ONING 
sina.com 
ture 
age texture is 
co-occurrence 
ment function 
h synthesizes 
tial reasoning. 
robability are 
vided, and the 
habitant area, 
ique based on 
fused features 
"n indicate the 
reliable, and 
HAFER'S 
TEXTURE 
re 
n from image 
a method of 
AX) is a real 
d self-similar 
nction having 
rown’s fractal 
E[| AX+0X)- AX) pL ox" c (3) 
Therefore 
lgEL LL X*0X)- fX) JUH-IgOOXO- lgC.— (4) 
Where H= a self-similar parameter 
C^ a rational constant 
Therefore Eq.(4) is a linear equation. If the Brown's fractal 
function AX) is used to simulate gray scale surface of image 
texture, the sum of squared errors is 
A maxi|AX| e 5 5 
e'- Y (gE[RX-AX)- fDOl]-H*lgll AX |-lgC)" (5) 
min|| AN 
where X x, y) E), 
AX)= gray scale at X. 
The fractal dimension f; can be obtained by the following steps: 
First of all, E[ | AX+11X)- AX)| ] (LUX=11120..., K) can be 
respectively calculated, where |[{X+X)-AX)|= Hata![ Jfoxy* 1 
PA) + fort Dx) Ax) Hx xp + p)Axp)| T; Then H 
and /gC of the isomorphic fractal model are calculated 
according to the least-square method; finally f; is obtained 
according to Eq. (2). 
2.2 Measurement of entropy feature based on gray co- 
occurrence matrix 
The co-occurrence matrix P(i,j,0,0) (or P(i,j,Ax, Ay)) of a 
image f(x, y) describes the probability for gray scale i and j (ijj 
i[0:g-1]) to occur at two pixels separated by distancedand 
directionO(or by displacement Ax and Ay), it can be written as 
P(ij,0,0)= P(ij, Ax, Ay) 
=P{f(x,y)=i and Ax+Axy+Ay)=7} (6) 
Separated co-occurrence matrices can be established for each 
combination of distance and direction. A set of 14 features 
based on a co-occurrence matrix was proposed by Halarick etc.. 
Once the co-occurrence matrix has been formed, texture feature 
can be computed. Since we are interested in rotationally 
invariant texture feature for classification, first, we specify the 
distanced, =max| |Ax| , JAy| ], and the co-occurrence matrixes of 
all directions (6=0 ,45 ,90 ,135 ) are computed; Then features 
are computed from co-occurrence matrixes; finally the average 
features can be obtained for texture classification. Some co- 
occurrence matrix-based texture features correspond to 
characteristics that are recognized by the eyes, but many do not. 
Experiments show entropy feature of gray scale co-occurrence 
matrix is one of the feature having the best discriminatory 
power. Here is given the entropy formula 
f I? 2 P(i,0,0)log; P(i,j,0,6) (7) 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
2.3 Feature fusion based on Dempster-Shafer reasoning 
theory for texture classification 
2.3.1 Dempster-Shafer reasoning theory: The Bayesian 
theory is the canonical method for statistical inference problems. 
Dempster-Shafer decision theory is considered a generalized 
Bayesian theory. It allows distributing support for a proposition 
not only to the proposition itself but also to the union of 
propositions. In a Dempster-Shafer reasoning system, all 
possible mutually exclusive context facts (or events) of the 
same kind are enumerated in “the frame of discernment ." 
Each texture feature / will contribute its observation by 
assigning its belief. This assignment function is called the 
“probability mass function” of the feature /, denoted by m;. So, 
according to feature fs observation, the probability that “the 
detected texture is A” is indicated by a “confidence interval”: 
[Belief; (A), Plausibility(A)] (8) 
The lower boundary of the confidence interval is the belief 
confidence, which accounts for all evidence A; that supports the 
given proposition “ texture 4”: 
Belief;(A)= D m;(A;) (9) 
A;GA 
The upper boundary of the confidence interval is the 
plausibility confidence, which accounts for all the observations 
that do not rule out the given proposition: 
Plausibility(A)=1- X m;(4,) (10) 
ó 
An A= 
For each possible proposition, Dempster-Shafer theory gives a 
rule for combining feature f/s observation m; and feature f/'s 
observation m;: 
2 m;(A;)m;(A;) 
a LT. H 
(m; € m; XA) » 2 m Am; (4j) un 
Aj Aj = 
(m; & m ;)(A) is called combined probability mass function. 
This combining rule can be generalized by iteration: if we treat 
m; not as feature f’s observation, but rather as the already 
combined (using Dempster-Shafer combining rule) observation 
of feature f; and feature f;. 
Compared with Bayesian theory, Dempster-Shafer theory of 
evidence feels closer to our human perception and reasoning 
processes. Its capability to assign uncertainty or ignorance to 
propositions is a powerful tool for dealing with a large range of 
problems that otherwise would seem intractable.