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.
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 )}.
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 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.
6.3.8 Generation and optimisation of fuzzy rule base by learning from
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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