Image Processing Reference

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are either too di
cult to choose due to the lack of understanding of the human approach

or cannot produce a satisfactory result. Membership functions generated from the large

amount of image data by a clustering technique is one way to tackle this problem.

Clustering is an unsupervised learning method, i.e. samples in the input set are unlabelled

(not classified) and, in many cases, the exact number of classes is unknown (Hoeppner et al.,

1999). Our task is to divide these samples into several groups according to a similarity

measure or inherent structure of the data. In order to build the membership functions from

the available image data we can use a clustering technique to partition them, and then

produce membership functions from the resulting clusters. Every generated cluster can be

interpreted as a single class. Therefore, during an object labelling step, a set of m desired

input-output data pairs are produced (the training data set) in the n-dimensional input

space:

{x
p
= (x
p1

, . . . , x
pn
), C
k
}, p = 1, . . . , m, C
k
∈C, k = 1, . . . , M

(6.22)

We have to distinguish between the training data set which is used for clustering and

building the classifier, and the test data set which is classified without influencing the

clusters. Given an input pattern from test data represented as the image feature vector, the

classifier determines its membership in all the classes (not clusters). These output values

can be interpreted as the result from classifier rules and can be considered as the “examples”

for the fuzzy rule base that is later created.

In the following, we describe the process of generating fuzzy rules by learning from ex-

amples. For this purpose we use a training data set (Eq. 6.22) and use the generated rule

to determine a mapping (Eq. 6.1). This approach was originally proposed in (Wang and

Mendel, 1991) to derive fuzzy rules for function approximation. In (Chi and Yan, 1995b,a,

1993) this idea was extended to solve problems related to image understanding. Therefore,

the method of generation of fuzzy rule base consists of the three main steps: (1) fuzzy

clustering of the input feature space (2) generation of a set of membership functions for

each input in the feature space and generation of the class labeling and (3) generation and

minimisation of fuzzy rule base by learning from examples.

6.3.6 Fuzzy clustering of the input feature space

In our methodology of fuzzy rule generation we focus on objective function-based clus-

ter analysis whose aim is to assign data to clusters so that a given objective function is

optimised. The objective function assigns a quality or error to each cluster arrangement

based on the distance between data and typical representatives of the clusters called cluster

prototypes. A large family of objective functions results from the following basic func-

tion (Bezdek J.C., 1992; Hoeppner et al., 1999):

J (U, V ) =
X

x∈X

X

µ
η
(x)(v) d
2
(x, v)

(6.23)

v∈V

where d(x, v) is a distance between the datum x and the cluster prototype v, µ(x)(v) is an

M×m matrix of the membership values, m denotes the number of data patterns, M the

number of clusters (classes) and η is an exponent weight factor called a fuzzifier factor. The

criterion for the optimisation of the objective function is obvious: it has to be minimised.

Because the values of function J are dependent on U and V , the clustering process is

related to iterative calculations of cluster prototypes v∈V .

and the membership values

∀
x∈v
µ(x)(v)∈U .

Based on the objective function (Eq. 6.23) several models with various distance measures

and different prototypes have been developed.

The most popular approach proposed by

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