in the corresponding cluster. Four
of the five pixels still stay inside this cluster while another is assigned
to the second cluster (Fig. 7.4b). After this iteration SSE decreases to
65.52 (Fig. 7.4c). At the end of the third iteration, a few pixels switch
their membership regimes. Consequently, cluster 1 comprises five
pixels, but cluster 2 encompasses only three (Fig. 7.4d). SSE continues
to decrease to 55.75. At the fourth iteration, the membership composi-
tion of both clusters does not change at all (Fig. 7.4e). However, SSE
decreases further to only 20.27. Thus, the process of clustering is ter-
minated after four iterations.