Image Processing Reference

In-Depth Information

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

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