The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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contrast, the prototype space makes it possible to analyse and 
study all bands based on the similarity of their physical 
properties with respect to all phenomena (classes). Since this 
kind of BS is conducted in prototype space, we call it prototype 
space BS (PSBS). 
The rest of this paper is organized as follows: in section 2, the 
components of the proposed BS are introduced. The description 
of data and our experimental results are discussed in section 3, 
and then the conclusions are given. 
2. PROPOSED METHOD 
2.1 Band Representation 
In general, feature vectors consist of a set of elements that 
describe objects. From a pattern recognition point of view, a 
space should possess some particular properties; so that a finite 
representation of objects can be characterized for the learning 
process (Pekalska, 2006). For hyper spectral images, a feature 
vector is defined in terms of spectral response of pixel 
x=[x lr x 2 ,...,JC„] T , where n is the number of bands. Hence, the 
pixels of an image are represented in the feature space (axes of 
this space are made by bands of hyperspectral data) to perform 
different types of analyses, such as clustering and classification. 
This representation is thereby appropriate for image 
classification tasks. 
The BS methods try to analyse similar bands for selecting 
effective bands to represent pixels in subspace with low 
dimensions. Intuitively, to study and find relevant bands, it is 
reasonable to express bands in terms of their properties. 
Accordingly, we propose to represent bands for this analysis in 
a new space called the prototype space (see Figure 2). In this 
space, a band is characterized in terms of the spectral response 
of different classes to pose reflect properties of classes to bands. 
Indeed, feature vectors in this space describe the band 
behaviour in terms of their reflectance in dealing with imaging 
scene phenomena. Let us assume that the pixels in an image 
belong to L classes and the spectra are given by n bands and 
that classes can be represented by a single prototype spectrum, 
e.g. the class mean. Figure 1 shows the spectra of L classes in 
spectral space. We can then denote the characteristic vector of 
band i hr=[m u ,m 2 i,..., »*//.,...,/w L ,] T in prototype space, where m# 
is the mean of class j in band 
Hence, the prototype space has L dimensions. We will use it to 
study and cluster bands based on the similarity of their 
behaviour. 
Figure 1. Example of the 3 prototype spectra for 3 classes 
In physical perspective, the reflectivity of phenomena in bands 
when situated in a block like 'R' in Figure 1 is the same. Hence, 
it can be argued the bands of this block are highly correlated 
and are redundance. These highly correlated bands are 
represented near each other in prototype space, and constitute a 
cluster. This situation can also occur for blocks 'p' and 'q', 
which are spectrally far away. A cluster analysis in the 
prototype space thereby finds spectrally similar bands. 
2.2 PSBS Method 
This method tries to distinguish highly correlated bands by K- 
means clustering and proposes optimum band subset as multi- 
spectral representatives of hyperspectral data. 
Let h, be an L-dimensional bands with components which 
represents the prototype space. We wish to cluster n bands in 
the prototype space to c clusters by k-means clustering. The 
final goal is to find a subset of bands to reduce the 
dimensionality n of the original spectrum to multispectral 
(Equation 1) in such a way as to maximize classification 
accuracy of data in reduced dimension c. 
s=W x (1) 
K-means clustering is an iterative clustering algorithm where 
for each data point hj, we introduce a corresponding set of 
binary indicator variables r ik e {0, 1}, where k = 1, . . . , c 
describing which of the c clusters the data point hj is assigned to, 
so that if data point hj is assigned to cluster k then r ik = 1, and 
= 0 for j = k. This is known as the 1-of-c coding scheme. 
We can then define an objective function, sometimes called a 
distortion measure, given by 
^=X£ r *|hi -t** II 
i =1 k =1 
which represents the sum of the squares of the distances of each 
data point to its assigned vector p k . Our goal is to find values 
for the { r ik } and the {p k } so as to minimize J. We can do this 
through an iterative procedure in which each iteration involves 
two successive steps corresponding to successive optimizations 
with respect to the r ik and the ji k . First we choose some initial 
values for the n k . Then in the first phase we minimize J with 
respect to the r ik , keeping the p k fixed. In the second phase we 
minimize J with respect to the p k , keeping r ik fixed. This two- 
stage optimization is then repeated until convergence.