Image Processing Reference

In-Depth Information

FIGURE 6.5: Illustration of definition of intervals r
1
, r
2
, . . . , r
q
for a single feature x

Step 2: Generate fuzzy rules from given training data set Produce a rule for

each input-output data pair included in the training data by assigning the given

inputs to the regions with maximum membership value, e.g. for a 2-dimensional

feature vector it is processed as:

x
(1)

1

= 0.3, x
(2)

1

= 0.8, C
K
= 1⇒

x
(1)

1

(µ(x
(1)

1

))(C
k
= 1) = 0.65 in r
(1)

(max)),

2

x
(1)

1

(µ(x
(2)

1

))(C
k
= 1) = 0.75 in r

(max))⇒

4

: IF x
(1)

1

is R
(1)

2

AND x
(2)

1

is R
(2)

4

Rule

THEN C is Class = 1

1

Step 3: Minimisation of fuzzy rules As mentioned above, each data pair from the

training data set generates one rule. Usually there are a large number (several

thousand) of available data pairs, so it is very likely that some conflicting rules

are produced. These conflicting rules have the same IF part but different THEN

parts. One way to solve this problem is to assign a soundness degree SD, and

then to select from the subset of conflicting rules that rule with the maximal

soudness degree. Two strategies are proposed to assign a soundness degree to a

rule R
j
, j = 1, 2, . . . , N where N denotes the number of rules.

Strategy 1 The soudness degree SD(R
j
) for j-th rule is determined by the ratio

of the number of training data pairs which supports the rule W
R
j

and the

total number of patterns which have the same IF part W
IF
j
:

W
R
j

W
IF
j

SD(R
j
) =

(6.34)

This strategy works better when a large number of training patterns are

available. Using this strategy we incorporate the statistical information

into the fuzzy system resulting in more reliable decision.

Strategy 2 The soundness degree is determined by the membership grades of

inputs and outputs:

SD(R
j
) = [µ(x
(1)

)(v = C)∗. . .∗µ(x
(n)

j

)(v = C)]∗µ(x
j
)(v = C)

(6.35)

j

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