flowing down the tree. 
DATA COMPRESSION 
At first, we select training data for the classification by 
assigning training areas. As these training data contain 
noise, we reduce the noise by compression of training data. 
We compress them by averaging pixel densities of 
neighboring 4 pixels. This averaging process achieves both 
reduction of processing time and stabilization of boundary 
for data division. After averaging, we assign category 
number to all training data as an identifier, and merge them 
into a group. 
PROJECTING DATA ONTO 1D SUBFEATURE SPACE 
In boundary search for binary division of training data, the 
increase of the number of spectral bands reduces efficiency. 
In order to reduce the quantity of data with the minimum 
loss of information, we apply principal component analysis 
(PCA) to the merged training data and obtain the first two 
principal components. We suppose that image data have p 
spectral bands. Using variance covariance matrix X, the 
PCA process is written by 
A 
70 
BX,B = | n 
D; 
RER NET (2) 
where, A {i = 1, 2, ..., p) are eigen values and 
À,2À,Z..2 À,, andb, (i 1, 2, ..., p eigen vectors. The 
first two principal components P, and P, are obtained from 
inner products between spectral density vector assigned to 
a pixel and eigen vectors b {i = 1, 2}, respectively. For 
abbreviation, we define PCA vector P as 
Ps Fh (3) 
After compressing the training data into 2D PCA vectors, 
we produce 8 histograms from inner product among PCA 
vector P and projection vectors 
W, = cos(kx/8) + jsin(kx/8) (k = 0, … 7). (4) 
Now, the merged training data are compressed onto 1D 
subfeature space with the minimum loss of information about 
data distribution, and we obtain 8 histograms. 
SELECTION OF DIVISION BOUNDARY 
We firstly select a candidate for the optimum boundary for 
the binary division in each of 8 histograms, then determine 
the optimum boundary among the candidates. Generally 
speaking, the optimum boundary in clustering minimizes the 
ratio of within-group-sum-of-squares to intragroup-sum-of- 
328 
    
  
Boundary 
Group 1 <= | => Group 2 
Fig. 1 Valleys and boundary selected in a histogram. 
squares. We adopt a clustering criterion for the selection of 
the optimum boundary. 
We suppose that number of training data is / and a histogram 
has total-sum-of-squares $,, and assume that histogram is 
divided into two groups which have /, and /, data and within- 
group-sum-of-squares S, and S,, respectively, as shown in 
Fig.1. The total-sum-of-squares S, is written as 
S, = 5 +5 +5 (5) 
where, 5, is intra-group-sum-of-squares. We select the 
candidate among valleys in the histogram minimizing an 
index 
Rz rS: (6) 
As histograms have own dispersion in abscissa, we use 
normalized index R / 5, for the selection of the optimal 
boundary among the candidates. The boundary in an one 
dimensional subfeature space corresponds to a hyperplane 
in the full feature space. These division procedures are 
applied recursively until all groups at terminal nodes have 
identical category number. The coefficient vector projecting 
spectral density vector onto the histogram on which the 
optimal boundary is selected and threshold (position of the 
boundary) are stored at the node of the binary division tree. 
CLASSIFICATION OF WHOLE IMAGE 
After production of binary division tree from training data, 
we classify whole image data by flowing pixel data down the 
tree. At a non-terminal node, we obtain inner product 
between pixel data and coefficient vector assigned the node, 
compare it with the threshold assigned, and determine the 
division path accordingly to the result of the comparison. 
PROCEDURES 
The following is procedures of MLDF. 
1)Select training areas for categories to be classified. 
2)Apply data compression process to pixel data in all training 
areas. 
3)Label all compressed data. We use category number as 
the identifier. 
4)Merge all compressed data into a group. 
5)Apply PCA process and obtain the first two components. 
6)Produce 8 histograms from the components. 
7)Select the optimal boundary for binary division and divide 
data group into two subgroups. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996 
  
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