Image Processing Reference
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(crisp) class to a given vector (x
1
, . . . , x
n
) is carried out by the following mapping, if
µ
(R)
C
k
(x
1
, . . . , x
n
) > µ
(R)
C
l
(x
1
, . . . , x
n
)
R(x
1
, . . . , x
n
) =
C
k
∀
C
l
∈C,C
l
6=C
k
(6.6)
unknown class /∈C
If there are two or more classes that are assigned the maximal degree by the rules, then
we do not classify it and assign it to an unknown class.
6.3
Fuzzy rule generation
The process of generating fuzzy If-Then rules, also called system learning, consists of two
steps: specification of antecedent part, and determination of a consequent class. Vari-
ous approaches have been proposed for the automatic generation of rules (Ishibuchi and
Nakashima, 1999b,a; Grabisch and Dispot, 1992). In this section, we describe two ap-
proaches: the first on is related to fuzzy rule generation with a grade of certainty and
weighted input patterns, while the second one performs rule generation through fuzzy clus-
tering with learning by examples (Wang L., 1991).
For image understanding problems, it is common that a rather large number of fuzzy rules
are produced. Therefore, we often want to optimise the generated fuzzy rule-base to make
recognition system more practical and as well as to speed up the process of reasoning. This
optimisation of the rule system can be achieved with or without reducing the number of
rules in the base. For every generated rule we assign a special value, called grade of certainty
or soundness degree, that can be used for optimisation of the rule base. We also present a
hybrid fuzzy classification system which incorporates a genetic algorithm to optimise the
rule base. Fuzzy rule-base optimisation can also be realised through rule splitting, and
removal or weighting of rules. All these techniques allow us to generate comprehensible
image understanding systems with high classification performance.
6.3.1
Fuzzy rule generation with grade of certainty and weighted input
patterns
In here, we present a strategy where each generated rule is assigned a grade of certainty.
The rule defined in Eq.6.3 is then modified into
x
i
is µ
(1)
and . . . and x
n
is µ
(n)
Rule R
j
:
IF
T HEN
class is C with CF
j
, j = 1, . . . , N
(6.7)
j
j
where CF
j
describes the grade of certainty of the j-th rule.
Let us assume that m training patterns x
p
= (x
p1
, . . . , x
pn
), p = 1, . . . , m are given for an
n-dimensional M -class pattern classification problem. The consequent class C
j
∈Cand the
grade of certainty CF
j
of a j-th fuzzy If-Then rule R
j
are determined during the following
two steps:
1. Calculate β
C
(j) for class C as
β
C
(j) =
X
x
p
∈C
µ
j
(x
p
),
(6.8)
where
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