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intensity values in a given spatial relationship. Co-occurrence matrices are based on the relative frequencies p(i, j) with 
which two pixels with a specified separation occur in the image, one with gray level I and another with gray level J. The 
separation is usually specified by distance vector @(d,c) . Pixel distance d and angular orientation « are parameters of a 
particular co-occurrence matrices. Using different parameters, several matrices can be derived. The matrices obtained are 
then used for extraction of texture features. 
For the image of N gray level, co-occurrence matrices cm can be obtained by estimating the pairwise statistics of pixel 
intensity. The size of cm is determined by the number of gray level N in the input image. The matrices cm are a functions 
of the angular relationship between the pixels as well as a function of the distance between them. The matrices can be 
illustrated as following: cm(d,o) =[p(i, jld,a)]. After introducing the symmetry (Burns & Smith 1996), we can 
only consider the angular o up to 180° rotation. The value d normally be chosen as 1. An example of a 4x4 image with 
four gray levels and the computation of the co-occurrence matrices for d=/ and o varying from 0° to 135 ° by 45° 
increments are shown in Figure 7: 
0 1 2 3 (Gray levels) 
  
1 1 #(0,0) #(0,1) #(0,2) #(0,5) 
] 11 
1 #(1,0) #(1,1) #(1,2) #(1,3) 
> 15 ; 
(a) image of 4 gray levels 
  
#(2,0) #(2,1) #(2,2) #(2,3) 
3 #(3,0) #(3,1) #(3,2) #(3,3) 
(Gray levels) 
(b) general form of any co-occurrence matrix (#(i,j) is the number of times gray level 
i and j have been neighbors) 
4:2 1: 0 4 1 0 0 6 0 2 0 2 1 370 
2: 47919 ) 4 aig 0°4 70 1 
cm(1,0°) = cm(1,45°) = cem(1,90°) = em 135) = 
1 0 6 1 0 2 4 1 2 2 2 ? 3 1 0.2 
0 0 1! 2 0 0 1 0 00.2 0 0 0 2 0 
Figure 7 Co-occurrence matrices for four given distance vectors (taken from Haralick[1973]) 
In texture classification, individual elements of the co-occurence are rarely used. Instead, features are derived from the 
matrix. A large number of textural features have been proposed starting with the original fourteen features described by 
Haralick, however only some of these are in wide use. The features we used are listed as following: 
1. Inverse Difference Moment 2. Contrast 3.Entropy 
f,» X PG, i G- jy LS PG f, == PG, j)log PG, j) 
i,j LJ] LJ 
4. Correlation 5. Energy (angular second moment) 
f, - MG-u)G-n,FG. j)o,o, fx S P 
y, and y, are the means and o,and o, are the standard deviations of i and j respectively. Rotating &œ, there are 4 
values for every texture feature. Then the minimum and maximum values of the texture features can be obtained. Since we 
only want to find tree areas, the number of samples for tree category is small. We use a min-max decision rule for 
classification of the image based on their texture features. The procedure is repeated for all the image blocks in the image. 
The decision rule is described by the following equation. If the equation is satisfied, the processed image block j can be 
assigned as catalogue K. 
N 1 X 1 
Il rR II e (2) 
n=l j 
b 
image blocks. n is the number of texture features 
aida D, Ad. A 
define the minimum and maximum texture feature values of the training samples and processed 
nj 
Figure 8 illustrates the delineation tress areas using the steps in Figure 6. Using image segmentation method described in 
Section 4.2.1, most of the tree areas can be delineated. Some areas obtained are not correct. Using the texture analysis and 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 525