International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B6. Istanbul 2004 
3.2 Fuzzy inference system 
Fuzzy inference is the process of formulating the mapping from 
a given input to an output using fuzzy logic. The process of 
fuzzy inference involves: membership functions, fuzzy logic 
operators and if-then rules. There are two types of fuzzy 
inference systems that can be implemented in the Fuzzy Logic 
Toolbox: 
=» Mamdani-type and 
=» Sugeno-type. 
Mamdani's fuzzy inference method is the most commonly seen 
fuzzy methodology and it expects the output membership 
functions to be fuzzy sets. After the aggregation process, there 
is a fuzzy set for each output variable that needs 
defuzzification. Sugeno-type systems can be used to model any 
inference system in which the output membership functions are 
either linear or constant. This fuzzy inference system was 
introduced in 1985 and also is called Takagi-Sugeno-Kang. 
Sugeno output membership functions (z, in the following 
equation) are either linear or constant. A typical rule in a 
Sugeno fuzzy model has the following form: 
If Input | = x and Input 2 — y, then Output is Z = ax + by + € 
For a zero-order Sugeno model, the output level z is a constant 
(a=b =0). 
3.2.1 Membership function 
Membership function is the mathematical function which 
defines the degree of an element's membership in a fuzzy set. 
The Fuzzy Logic Toolbox includes 11 built-in membership 
function types. These functions are built from several basic 
functions: 
=» piecewise linear functions, 
=» the Gaussian distribution function, 
=» the sigmoid curve and 
=» quadratic and cubic polynomial curve. 
Two membership functions are built on the Gaussian 
distribution curve: a simple Gaussian curve and a two-sided 
composite of two different Gaussian curves (Figure 3.) 
  
  
  
  
  
  
  
  
o 2 à 6 $ © ° : 1 
4 6 + $ 
LEE guas3nl P zt 234] gbsiml P 1248 
gbellmf 
built on the Gaussian 
gaussmf gauss2mf 
Figure 3. Membership functions 
distribution curve 
This type of membership function will be used later on, 
according to the results coming from PCI. 
3.2.2 Fuzzy logic operators 
The most important thing to realize about fuzzy logical 
reasoning is the fact that it is a superset of standard Boolean 
logic. In other words, if the fuzzy values are kept at their 
extremes of | (completely true) and 0 (completely false), 
standard logical operations will hold. That is, A AND M 
operator is replaced with minimum - min (A,M) operator, A OR 
M with maximum - max (A,M) and NOT M with 1-M. 
85 
3.2.3  If-Then rules 
Fuzzy sets and fuzzy operators are the subjects and verbs of 
fuzzy logic. Usually the knowledge involved in fuzzy reasoning 
is expressed as rules in the form: 
IfxisA Theny is B 
where x and y are fuzzy variables and A and B are fuzzy 
values. The if-part of the rule "x is A" is called the antecedent or 
premise, while the then-part of the rule "y is B" is called the 
consequent or conclusion. Statements in the antecedent (or 
consequent) parts of the rules may well involve fuzzy logical 
connectives such as ‘AND’ and ‘OR’. In the if-then rule, the 
word "is" gets used in two entirely different ways depending on 
whether it appears in the antecedent or the consequent part. 
3.3 Classification procedure 
Since the goal of both procedures, maximum likelihood (ML) 
and fuzzy logic, is to classify the image, input data must be the 
same. That is, three SPOT channels are used as the starting 
point for the image classification based on fuzzy logic (Figure 
1) 
The Fuzzy Inference System (FIS) Editor displays general 
information about a fuzzy inference system: a simple diagram 
with the names of each input variable (green, red and NIR 
channel) and those of each output variable (water, urban area, 
crop 1, crop 2 and vegetation). There is also a diagram with the 
name of the used type of inference system (Sugeno-type 
inference). 
The Membership Function Editor is used to display and edit all 
membership functions associated with all of the input and 
output variables for the entire fuzzy inference system. 
Because of the smoothness and non-zero values, in order to 
define a membership function, in the process of image 
classification simple Gaussian curve (gaussmf) is used (Figure 
3a). In this case, Matlab's Fuzzy Logic Toolbox needs two 
parameters for the valid membership function definition: mean 
and standard deviation values. Values given in the Table 1 
(mean gray value and standard deviation for each class in green, 
red and near infrared channel) come from PCI’s ‘Signature 
statistics’ panel. These values are used as the pattern 
(parameters) in FIS (‘fuzzy inference system’) membership 
function design. In this table, values in cursive (mf;) represent 
membership functions. That is, mf] represents membership 
function for water in green input variable. For some reasoning, 
sampled areas used for testing showed that results are much 
better if in membership function definition half of standard 
deviation values is used, instead of values given in the Table 1. 
Reason can be found in large overlap (Figure 4.) between very 
close range of membership functions (mfl, mf2, ..., mf5). This 
close range was also the reason why specific names for 
membership functions (linguistic hedges) like: not very light, 
light, middle tone, dark, very dark,... are not given (wider range 
may be found just in NIR channel). The names of membership 
functions remained the same: mfl1, mf2, ... , mf5.