Image Processing Reference

In-Depth Information

In image understanding tasks the system learning can achieved in several ways. The first

step is related to feature selection which in practice is always task-dependent. Moreover, in

many cases only the feature set is defined. The number of classes is di
cult to estimate or is

completely unknown. In these cases the learning process is “supervised” by the knowledge

expert. In our approach, expert validation can be performed in two ways: (1) the expert

defines the number of classes or (2) the expert accepts the number of classes proposed

by the fuzzy clustering algorithm. In the second case, the suggestion is based on cluster

validity measures (Zheru et al., 1996) and classification results on the images used for system

learning. We have used the following three cluster validity measures, defined for m input

patterns and M clusters (classes), µ(x
j
)(v), denotes membership of the j-th object to the

cluster and v
k

denotes a prototype of the k-th cluster:

1. Partition coe
cient

M

m

X

X

1

m

(µ(x
j
)(v
k
))
2

P K =

(6.31)

k=1

j=1

Additionally, suppose that Ω
M
∗
represents the clustering result, then the optimal

choice of M
∗
is given by max
M
∗
{max

P K(M
∗
)}, M
∗
≥2.

Ω

∗
M

2. Separation coe
cient

(µ(x
j
)(v
k
))
η
(||x
j
−v
k
||
2
−||
P
l=1

M

m

X

X

x
l

m
−v
k
||
2
)

S =

(6.32)

k=1

j=1

The optimal choice of M
∗
is given by min
M
∗
{min

S(M
∗
)}.

Ω

∗
M

3. Separation and compactness coe
cient

CS =
P
k=1
P
j=1

(µ(x
j
)(v
k
))
2
||v
k
−x
j
||

min
k,l
||v
k
−v
l
||
2

(6.33)

Here, the optimal choice of M
∗
is given by min
M
∗
{min
Ω
∗
M
CS(M
∗
)}.

The clustering procedure can be useful when the partition of the feature space is as crisp

as possible. It means that clustering with optimal number of clusters should make all input

patterns as close to their cluster prototypes as possible and all cluster prototypes should

be separated as much as possible. In this case, the possible loss of information is relatively

small.

examples

The next task is to generate a set of fuzzy rules from the training data defined in Eq. 6.22

and use these fuzzy rules to determine a mapping in Eq. 6.1. The method of learning by

examples consists of the following three steps:

Step 1: Find the intervals for each input of the feature space Find the do-

main intervals for each input by finding the crossing points between the generated

continuous membership functions. It makes partition of each input into q regions

denoted as r
1
, r
2
, . . . , r
q
and generate a crisp set of hyper-cubes in the feature

space
(
see Fig. 6.5)
.

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