d network, 
4 and n^ 
sformation 
(4) 
the current 
exception: 
ed as 
(5) 
d into all 
important 
etworks is 
a single 
networks 
to collect 
of several 
ion of the 
nberships. 
infinitely 
set theory 
nd for all 
he mostly 
(6) 
of fuzzy 
AND-type 
e is the 
nation of 
B (x))} ; 
(7) 
  
The y parameter controls how strong is the AND-feature. 
In the case of y = 0, the function is the same with the 
algebraic AND. 
In fuzzy decision there are some modifiers called hedge 
values. These hedges can be interpreted linguistically as 
VERY, NOT SO etc. The simplest way of using them is 
calculating the exponents of the memberships. 
Let’s see the steps of fuzzy decision making! (Figure 2) 
  
« reaching the desired network accuracy 
+ reaching the maximal number of epochs (iterations). 
In the second case the necessary network accuracy isn’t 
fulfilled so the training must be started again. The design 
of the networks follows some rules; these are the 
conditions to get the minimal but adequate network. It's 
possible that the number of neurons in a layer must be 
increased in order to give much "flexibility" to the 
  
Rule 1 
; Rule 2 LM 
Inputs pi — Implication — Inference | —»| 
[E 
  
  
Rule n 
  
  
Output 
  
  
  
  
  
  
  
  
  
Figure 2 
Flowchart of the fuzzy decision making 
In the first steps the input values must be eventually 
fuzzified and the fuzzy rules must be evaluated. These 
rules can contain constrains and hedges. In the 
implication phase the results of rules are accumulated. 
Also the importance of rules can be taken into 
consideration. Inference means the decision so we'll have 
the class belonging. In the praxis the max function is used 
for inferencing. 
3. THEMATIC CLASSIFICATION 
3.1. Preparation of the classification 
In the thematic classification the first step is the selection 
of the training areas. The training areas deal to bring 
terrain information into the classifier so the method is 
supervised. 
Neural networks must be learned directly with pixel 
intensity — this is a difference to the traditional statistical 
methods where some (statistical) measures are derived 
from the pixels and are integrated in the classifier. The 
usual measures are the mean vectors and covariance 
matrices — using a maximum likelihood classifier — or 
just a mean vector if the minimum distance method is 
applied. 
It was yet proved that the artificial neural networks could 
learn directly from the pixels and have acceptable 
accuracy; while they get only statistics the result won't be 
night. 
With the usual masking technique the training areas can 
be marked. After the selection the training material of the 
independent networks is to be prepared: all the class-own 
pixels and a resampled set of the rest pixels are chosen. 
This step is to be done for every thematic class. 
The training of the neural networks is executed 
independently. As it's known the training of a net is 
iterative. There are two criteria to stop the iteration: 
  
  
network. 
When all independent networks are trained the previously 
described transformation is executed. At the end we'll get 
a single neural network which contains all the features of 
the original nets. 
3.2. The neuro-fuzzy method 
Combining the neural network technique with the fuzzy 
decision making we get the neuro-fuzzy classifier. Let's 
. see how! (Figure 3) 
As in chapter 2.1. is mentioned, neural networks give an 
output in the range of 0 and 1. This continuous interval 
can be interpreted as "neural probabilities". The inputs of 
the fuzzy decision making are to be fuzzy or to be 
fuzzified. Fuzzification means that we have to calculate 
class memberships. In the thematic mapping it's very 
luckily that during the classification yet the class 
belongings are computed. If the network gives an output 
of 0.9 for a pixel that can be understood as it belongs to 
the category with a membership of 90 %. That’s why 
these values are fuzzy inputs for the decision making! 
Because of having several thematic classes the 
transformed neural network produces a membership 
vector. These memberships are taken into consideration 
in fuzzy rules. These rules mirror the human knowledge 
of the nature. The rules are easily coded in table form, 
this is the knowledge table (KT). The knowledge table 
can contain any terrain information: texture, digital 
elevation data, intensity data etc. We can interpret the 
knowledge table also a projection of the input data into 
the (output) categories. Having n inputs and m classes the 
evaluation of the rules is the following: 
-— h, 
IPM, = and(o", KT,) 
i=] 2..n 
12m] ] ..m 
(8) 
where IPM is the implication matrix, o the input vector 
(same as the output of the neural network eventually 
extended with further terrain data), h the hedge vector. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXII, Part 7, Budapest, 1998 325