ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
to class i is p i (X) . The task of classification is to classify 
the object pixel j(j EE {1,2,* * *, TV}) into a certain category 
i according to its observation values vector X . The 
classification should obey a certain rule. And minimizing the 
probability p (/) of mistaken classification is generally 
required. That is to say, 
A A 
p mc {i) = p r {c(x) * i,c(x)e {1,2, —,c}|/ =/} is a 
A 
minimum. In this equation, I is the actual class and C is the 
classifier having the function shown as follows. 
A 
c : {XjJ = 1,2,**-, A^} -> {1,2,---,c} (Eq.1) 
According to the Bayesian theorem, the posterior 
probability p(i / a:) can be calculated. Under the condition of 
A 
minimizing the mistaken classification probability, C can be 
described as: 
C = / , if p(i/ x) = max(// x) (Eq.2) 
!<c 
To evaluate the classifier precision is to calculate the probability 
of correct classification. 
p(correct / X = x) = p(c{x) ^ 
= max p(i / x)\X = x) - max p(i / x) 
/ 1 / 
In Eq.3, picorrect IX — X) stands for the wholly 
A 
accessible probability expectation value of c(x) theoretically. 
As for a certain classification procedure, the strict precision 
evaluation needs the error matrix to be calculated. The error 
matrix E can be described as follows. 
E - {e tJ ) i,je{l,2,---,c} (Eq.4) 
In Eq.4, 6 tJ refers to the number of objects or pixels that 
have been classified into class i but actually belong to class J . 
According to it, the indexes concerning the classification 
precision can be calculated, such as the whole classification 
precision and Kappa coefficient, etc. 
From the above, the major mathematical procedure of a 
typical image classification method can be seen. We may 
expand it to the fully fuzzy aspect. That is to say, the fuzzy 
characters should be efficiently described and processed during 
the whole classification procedure including training, classifier 
design and precision evaluation of its products. 
Designing the fuzzy classifier is comparably easy to come 
true in the whole fuzzy process. On the one hand, the popular 
method is fuzzy clustering for the unsupervised classification. On 
the other hand, the supervised classification method can 
theoretically be made fuzzy, which is an important part of this 
paper. The different fuzzy methods may be taken according to 
the their different classifier. But they have something in common 
that is the middle results closest to the output disperse 
categories, such as p(i / X) shown in Eq.2, act as a direct or 
indirect evidence for calculating the fuzzy affiliation degree 
Hi (x). 
The precision evaluation of the fuzzy classification products 
uses the routine ways, which are not fuzzy and fuzzy 
respectively. The precision evaluation indexes of fuzzy 
classification include entropy, cross-entropy, disperse degree, 
correlation coefficient, dot metrix and Euclidean distance. This 
paper only describes the calculation equations of entropy and 
cross-entropy simply. 
entropy - - (■*) l°g2 Ei (•*) ( Ec ^- 5) 
ie{l,2,-”,c} 
In Eq.5, the entropy stands for the out-of-order degree of the 
classification results. The higher it is, the higher the fuzzy degree 
is, which indicates the lower the classification precision is. 
Actually, the value of the entropy only stands for the fuzzy 
degree. As for a natural phenomenon that is originally very fuzzy, 
maybe any of the fuzzy classification methods could describe its 
fuzzy characteristic rather precisely. So there is no clear reverse 
one-to- one correspondence between the value of entropy and 
precision. In this case, the more objective and fair-minded 
precision evaluation should be based on the cross-entropy. 
cross - entropy = - ^/Z f U)log 2 g, (x) 
/e{l,2,---,f} 
+ ^Mi(x) log 2 //,.(x) 
I€{ 1,2,- 
(Eq.6)