2.2 Cluster Assignments Using Bayes’ Rule
After each class training data has been represented by the given set of clusters, clusters can be recognized as an
associated spectral classes of information classes.
The cluster vector is identified by its index number k, k= 1,2,3, . . .K.
H
i(k), k=1,2,3, . . .K represents the probability
distribution for class
wi in the cluster-space.
H
i(k), i=1,2,3, . . .M gives the relative likelihoods that the cluster vector
k belongs to each defined class. Cluster index k can be classified into the information classes using Bayes’
Theorem. Assuming that prior probabilities are equal for all the classes, the decision rule is:
kÎ wi, if Hi(k)>Hj(k)
for all j>¹i (1)
This cluster assignment process is simple, fast and reliable (assume that the training data is
reliable) for the following reasons: (i) This is a one dimensional problem; (ii) The number of
data to classify (K) is small; (iii) The true probability distribution shapes are used.
The clusters labels,
wi, are stored for later use in classifying
unknown data. If the probability
values H
i(k) are recorded as well, it will be possible to provide information on how reliable the
classification results are.
Since several clusters may be labeled as the same class generally, each class has been defined by
a group of clusters. In other words, instead of using a single mean position as a class signature,
several cluster mean positions are used. This multi-signature structure generates a flexible and
adaptable representation which is wholly determined by the training data itself.
2.3 Unknown Data Classification
There are two steps in labeling an unknown pixel vector.
An unknown pixelvector x will be first classified into one of the given set of clusters (k) using
the minimum distance classifier. The dicision rule is:
xÎk,
if d(x, ck)2<d(x,cj)2
for all k¹j (2)
where c
k is the mean vector of the cluster k data and
d(x, c
k)
2=(x –c
k)
t (x –c
k).
The second step is to label the data into the classes with which the cluster is associated.
xÎ wi,
if kÎ wi. (3)
3. Experiments
An AVIRIS data set covering an area of mixed agriculture and forestry in Northwestern Indiana, USA, was used in
the following tests. It was recorded in June, 1992 with 220 bands. Water absorption bands, 104 to 108 and 150
to 162, were removed, leaving 202 bands. A traditional principal components transformation was applied to the
data and the first 40 new features, which contained 99.93% of total variance in the original data, were used for the
tests. 4 classes were selected as shown in Fig.1.
| Class Name |
Training Fields |
No. of Training Pixels |
Testing Fields |
No. of Testing Pixels |
| Soybean |
1, 2 |
121 |
3 |
156 |
| Corn |
4 |
105 |
5, 6 |
75 |
| Grass |
7 |
56 |
8 |
60 |
| Hay |
9 |
95 |
10 |
96 |
Fig. 1. The Image and the Reference Data.
The ISODATA clustering algorithm was run using the software package MultiSpec (Landgrebe and Biehl, 1999).
20 clusters were generated from the training data for the four classes. The training samples were then labeled into
the defined clusters. The results have been normalised to show the percentage (density) of the training pixels
which were labeled into each of the clusters. They are plotted in Fig. 2.
It can be seen from Fig. 2 that three clusters, Clusters 2, 4 and 6, were shared with more than one class. The rest
of clusters has been fully dominated by one class only, ie., Hj(k)=0 for j.i. Nevertheless, all the 20 clusters have
been assigned to one of the 4 classes, using the decision rule given in (1). The allocation results for all the clusters
are given in Table 1.
Each class has been represented by several clusters. The cluster vectors form the multiple signatures for each
class.
Fig. 2. The Cluster-Space Training Data Representation.
Table 1. The Subclasses of the Information Classes
| |
Soybeans |
Corn |
Grass |
Hay |
| Clusters (Subclasses) |
1,3,5,9,11,13 |
2,7,14,15,17,19 |
4,6,10 |
8,12,16,18,20 |
To assess the algorithm quantitatively, the training data and the testing data have been classified using the two-step
decision rules given in (2) and (3) and the conventional minimum distance classifier (single signature) for
comparison purposes. Table 2 shows the classification accuracy obtained from the two methods. The
classification results on the training data and the testing data have both been improved significantly.
Table 2. Overall Classification Results Comparison
| |
Training Data |
Testing Data |
| Conventional Single Signature |
78% |
59% |
| Multi-Signature |
97% |
74% |
