Image Processing Reference

In-Depth Information

(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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