Image Processing Reference

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measure. These measures include summing up fuzzy membership values of all objects from

the given set, in our case objects of (generalized) lower and upper approximations. Other

choice present different types of fuzzy membership functions.

Probabilistic and Non-Probabilistic Entropy

During calculation of rough entropy value (Pal et al., 2005), many type of entropy are

possible to be taken into account, for example Shannon entropy, Renyi entropy, Tsallis

entropy. All these entropies are generally probabilistic entropies, meaning that probability

of all possible states equals to unity.

Fuzzy Rough Entropy

= Fuzzy Rough Entropy−
e

2

·roughness(l)·log(roughness(l));

In the presented solution, this condition has not been considered and roughness vales

are not equalized to probability distribution. In case of probabilistic distribution, given

n possible states, entropy attains maximal value in situation with all states having equal

probability of 1/n. In rough entropy framework, roughness value for each cluster is contained

in the interval [0, 1]. In such manner, total rough entropy depends on summing up all partial

rough entropies. Total rough entropy attains the maximum in case of all partial entropies

equal its maximum. Maximum value for partial (cluster) entropy

−R·log(R)

This partial cluster entropy reaches the maximum in case of roughness value equal to 1/e.

Rough entropy framework searches for optimal clustering solution that boundary region is

approximately equal 1/3.

11.4.3

Crisp-Crisp Distance RECA

Standard Crisp - Crisp Distance RECA algorithm as proposed in (Malyszko and Stepaniuk,

2008) incorporates computation of lower and upper approximations for the given cluster

centers and considering these two set cardinalities during calculation of roughness and

further rough entropy clustering measure. Rough Entropy Clustering Algorithm flow has

been presented in Algorithm 1. Rough measure general calculation routine has been given

in Algorithm 2. In all presented algorithms, before calculations, lower and upper cluster

approximations should be set to zero. For each data point x
i
, distance to the closest cluster

C
l
is denoted as d
min

dist
= d(x
i
, C
l
) and approximations are increased by value 1 of clusters

C
m
that satisfy the condition:

|d(x
i
, C
m
)−d
min

dist
|≤
dist

11.4.4

Fuzzy-Crisp Difference RECA

In Fuzzy-Crisp Difference RECA setting, for the given point, lower or lower and upper

approximation values are incremented not arbitrary by 1, but are increased by this point

cluster membership value. In this way, fuzzy concept of belongings to overlapped classes

has been incorporated. Taking into account fuzzy membership values during lower and

upper approximation calculation, should more precisely handle imprecise information that

imagery data consists of.

Fuzzy membership value µ
C
l
(x
i
)∈[0, 1] for the data point x
i
∈U in cluster C
l
(equiva-

lently X
l
) is given as

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