Applications of Fuzzy Rule-based Systems in Medical Image Understanding - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 99

Let us define the fuzzy classification task for image understanding as follows (Hoeppner,
Klawonn, and Runkler, 1999). We have n real variables (image features) x 1 , . . . , x n with
domains X i = [a i , b i ], a i ≤b i , i = 1, 2, . . . , n, a finite setCof M classes and a partial
mapping

class : X i ×···×X n →C (6.1)

that assigns classes to some, but not necessarily to all, vectors (x , . . . , x n )∈X ×···×X n .

1 1

The aim is to find a fuzzy classifier that solves this classification problem.
A fuzzy classifier is based on a finite set R of rules of the form R j ∈R, j = 1, 2, . . . , N :

x i is A (1)
j

and . . . and x n is A (n)
j

Rule R j : IF T HEN class is C (6.2)

where C∈Cis one of the classes A (1)
j

, . . . , A (n)
j

are antecedent fuzzy sets described by fuzzy

membership functions µ (1)
j

, . . . , µ (n) . In order to keep the notation clear, we incorporate

n

these functions µ (i)
j

, i = 1, 2, . . . , n directly in the rules:

x i is µ (1)
j

and . . . and x n is µ (n)
j

Rule R j : IF T HEN class is C (6.3)

In real image understanding tasks one would replace them by suitable linguistic values
like “rather dark”, “well contrasted”, “highly patterned”, etc. and associate these linguistic
value with corresponding fuzzy membership functions. Fig. 6.1 shows a simplified block
diagram of a fuzzy rule-based system (Yager and Filev, 1994) that realises process of fuzzy
reasoning. The fuzzy result as output of the fuzzy rule base indicates the degree to which
an input pattern X 1 ×. . .×X n satisfies our decision criteria. The deffuzification problem
defines the strategy of using the fuzzy result in the selection of one representative class of
the setC.

FIGURE 6.1: Block diagram of fuzzy rule-based classification system

Fuzzy reasoning is realizable via a number of strategies such as max-min reasoning, max-
matching, max-accumulated matching, and centroid defuzzification (Bezdek J.C., 1992;
Zimmerman, 1991). Here, however, we restrict our attention to the simplest max-min
strategy, i.e., we evaluate the conjunction in the rules by the minimum and aggregate the
results of the rules by the maximum. Therefore, we define

{µ (i) R j (x i )}

µ R j (x , . . . , x n ) = min

i∈{

(6.4)

1

1,...,n}

as the degree to which the antecedents of rule R j are satisfied, and

µ (R)
C k

(x 1 , . . . , x n ) = max{µ R j (x 1 , . . . , x n )|C = C k } (6.5)

is the degree to which the vector (x 1 , . . . , x n ) is assigned to class C k ∈C, k = 1, . . . , M . The
defuzzification, the second and final process, which provides an assignment of a unique